QUEUES AND STORES WITH NONHOMOGENEOUS INPUT
by
Donna Katzman McClish
A Dissertation submitted to the faculty of
The University of North Carolina at Chapel
Hill in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in the Department of Statistics.
Chapel Hill
1979
This research was partially
supported by the Office of
Naval Research under Grant
N00014-76-C-0550.
McCLISH, DONNA KATZMAN.
Queues and Stores With Nonhomogeneous Input.
(Under the direction of WALTER L. SMITH.)
•
Virtually all the literature concerning queueing and storage theory
assumes that the input to a system has an arrival rate which is constant
over time.
This research is concerned with extending results to the
more realistic case of nonhomogeneous input.
It is shown that under a
variety of circumstances, the presence of a periodic pattern of input
is enough to insure that a quasi-limiting distribution of store content
exists.
For stores with nonhomogeneous Compound Poisson input and
general release rule a probabilistic approach is developed which allows
equations to be written down easily for such quantities as the mean
store content and the probability of an empty store.
To do this it is
assumed that the intensity can be written in the form A(t)=A o(l+E¢(t)).
The formulas are series expansions in E. A more in-depth look at the
first two terms of the E-expansion shows that for periodic ¢(t), a
general expression can be found for an approximation to the probability
of an empty store.
Applications of the formulas for stores with release
rules r(x)=c, r(x)=a+bx and r(x)=aX are examined.
For at least some
instances of the M(t)/M/l queue it is shown that the approximations
developed are very good.
The problem of a stochastic intensity function
is also briefly considered .
•
TABLE OF CONTENTS
Acknowledgements
iv
Conventions
v
Chapters
l.
Introduction and Review of the Literature.
2.
Limit Theorems for Storage Systems
12
2.1 Some Results in the Storage Theory Literature
12
2.2 Some Basic Tools •
..·
····
····
2.3 A General Limit Theorem
e
3.
4.
y
e
1
16
22
2.4 Limit Theorems for Type A Processes
with General Periodic Input · · · ·
···········
26
2.5 A Limit Theorem for Type B Processes
with General Periodic Input · · · ·
···········
41
2.6 A Limit Theorem for a Homogeneous Storage System.
44
Perturbation Approach to the Store
with Nonhomogeneous Input . ·
55
·····
3.1 A Probabilistic Method.
55
3.2 Extending the Approach to General cj>(t)
59
·········
3.3 An Exact Limiting Result
····
·····
3.4 Periodic Intensity . . ·
····
3.5 An Example: cj>(x)=cos2TIX . . · ·
····
Limiting Results for the Standard Store.
····
4.1 The Limiting Probability of Emptiness
····
4.2 The Mean Store Content • .
····
4.3 Special Case: cj>(t)=cos2TIt
·····
ii
66
72
81
83
83
88
91
4.4 A Queueing Theory Example.
·· ········
92
4.5 General Service Oistribution-An Approximation.
97
4.6 A Special Case: cm=( 1-0'. ) 0'.m-l
4.7 Another Example · · · · · ·
5.
·····
102
····
104
·········
110
·
Limiting Results for some Stores with
General Release Rules · · · · ·
5.1 The Limiting Probability of Emptiness;
r(x)=a+bx
· ·
·
110
5.2 A Special Case: Computational Aspects of nO
114
5.3 Limiting Value of the Mean Store Content;
r(x)=a+bx
··
117
5.4 A Speci a1 Case: ep(u)=COS21TU
118
···
·
·····
····
..··
B Process · · · ·
5.5 Some Graphs
5.6 A Type
5.7 An Example: ep(t)=COS21Tt
6.
····
····
7.
137
A Matrix Approach to the Nonhomogeneous Queue
····
·
6.2 Construction of the Transition Matrices
123
132
·····
6.1 A Step Function Approximation to A(t)
121
····
137
139
6.3 An Example: A(t) a Step Function
142
6.4 Another Example: Periodic A(t)
151
Storage Systems With Stochastic Input Intensity
156
7.1 The Method
..··········
7.2 An Example: A Queue with Two Phases
····
······
Appendix
156
158
163
Bibliography.
····
iii
166
e
ACKNOWLEDGMENTS
I am grateful to Professor Walter L. Smith for serving as my
adviser during this research.
He provided a never ending source of
ideas and directions.
I also wish to thank all those who were on my committee, Professors Leadbetter, Shachtman, Ehrhart, Carroll, Kelly and Wegman.
I have received much support from friends and colleagues through
the years which has enabled me to reach this point.
Special thanks
must go to the people on the SENIC project with whom I have worked
for the past 3 years.
Their encouragement, particularly in the final
stages of writing, was greatly appreciated.
Finally, I must acknowledge the love and understanding extended
to me by my family and particularly my husband, Tom, who was always
there to reassure me, even while wrestling with his own research.
iv
CONVENTIONS
Numbering and Referencing of Equations
Equations will be numbered consecutively throughout each chapter.
References to equations within a chapter will be by number only.
References to equations in earlier chapters will include both chapter
and equation number.
Thus a reference to equation 25 in Chapter 3 would
be to equation 3.25.
Notation
The Laplace-Stieltjes Transform of a function F will be denoted by
F*(s); i.e.
F*(s)
=
[ooe-sxdF(x).
o
The (ordinary) Laplace Transform of a function F will be denoted
FO(x); i.e.
FO(s)
=
fooe-sxF(x)dx.
°
The Stieltjes Convolution of functions F and G will be denoted
F*G(x) = fX F(x-u)dG(u) = fXG(x-u)dF(u).
o
0
We note that the above functions F and G are assumed to be zero on
(-00,0).
v
CHAPTER 1
INTRODUCTION AND REVIEW OF THE LITERATURE
The problem this research considers is the modeling of and effect on
queueing and storage systems which have input that is not homogeneous in
time.
Most work concerning queueing and storage processes has been for
the case where the times between inputs to the system are i.i .d. random
variables.
This implies that the probability of an input in one time
interval is equal to the probability of an input in any other time interval of the same length.
But in many practical situations the input
process is of a more general nature, often varying with time.
the arrival rate seems to fluctuate randomly over time.
Sometimes
More often,
though, the dependence on time is of a more deterministic nature, with
the arrival frequency depending on, say, the time of day, month or year.
Automobile traffic, airline departures and landings, restaurant and resort
business, telephones--all these exhibit a cyclical or periodic pattern of
input and are quite typical of the types of processes one often finds.
A common method that has been used to deal with the problem of time
varying input (when it is recognized) is to treat periods of widely
differing inputs separately.
For example, in queueing theory much has
been written on the problem of the rush hour-referred to as heavy traffic
and non-stationary queues.
Such work focuses on the time when the rate
of arrival to a queue is almost equal to or even greater than the service rate--not the typical situation the majority of the time.
If
reasonably practical results could be obtained for a process with
2
nonhomogeneous input, then we could model the entire system simultaneously, which would certainly be preferable.
To understand the complex-
ity of the problem, we should first look at what has been discovered in
the literature to date.
The literature of queueing theory has developed quite separately from
that of storage and dam theory, although the
ent aspects of the same problem.
~sider
t~/O
areas are often differ-
This can readily be seen if we con-
that the input to a store (or dam) in (D,t] is equivalent to the
total service time, or work load, of all customers to arrive in (G,t];
the content of a store at time t corresponds to the (virtual) waiting
time of a customer who arrives at a queue at time t, and the release
rate of the content of a store, per unit time, is equivalent to the
amount of work load processed per unit time by the server of a queue.
Thus, in many cases, what has been discovered about one process can be
applied to the other.
Still, the work done in queueing theory on non-
homogeneous arrivals has not been picked up by those concerned with
storage theory.
There has been relatively little published in the area
of nonhomogeneous processes and virtually all that has been done has
been in the area of queues.
The first major work in this area was done by Takacs (1955) who
investigated the virtual waiting time process of a single server queue.
The virtual waiting time is defined as the timE! a customer arriving
would have to wait to be served.
If the system is empty at the time of
arrival then the customer is served immediately and the waiting time is
zero.
For the single server queue with nonhomogeneous Poisson arrivals
(intensity A(t)) and general service time distt'ibution, Takacs derived
3
the integro-differential equation for the distribution of waiting time
at t, F(t,x):
aF(t,x) = aF(t,x) - A(t)F(t,X)+A(t)~H(x-y)dyF(t,y).
at
ax
From this he derived an expression for
~(t,s),
the Laplace-Stieltjes
Transform of the waiting time:
~(t,s) = e
st - [l-1jJ(s)]A(t) [
(t -su+[I-1jJ(s)]A(u)
~-sJoe
1
F(u,o)d~ •
where 1jJ(s) is the Laplace-Stieltjes Transform of the service distribution
and A(t)=/A(u)du.
o
If the probability the system is empty can be determined uniquely,
then the distribution of waiting time can be determined.
As Takacs ad-
mits, the determination of this probability is generally quite difficult.
In the particular case where A(t) converges to a positive constant A for
large values of t, and Au<1 (where
U
is the expected service time) the
limiting distribution of waiting time exists, independent of the initial
distribution.
The probability of emptiness in this case is I-Au, which
agrees with the results for the queue with homogeneous arrival rate A.
The problem of determining the probability of emptiness has been pursued by a few authors with varying degrees of success.
Reich (1958,
1959) was able to show that the probability a queue is empty at time t
is the unique solution of a Volterra equation of the first kind.
With
this he was able to examine the behavior of the probability for large t,
culminating in a result which gives the time average of the probability
subject to an error of magnitude 0(1).
4
Gani (1962) approached the problem in the context of dam processes.
He showed that the probability of emptiness is a function of the first
empti ness probabil ity, whi ch in the case of inputs of constant unit size
reduces to a simple recurrence relationship.
Hasofer (1964, 1965) was able to use the results of Reich to get
explicit results for the M(t)/G/l queue by assuming a particular form
for the Poisson parameter A(t).
Taking jtA(u)du=t-zb(t)
Hasofer was
o
able to write the probability of an empty queue at time t as a power
00
series in z, p(x,t)= n~o ZnFn(t).
Using Reich's equations, some intri-
cate complex analysis and the additional assumption that b(t) and
b'(t) are uniformly bounded, Hasofer was able to find a general expression for the Laplace Transform of the Fn:
where
G~(p)= L[l-ljJ(n)t; g~(p)= ; (n)[l-:jJlilln ; b~(p)= 6~-pt[b(t)J"dt
n![l+ljJ'(n)]
n!
[l+ljJ'(n)]
and n is the unique solution of s-l+ljJ(s)-p=O.
5
Much neater results were obtained when the Poisson parameter was
n
assumed periodic. For A(t)=t-z n~lansin(nwt+~n) Hasofer showed that the
functions Fn have the asymptotic form
00
Fn(t) ~ k~l [A kn cos(kwt)+Bknsin(kwt)]
and the Laplace-Stieltjes transform of the waiting time has a similar
asymptotic form.
The coefficients in the Fourier expansion were not
given explicitly, although for the special case A(t)=l-zwcos(wt) Hasofer
derived Fo and Fl.
A number of authors have approached the problem of nonhomogeneous
arrivals by employing the familiar Kolmogorov difference-differential
equations for the queue size distribution.
Clark (1956) explored the
queue size distribution of a single server queue with Poisson arrivals
and negative exponential service times, where both parameters are continuous functions of time.
While no exact representation for the dis-
tribution of queue size was obtained, the problem was reduced to finding
the solution of an integral equation.
Much simpler results were pre-
sented for the mean and variance of the queue size, which only depend on
the (unknown) probability of emptiness.
In the special case where
traffic intensity is constant the distribution of queue size was found
explicitly.
It can readily be seen that in this case the queue is
equivalent to one with constant arrival and service rates.
Luchak (1956) treated a similar problem, but with a more general
service distribution.
He approximated a general service distribution
by a Weighted-sum Erlang distribution.
~
Starting again from the Kolmogorov
equations, Luchak found that if the traffic intensity could be expressed
as a polynomial in time, then the distribution of queue size could be
6
found recursively, but only in selective cases could a closed form be
found.
At the end of his article, Luchak noted that periodic arrival
rates could be dealt with more easily than most, since the queue size
distribution need only be found for the initial period.
The state at
the end of each period can then be used as the initial condition for the
next period.
He suggested that if the initial state of the system were
taken to be the steady state solution of a queue with constant traffic
intensity equal to the average intensity over a period, then the "quas istationary" steady state would be reached in only a few periods.
We
will make use of.this suggestion in later chapters.
In recent years the work of Luchak and Clarke has been expanded to
include more generality.
Koopman (1972) was interested in arrivals and
departures of ai rp1anes at an air termi na 1.
HE~ i ncorpora ted certa in of
the peculiarities of the airport into his mode·l.
e
These included periodi-
city (24 hours) and a limit on the maximum queue size.
He also made
arrival and service time parameters dependent both on time and queue
size.
The method of solution suggested by Koopman was a direct computer
solution of the (finite) system of differential equations.
Moore (1975) like Luchak, used a Weighted-sum Erlang distribution
for service times and assumed customers could arrive in groups.
He
developed a technique, in the stationary case, which focused on the
embedded Markov chain formed by looking at departure times.
Then by
approximating the continuously varying arrival rate by a step function
he was able to extend this, developing a computer algorithm solution.
E. L. Leese (1967) wrote a paper in which he evaluated the computati ona 1 feasabil i ty of a number of proposed sol ut ions for the nonhomogeneous queue.
Leese considered direct solution of the Kolmogorov
e
7
differential-difference
equations~
generating function
Taylor series expansion of the queue size
techniques~
probabilities~
which uses a step function approximation to the traffic
a
a matrix approach
intensity~
and a
method developed by Wragg (1963) which involves solving an integral equation.
Most of the objections raised by Leese revolve around the excessive
number of computational steps required.
It should be noted that over ten
years of rapid advances in computer technology has abrogated some of his
original objections.
In fact, we will later investigate the uses of the
matrix approach when the arrival pattern is periodic.
A number of authors have investigated a queueing problem where the
arrival (and possibly service) rate is not a deterministic function of
time.
Rather~
the arrival rate is a heterogeneous Poisson process
governed by an "extraneous phase process" which is a continuous time
Markov chain, i.e. customers arrive in a Poisson stream with the arrival
rate depending on the "phase" of the
queue~
and the amount of time spent
in a phase is a negative exponential random variable.
Service times may
or may not vary with each phase.
Eisen and Tainiter (1963) and Yechiali and Naor (1971) looked at such
a process with two phases, where service also followed a negative exponential distribution.
Eisen and Tainiter found a formula for the mean
number in the system and mean waiting time in steady state.
and Naor looked at a number of special cases.
Yechiali
They found that if the
traffic intensity in each phase of the queue is the
same~
then the dis-
tribution of queue size in the steady state is what would be expected in
the MIMl1 queue.
They also claimed the above result would hold only in
the case of equal traffic
true.
intensities~
which we will show later is not
8
Yechiali (1973) later extended his result to include an m-phase process.
To find explicit results requires finding the roots of a difficult
equation which yields that nemesis, the probability of an empty queue.
No closed form solutions were presented except in the special case where
traffic intensity of each phase are all equal, with results as before
for m=2.
Neuts (l971) extended th is work even
vice time distributions in each phase.
furthE~r
by all owi ng general ser-
By using very powerful tools
along with a Markov renewal branching process approach Neuts was able
to find both transient and steady state results.
Three separate papers have been published concerning the existence
of limiting distributions when arrivals to a queue are from a nonhomogeneous Poisson process--one by Sakr (l971), another by Yevdokimova
(l974) and a third just recently published by Harrison and Lemoine (l977).
All three assume that the intensity of the arrival process, A(t), is
periodic, i.e. A(t+T) = A(t) for some period T and every t.
Sakr proved that for a queue with m servers, if the traffic intensity
per period, A/~ is less than one (where A = ~ J~A(u)du is the average
intensity per period and
1/~
is the mean service time) then the distri-
bution of queue length converges to a "quasi-stationary" limiting distribution, where the distribution of queue len9th is a function of the
time within the period.
Yevdokimova expanded upon the work of Sakr to
give a representation of the waiting distributi'on for the single server
queue.
He showed that if
A/~<1
then the waiting time is the sum of two
functions, one of which is periodic and independent of the initial distribution, while the other goes to zero as t goes to infinity.
9
The methods used in both articles are virtually identical.
Both
authors used two queues with related arrival processes to bound the original process.
They were then able to use classical theory for the homo-
geneous queue to get limiting results.
To get his more explicit results,
Yevdokimova also made use of the fact that the time points {nT}n:l such
that the system is empty at nT form an embedded renewal process.
Harrison and Lemoine, with assumptions identical to those of the first
two articles, proved the existence of both time average limits and limits
in distribution for the virtual waiting time and waiting time process.
They also derived relationships between the limiting distributions.
The
method of proof used is similar to that of Yevdokimova, focusing on empty
times as a regenerative event.
As this brief literature review demonstrates, very little progress
has been made in developing practical formulae for determining virtual
waiting time when nonhomogeneous input is considered.
In fact, no work
at all has been done for storage processes with time varying input and
general release rules beyond the queueing theory results corresponding
to r(x)=c.
This research will attempt to develop practical formulae
for the probability of emptiness and expected content of stores with nonhomogeneous Compound Poisson input and general release rate.
In Chapter 2 we are concerned with the existence of the limiting or
quasi-limiting distribution of store content under rather general conditions.
In particular, we show that under a variety of circumstances the
presence of a periodic pattern of input is enough to ensure that a limit
exists.
A probabilistic approach is developed in Chapter 3 which allows us
to quickly and easily write down equations for such quantities as the
10
distribution of store content, probability of E!mptiness, etc.
To do
this we assume that the intensity has the form A(t)=A o(l+E¢(t)). The
formulae are series expansions in E. A more in depth look at the first
two terms of the expansion shows that for periodic ¢(t), a general
expression can be found for an approximation to the probability of an
empty store.
Chapters 4 and 5 focus on applications of the formulae developed in
Chapter 3 for specific release rules.
rule, r(x)=l,x>O.
Chapter 4 treats the most common
Specific formulae are given for an approximation to
the mean store content and probability of an empty store when the input
size distribution is negative exponential or Weighted-sum Erlang.
Chapter 5 considers the release rules r(x)=a+bx and r(x)=cx,x>O.
The
latter is representative of a class of storage processes which never
empty.
We show that the expected content of sUich a store can be written
as a second degree polynomial in E.
Chapter 6 presents an alternative matrix approach to the study of
the M/M/1 queue with nonhomogeneous input.
case of periodic input is pursued.
Its particular appeal in the
We also use this method as a means
to investigate the adequacy of the E-approximations in the special case
of negative exponential service time.
In Chapter 7 we consider a slightly different problem from earlier
chapters.
There we are concerned with the situation where A(t)=A(t,W)
is a stochastic process.
We show that it is sufficient to know the mean
function of A(t,W) to be able to investigate the behavior of the store
content.
The problem presented by Yechiali and Naor (1971) is also
examined, with the surprising result that when the mean service time
11
does not vary, the formula for the probability of emptiness is quite
simple and only involves the first two terms of the E-expansion.
also correct a mistake by Yechiali in his papers.
We
CHAPTER 2
LIMIT THEOREMS FOR STORAGE SYSTEMS
In this chapter we will prove the existence of certain limiting
probabilities for storage systems with nonhomogeneous input.
We will
be concerned here with the distribution of store content, Z(t).
A
limiting distribution for Z(t) exists which is independent of the
initial content distribution if
lim
P{Z(t) -< x
t~
for some random variable Z*.
I Z(O)} = P{Z*
_<
x}
If instead the limiting probability is not
completely independent of time, but rather exhibits a cyclical pattern,
then we say we have a quasi-limiting distribution, i.e. we have a quasilimiting distribution if
~: P{Z(kw+T) ~ x) I Z(O)}
=
P{Z*(T) < x}
for some wand random variables Z*(T), 0
~
T < w.
2.1 Some Results in the Storage Theory Literature
Limiting distributions were first shown to exist for the queue with
homogeneous Poisson input and negative exponential service time distribution.
The general method employed focused on the states {E j
}
which
denote the queue size.
When service is negative exponential these states
form a Markov process.
When service time is not negative exponential
this is not the case.
But Takacs, in 1955, showed that this was not
necessary to prove the existence of a limit, since the waiting time is
13
a Markov process if arrivals from a Poisson process.
Even if inter-
arrival times are merely from a renewal process, the arrival times are
regeneration points, and this is sufficient for the proof.
Thus Takacs
was able to prove results such as the following theorem.
Suppose A(t} is the intensity of a nonhomogeneous Poisson process,
the random variable ~ is the service time of the nth customer, and
F(x,t} is the distribution of waiting time.
Let the expected value of the random variable xn
im A(t}=A be a positive constant.
be a = oJXdH(x}= 0/"(l-H(x}}dxand let lt ~
If Aa<l then the limiting distribution i~ F(t,x}=F*(x} exists, is inde-
Theorem 3 (Takacs):
pendent of the initial distribution F (x) and is uniquely determined by
o
the equations
F*(O}=l- Aa
and
dP"(x}
dx
where dF*
dx denotes the right hand derivative.
If Aa>l then the limiting
distribution F*(x} does not exist, however ~~ F(t,x}=O for each x.
Of course this result can also be applied to standard storage and
dam theory.
No new limiting results appeared concerning these processes
until interest in storage processes with general release rules was
aroused in the late 1960's and early 1970's.
A release rule (or rate) r(u} is a function such that in any time
interval (t,t+dt) the amount of content in a store which is released is
r(Z(t}}dt+o(t}; i.e. the amount of content that will be released from
the store per unit time is a function of the content of the store at
that time.
In queueing theory this is equivalent to saying
14
that the amount of work load processed by the server per unit time is
dependent on the waiting time.
and r(O)=O.
The standard release rule is r(u)=c, u>O
This release rule is essentially independent of the content,
as the store content is released at a constant rate as long as the store
is not empty.
Some general release rules which have been considered
which are dependent on store content are r(x)=a+bx and r(x)=cxa .
Storage processes with general release rules which depend on the content of the system have more recently become a subject of interest.
In
1971 Cinlar and Pinsky opened up the field with a landmark work in this
area.
They focused on a storage process with input which has stationary
and independent increments and finite jump rate, and a release rule r(x)
which is a Lipschitz continuous, strictly increasing function.
They were
able to show that if the mean input is less than the supremum of r(x)
over all x, then the storage content has a limiting distribution which
is independent of initial conditions.
Brockwell (1977) extended this work to show that under certain conditions, a store with input which is a pure jump Levy process (with perhaps
infinite jump rate) will admit a stationary distribution of store content.
In particular, in Lemma 5.4 he proved that for r(x) strictly positive,
nondecreasing and continuous on (0,00), if r(O+»O, expected input in
(O,t) At finite, and s~p r(x) >EA then a stationary distribution
1
exists and is equal to the limiting distribution.
In 1976 Harrison and Resnick derived necessary and sufficient conditions for the existence of a stationary and limiting distribution of
store content in the case where the input process is Compound Poisson
~
with finite jump rate, and the release rule meets the following conditions:
15
i) r(x) is strictly positive for x >
°
ii) r(x) is left continuous
iii) r has a strictly positive right limit everywhere in (0,00)
iv) R(x)= ft ~ < 00
o rlY)
°< x
<
00
With these assumptions not only were Harrison and Resnick able to find
necessary and sufficient conditions for the existence of a stationary
(and limiting) distribution of store content, but also a representation
of the distribution.
They proved the following two theorems:
Suppose A is the input rate and F( ) the distribution of input
size.
Define
Q(x)= A (l-F(x))
K(x,y)=Q(x-y)/r(x)
Kn+l(x,y)=~x K(x,z) Kn(z,y)dz
K*(X,y)=n!l Kn(x,y)
Theorem 1 (Harrison and Resnick):
The contents process Xt has a station-
ary distribution iff
1 = 1 + foo K* (x,o) dx < 00
Yo
0
in which case the unique stationary distribution y has density
g(x) = Yo K*(x,O) on (0,00) and y(O)=Yo.
Theorem 2 (Harrison and Resnick):
00
Xt is positive recurrent iff
1 K*(x,O)dx < 00 iff X has a stationary distribution y. In this case
t
o
.
the limiting distribution IT coincides with the unique stationary distribution in Theorem 1.
16
While the input distribution considered by Harrison and Resnick is
more restrictive than before, the release rule considered here is certainly more general than that of either Cinlar and Pinsky or Brockwell.
Unfortunately, as they pointed out, assumption (iv) rules out any
release rule which makes it impossible for the store to empty.
As mentioned in Chapter 1, the problem of proving the existence of
a limit when the input process is not homogeneous in time has only been
examined for the M(t)/G/1 * queue.
There we have two results.
The first,
stated earlier in this chapter, is for an intensity which approaches a
finite limit.
The second, proved separately by Sakr (1972), Yevdokimova
(1974) and Harrison and Lemoine (1977), shows that for a queue with nonhomogeneous Poisson arrivals with periodic intensity, a limiting distribution of waiting time will exist if the average traffic intensity per
period is less than one.
2.2 Some Basic Tools
We extend these results here to storage processes under a variety
of conditions on the input process and the release rule.
For the most
part, the input processes we consider will be much more general than
those that have been considered in the past.
One restriction we add is
the assumption that each realization of the storage process can be represented by the infinite sequence of inputs
= {Xl'
U1 , X2 , U2 , ••. }
where Xi is the time between the i_1 th and i th input and Ui is the size
of the i th input. We call such input discrete input.
W
*We adapt the standard queueing theory notation to let M(t) denote
a nonhomogeneous Poisson arrival rate.
17
A method of proof that will be used often throughout this chapter is
to compare two storage processes with the same input w, i.e., we will
compare
Zl(t)=Z(t,w,~,r,)
and
Z2(t)=Z(t,w'~2,r2)
where
Z(t,w,~,r)
denotes
the storage content at time t of a store with release rule r, initial
content
~,
be made on
and input w.
In forthcoming proofs various assumptions will
and r 1..
But by assuming that for a given realization both
~.
1
processes receive inputs at precisely the same moment and have the same
size, we remove any differences which could be caused by the input
structure.
Thus we can more readily compare the contents of the two
processes.
We will consider here two types of storage processes.
A Type A pro-
cess is defined as a storage process with release rule r(x) such that
for some "a", r(a)=O and
(1)
R(a,x) = f
x du
rruT
a
<
00
o< x
<
00
The function R(a,x) represents the amount of time it would take the content of the store to go from level X to level a if there were no input.
Thus (1) implies that for a Type A process ther'e exists a level a which
can be reached from any level X above a (with positive probability) in
a finite amount of time.
A Type B process is a storage process with release rule r(x) such
that for some level a, r(a)=O and for every r::
>
0
a + r:: < x
<
00
(2)
R(a+r::,x) <
00
but there is an X such that
(3)
R(a,x) =
00
18
Thus for a Type B process there is a level a which cannot be reached
from above in a finite amount of time, even if there is no input, but
the level of the store content can get arbitrarily close to a.
Note that without loss of generality we can assume a=O.
When a>O
we simply have the situation of a store with a IIfa1se bottom ll •
For sake
of simplicity, in the future we will assume a=O unless otherwise
specified.
At this point it is convenient to derive a few properties of Type A
and B storage processes with general release rules.
We define a con-
sumption curve as the graph a realization of store content Z(t) would
follow if there were no input.
These curves are described by the differ-
ential equation
dZ(t) = -r (Z(t))
dt
(4)
We take Z(t) to be right continuous, i.e. Z(t+O) = Z(t).
Thus (4) implies
that Z(t-) satisfies
(5)
From (5) we can see that Z(t) is continuous except where inputs occur,
since if there is no input in (t o- ~, to)
1 im
n~
Z(t - 1)
0 du n= 1 im
J
rruT
Z(t o)
i.e.
Z(to-O) = Z(to).
n~
[t
o
-
(t o
1)]
= 0
n
19
Thus we have that the consumption curves are continuous and strictly
decreasing functions of t passing through the point (to' Z(t o)) and every
positive level less than Z(t o)' The latter is true since for a Type A
or B process
b
f
a
dx
rrxT
<
a
00
<
a < b
We prove now some Lemmas concerning the storage processes.
Lemma 2.1.1:
Let ZI(t) = Z(t,w,
content of two storage processes.
~1,r)
and Z2(t) = Z(t,w,
~2,r)
be the
If for some t o Z2( t 0 ) -> ZI (t 0 ) then
Proof:
Let t 1 be the time of the first input after to' We first show
that Z2(t)~ ZI(t) for to ~ t <t 1· In the case ZI(t o ) ~ Z2(t 1) this is
clear, since ZI(t 1) ~ ZI(t o) ~ Z2(t 1). Thus we need to look at the
case Z2(t 1) < ZI (to)' Suppose the conclusion we seek does not hold and
Z2(t) < ZI (t) to ~ t<t 1. Then
But we also have
which implies that
e
20
This is impossible since r(u»o and Z2(to »Zl(to) and Zl(t»Z2(t).
we can conclude that Z2(t) ~Zl (t) for to ~ t<t 1.
Suppose at t
there is an input of size U.
1
Thus
We know
Then
Thus we know that Z2(t) 2Z1(t) both between inputs and directly after
inputs, which is at all times; i.e. we can conclude
for all t
Lemma 2.1.2:
Suppose r(u) is nondecreasing.
i
b
b+C
r rTuT
J
J rTuT
du <
a,b,c>O
du
rTuT -
a+c
Proof:
Then
a
For the integra l
b+c
du
we make the substitution y=u-c.
a+c
gives
i
b+~U _
rTuT -
a+c
rb~
J,.a r(y+c)
then
r+
du
rTuT
e
a+c
b
c
-
J. r1~)
b
=
=
d
Ja r(y~c)
-
b d
Jrfu
a
[b r~y~-r~y+c~ dy
r y r y+c
a
This
21
Since r(y) is nondecreasing r(y)
b+C
i
du
Lemma 2.1.3:
r(y+c).
Thus we have
b d
<
. rruTa+c
~
J r(~)
a
Let ZI(t) = Z(t,w'~l'rl) and Z2(t) = Z(t,w,~,r2) be the
contents of two Type A or B storage processes, with
rl(x)~r2(x).
If
for some to' ZI(to)~Z2(to) then ZI(t)~Z2(t) foY' all t~to'
Proof:
Let t 1 be the time of the first input after to'
t - to
Suppose that ZI(t)<Z2(t).
Then
Thus for to<t<t 1 , ZI (t)~Z2( t).
Suppose at t 1 there is an input of size U. Then
a contradi cti on.
For t o<t<t 1
22
Since we have Zl(t»Z2(t) in between times of inputs and directly after
•
an input, it is clear that Zl(t»Z2(t) for all t>t o .
2.3 A General Limit Theorem
We now proceed to results concerning the limiting value of store
content.
The first theorem shows that under certain circumstances the
existence of a stationary distribution is sufficient to insure that a
limiting distribution will exist which is independent of the initial
distribution.
Theorem 2.1:
Let Z(t) be the content of a Type A or B storage process
with nondecreasing release rule r(x) and discrete input. Suppose further
that for any E>O and initial content
variable
T(~,E)=T
such that Z(T)=E.
~
there exists a proper random
(If
~~E
then define T=O).
If the
process has a stationary distribution then it has a limiting distribution which is independent of the initial distribution, and the two
distributions are equal.
Proof:
Suppose we have two storage processes of Type A or B, Zl(t)=
Z(t,w'~1,r1)
and
Z2(t)=Z(t,w'~2,r2)
where r 1(x)=r 2(x) is nondecreasing
and both processes satisfy the assumptions of the theorem. Let ~1 ~ ~2.
By Lemma 2.1.1
for all t
(6)
It follows that
T(~1'E) ~ T(~2,E),
which together with (6) ,implies that
23
Suppose that the first input after T(E;l,E) occurs at t l . If we can show
(7) holds for all T(E;l,€)~t<tl then clearly it will hold for all t~T(E;l'€)'
Let T(E;l,€)<t<t l , Since there have been no inputs in this interval
we sti 11 have
i=1,2
(8)
so that
Now suppose at t l we have an input of size U.
Then
~
and we have from (6) and (8) that
Hence we can conclude that for all t >T(E;l'c)
Similarly, if E;{ E;2 ·we have that
From which we can conclude that
IZ I (t)-Z2(t)
I~
€
for t>max [T(E;l'c)' T(E;2'c) ] = T*
We can now easily show that
Zl(t)-Z2(t)~
in probability for
24
o + P{t<T*}
But
T(~i'£)
Let
F~
"'i
is proper for each i, and thus so is T*, which means
(x,t) = P{Z.(t) < x
'
-
I
Z.(o) =
,
~.}
,
i=1,2.
Since Zl(O) is independent of Z2(O) we have
i.e.
Suppose now that
~i
has distribution Gi . From (9) we have, taking
expectations
where Fi (x,t) = E
F~1
(x,t)
;=1,2.
25
We note that by dominated convergence
Thus
(10)
lim FI(x,t)
~
t -7<JO
lim F2 (x+£,t)
t -7<JO
Similarly, by interchanging the roles of ZI and Z2 and substituting x-£
for x we can get
By assumption a stationary distribution H exists for each storage
process.
Thus we may let G2(x)
t, and from (10)
lim FI(x,t)
t-7<JO
~
=
H(x).
Then
H(x)
H(x+£)
Letting £ go to zero we can get
lim FI(x,t)
t-7<JO
~
H(x+O)
Simi larly
H(x-O)
<
lim F1(x,t)
-t-7<JO
Thus
H(x-O)
~
lim FI(x,t)
t-7<JO
~
lim FI(x,t)
t
At each continuity point of H we have
lim FI(x,t) = H(x)
t-7<JO
the desired conclusion.
~
H(x+O)
=
F2 (x,t)
for all
26
2.4 Limit Theorems for Type A Processes with General Periodic Input
Theorem 2.1 requires a very strong assumption, the existence of a
stationary distribution, which in itself is often difficult to verify.
The next theorem, and those which follow, will not require such a strong
In exchange, more structure will be put on the input
assumption.
process.
We suppose there is a number w (called the period) and in each interval [kw, (k+1)w) there are a finite number of times where there is a
positive probability of an input to the system.
Let these points be
denoted.
kW+8 1 <kw+8 2< .•. <kw+e p
where the 8. are the same for each interval.
1
Each point has associated
with it a distribution of input size Gj(x), where we allow Gj(O+»O.
Laplace-Stieltjes Transform of these inputs in (O,t] t=kW+T,
M1(S,t) =
[j~l G;<Sl]k
j;l
O~T<W,
The
is
G;(S)
where
and
1
*
GJ.(s) is defined to be 1 if T < 8 1.
j=l
IT
We also suppose that in any interval (t,t + dt) there is a probability
A (t) dt + o(dt) that an input will occur, where A(t) is taken to be a
periodic function, i.e. A(kw+T) = A(T),
O~T<W.
The size distribution of
these inputs B (x,t) is also taken to be periodic.
The Laplace-Stieltjes
27
Transform of these inputs into the system in (O,t) is
M2(s,t)
=
exp {- f ot (1-8 * (s,u)) A(u)du}
The total input to the storage system is a convolution of these two
inputs, and has Laplace-Stieltjes Transform
We will call any input process which is characterized by this LaplaceStieltjes Transform, general periodic input or g.p.i. for short.
i th moment of input size will be denoted by S..
1
The
The g.p.i. is really much more general than we have encountered in
the literature.
Some storage theory assumes discrete-time input, some
have continuous time input, but none have both in combination as we have
here.
Another important feature of the g.p.i. is that it is not homogeneous
in time.
In particular, both the input size distribution and the rate of
occurrence of inputs is periodic.
Most work in this area has required
that inputs be independent of time, with stationary increments.
This
implies that the distribution of input in one time interval is equal to
the distribution of input in any other time interval of the same length,
as mentioned earlier.
With periodic input, we have that the distribution
of input in one time interval which begins at some point
start of a period, 0
~ T<
Wis the same for all periods.
T
from the
This fact allows
regenerative events to be of use here, as they have been of use in the
past for the homogeneous case.
It is clear that a change of time scale can transform the g.p.i. to
have period
w=
1.
Thus, without loss of generality we will assume
throughout the remainder of this work that
w = 1.
28
Theorem 2.2:
Let Z(t) be a Type A storage system with general periodic
input and nondecreasing release rule r(x).
Y(U)
= fU
o
Define
dx
rrxr
If
~
=
y
p
00
1~1 f
-
0
1
00
Y(u)Gl(du) +f f Y(x)B(dx,t)A(t)dt<1
0 0
then the store content has a quasi-limiting distribution which is independent of the initial distribution.
Proof:
Let Z(t) be the store content at time t and define the event €T
o < T < 1 as
Clearly,
that
£
T
£
T
follows:
£
T
occurs at t=n+l (n an integer) if Z(n+T) = O.
is a regenerative event, as defined in Smith (1955).
is periodic with period 1.
Z(n+~),
exist for
0
~ ~
Note
For a quasi-limiting distribution to
< 1 we must show, for a useful class of sets A,
that
AI)
€T is certain, i.e.
£
T
will occur a first time with probability
one.
00
A2)
r
~(n+~)
n=o
where
(I-F(n+~))
~A(t-m)
<
00
= P{Z(t)€A I Z(O),nt>O, Tnt = m}
nt = number of occurrences of €T in [O,t]
T = time of last occurrence of €T before t
nt
To prove Al and A2 we will use a series of Lemmas.
Before looking at
the Lemmas, though, we need to introduce the concepts of empty time and
~
busy time.
We define the empty time of the store in the interval [O,t),
denoted e(t), as the Lebesgue measure of the set {s < t: Z(s) = OJ.
29
In a corresponding manner we define the busy time in the interval [O,t),
denoted b(t), as the Lebesgue measure of the set {s
Notice that e(t) + b(t) = t.
<
t: Z(s) > OJ.
•
In addition, we can define a busy cycle.
A busy cycle is defined to begin when the store empties, and ends when
the store next empties.
The initial busy cycle begins at t=O and ends
when the store first empties.
Lemma 2.2.1:
Suppose Z(t) is the content of a Type A storage process as
described in Theorem 2.2.
lim E Ie(n)]
n~
Proof:
We now proceed to the Lemmas.
Then
>
0
n
To prove this lemma we need to construct an alternative storage
model which we shall call a warehouse model: The warehouse system will
receive the same input w as the original system.
~
If an input arrives
while the alternative store is not empty then the input is stored in a
separate warehouse.
When the alternative storE! is about to empty the
next input waiting in the warehouse is immediately put into the store to
prevent emptiness unless an input arrives at that moment, in which case
this input is added to the store, preventing emptiness.
Let ZA(t) be
the store content in the alternative warehouse model, and ZW(t) be the
warehouse content.
We claim that the total content in the ware'house system (both store
and warehouse) is at least as great as the content in the original model.
i.e. ZA(t) + ZW(t) ~ Z(t).
We prove this by induction on the number of
Suppose inputs occur at times t 1 , t 2 , ... with size U(t 1), U(t 2),
For 0 < t < t ZW(t)=O and ZA(t)=Z(t).
1
inputs.
4It
30
Define To = inf {t: ZA(t) = O}.
Suppose t l < To' At t l ZW(t l ) = U(t l ) and ZA(t l -) = zA(t ).
l
Z(t l ) = Z(t l -) + U(t l ). But we know Z(t l -) = ZA(t l -). Thus
Suppose, on the other hand, that To
and ZW(t l ) = O.
~
Also
tl.
Also
since the arriving input is put directly into the store, giving
Thus
Now suppose we know that ZA(t) + ZW(t) ~ Z(t) for all t ~ t k. We
want to prove that
for t
~
t k+l
For definiteness, let U(t l ), U(t l +l ), .•• U(tm) be the inputs stored in
the warehouse and define
There are a number of cases to consider.
31
Suppose t k < t < t k+1
Recall that
r
(t~~
=
rruT
Z( t-)
From Lemma 2.1.2 we have
and we know by assumption that Z(t k) ~ ZA(t k) +
Thus
tJ (t k)
which yields
I
Z(t k )
du
>
-
rruT
Z(t-)
For the inequality to hold we must have Z(t-) ~ ZA(t_) + ZW(t k).
Since t F T , ZA(t_) = ZA(t).
k
Also, for t £ (t k , t k+ 1) Z(t-) = Z(t) and ZW(t k) = t~(t).
Thus we have
32
= t k+1
Z(t k+1) = Z(t k+1)
At t
+ U(t k+1)
+
.::. ZA(t k 1 ) + ZW(\+l) + U(t k+1 )
= ZA(t k+1)
W
+ Z (t + )
k 1
The latter is true, since if t k+1
Z (t k+1)
= ZA(t k+1)
ZW(t k+1)
= ZW(t k+1)
A
and
A
Z (t k+1)
<
Tk then
+ U(t + )
k1
= U(t k+1)
+
ZW(t k 1) = ZW(t k+1)
and
since if an input arrives just as the alternative store empties, this
input goes into the store.
Suppose
Z(t k )
du
t < Tk• As we did for Case I we can get
=
i rrur
Z(t-)
Thus we have
33
which implies
Again since t
since t < T ,
k
E
(t , t k+1), Z(t-) = Z(t) and
k
ZW(t ) = ZW(t), which gives
k
ZA(t_) = ZA(t).
Also
we have
Thus
Case III:
t k < Tk ,l < ... <Tk,j < t < Tk ,j+l < .... <Tk,l <\+1 <T k,l+l
There can be one or more times before t k+1 where the store in the warehouse system will empty. We look at the interval Tk ,1 < t ~ Tk ,2 which
is typical of the others. Again, for t < Tk ,2
Thus we have
34
But Z(t) = Z(t-), ZW(T k,l) = ZW(t) and ZA(t_)=ZA(t), since no inputs or
emptiness occur at t. Thus
for
Tk,l
<
t
<
Tk ,2
which means
Z(T k ,2) = Z(T k:2) ~ ZA(Tk~2) + ZW(Tk~2) = ZA(T k ,2) + ZW(T k ,2)
So for Tk ,l < t
~
Tk+1
Clearly, it will also be the case that
for
Tk ,J.
< t <
Tk,J·+1
We now look at the last case.
Case IV:
Tk,l
Suppose Tk,l
<
<
t
~
t k+1 ~ Tk ,l+l
t
<
t k+1• As usual
which implies
Z(t-) ~ ZA(t_) + ZW(Tk,l).
~
For t < t k+1 we have Z(t-) = Z(t) and ZW(Tk,l) = Z(t).
35
Also, since t
(Tk,l' Tk ,l+l) we have ZA(t_)
£
= ZA(t).
This gives
For t = \+1
Z(t k+1) = Z(t k+1) + U(t k+1) ~ ZA(t k+1) + ZW(t k+1) + U(t k+1)
•
But we know that elther
ZA(t k+1) = ZA(t k+1) and ZW(t k+1) = ZW(t k+1) +
U(t k+1) if t k+1 < Tk ,l+l' or if t k+1 = Tk ,l+l then ZA(t k+1) = 0,
ZW(t k+1) = ZW(t k+1), ZA(t k+1 ) = U(t k+1).
In either case
Thus
With these four cases we have shown that
and by induction this then holds for all t.
Recall that we denote the busy time and empty time by b(t) and e(t)
respectively.
For the alternative warehouse model we will use bA(t) and
Since Z(t) ~ ZA(t) + ZW(t) we have e(t) ~ eA(t) and b(t) ~ bA(t).
U(t j )
We call Y(U(t.)) = f
du the standard busy time. We can think
eA(t).
J
0
rruT
of Y as the amount of busy time associated with the input U(t j
warehouse model.
)
in the
We note that the busy time associated with U(t j
original model will be no bigger than this standard busy time.
)
in the
36
For the warehouse model we have the inequality
The first term on the right hand side of (11) represents the standard busy time associated with the initial content Z(O).
The second
term represents the standard busy time from all inputs to arrive at the
warehouse system in (O,n).
The inequality indicates that possibly all
inputs arriving in (O,n) are not used up during that time (i.e. possibly
ZW(n) > 0) so the right hand side of (11) may be an overestimate.
Since eA(t)
=t
- bA(t)
we have
E [e(n)] ~ E [eA(n)]
=n-
E [bA(n)]
~ n - n~y -
E [V(Z(O))]
Then
E ~ ~ 1 - ~ - E .[V(Z(O))]
y
n
n
and since by assumption 1 lim
n-+oo
Lemma 2.2.2:
exists
T,
0
E
(e(n)]
Let
> 0
~y
>0
With a storage process as defined in Theorem 2.2, there
~ T <
1 such that £T
mean recurrence time
Proof:
> 1-
n
~y
l;t =
~
{~
T
will recur with probability 1, and the
is finite.
if Z(t) = 0
if Z(t) > 0
37
Then
J
n
l:t dt = e(n)
0
and
I(n) = E [e(n)]
n
= J P {Z(t) = O} dt
0
= oJ
n
n-1
E
j=o
) = O} dT
P {Z (
j+T
From Lemma 2.2.1 we know that lim 1
I (n)
n--)<lO n
>
O.
Then
1
lim
n--)<lO
J
0
nE1 P {Z(j+T) = O}
j=o
We can think of the event
E:
T
d T> 0
as a delayed recurrent event where
E:
T
stands for "a success occurs" (system is empty at t=n+T) in a sequence
of Bernoulli trials.
Pursuing this (in the manner of Feller) we note
then that
is the expected number of renewals in [o,nJ.
Wlhat can we say about
U (n)/n?
T
If
II
T
=
00
or the probability
1im
n--)<lO
If
E:
T
U (n)
.!...-
E:
T
recurs is less than 1 then
= O.
n
recurs with probability one and the first occurrence of
probabil i ty ph) > 0
then
U (n)
T
-n-
=p(T)
llT
E:
T
has
a
~
38
In any case, it is always true that UT (n)/n coverges to a limit as n
goes to infinity.
We will call this limie e(T).
Since e(T) -< ~1T we
have, by dominated convergence, that
>0
Thus we must have that e(T) > 0 on a set of positive Lebesgue measure.
Let e
= {T: e(T)
which implies that
Lemma 2.2.3:
>
~
a}.
T
<
For T £ e we have that
00
,
Z2(t)
recurs with probability one and p (T) > O.
T
For a storage system as described in Theorem 2.2, if
then the first occurrence of
Proof:
£
£
T
T £
e
is certain.
Suppose we have two storage systems, Zl(t) = Z(t,w,sl,r) and
= Z(t,w,
and s2 > sl
S2,r) which are identical except for initial store content,
= O.
By Lemma 2.1.1 we know that Zl(t)
~
Z2(t) for all t.
If there is a to = to (s2'w) such that Z2(to) = 0 then Zl(t o) = 0 and for
t ~ to Zl(t) = Z2(t). We will prove that there must exist such a to'
Then e 2(t) = 0 for all t. But this implies that
I 2(t) = E2 (e(t)) = 0, a contradiction as long as ~y < 1 (Lemma 2.2.1).
Hence there must exist a to'
Suppose not.
39
Suppose now that we have two processes, Z3(t) = Z(t,wl'~3,r) and
Z4(t) =
Z(t,wl'~4,r)
such that Z3(T) = 0 and Z4.(T) = ~ for some T £:
G.
Construct two new processes with T as the new origin:
*
Z3(t)
= Z3(t+T)
*
and Z4(t)
= Z4(t+T).
such that
*
*
Z3(t)
= Z4(t)
for
*
to(~,wl)
We know there exists a
t
*
~ to(~,wl).
By Lemma
2.2~.2
we know that for some
k>- t *0 (~,wl)' Z3(k+T) = 0, since for T E: G,E: T y'ecurs with probability
*
But Z3(k+T) = Z3(k)
*
*
= Z4(k)
since k ~ to(~,wl)'
and
*
Z4(k)
= Z4(k+T), i.e. Z4(k+T) = O. Thus we have shown that
one.
if we have
a storage system which meets the requirements of the Lemma, and T
then
E:
T
E:
G,
is certain.
Proof of Theorem 2.2 continued:
Choose T
E:
G.
From Lemma 2.2.3 we
00
know £:T is certain.
n~o ~A (n+~) (1 - F(n+~))<
To show that
00
note
that
00
l:
n=o
by Lemma 2.2.2.
Thus in conclusion, by Theorem 3 of Smith (1955)
The next result follows directly from Theorem 2.2, but now the release
rule is not constrained to be nondecreasing.
Instead, we assume that the
release rule is always bounded away from zero.
Theorem 2.3:
Let Z(t) be the content of a Type A storage system with
general periodic input and release rule r(x) such that
r(x) > a >
P
·l:1
J=
00
f
o
00
x d G.(x) + fl
J
0
f
o
x B(dx,u) A (u) du
for each
40
x >0.
Then there exists a quasi-limiting distribution of store content
which is independent of the initial content distribution.
Define two Type A storage processes with g.p.i.
Proof:
Z(t,w,~,rl)
and Z2(t) = Z(t,w,s,r 2) where r 1(x) is as described
in the theorem and r 2(x) = a, x > o. We will denote functions of the
ZI(t) =
. 1,.
2
. t·1, 1+
1. th process W1. th su bscr1p
Le t th e regenera t 1ve
. even t
be the event which occurs at time n+1 if Zi(n+T) = 0, T E
()
i T
0, i=I,2.
E
Since r 1(x) ~ r 2(x) we have Zl(t) ~ Z2(t). In particular,
Z2(t) = 0 implies ZI(t) = O. Thus if we can show that E2(T) is certain
we will have shown that E 1(T) is certain and ~1 <
From
Lemmas 2.2.1 - 2.2.3 we know that E 2(T) is certain and ~2 < if we can
and
~2 <
00
00.
00
show that
This follows from the definition of Y(u).
=
1
o
u
Since
dx - -1 1u dx = -1 U
r {x)
a 0
a
2
we have that
p
00
=
<
~
L
1=1
1
00
1 _ u G.(du) + 11 1 1 u B(du,t) A (t) dt
0
a
J
0
0
a
1 . a = 1
a
by the assumption of the Lemma.
From Theorem 3 of Smith (1955) we can con-
clude that a quasi-limiting distribution exists.
41
2.5 A Limit Theorem for Type B Processes with General Periodic Input
In looking at a Type A storage process we have the advantage of being
able to focus on the times the store is empty, a convenient event which
is regenerative in nature.
(1976) require
The limit theorems of Harrison and Resnick
assumption (iv) to rule out any process which cannot
empty for this very purpose.
But Type B processes are quite common and
one would hope that results similar to Theorem 2.2 or 2.3 should exist.
And indeed such is the case.
The next theorem is for a Type B process with periodic input--a
partner to Theorem 2.2.
The fact that a Type B process, while not
emptying, can have content arbitrarily close to zero is used to set up
a IIfalse bottom near zero where Theorem 2.2 can apply.
ll
Theorem 2.4:
Suppose Z(t) is the content of a Type B storage system
with general periodic input and nondecreasing release rule r(x).
Define
u dx
Y(u,E:) = { rTx).
If, for every E:
p
~E:
00
100
6
= lEI ~ Y(u,E:) Gl(du) + & Y(u,E:) B(du,t) A (t) dt
<
1
then the store content has a quasi-limiting distribution which is independent of the initial content distribution.
Proof:
theorem.
let Z(t) = Z(t,E:,l;,r) be a Type B process as described in the
In addition, define a sequence of storage processes
Zn(t) = Z(t,w,l;,r n) which have the same input wand initial input
the Z(t) process, but release rule rn(x), where
~
as
~
42
x > l/n
x ~ l/n
r (x) = {r(x)
a
n
Note that each Zn- process is a Type A process with "bottom" l/n.
~1/n
<
cess.
Since
1 the assumptions of Theorem 2.2 are satisfied for each Zn- proThus we can conclude that each Zn- process will have a quasi-
limiting distribution; i.e., if Hn(x,t) = P {Zn(t)
= -H (X,T)
lim Hn(x,k+T)
k~
n
~
x} then
-
where H is the quasi-limiting distribution.
n
Since {r n} is a nondecreasing sequence, by Lemma 2.1.3 {Zn(t)} is
nonincreasing. Thus {Hn(x,t)} is a nondecreasing sequence of distribution functions for each t.
This implies that {Hn(X,T)} is also a non-
decreasing sequence, since if m>n implies
then
-H (X,T)
n
=
lim H (X,k+T) < lim H (X,k+T)
k~ n
- k~ m
=
-
H (X,T).
m
Thus we know that a limit of the distribution functions exists.
-
Let
-
H(X,T) = lim H (X,T).
n~
n
-
-
-
Since {Hn(X,T)} is a nondecreasing sequence we have H1(x,T) ~ Hn(X,T)
for all n. H (x,T) is a distribution function, so lim H1(x,T) = 1. Thus
1
_
for every £ there is an A such that H1(A ,T) > 1 -E. Then we know that
-
x~
H (A
n
£
£
,T) > 1-£ for all n.
£
-
This means that {H n (X,T)} is a tight
sequence, since for all n
-Hn(A£,T) - -Hn(-A£,T) = -Hn(A£,T)
> 1 - £.
43
-
Thus we can conclude that H(X,T) is a distribution function and Hn(X,T)
-
converges weakly to H(X,T).
Next we show that for each n, 0
<
-
Zn (t) - Z(t)
< 1,
- n
for all t.
£nT be the event: £nT occurs at t=k+1 if Zn (k+T) = lIn, 0
Zn (t) is the content of a Type A process and
2.2.3 that for some T, £T is certain.
lJ : <
E
<
-
T<
1.
Let
Since
1 we know, from Lemma
Thus if we let Tn be the first
time Zn(t) = ~ then Tn is a proper random var'iable.
For t
<
Tn'
Z(t) = Zn(t).
Tn
t
<
t
<
Let t 1 be the time of the first input after Tn. For
t 1 we have Zn(t) = 1
< 1 for
n and Z(t) < 1,
n so 0 -< Zn(t) - Z(t) -n
At t 1 , there is an input of size U, say.
Zn (t 1 ) = U + Zn (t 1-) and Z(t 1) = U + Z(t 1- ) . So
<
t 1.
Then
o ~ [Zn(t 1-) + U] - [Z(t 1-) + U] ~*
.
But this is equivalent to
Thus for t
(12)
~
t1
o -< Zn (t) - Z(t)
<
1.
- n
Since (12) holds both between inputs and directly after an input, we can
conclude that it holds for all t.
From (12) it must hold that for every n
(13)
P {Z n (t) -< x} -< P {Z(t) -< x} -< P {Z n (t) -< x + l}
n
Let H(x,t) = P {Z(t) < xl
Then (13) can be rewritten
44
Hn(x,t) -< H(x,t) -< Hn(x + 1,
n t)
for all t.
This implies
(14)
Hn(X,T) ~ lim H(x,k+T) ~ lim H(x,k+T) ~ Hn(X + *.T)
k-+oo
k-+oo
For any E, let n be large enough that E > 1n·
-
1
-
Hn (x + -n,T) -< Hn (X+E,T).
-H(X,T)
Letting n go to infinity in (14) we get
< lim H(x,k+T)<lim H(x,k+T)
- k-+oo
Then
-k-+oo
~
-H(X+E,T)
Letting E approach zero we have
-H(X,T)
< lim H(x,k+T) < lim H(x,k+T)
-k-+oo
-k-+oo
~
-H(X+O,T)
-
-
At every continuity point of H we have lim H(x,k+T) = H(X,T).
k-+oo
-
Thus we have shown that H(x,k+T) converges weakly to H(X,T), the
desired conclusion.
2.6 A Limit Theorem for a Homogeneous Storage System
We close the chapter with a theorem concerning homogeneous storage
processes.
The result was originally developed as a response to the
paper by Harrison and Resnick (1976), whose conditions for the existence
of a limiting distribution are not at all obvious or intuitive.
Also,
it appeared that the conditions would often be difficult, if not impossible to verify.
In addition, the paper unnecessarily eliminated a
rich class of processes--Type B processes.
It seemed apparent that
some other level than the zero level could be used as a basis for a
45
regenerative event, allowing Type B processes.
For these reasons,
Theorem 2.5 was developed.
Theorem 2.5:
Let Z(t) be the content of a stor'age system with homo-
geneous Compound Poisson input, intensity A, and release rule r(x)
which satisfies the following criterion:
There exists a level c such that
r(x) > a
for x > c
r(x) -< Mc for x < c
for some constants a and M.
c
If the first moment of the input size distribution (8 1 ) is finite and
a > A 6 then there exists a limiting distribution of store content that
1
is independent of initial conditions.
Proof:
For the store content Zl(t) =
Z(t,w,~,Y'l)
as described above,
let us say E~ occurs at t if Zl(t) = c and Zl(t-) > c.
crosses c from above at time t).
(i.e. Zl(t)
We do not consider t,:=c an occurrence
of E: A.
Clearly E: A is a regenerative event.
1
If we can show the following
three criteria are satisfied then the desired liimiting distribution will
ex i st.
(81) E:~ is certain
(B2) The time between successive occurrence of E: A
1 has finite
expectation
(B3) For some useful class of sets C, ¢c(t)(l-F(t)) is of bounded
variation in every finite interval, c
¢c(t)(l-F(t)) = P{Zl(T~ + t)E:C,t 1>t
of the first occurrence of E:~.
E
C, where
A
A
~,T~}and T~
is the time
46
If c=O then we have r 1(x) ~ a > AS 1 for x > O. Define
Z3(t) = Z(t,w,~,r3) where r 3(x) = a ~ r 1(x). From Lemma 2.1.1
ZI(t) ~ Z3(t). Thus if Z3(t) = 0 then ZI(t) = O. From standard storage
theory (see Appendix) we know that the Z3 - process will empty with
probability one and the expected time between emptying is finite.
Thus
we have (81) and (82) satisfied for the case c=O.
To prove (81)-(82) for c > 0 we will need to define some additional
Let us say E~ occurs at t if ZI(t-)<c and
events and random variables.
A
B
Let U1 be the time from an El to the next occurrence of E1 ,
VI be the time from an E~ to the next occurrence of E~ and WI be the
ZI(t»c.
time an E~ to the next occurrence of E~.
r 2(x)=M c ' x >
o.
Define Z2(t)=Z(t,w,~,r2) where
All the above events and random variables apply to the
Z2 - and Z3 - process (defined above), denoted by appropriate subscripts.
Note that the Z2 and Z3 systems have homogeneous Compound Poisson input
and constant release rule.
Results on such processes are well known.
In particular, we know that Z2(t) and Z3(t) are type A processes and can
empty.
Thus for Z2(t) and Z3(t) we can introduce the event E j which
occurs at t if Zj(t)=Q and Zj(t-»O, j=2,3. The random variable Xj will
be the length of the busy cycle in the Zj - process, j=2,3.
To begin, we show that E~ is certain.
Suppose ~ > c.
For t ~ T~
we have Z3(t) ~ ZI(t), since r 1(x) ~ a = r 3(x). Thus E~ will be certain
if E~ is, and E~ will be certain if E3 is. But A SI < r 3(x) = a implies,
A
by standard storage theory, that E3 is certain. Thus we have that E 1 is
certain if
~
> c.
Let us next suppose ~ ~ c.
The event E~ will be certain if E~ is
certain and VI is a proper random variable.
Let T~ be the time of the
47
first occurrence of E~.
For t ~ T~ we have Z2(t) ~ Z1(t).
This is true,
since while Z1(t) < c, r 1(x) ~ Mc = r 2(x). Thus E~ will occur before E~
does, which implies that E~ is certain if E~ is.
Let
p = P{Z2(t) jumps above c during a busy cycle of the Z2 - process}
p
= P{Z2(t) jumps above c during the initial busy cycle of the
Z2-process}
q
=1
~
- p,
q
=1
~
- P
If we assume p > 0 then
P{T~
< oo} =
P{E~ first occurs during the initial busy cycle}
+ kI 1
P{E~
= p+
q p + q qp
= p+
q P (1
first occurs during the kth busy cycle}
+
q (q)2
p+
+ q + q2 + ... )
~
=p +
q
pip
=1
i.e., if p>O then the first occurrence of E~ is certain.
Now, in any
interval of length y we know that the output of the Z2 - process must
be less than or equal to McY' since r 2(x) = Mc " We know that with Compound Poi sson input there is a pos iti ve probab-j 1i ty of input greater
. .1mp 1·
th an c + Mc y. Th us we have p> 0 an d £2B.1S certa1n,
y1ng £1B.lS
certa in.
48
Note that ZI(T~) > c. Thus showing that VI is a proper random
variable is equivalent to showing that £~ is certain if ~1 = ZI (T~),
which we have already shown.
Hence we have that £~ is certain.
To show that (82) is satisfied we need the following lemma.
Lemma 2.5.1:
Let Zl(t) be the content of a storage system as described
in Theorem 2.5.
Let £~ occur at t if ZI(t)=c and ZI(t-»c, and WI be
A
the time between successive occurrences of £1.
E(W )
1
Proof:
<
If 82 <
00
then
00.
We employ the same processes and random variables as in the proof
of Theorem 2.5.
Define further the following random variables:
n is the length of a busy cycle of Z2(t) during which the store
content does not cross level c during the busy cycle.
s is the length of a busy cycle of Z2(t) during which the store
content does cross level c during the busy cycle.
Since WI = UI + VI we need to look at U1 and VI. Suppose ~=c. Then
U1 ~ U2. This is true since when Zl(t) ~ c,r 1(x) ~ r 2(x), so Z2(t)
will take at least as long as Zl(t) to rise above c. Now
where N, a random variable, is the number of times the process does not
cross level c during successive busy cycles, before finally crossing c.
Let SN = nl + n2 ... + nN' so that e = SN + S, a random sum.
n,s and N are mutually independent.
= E(N)E(n) + E(s)
Using Walds equation
Note that
49
We also have
Since S, nand N are independent
What is E(S~)?
E(S~) = E[E(S~ IN)]
= k!O P(N=k)E(S~)
k=l k
k
= ~ P(N=k){.~ E(n~) + 2 j~l l~j+l E(njn )}
k=o
J=l
J
l
00
2
= kEl P(N=k) {kE(n ) + k(k-l)[E(n)]2}
2
= E(n )
k~l
kP(N=k) + (En)2 {kEl k2p(N=k)-k E1kP(N=k)}
= E(n 2)E(N) + (En)2 LE(N 2)-£(N)]
Thus
We know that P(N=k) = qk p, where p and q are as defined in the proof
of Theorem 2.5.
Thus E(N) = q/p
Theorem 2.5 we showed that p>O.
E(X 2 )
<
00
if p>O.
But in the proof of
Both E(n) and E(r,) will be finite if
and p>O, since E(X 2 ) = q E(n) + P E(s). By standard storage
theory results, E(X 2) < 00 if Sl < 00 and A Sl < Mc ' both of which are true
by assumption. Similarly, E(n 2) and E(s2) will be finite if S2 < and
<
00
00
A Sl < Mc .
Thus we have that for
~=c
4It
50
E(U I )
~
E(U 2)
~
E(e) <
00
2
E(Ui) ~ E(U~) ~ E(e ) <
00
Suppose ~1c and define ZI(t) = ZI(t+T~,w,~,rI).
E(U 1* ) <
00,
where U1* is as defined earlier, but with reference to the
ZI * - process, rather than Zl(t).
A
ZI(T~
+ U1* ) > c, i.e. U1
for all
We know that
~
U1* .
Note that ZI * (U 1* ) > c implies that
Thus we have that E(U 1 )
~
E(U 1* )
~.
We still need to consider VI.
Suppose
that V1<V 3 ' and clearly we know that
~>c.
V3<~.
We have shown earlier
Thus we have that
E(V 1 ) < E(d
E(Vf) < E(~2)
•
and we know that these both will be finite since 61 < 00,6 2 <
00
and
A 6 < a.
1
Next suppose we drop the assumption that
Zl** (t) =
~=c.
Define
We know that E(V **
1 )<oowhere
VI ** is as defined earlier, but with reference to the Zl** - process. Note
B
Zl(t+T~,w,~,rl).
Then Zl** (O»c.
B
that ~l** (VI** )<c, which implies that Zl(V **
1 + T~)<C,
we have E(V 1 ).::. E(V **
)<oofor all ~.
1
i.e. VI
~
**
VI.
Thus
Since WI = U1 + VI we have E(W I ) = E(U 1 ) + E(V I ) < and since U1 and
VI are independent
00
and the lemma is proved.
51
To finish the proof of the theorem, we must prove (B3).
This can be
done using the following adaptation of Lemma 2 of Smith (1955).
Lemma 2 (Smith):
If (i)
I
=
(to+a, to+b] is some finite t-interval,
and if y. is a nonnegative random function of half-closed subintervals
I. of I with the properties
J
(a) YI
1
+ YI
=
2
YI
I for any adjacent subintervals II' I 2 c:.. I
1 2
(b) with probability one E(YI Ito, Z(D))
< ()
(ii) ItJith probability one, conditional upon to" Z(D), for any Ijc- I,
T1{Ijr\TS A} ~YI
~A(t) =
Then
~A(t)(I-F(t))
is of bounded variation in (a,b).
For the above lemma, to is the time of the first occurrence of some
event
E,
A is some useful class of sets, SA =
{t:
Z(t)£A},
~SA
denotes
the fronti er or boundary of SA and 11 denotes the cardi na 1ity of a set.
The original lemma required that (a) hold for any disjoint intervals
II' I 2 C.I, but since the additivity property was only used in a binary
manner over the disection of I, the property need only hold for adjacent
subintervals.
Since we are only concerned with showing that the distri-
bution of store content has a limit, we can restrict ourselves to the
consists of sets Ac = {x: x ~ c}.
In order to prove (B3) for the class-*we must find a set function
class of
sets~which
y. which satisfies (i) and (ii) of Lemma 2 above.
(1955) we define the random variable
For any interval (T. l' T.]
J-
J
= I.J
As suggested in Smith
o. as follows:
52
, =t
if Z(T j _1 ) EA and Z(T j) EA
or Z(Tj_1) ¢A and Z(Tj) ¢A
= 1
if z(Tj-1) ~A and Z(T) EA
=-1
if
01.
and Z(T j ) ¢A
Z(Tj_1) EA
We also define a second set function n1 . as the number of inputs into the
J
system in the interval 1j , and take y. = 2 n. + 0.
Additivity is clear for n.
easier to think of 0.
. To see this for 0.
as a sum of two functions a. and S.
1. = (T. 1,T.]
J
JJ
a
"
e
1.
J
=
S1. =
{ -1/2
if Tj-1 EA
1/2
if Tj_1 ~
1/2
if T. EA
-1/2
if T. ~A
J
{
J
J
Clearly ° 1 . = a 1 . + S1 .•
J
°I
j
J
For additivity we examine
J
+ °1 + = (a1j + S1 ) + (a 1j + + S1 + )
j 1
j
1
j 1
= a 1 . + (S1 + a 1 ) + S1
J
j
j+1
j+1
=-
since S1.
J
a
1j
a
But
1j +1 "
+S
it is somewhat
-a
1j +1 -
=0
1j Ul j +1
IjU1 j+1
+S
1j U1 j +1
where for
53
So we have 01 . + 0
=0
I jU1 j+1
J
I j +1
That (b) is true is obvious, since E • < 2 E n. + 1 and E(n.) <
Y since we are dealing with Compound Poisson input.
00
To show that (ii) is satisfied we must determine how many boundary
points there are in SA which are also in 1j . Let Ac = [O,cJ. Then
Z(t) E: Ac if and only if Z(t) 2. c. Note that if Z(t) E: Ac then
Z(t+s) E: Ac if there are no inputs to the sys tern in (t, t+s J .
is an input there are two possibilities
If there
(1) The process remains in Ac (hence creates no boundary points)
(2) The process jumps above c. Then we have a boundary point, and
possibly another if the process reenters Ac .
Thus an input during I. = (T. 1,T.J results in creating at most two
J
JJ
boundary points of SA -- the first if the input is large enough for Z(t)
to jump above c, followed by a second if the py'ocess falls back below c
before Tj . Let J k = (t k_1 , tkJ where t 1 , t 2 , ..•• , tN are times where
inputs occur in 1j , to = Tj _1 , t N+1 = Tj and N = n1 .. (Note, if t N =T j
then we say I N+1 = ¢.) We can write 1j = ~~i llk. Certainlyl\(1/'TS A) =
N+1
L: -n(J () ~SA)' We have
k
k=1
c
,
lI(J j
(\
-;-SA ) <
c
{~
if Z(T j -1) E:A c
if
Z(T. 1) ~A
J-
k
= 2,
... , N
if
Z(T j
if
z(T j) ¢Ac
)
.
c J
cA c
4It
54
Thus
.
•
1+2{N-1)+1
if
2+2{N-1)+O
if
2+2{N-1)+1
if
1+2{N-1)+O
if
We have shown that y. = 2 n. +
Z{T j _1) EA c and Z{T j
Z{T j _1) ¢Ac and Z{T j
)
EA
)
¢Ac
Z{T j _1) ¢A c and Z{T j
Z{T j _1) EA c and Z{T j
)
EA c
)
¢A c
o.
satisfies the assumptions of
Lemma 2 of Smith (1955), which implies that
variation for sets A of the type [O,c].
c
~A{t)
{l-F{t)} is of bounded
This is sufficient to complete
the proof of our theorem, since with (B1)-{B3) satisfied, theorem 2 of
Smith (1955) states that the limiting distribution of Z{t) exists.
In
fact, the theorem specifies that
lim P {Z{t) ~ c}
t~
00
=
~ ~ ~A (v) (l-F{v)) d v
where A = [O,c].
Sometime after this theorem was completed the paper by Brockwell (1977)
appeared.
His limit theorem is similar to Theorem 2.5, but Brockwell
assumed from the outset that r{x) was nondecreasing and continuous, while
this theorem does not.
•
CHAPTER 3
PERTURBATION APPROACH TO THE STORE
WITH NONHOMOGENEOUS INPUT
"
3.1 A Probabilistic Method
In this chapter we present an approach which will allow us to develop
formulae for various functions of a storage system with time dependent
input.
The method is direct and probabilistic in nature, making use of
the structure of the input process.
We will be interested, from this time forward, in a process with
general release rule, where the number of inputs into the system in any
interval (O,t] is a nonhomogeneous Poisson random variable, i.e.
P{l input in (t,t+dt)} = A(t)dt + o(dt)
P{more than 1 input in (t,t+dt)} = o(dt)
Let N(t) be the number of inputs in (O,t].
(l)
P{N(t) = n} = [A(t)]n e-A(t)
Then we have
A(t) =
n!
6t
A(u)du
Note that if A(t)=A, then A(t)=At and (1) is just the familiar homogeneous Poisson probability.
If we let B(x) be the distribution function of
input size, then the total input in (O,t], A(t), has nonhomogeneous Compound Poisson distribution, with
P{A(t) ~ x} = n~o e-A(t) [~ B*n(x)
n!
•
where B*n is the n-fold convolution of B with itself .
56
The key to the development of our approach will be to assume that
the intensity A(t) = Ao (l+E¢(t)) where A0 >O,E>O and ¢(t) is some measurable nonnegative function. The resulting input process can thus be
•
•
regarded as the sum of two independent Compound Poisson processes: one
with intensity A1(t)=A o ; the other with intensity A2(t) = AOE¢(t). We
will call the inputs into the system originating from these two sources
Type I and Type II inputs respectively.
time homogeneous.
Notice that the Type I input is
This will play an instrumental role in the develop-
ment of formulae, since, conditional on knowing when the Type II input
occur, the storage process has homogeneous inpUit.
In what follows, we will focus our attention on the probability of
The method developed here can be applied directly
emptiness, P{Z(t)=O}.
to other quantities, such as the mean store content and Laplace-Stieltjes
Transform of store content.
Let Nj(t) be the number of inputs of Type j
which arrive at the store in (O,t], j=1,2.
Then the probability the
store is empty can be written
00
P{Z(t)=O} =
k~o
P{Z(t)=O and N2(t)=k}.
We will examine this series term by term.
To begin, we note that
P{Z(t)=O and N2(t)=O} = P{Z(t)=O I N2(t)=O} P{N 2(t)=O}
If we know there have been no Type II inputs up to time t then the process (up to that time) is equivalent to the homogeneous store with
intensity AO. Let rr(t,A(·),VJ) denote the probability the store is
empty at t, for a store with Compound Poisson input, intensity A(·), dnd
initial content distribution W.
Then we have
e
57
and thus
-£A ~(t)
P{Z(t)=O and N2(t)=k} = rr(t'A o ,W)e
0
0
t
where ~(t)= 6~(u)du and Wo(x) is the distribution function of Z(O).
Next, suppose N2(t)=1.
in the interval (O,t].
The Type II input will occur at some point u
Let Ti be the time of the i th Type II input.
Then
t
P{Z(t)=O and N2(t)=1} = ~ P{Z(t)=O and N2(t)=1 and T1£du 1}
which is just
tp {Z(t)=O
6
I
N2(t)=1 and T1=u 1} P{N 2(t)=1 and T1£du 1}.
The probability of a Type II input in dU 1 is £Ao~(ul)dul + 0(du 1) and
the probability that this is the only Type II input in (O,t] is
t
t
£A ~(x)dx
_f , ~(x)dx
_f £A ~(x)dx
o 0
u £AO~
0 0
e
£Ao~(ul)dule 1 + o(dul)=£Ao~(ul)e dU 1 + 0(du 1)
u1
•
_f
~
We need to determine the probability the store is empty at t, if we
know that the first Type II input occurs in dU 1 and no more Type II inputs occur before t. The input up to time u1- is all homogeneous (Type
I) input, so at u1- the distribution of store content is W(x,u1-,Ao'W o)'
where W(X,t,A,W) denotes the distribution at time t of the content of a
store with Compound Poisson input, intensity A, and initial distribution
W(x).
At time u1+ there is an input to the store, and the size of the
input is independent of the content at u1. Thus the distribution of con-
tent at u1+ is the convolution W(X,U,AO'WO)*S(x). Let W(X,t,A/U) be the
content at time t of a homogeneous store which has Compound Poisson
input with intensity A,
occurs in duo
conditional on knowing that a Type II input.
Then we can write W(x,u1,Aolu 1) = W(x,u1,Ao'W)*S(x).
58
We use the shorthand notation WI(u l ) for this distribution function at
time ul .
Since there are no more Type I I inputs in (u I' t], the probabil ity
the store wi 11 be empty at tis jus t the probabil ity that a store wi th
homogeneous Compound Poisson input, intensity
)\0'
and initial distribu-
tion W(x,ul,Aolul) will be empty t-u l time units later, or
rr(t-uI,Ao,WI(u l )). We are therefore led to the equation
-£:A ¢( t)t t
P{Z(t)=O and N2 (t)=I} = e
o
-6 <j>(ul)rr(t-uI"Ao'W1 (ul))du 1 ·
In a similar fashion, we can see that for N2(t)=k, if the k Type II
inputs occur in du l , du 2 , ... , dU k then
P{Z(t)=o and N2(t)=k} =
The notation Wk(u ) represents the distribution function
k
W(x,uk,Aolul, •.• ,uk) where W(x,t'Alu l , ... un) is the content at time
t of a store which has homogeneous Compound Poisson input with intensity
AO ' conditional on knowing that Type II inputs occur in dU 1 , dU 2 , ... ,
We assume that W(X,O,A o lUI' ... ,u n ) = W(x)
for all n.
(I
59
We have thus derived the following formula for the probability a
store will be empty at t:
-EAO~(t)
n(t,A(.),Wo)=e
{n(t,Ao,Wo)+EA o
t
&
~(ul)n(t-ul'Ao,Wl(ul))dul
(2)
An alternate form of (2) expresses n(t,A(.),W o) as a power series in E.
To do this, simply note that
t
Then if we let Ik(t)= ~ ~
uk u2
~ ~(ul) ..• ~(uk)n(t-uk,Ao,Wk(uk))dul. .• duk,
t
k>l, Io(t)=n(t,Ao'Wo) and I 1(t)=
rewrite (2) as
&
~(ul)n(t-ul'Ao,Wl(ul))dul' we can
j
= J=O
.E rr.(t)E
J
(3)
where
Similar equations to (2) and (3) can be written for such quantities as
the distribution of store content and mean content.
3.2 Extending the Approach to General p(t)
All the work so far in this chapter has required that
~(t)
be a non-
negative function, since E\,~(t) was taken to be the intensity of a
60
Compound Poisson process.
sirable in general.
But this assumption is restrictive and unde-
Unfortunately we have been unable to prove a general
theorem which applies to a wide range of processes.
We feel that, at
the very 1eas t, the fo 11 owi ng can be shown to hold.
Conjecture 3.1:
For a storage system with nonhomogeneous Compound
Poisson input which has intensity A(t)=A o (l+E¢(t)) and a release rule
r(x) which is a finite degree polynomial in x, the Laplace-Stieltjes
Transform of store content can be written
(4)
where M(S,t,A('),W) is the Laplace-Stieltjes TY'ansform of the content Z(t)
of the store with intensity A(') and initial dlistribution W.
Outline of Proof:
We can show that M(S,t,A('),W O) satisfies the following
partial differential equation:
a M(S,t,A(') , W) + A(t)(l-B*(s)) M(S,t,A(') , W) =
---O:-a-:-t--(5)
m
s k~o (-l)k ak akM(s,t,A(')'W) - aosrr(t,A(')'W)
ask
where r(x) =
m
k~o
akx
k
find useful later also.
We do this in the following Lemma, which we will
4It
61
Lemma 3.1.1:
For a storage system with nonhomogeneous Compound Poisson
input which has intensity A(t) and release rule r(x), M(S,t,A(.),W)
satisfies the equation
(6)
aM(s,t,A(·),W) + A(t)(I-B*(s))M(s,t,A(·),W)
at
In addition, if r(x) =
m
k~O
k
akx then
(6)
= sE[r(z(t)~-sZ(t)].
becomes
aM(s,t,A(·),W) + A(t)(I-B * (s))M(S,t,A(·),W)
at
m
= s k~o (-I)m aka k M(S,t,A(·),W) - aosrr(t,A(·),W).
ask
Proof:
We need to consider
aM(s,t,A(·),W)
at
= lim M(S,t+T,A(·),W) - M(S,t,A(·),W)
T+O
T
Let Ao = AO(t,T) = {W:N(t,T,W) = O}
Al = A1(t,T) = {W:N(t,T,W) = I}
A2 = A2(t,T) = {W:N(t,T,W) > I}
where N(t,T,W) is the number of inputs to arrive in (t, t+T].
Dropping
the parameters A and Wfrom the notation for the present, we write
M(S,t+T) as
62
We will examine each integral separately.
Note that for small T,
the output of the system in (t,t+T] is r(Z(t))J.
If there is no input
in (t,t+T] then the content at time t+T can be written
Z(t+T) = Z(t) - r(Z(t))T if T is sufficiently small.
For such T
= Al e-sZ(t)dP + AISTr(Z(t)) e-sZ(t)dP + O(T)
o
0
(8)
where XA is the indicator function of the set ,0...
Since AO(t,T) is inde-
pendent of Z(t), (8) can be written
For a process with nonhomogeneous Compound Poisson input, if T is small,
Suppose there is an input in (t,t+T] of size U.
time t+T can be written, for small
this to write II as
T,
Then the content at
as Z(t+-r)=Z(t)+U--r(Z(t)h.
We use
.
63
I = A Je-sZ(t)-su+sTr(Z(t))dP
1
1
(11 )
Since A1(t,T),U and Z(t) are mutually independent, (11) becomes
Recall that P{A 1(t,T)} = A(t)T + O(T).
(12)
~
Thus
II = A(t)TB * (s)M(s,t) + O(T)
The third integral, 12 , is nonnegative and bounded above by a function of order O(T). We see this as follows:
We can combine (10), (12), and (13) to show that
aM(s,t)= -A(t)M(s,t)+s E{r(Z(t))e-sZ(t)} + A(t)B*(s)M(s,t)
at
-
Suppose r(x) =
00
k~o
k
-
akx , and let r(x) = r(x) for x>o.
E{r(Z(t))e-sZ(t)} = tr(Z(t))e-sZ(t)dP
n
= Jr(Z(t))e-sZ(t)dP
{Z(t»o}
Then
64
= 0,ir (Z(t) )e- sZ(t )dP =
r(°)II (t )
kE a £Zk(t)e-sZ(t)dP - aoII(t)
o k
m
= k~O
a k(-I)k aMk(~,t) - aoII(t)
as
Substituting (14) into (5) gives the desired conclusion.
(14)
We also know that (4) holds for ¢(t»o.
Outline of Proof continued:
This can be shown by using the perturbation argument presented earlier
in this chapter.
Thus M(S,t,A(·),W) as defined in (4) must satisfy (5).
It seems clear that (4) will formally satisfy the partial differential
equation, regardless of the range of values assumed by ¢(t).
If we can
show that (5) has a unique solution which satisfies the boundary conditions
M(O,t,A(·),W)
=
1
(15 )
<X>
M(S,O,A(.),W)
=
J e-sxdW(x)
o
then we will have shown that (4) is the
LaplacE~-Stieltjes
Transform we
seek.
For the store with r(x):::I,x>O, Reich (1958) has shown that there is
a unique probabilistic solution to (5).
For higher order polynomials,
we have been unable to find appropriate criteria for the existence of a
unique solution.
Corollary 3.1.1:
ing hold:
For a store as described in Conjecture 3.1 the follow-
65
where
~
Proof:
is the expected store content.
Recall that lim M(S,t,A(·),W )=n(t,A(·),W).
s~
0
0
as s goes to infinity in (4) yields (16).
lim aM(s,t,A(·),W
s~o
0
)/as=~(t,A(·),W).
0
Taking the limit
Note also that
Hence using (4) we can get (17).
Some comments are in order about our chief assumption--that the intensity of our input process can be written in the form
A(t)=Ao(l+£~(t)).
At first glance this might be thought to be rather restrictive.
it is not.
Suppose we have an intensity A(t»O.
and £ we can define
~(t)
as
Actually,
For any choice of AO
~(t)=[A(t)-A
o]/£A 0 •
Of course, more suitable choice of A '£ and
O
~
can be made.
If
A(t) is periodic, we would want AO to be the average intensity per
period.
-
Then A = jWA(U)du
o 0
and
-
~(t)=A(t)- jWA(u)du
0
£t\(u)dU
where
wis
the period.
for AO would be L.
If we know that
A(t)~L
as
t~
then a good choice
66
3.3 An Exact Limiting Result
It is well known that in general, the transient probability
IT(t,Ao'W k) is difficult to calculate.
Thus determining IT(t,A(.),W o)
will be even more difficult, as it involves an integral of which
IT(t,Ao'Wk(u k)) is just a part.
For this reason, we will concentrate on
results concerning the limiting value of IT.
Suppose we know that A(t) converges to a finite limit.
let
A(t)=Ao(1+E¢(t))~Ao.
In particular,
We might expect intuitively that
o)~IT(oo.A 0 ). With only some minor assumptions on ¢(t) we can
show that this is indeed the case. Theorem 3.2, which proves this, is
IT(t,A(.),W
essentially an extension of part of Theorem 3 of
Tak~cs
(1955) for the
M/G/1 queue.
Before provi ng the theorem, we wi 11 fi rst prove a few 1emmas whi ch
will be useful both here and elsewhere.
The first shows that if the in-
tensity of the input process is bounded, then the probability of emptiness
and the content moments are bounded.
Lemma 3.2.1:
Suppose we have a storage system which has nonhomogeneous
Compound Poisson input with intensity A(t) whet'e
M1(t)~A(t)~2(t)
for
some positive functions M1 and M2 . Let Zi(t)=Z(t,wi,~,r) denote the
store content with intensity Mi(t), i=1,2 and Z(t)=Z(t,w,~,r) denote
the content with intensity A(t).
Then
e
67
Proof:
Suppose we have a process
Z3(t)=Z(t,w3,~,r)
where the Z3 - process
has two independent input sources, both with Compound Poisson input and
intensities A1(t)=A(t)-M 1(t) and A2(t)=M 1(t). Let Ni(t) be the number of
inputs in (O,t] from the source with intensity Ai(t), i=1,2. If, for the
Z3 - process, we know that N1(t)=O then
P{Z3(t) = 0
and
I
N1(t) = O}
~
P{Z3(t) = O}
E{Z~( t) I N1(t) = O} ~ E{Z;( t)}
But if the Z3 - process only has input from its second source, then
since A2(t) = M1(t)
P{Z3(t) = 0
and
E{Z;(t)
I
I
N1(t) = O} = P{Zl(t) = O}
N1(t) = O} = E{Z~(t)}
Note that the overall input rate of the Z3 - process is (A(t)-M 1(t))+
M1(t) = A(t), which implies that
P{Z3(t) = O} = P{Z(t) = O}
and
E{Z;(t)} = E{Zk(t)}
Thus we have
P{Z(t) = O}
and
~
P{Zl(t) = O}
E{Zk(t)} ~ E{Z~(t)}
Similarly we can obtain the other inequalities by introducing a pro-
~
cess Z4(t)=Z(t,w4,~,r) where the Z4 - process has two independent input
68
sources, both with Compound Poisson input and "intensities A (t)=M 2(t)-A(t)
1
and A2(t)=A(t).
Then
P{Z(t)=O}=P{Z4(t)=O}~
P{Z4(t)=O
E{Zk(t)} = E{Z~(t)} ~ E{Z~(t)
I
N1 (t)=O} = P{Z2(t)=O}
I N1(t)=O} = E{Z~(t)}.
This second lemma is a basic limiting resuH which will be used often.
Lemma 3.2.2:
A)
For any
~ELI
Ifl~k
If f(x) is a function such that
t
and f(x)-+L as x-+oo then
00
lim ~ ~(u)f(t-u)du = L J ~(u)du
t-+oo
0
B)
If f(x) is a periodic function, period
k-+oo, O~T<W, then lim J
k-+oo 0
Proof:
kW+T
~(u)f(kw+T-U)du
To prove part (A) we note that since
I~(u)f(t-u)
I~
KI~(u)
I, and
w,
K~(u)ELl.
Also,
and f(kw+T)-+f * (T) as
*
00
= J ~(u)f (T-u)du
Ifl~k
0
we have
~(u)f(t-u)-+L~(u)
as t-+oo.
Thus by dominated convergence
lim J
t-+oo 0
t
~(u)f(t-u)du
00
= L J (u)du.
0
Part (B) follows in a similar fashion.
In this case we note that
~':~(U)f(kW+T-U)=~(U)f(T-U). Thus by dominated convergence
1.1m
k-+00
kW+T
00
J ~(u)f(kw+T-u)du = J0 ~(u)f(T-u)du.
0
Theorem 3.2:
Let Z(t) be the content of a Type A storage process with
nonhomogeneous Compound Poisson input which has intensity
A(t)=A o (l+E<jl(t)) ,c>O where tll(t»O,
</lLL, and
-
<~(t)->{)
as
t}
<Xl.
If the
69
corresponding homogeneous storage process with intensity AO has a limiting distribution of store content, then
lim rr(t,A(·),W) = rr(OO,A )
t~
Proof:
(18)
0
By Lemma 3.2.1, since
A~A(t)
we know
rr(t,A 0,W0»- rr(t,A(.), W)
0
Taking upper limits of both sides of (18) yields
~
rr(OO,A O)
lim rr(t,A(·),(Wo)
t~
where rr(OO,A ) = lim rr(t,A ,W ) exists by assumption.
o
t~
0 0
We can use (16) to obtain the other inequality.
By Lemma 2.1.1 we
P{Z(t)=OIZ(O»O} and hence rr(t-uk,Ao'W k) ~
rr(t-uk'Ao'O) where O(x) is the degenerate distribution function 0(0)=1.
have P{Z(t)=O/Z(O)=O}
~
Then we have the inequality
~(t)
-EA
=e
0
t
{rr(t,Ao'Wo)+ ~ EAo~(u)rr(t-u,Ao,O)du
The summation can be simplified using the identity
70
=
[ f xk (p(u)du ]k
o
k!
This identity can be easily proved by induction.
Assume (20) is true for k=n.
Xn+l
~n+l(xn+l)= fo
For notational convenience, let
xn xl
f0 ... 1
¢(x 0 ) ... ¢(x n- l)dx 0 ... dx n
0
X
= 1 n+l¢(x )~ (x )dx
o
n n n n
since
d~l(xn)
dX n
The case k=l is obvious.
= ¢(x n) we have
~n+l(xn+l)= 6xn+I~~(Xn)d~I(xn)
n!
n+l( x + )
n1
(n+ 1) !
=~1
as claimed.
Using (20) we obtain
71
rr(t,A(·),W»e
o-
-EAO~(t)
t
{rr(t,Ao,W)+EA
f
¢(u)rr(t-u,A ,D)du
0o
0 0
kk t
k1
+ k~2 E AO ~ ¢(u)¢ - (u)rr(t-u,Ao,D)du}.
(k-l) !
00
Since ¢EL I we have that
lim f t ¢(u)~k-l(u)du
t~ 0
( k-l) !
¢~
= lim
[ t
f ¢(u)du
t~
]k<
00
k!
0
k-l ELI'
In addition, as rr(t,Ao'Wo) is bounded and approaches a
limit, we can use Lemma 3.2.2 to show
i.e.
k-l
t
lim ~ ¢(u)~JJil rr(t-u,A ,D)du =
t~
~!
0
k-l
6¢(u)¢ iuldu
00
rr(oo,A o)
~!
=
This yields
kk
k
+ k~2E Aorr(oo,A )~ (oo)}
0
k!
00
=e
-EA
~(oo)
o
rr(oo,A)e
o
EA
~(oo)
0
72
From (19) we have lim n(t, (.) ,W o ) :::.
n~,~)
and from (21) we have
t~
lim n(t,A(.),Wo ) < n(OO,A),
0
t~
n(oo:~) ~
so we have
lim II(t,A(·),WO )
t~
~
lim n(t,A(.),W o )
t~
~II(OO'AO)
which gives us our conclusion.
Note that we have not assumed here that E<l.
Thus Theorem 3.2 can
apply to any function A(t) which can be written in the form a+b(t) where
a>O,b(t»O and bEll' since we can write A(t)=a(l+ ~ b(t)).
that <p(t)+l as t~, l>O.
Or, suppose
If <p(t»l and ~(t)=<p(t)-l then we can write
A(t)= i(l+~<p(t))where i=A o (l+El) and ~=E/(l+El). If in addition ~El,
then Theorem 3.2 applies and II(t,A(·) ,wo)+n(OO,Ao(l+El)) as t~.
3.4 Periodic Intensity
The situation in the preceding section is the exception rather than
the rule.
In most cases we will not be able to get such simple answers.
It is impractical to attempt to evaluate all of the terms in the expan-
sian of II, or in fact many at all.
We can perhaps hope to calculate a
few terms, leaving a remainder term of order O(E k) as E:->{). To find
the probability the store is empty at t, up to an error of order of
magnitude O(E k) we need to consider at most k-l inputs of Type II occurring in (O,t].
To see this, recall that
73
00
P{Z(t)=O}=
n~o
P{Z(t)=O and N2(t)=n}
k-I
= E P{Z(t)=O and N2(t)=n}
n=o
and
00
P{N2(t)~k} = n~ke
~(t)
-£A
0
n
[£AO~(t)] = O(£k).
n!
vie will only look at the first two terms of the £-expansion, which
will give us the probability the store is empty at t, up to an error of
0(£2); i.e. we look at
(22)
TI(t,A(.),Wo)=IT(t,Ao,Wo)(I-£Ao~(t))+£Ao
t
6¢(u)IT(t-u,A o ,W,(u))du+R 2
where R2=O(£ 2 ).
This approximation, while appearing simple enough, is not easy to
evaluate.
The difficulty lies in the fact that WI is a function of u.
Since we are interested in the limiting distribution of IT, this problem
can be alleviated.
As we have seen in Chapter 2, under many circumstances
a limiting distribution of store content will exist, independent of the
initial content distribution.
Thus an advantageous choice of initial
distribution could simplify things considerably.
We will see that very
often a good candidate for this is the limiting value of W(X,t,Ao'W O)
which we will denote W(X,A O)' If this is taken as the initial distribu~
tion, then W(x,uI,Aolul)=W(x,Ao)* B(x).
Note that this is not dependent
74
on u1. For this reason, we will henceforth write W1(u1)=W 1 in this
situation. This will prove helpful, as we will soon see.
In order to be able to take advantage of the limit theorems of
Chapter 2, we assume here that we have periodic intensity, with
¢(t+k)=¢(t) and ¢(t) is Riemann integrable.
Then ¢ can be expressed
as a Fourier Series
00
(23)
¢(u)=a o+m=-oo
L: (b cos2 1rmu+c sin2'rrmu)
m
m
= ~ a e 2TIimu
m=-oo m
It will be convenient to assume that
not, then we can define a new function
61¢(u)du=O;
i.e.
~(u)=¢(u)-tl¢(X)dX.
a =0.
o
If
In this case
_
1
1
A(U)=A(l+~~(u)) where ~=Ao(l+E ~ ¢(x)dx) and ~=E/(l+E ~ ¢(x)dx).
It
will also simplify matters to assume ¢ can be expressed as a finite series.
We can substitute (23) into (22) to get the approximation
A
Making the substitution IT(t,Ao,Wl)=IT(t,Ao,Wl)-IT(oo,Ao) and noting that
IT(t,A o ,W)=IT(oo,A 0 ) we can rewrite (24) as
e
75
If the storage process satisfies appropriate conditions, such as
those of Theorems 2.2-2.4 then we should be able to find
rr(T,A('))=lim rr(n+T,A(')'W), O<T<l.
We need to evaluate terms of the
n-+oo
form
(25)
. ( ) n+T 2 .
lim e21I'1m n+T fo e- 7Tlmu n(u , A0' W)du
1 .
n-+oo
27T imT",0
We claim that this term equals e
rr (2 7Tim,A o 'W l ).
First we note that if ITEl l then since
by Dominated Convergence
lim In+;-2 7Timu IT(u,Ao,Wl)du = tOO e27Timu IT(u,Ao,Wl)du
n-+oo
To show that this integral is meaningful, we look at
We have Ie
-(l/n+iS)x
gence again,
'"
IT(x,Ao'Wl)I~lrr(x,AO,Wl)l,
so by Dominated Conver-
76
Havi ng detenni ned the 1imi tin (25), we can now wri te
n(-r,A('))
(26)
"-
Of course, the above limiting formula is contingent on nEll'
We
prove this in the following theorem.
Theorem 3.3:
Consider a Type A storage process with Compound Poisson in-
put and release rule r(x) such that there exists a level c where
r(x)~ex>Al3l
for x>c
r(x)<M
-c
for x<c
for some constants ex and
~1c'
let n(t,A,W) = n(t,A,W) - rr(OO,A).
If the
first two moments of the input size distribution (13 1 , 13 2) are finite,
then nEll'
Proof:
let EC be the event: EC occurs at t if Z(t)=c and
Z(t-)~c.
Define the following:
q(t,u) = probability the store is empty at t, given
E
C
occurred at
u, and EC has not yet recurred.
q(t) = probability the store is empty at t and
f
C
has never occurred.
e
77
G(t) = distribution of the time to the first occurrence of £c
F(t) = distribution of the time between successive occurrences of
Note that since £c is regenerative, q(t,u) is a function of t-u.
We will
write q(t,u)=q(t-u).
We have the following equation:
t
TI(t,A,W) = q(t) + ~ q(t-u)dG(u) +
t
t
q(t-u)d[G* F(u)] + I0 q(t-u)d[G* F* F(u)] + ...
o
I
t
= ij(t) + Io q(t-u)d H(u)
where H(u)=G(u) + G* F(u) + G* F* F(u) +
fies the renewal equation H(u)=G(u) +
~
Notice that H(u) satis-
u
H(u-x)dF(x) and thus is a
renewal function.
We need to determine TI(m,A o)' We have shown previously (in the proof
of Theorem 2.5) that £c is certain, so q(t)<P{£
c has not occurred by
t}~
as
t~.
If q(t) can be shown to be of bounded variation on every
finite interval, and q£L 1 , then by the Key Renewal Theorem we will have
lim I
t~ 0
t
1 00
q(t-u)dH(u) = - I q(u)du
~1 0
00
where ~1 = 1o udF(u). We have q(u)<-u~ dF(u) = 1-F(u) and
00
00
I q(u)du < I (l-F(u))du = ~1<00, by Lemma 2.5.1. Thus q£L •
1
o
- 0
To prove bounded variation we refer back to the proof of Theorem 2.5.
There we saw that for the class ~ of intervals [O,a], we can use Lemma
78
2 of Smith (1955) to show that the function
~A(t)=¢A(t)(I-F(t))
bounded variation in every finite interval.
Let A={O}.
is of
We have At:::+-
Note that if we suppose t: c occurs at to and its next occurrence is at
t o+t 1 then
is the function we are interested in.
The function I-F(t) is monotone, so (I-F(t))-1 is also monotone.
Then
(I-F(t))-1
is of bounded variation on every bounded interval.
This implies that
~A(t)/(l-F(t))
is also of bounded variation, since
the product of two functi ons of bounded vari at"j on is agai n a functi on
of bounded variation.
But
~A(t)/(I-~t»=¢A(t)=q(t).
Thus we have
shown that the conditions of the Key Renewal Theorem are met.
With this we have that
t
1 co
= q(t) + ~ q(t-u)dH(u) - -. J q(u)du
l1
r
0
t I l 00
= q(t) + J q(t-u)d(H(x) - - ) - - J q(u)du.
o
11
11 t
1
1
00
J dG(v)
u
Now q(u) ~
but vI
~ 11(00;
00
J J
o t
00
i.e.
= I-G(u), so 0Joo q(u)du = 0Joo (I-G(u))du = vI' say,
qt:L 1.
00
q(u)dudt < J Jt
- 0
Looking at the th"ird term of (27) we have
00
(l-F(u))dudt
= Joo JU (I-F(u))dtdu
o 0
79
00
= fo u(l-F(u)du
1
= 2" 112
<
00
by Lell111a 2. 5. 1.
For the second term in (27) to be in L , we must have
1
00
(28)
~
f
o
t
I 0f
I
q(t-x)d(H(s) - ~)
111
dt <
00
We have the inequality
t
00
f If
o
0
f
dt<
111
t
00
o
q(t-x)d(H(x) - ~)
q(t-x)
f
0
d(H(x) - ~)
111
I dt.
By Fubinis Theorem this can be written
00
f
o
00
f
x
q(t-x)dt
I d(H(x) -
00
~ )1.
111
00
The inner integral is ~ q(t-u)dt= ~ q(v)dv<
00
since q£L 1.
We know that H(x) is a renewal function of a delayed renewal process,
where the distribution function of the time until the initial event is
80
G(x).
Let Ho(x) be the renewal function of the ordinary renewal process.
Then H( x)=G( x)+G * H (x). Let O(x)=Ho(x) - -x we have
o
lJ 1
G * O(x) = [ G * Ho(x)
-
x
- ] +(
lJl
~1 -
By Theorem 8 of Smi th (1954), if lJ 1 <
00
-1
lJl
j
0
x
G(y)dy
and lJ2 <
00,
]
then
00
j IdH (x)- ! I <
as long as F(x) has an absolutely continuous compoo
0
lJl
nent. We would like to write F(x)=L(x)+A(x) where A(x) is an absolutely
00
Let X be the time between successive occurrences
continuous function.
of
E '
C
~
and let A(x)=P{X
x and no input in (O,t*J} where t* is the num-
ber which makes the following hold:
t*
/
dx
r(x)
= 0
Thus t* is the time the store will first empty if we start at level c
and there are no inputs in (O,t*J.
We have A(x)=O, x
~
t*.
Let Kc(x) be the distribution function of
the time from the first input to an empty storie until the content next
crosses c from above.
A(x) = e
Then for x > t*
_At*jX-t* -AU
0
Ae
Kc(x-t*-u)du.
Let A1(x) = l-e -AX and A2( x) = e -At* Kc ( x).
Then A(x) = Al * A2(x-t*).
00
Since Al is absolutely continuous, so is A.
Since G(x)
<
1 I
I dG * O(x) I
2.
Thus oI
II dO(x) I
< "'.
I dD(x) I
If
<
00.
we can show
81
IX
b I d(-x - -1
00
]..11
00
then we will have I
o
I
]..11 0
G(y)dy) I
<
00
d(G * H (x) - ~ )1
0]..11
<
1
=-
. But
00
00
I (l-G(x))dx <
00
]..11 0
since
I
I d(H(x) -
~)
]..11
I .::.
I
I dG (x) I +
I
I d(G * H0 (x) -
~ )I
]..11
00
we thus have I
o
I
d(H(x) - ~)I <
so Itq(t_X)d(H(X) o
]..11
~]..11 )
ELI'
00.
All this implies that (28) holds,
We therefore have shown that IT(x,A,W)EL 1.
3.5 An Example: p(x)=cos2TIX
The formula in (26) is not as formidable as it seems.
simple case.
Suppose
~(t)=cos2TIt.
This can be written
~(t) = ~ e- 2TIit + ~ e2nit . This makes (26) look like
We look at a
82
TI(T,A(·))=TI(OO,A
o
)~21 A0
[e-2niTfto(_2ni ,A ,W1)-IT(oo,A ) e- 2niT
0
0
·-2nl
+e
2niT
fi O(2ni >Ao,~Jl)-IT(oo,Ao) e2niT ] + R2·
2ni
Let TI o (2ni,A o 'W 1) = x+iy.
(30)
Then we can simpli~f the above equation to
TI(T,A(·)) = TI(OO,A ) +£A {xcos2m-ysin2m·-TI(oo,A )sin2n-d+ R
2
o
0
0 2n
The values of x and y will depend on the specific storage process we
are interested in.
Both the input size distribution and the release rule
are essential ingredients.
Chapters 4 and 5.
r(x)=cx, x>O.
We will explore this more extensively in
There we treat the release rules r(x)=c,r(x)=a+bx and
The expected content will also be examined in these cases.
e
CHAPTER 4
LIMITING RESULTS FOR THE STANDARD STORE
The standard release rule in storage theory is r(x)=c, x>O.
For
such a process, the content of the store is released at a constant rate
as long as the store is not empty.
at some length this storage process.
In this chapter we will investigate
We focus on the particular case of
nonhomogeneous Compound Poisson input with periodic intensity
A(t)=Ao(l+£~(t)) where 61~(U)dU=O.
that c=l.
Without loss of generality we assume
A simple change of time scale can transform a system with
c!l to one with c=l.
4.1 The Limiting Probability of Emptiness
A store with r(x)=l is a Type A storage process, since R(x,O) =
IX dy = X < 00 for all x < 00. Thus Theorem 2.3 can be used to establish
o
the existence of a quasi-limiting distribution of store content. We
need only assume that SlAo = Sl &lA(U)dU < 1.
In order to determine limiting values of either the probability of
an empty store or the mean store content we will need to look at the
homogeneous store.
The approximations derived in Chapter 3 require values
for rr(OO'Ao,W),~(OO'Ao'W) and rro(i,Ao'W).
The first two quantities are
standard results in storage theory: rr(OO,Ao'W) = 1-A oS1 and
AoS2/2(1-A oS1). (See Appendix.)
~(OO,AO'W)
=
A value for the Laplace Transform rro(~'Ao'W) can be determined
through the aid of the differential equation for the Laplace-Stieltjes
84
Transform of store content.
(1)
From Lerrma 3.1 thi;s equation is
aM(s,t,Ao'W) + Ao(l-B*(S))M(s,t,A o 'W) = sM(s,t,Ao,W)-srr(t,Ao'W)
at
Taking Laplace Transforms, with respect to t, of both sides of equation
(1) yields
or
(2) MO(S,Z,Ao'W) = M(s,O,Ao,W)-srro(z,Ao'W)
z-s + Ao (l-B*{s))
Since M(s,t,Ao'W) is a Lap1ace-Stieltjes Transform it is analytic for
R(s»O.
The Laplace Transform of Mis also analytic in the right half
plane, because an analytic function of an analytic function is analytic.
Thus any zero in R(s»O of the denominator in (2) must also be a zero of
the numerator.
Let ¢(s)=z-s + Ao (1-B*(s)).
that ¢(s) has a zero in R(s).::-..O.
sequence of real numbers Yk'
ger k, let
We will use Rouches Theorem to show
Fix z, R(z).::-..O .
Define an increasing
° < Yk < 1 such that Yk
t 1.
For each inte-
k be the contour {u: lu-al = Ao-ckJ where a=>-o+z and
chosen such that E k<A o(l- Yk)' We also let f(s)=z-S+A o and
gk(s) = -AoYkB*(s).
f
I
k is
,.
85
Clearly, both f and 9k are analytic in and on rko
If(s) I = Ao-£k and 19k(s) I ~ AoYk
I
B*(s)
I
On r k,
~ AoYk o By assumption
Hence by Rouches Theorem, the number of zeros of f in r
is equal to the
k
There is only one zero of f in r k
number of zeroes of f+g k in rko
(at S=AO+Z)° Thus there is one zero of f+g k in rko Call the zero sko
Let r be the contour {u: I u-a I = AO}o Certainly sk is inside ro
Since {sk} is a bounded infinite sequence there is an s* and a subsequence sk such that sk
n
+
s* as
n~.
The point s* is in or on ro
We
n
want to show that
~(sk
n
~(s*)=Oo
Let 9(x) = -A B*(s) = lim 9k(x).
o
k~
) = (f + 9k - 9k + 9)(sk )
n
n
n
= (f + 9k )(sk ) + (g
n
n
9k )(sk )
n
n
Now
and
as
n
+
00.
We have
86
Thus
lim
~{sk
n~
i.e.
}
=
n
a=
there is a zero of
~{s*};
~{s}
in or on r.
R{s*}>O then MO{S*,Z,AO'W}
If
M{s* ,O,AO'W}-S* rro{z,Ao'W}
=
O.
<00,
so we must have
If R{s*}
=
a then
we must check to see
But I M{S*,t,A 0 ,W}I -< 1, so MO{S*,Z,A (I ,W) -z
< 1 < 00,
o
and again we must have M{S*,O,A o ,W}-s* rro{z,A 0 "W) = O. Thus we have the
relationship:
if MO{S*,Z,A ,W} < 00.
{3}
rro{z,Ao'W} = M{S*,O,AO'W}
s*
The actual value of rro will depend on the initial distribution of
content.
Looking at equation 3.26 we see that the distribution of
interest is W {x} = W{x} * B{x} where Wis the limiting distribution of
1
content of the homogeneous store, intensity AOl' and B is the distribution
of input size.
For every t, M{S,t,AO'W} = M{S!,OO,AO'W}.
M{s,0,A o 'W 1} = M{S,OO,AO'W} * B*{s}.
{4}
M{S,OO,AO'W}
Thus
Smith (l%3) showed that
= s{I-A081}
s - Ao 0- B* {s }}
Recalling that z - s* + A (I-B*(5*}}
o
o
rr {z,A O'W 1}
= {1-A081}
z
=
B*{s*}
0, we can now write
87
The value of B*(s*} can be computed in two ways.
~(s).
way is to find s*, the zero of
to consider the following.
period.
The most obvious
Another, more illuminating way, is
Let G(X,A O} be the distribution of the busy
It is well known that the relationship
=u
holds.
- z - A0
- AO
Substituting into (5) gives
B*(u} = u - t - AO
- AO
•
or
(6)
-A oB*(u} + Aa +
A zero of (6) is u = s*.
(7)
B*(5*}
Z -
U
=0
Thus
= G*(Z,A O}
•
Returning to the original problem, we can now write
rro(z,A O 'W 1} = (1 - AoS1) G*(Z,AO}
z
and we thus have
Finally, we can write down an equation for the limiting value of the
probability of emptiness as
88
-
M
2 •
2
~im rr(n+T,~(.),W)=(I-~ 61)+E~ (1-~ 61)m~-M arne TIlmT[G*(2TI!m)_I]+O(E )
(8)
n~
0
~(t)
where
'I'
=
0
2TIlm
0
~ a e2TIimt
m=-M m
4.2 The Mean Store Content
An approximation to the mean store content can be obtained by using
the storage equation:
N( t)
(9)
Z(t)=Z(O) + j~o
t
Uj -
6 r(Z(u) )du.
Taking expectations of (9) we have
_
l1(t,~(.),W)
_
=
l1(O,~(·),W)
t
+f1EN(t)-6
t
_
(l-rr(u,~( .
t
),W))du
_
= l1(O,~(.),W) + 6 oJ ~(u)du-t + 0J rr(u,~(·),W)du
1
We can
sub~titute
for
rr(u,~(·),W)
into equation (10) to obtain
JtJu e2TIim(u-x)rrA(x' W)dxdu = JtrrA(x' W)[ p2TIim(t-x)1 ] dx
o 0
'1\0' 1
0
'1\0' 1
.-.
,
2mm
89
we have
a
k
= am/2rrim
kfO
M
-m~-M
(am/2rrim)
k=O.
Then
~(t,A(.),W)=~(oo,AO)+£AOSl~(t)+£Ao
mr_Mam6te2rrim(t-k)fi(x,AO,Wl)dX
mfO
As in Chapter 3, we have for mfO,
When m=O
n+T
lim 1
n~ 0
ooA
A
rr(x,A ,W1)dx=1 rr(X,A ,W1)dx
0
0
0
00
=1
(l-A Sl)(G(x,A )-l)dx
000
= -{l-A 011
S }y
where Yl is the expected length of the busy period. Thus we can finally
~
write
90
limv(n+T,A(·) ,W) = ~(00'Ao)+£AoB1~(T)-£Ao(1-AoB1)Y1aO
tr+oo
(12)
We shall henceforth denote the approximations to the limits in (8) and
(12) by n(T,A(·)) and 0(T,A(·)).
It does not follow immediately that the limit of
~(n+T,A(.),W)
equal to the expectation of the quasi-limiting random variable.
is
It is
still necessary to prove a condition such as uniform integrability.
We
do this by proving uniform integrability for a sequence of random variables which bounds our store content from above.
According to Lemma
3.2.1, for two processes with the same release rule and Compound Poisson
input
where ¢(t) ~ Mand ~k(t,A(·),W) denotes the kth moment of store content
at time t of a store with intensity A(·) and initial distribution W.
we can show that
sgP~2( n+T ,A
O(l+£M) ,W)< 00 then
which implies uniform integrability.
s~P~2(
n+T ,A(·) ,W)<
00
,
In standard queueing and storage theory it has been shown that as
long as Ao(1+£M)B <1 and B2<oothen, if A1=A o(1+£M),
1
(13)
lim
n~
~1(n+T,A1'W)
= A1B2/2(1-A 1B1 )
where A1B2/2(1-A181) is the expectation of the limiting value of store
content.
Since store content is nonnegative, (13) implies uniform
i ntegrabil ity for the homogeneous process.
Thus
If
91
and we have uniform integrability of the nonhomogeneous process.
This
implies the anticipated conclusion.
4.3 Special Case: p(t) = cos2'ITt
Let us now look at a specific example in more detail.
p(x)=cos2'ITX =
i e- 2'ITix + ~ e 2'ITix.
Suppose
Equation (8) becomes
Equation (14) has an alternative form without any complex numbers.
G*(2'ITi ,Ao)=x+iy.
Then
G*(-2'ITi,~)=x-iy
Let
and (14) can be written
(15) II( T,A (.) )=(1-A oB1)+E:A O( l-AoS1 ){xs i n2'ITT+YCos 2'ITT-S i n2'ITT}
2'IT
To determine
~(T,A(·))
in this case, first note that
~(T) = 4'I1Ti [2'ITiT
-2'ITiTJ. Since uo=O we do not need to know the
e
- e
'¥
expected length of the busy period here.
in this particular case is:
The approximation to the mean
92
(16)
= AoS2 - E:A o(l-A oS1) (x-l)cos21TT + E:A O. [21TS 1+(1-A OS1)Y]sin21TT
2( l-A oS1)
41T 2
4;2
4.4 A Queueing Theory Example
The actual value of IT and ~ depend on the distribution B(x).
Suppose that we have the most basic queueing situation-negative exponential service time with average service time
1/~.
It is well known that
the Laplace-Stieltjes Transform of the busy period of an MIMII queue is
G*(Z,A O)= ~ + AO + z - J(~+Ao+z)2_4AO~
2A O
and the mean waiting time in steady state is
Ao/~(~-Ao)'
Figures 4.1-4.3 present graphs of the approximation (15) of the
limiting probability of emptiness in an M(t)/M/l queue with
A(t) = Ao(I+E:cos21Tt).
1..
0
=
l.O~
We take
~=10
for all three figures, while
5.0, 9.5 in Figures 4.1-4.3 respectively.
In each figure, E:
takes on the values .1, .25, .5 and .9.
The values of rr(oo,A(')) fluctuate symmetrically about
rr(oo~Ao)
=
l-Ao/~~
on the size of E:.
(15).
for each E:, with the amount of fluctuation dependent
This was to be expected from the form of equation
It is also intuitively reasonable that the probability should
vary about the emptiness probability of the M/M/1 queue which uses the
average intensity per period as the arrival intensity.
In fact, the
M(t)/M/l queue with A(t) = A (1+E:¢(t)) is usually approximated by the
o
M/M/1 queue with A(t)=A o '
93
It is interesting to note how the extreme values of IT{T,A{')) differ
from IT{oo,A O). This gives an indication of how much the process is
smoothed by using a homogeneous intensity rather than a nonhomogeneous
one.
The extrema of IT{T,A{.)) can be determined by differentiating (15)
with respect to T:
aTI{T,A{')) = £A {l-A Sl)2TI{xcos2TIT-ysin2TIT-cos2TIT}.
aT
0
0
Setting the derivative equal to zero, we find that the extrema occur at
points TO in the period such that
tan2TIT o={x-1)/y.
Note that the extrema occur at the same point in each period, irrespective
of the value of £, which only effects the value at that point.
trema for each graph are shown in Table 4.1.
The ex-
If we consider the percent-
age deviation of the extrema from IT{oo,A O)' we see that the percentages
increase as A increases. These values are in Table 4.1.
O
It is also of interest to note the time from the maximum (minimum)
value of A{t) to the corresponding minimum (maximum) value of n{T,A{'))'
We shall call this the response time, as it measures how long it takes
for the queue to fully respond to extremes in the arrival pattern.
of response time are found in Table 4.1.
Note that as AO increases, the
response time increases.
Table 4.1
AO
Extreme
Values
1.0
5.0
9.5
.90+.08£
.50+.23£
.05+.045£
Values
% Deviation
from IT{oo,A )
O
Response
Time
9£
46£
90£
.0995
.1355
.1655
94
FIGURE 4.1
Probability of an Empty Queue
r(x)=l
AO=1.0 ~=10.0
A(t)=Ao(1+€cOs2nt)
1.
0.9 rL===============:::::::::~~~~============:==~~i.1lo.....,,~::::::~
0.8
0.7
0.6
>-
I-
~
......
0.5
OJ
et:
OJ
~
O. It
0-
0.3 0.2
:.1 t
I
I
0
0.1
0.2
1_--.1
0.3
O.Lt
I
I
I
0.5
0.5
0.7
TAU
0.8
0.9
1.
£=.10
£=.25
£=.50
£=.90
95
FIGURE 4.2
Probability of an Empty Queue
r(x)=l
Ao=5.0 11=10.0
A(t)=Ao(1+£cos2nt)
1.
0.9
0.8
0.7
>-
I-
......
--l
......
~
c:e
o
c::
0.6
0.5
e:=.10
e:=.25
e:=.50
e:=.90
0...
0.2
0.1
o
o
0.1
0.2
0.3
O.~
0.5
TAU
0.6
0.7
0.8
0.9
1.
96
FIGURE 4.3
Probabi 1i ty of an
Empty QueUl~
r(x)=l
1. =9.5
0
11=10.0
A(t)=Ao(1+Ecos2TIt)
1.
0.9
0.8
0.7
>-
I-......
0.6
--J
......
co
c:(
co
0.5
0
0::
0-
O.if
0.3
0.2
0.1
0
0
0.1
0.2
0.3
O.if
0.5
TAU
0.6
0.7
0.8
0.9
1.
97
Figures 4.4-4.6 are graphs of the mean waiting time of the
A(t)=Ao(I+£cos2~t).
.25, .5, .9.
Again, we let
~=10
and Ao=1.0, 5.0,
9.5 respectively for the three figures, and £ takes on the values .1,
M(t)/M/l queue,
The graphs of the mean waiting time are similar to those
of the emptiness probability.
metrically about
~(OO,AO).
The values of
~(T,A(·))
fluctuate sym-
The extrema occur at points TO such that
tan2~To=2~Bl+(I-AoBl)Y
(I-A oB1)(I-x)
Table 4.2 gives the extrema, percentage deviation of the extrema from
~(OO,AO)'
and the response time for each graph.
Table 4.2
AO
1.0
5.0
9.5
Extreme
Values
.011+.009£
.10 +.06£
1. 90 +.14£
% Deviation
from ~(OO,AO)
Response
Time
82£
60£
7£
.1071
.1768
.2434
We see that while response time increases as AO increases, the percentage
deviation of the extrema from ~(OO,AO) decreases, in contrast to what
happened with n(T,A(·)).
4.5 General Service Distribution-An Approximation
As mentioned earlier in the chapter, to determine explicitly the
value of n(T,A(·)) and ~(T,A(·)) for a store, we need to either know the
busy period distribution or be able to find the zero of the equation
z+A(I-B*(u))-u=O, R(u»O.
distribution.
This may not be an easy task for an arbitrary
An approximation to many general distributions which often
98
FIGURE 4.4
Mean Waiting Time
r(x)=l
;\0=1.0
11=10.0
;\(t)=;\
(1+Ecos2nt)
, 0
0.020
0.018
0.016
O.Ollt
0.012
0.010
2:
a:
w
0.008
L.
0.006
0.00q.
0.002
i-~
a
0
0.1
0.2
0.3
__lC-.--_L.__l_--L__-.-L
O.q
TAU
0.5
0.6
0.7
L _
0.8 ·0.9
1.
99
FIGURE 4.5
Mean Waiting Time
r(x)=l
1.. =5.0
0
11=10.0
A(t)=AO(l+£cos2TIt)
0.18
e
0.16
O. 1q
:z:
~ 0.12
0.10
0.08
0.06
O.Oq
0.02
e
0
0
0.1
0.2
0.3
O.q
0.5
TAU
0.6
0.7
0.8
0.9
1.
100
FIGURE 4.6
Mean Waiting Time
r(x)=l
A =9.5 1l=10.0
o
A(t)=A o (l+cos2nt) .
101
is simpler to deal with is the Weighted-sum Erlang distribution which
has density
00
(17) B'(t)=b(t)=k=~ c ke- lltll(llt)k-l; O<c k 2. 1, k~1
(k:r)!
With proper choice of {cm} and
tributions.
II
C
k=1
it can approximate a wide range of dis-
The Weighted-sum Erlang distribution can be seen to be related to
the more basic negative exponential distribution (which it contains as a
special case, when c 1=1), deriving from the relationship its advantage
as an approximation. This is easier to understand in the context of
queueing theory.
Suppose we let A(t)cmdt+o(dt) be the probability that
during (t,t+dt) a customer requiring m phases of service arrives at the
queue.
If the distribution of each phase of service is negative exponen-
tial with mean 1111 then B(t) as in (17) is the resulting (total) service
time distribution for each customer.
It is also interesting to note that this same approach can be used
to model a bulk queue.
In this case A(t)cmdt+o(dt) is to be interpreted
as the probability that during (t,t+dt) m customers will arrive at the
queue, each with negative exponential service time.
The Laplace-Stieltjes Transform of B(t) is easy to find.
00
=k~1
e
(18)
c llk foo e-(s~)t(S+ll)k t k- 1 dt
k
o
(k-l)!
(S+ll)k
00
k
= E c )J
k=1 k(S+ll)k '
= C (~)
We have
102
where
C(x) is the probability generating function of {cm}. Since we
know that G*(z,>..o) = S*(z+>"o->"oG*(z,>..o)) we cain solve for G*(z,>..o) once
we determine the probability generating function of {cm}.
4.6 A Special Case: cm=( I-a )am-l
Suppose cm = ( I-a )am-l , m>O.
C( x)
=
Then
x (l-a)
I-ax
Thus
= (l-a)z+>.. ->.. ~*(z >.. )+~
o
1 -
=
0
' 0
a~
z+>.. 0
->.. 0
G*(z,>..
)+~
0
~ (l-~)
z+>.. o -A 0 G*(z,>.. 0 )+~-a~
This can be transformed to the equation
Solving this for G* gives
The correct solution is the one which makes G*(O,>"o)=I.
103
We still need to determine IT(OO,A o) and ~(OO,A 0 ) to be able to find
n(T,A(o)) and ~(T,A(o))o We know that IT(OO,A O) = l-Ao81 and
81 = - ~s B* (s)
= -d
ds
I s=0
C(-lL)
s+~
Is=O
In the particular case c =(I-a)am- 1, m>O,
m
- d
as
1 - S!lL
s+~
s=O
=
Thus
In order to determine
d
2
82 = -:2 B*(s)
ds
Is=O
~(OO,AO)
we need to know the value of 82 , where
104
Again, for our example we have
S2 = -- v(1-a)
d (
ds
(s+~-a~)
2 )
Is=o
I
= 2~(1-a)
(s+~_a~)3
s=o
Thus
A
o
= ---'------
4.7 Another Example
In Figures 4.7-4.10 we have graphs of the probability of emptiness
and mean waiting time of the MX(t)/M/1 queue where A(t)=Ao(1+Ecos2nt)
t
and c = (~)m-1. We take ~=10 for all figures, and A = 1, 5.
m
o
not have any graphs for Ao = 9.5, since for this value of Ao '
We do
Ao131 = 9.5 x ~o > 1, and no quasi-limiting value of waiting time exists.
As before, in each figure E takes on the values .1, .25, .5, .9.
These graphs are similar to the others we have seen.
We see that
the response time increases as Ao increases, and the percent deviation
from the average increases for the probability of emptiness, but decreases
for the mean waiting time.
Tables 4.3 and 4.4 give these values.
105
Table 4.3
Ao
Extreme
Values
1
5
.85+.093£
.25+.135£
% Deviation
from II(oo,A )
o
Response
Time
11£
54£
.0689
.1589
% Deviation
from ].l(oo,A )
o
Response
Time
62£
22£
.1321
.1666
Table 4.4
e
Ao
Extreme
Values
1
5
.026+.016£
.45 +.10£
106
FIGURE 4.7
Probabil ity of an Empty
QueUl~
r(x)=l
2 (lJm-1 m>O
m=3 l3
'
C
>.. o=1. 0 ]1=10.0
>..(t)=>"o(1+Ecos2nt)
1.
0.9
:=:=========:=::::::::::~~~~::::::===========:::::::=s~~~:::Je:=.e:=. 10
25
e:=.50
e:=.90
0.8
0.7
0.6
>-
~
......
-l
......
co
c:e:
0.5
co
0
0:::
c..
O.lf
0.3
0.2
O. 1 --
o ____-l
o 0.1
L
0.2
L.__J.
0.3
O.lf
L.__.J
0.5
0.6
J ..
H
••••
0.7
TAU
L__.ul
0.8
0.9
1.
107
FIGURE 4.8
Probability of an Empty Queue
[
I
II
r(x)=l
c _2 (lJm-l m>O
m3 3
'
A =5.0
o
].1=10.0
A(t)=Ao(l+£cos2TIt)
1.
II
II
0.9
0.8
0.7
>-
r.....
-J
.....
Ct:l
c:e
Ct:l
0.6
0.5
0
0:::
0-
O.lf
0.3
0.2
:-------:::::::::==::~~::::::::=--------=:::::::::::::;~~=d
0.1
0
'---_.1..--_L_-L--_-.L-_---ll---_..J-_......1.-_-J-_---J.._--'
0
0.1
0.2
0.3
O.lf
0.5
TAU
0.6
0.7
0.8
0.9
1.
e:= .10
e:=.25
e:=.50
£=.90
109
FIGURE 4.10
Mean Waiting Time
r(x)=l
cm=~ (~)m-l ,m>O
~\=5.0
~=1O.0
A(t)=A
(1+£cos2nt)
•
0
1.
0.9
0.8
... e
0.7
z
a::
w
0.6
L
0.5
£=.10
£=.25
£=.50
£=.90
O.lI-
0.3
0.2
O. 1
0
e
0
0.1
0.2
0.3
O.lI-
0.5
0.6
0.7
TAU
0.8
0.9
1.
CHAPTER 5
LIMITING RESULTS FOR SOME STORES WITH GENERAL RELEASE RULES
There are many release rules that are of interest in storage theory.
The most common, r(x)=c,x>O was considered in the previous chapter.
output of such a
stor~
r(Z(t))=Oif Z(t)=O).
The
is not a function of the content (except that
In the class of IIgeneralll release rules, where
output is dependent on the content of the store, the release rules
r(x)=a+bx,x>O and r(x)=cx,x>O are the most popular in the literature.
Both these release rules will be discussed in this chapter.
5.1 The Limiting Probability of Emptiness; r(x)=a+bx
We look first at a storage process with nonhomogeneous Compound
Poisson input, periodic intensity A(t), and linear release rate
r(x)=a+bx ; a>O, b>O.
We would like to use Theorem 2.2 to show that a
quasi-limiting distribution of store content exists.
need to look at Y(u).
Recall that
u
Y(u) = [o dx
r{x)
=[
u dx
-
o a+bx
= -b1
, in the present case,
b
a u)
In(l~
The assumptions of Theorem 2.2 are met as long as
To do this, we
111
Thus we can conclude that a quasi-limiting distribution of store content
exists as long as (1) holds.
Suppose
~(t)=~o(l+£¢(t))
where ¢ is periodic and
~
1
¢(u)du=O.
The
formula for the limiting value of the probability of emptiness is dependent on the existence of a limiting distribution of the content of the
corresponding homogeneous store with intensity
~o.
From Theorem 2.5
this limit will exist if there exists a level c such that
x~c
and
r(x)~Mc
for x<c.
If
a>~oSl
is bounded on all finite intervals.
then
r(x)~AoSl
r(x»~oBl
for
for all x>O and r(x)
1
If a<~oSl we can choose c=b(~oBl-a).
oSl for x>c
- and r(x)<~
- 0 Sl=M c •
To be able to compute rr(T,~(·),W) we need to determine rr(oo,~o'W)
o
and rr (21Tim '~o ,WI) for m=~l ,~2, . . . ,+M and WI (x)=W(x)*B(x) where Wis
Then
r(x)=a+bx>~
the limiting distribution of store content for the homogeneous store.
An examination of the Laplace-Stieltjes Transform, M(s,t,~o'W), will be
helpful.
From Lemma 3.1 we know that this satisfies the following
equation:
(2)
aM(s,t) +[A (l-B*(s))-sa] M(s,t)+bsaM(s,t)=-asrr(t)
at
0
as
where we write M(S,t,A ,W)=M(s,t) and rr(t,~ ,W)=rr(t).
o
0
Let M(s) be the Laplace-Stieltjes Transform of the limiting value
of store content.
(t~)
M(s) satisfies (2) with aM(s)/3t=O.
we have the differential equation
Then in the limit
112
The solution to this is
(3)
M{s) = e 6{s)
[c-~
rr{oo)£s e- 6{X)dX]
a Ao s 1-B*{ )
where 6{S)=b s-lb
£
x x dx, and c is some constant.
M(0)=1.
This implies that c=1. As s approaches infinity, M{s) must
approach rr{oo), which is finite.
But
exp{6{s))~
as
that
or
(4)
rr{oo) =
b
a foo e- 6{x)dx
o
Substituting (4) into (3) gives
r e-8{x)
o
dx
We also note that
-{ )
a
a
. Ao(l-B*{s))
lim _dMs =~(oo)=- rr{oo)-- + 11m
s~
ds
b
b s~
bs
An application of 1'Hopita1 s rule yields
l
(6)
~(oo)
When s=O,
= arr{oo) - a + AoS1
b
M{s)
s~,
so we must have
113
It still remains to determine rro(z,AOW 1). We go back to equation
(2), where we take the Laplace Transform with respect to t on both sides
of the equation, to transform the partial differential equation into a
differential equation in MO(s,z).
This gives:
Reorganizing the equation into a more familiar form we have:
(7) aMo(s,z) + [Z+A (l-B*(S))-sa]MO(S,Z) = M(s,O)-sarro(z)
as
0
bs
bs
Solving (7) we have
MO(s
,
z)=exp{-~b JSdx
+ e(S)}{c_Jsexp[~ JX~ _e(x)][M(X'O)_~ rrO(Z)]dX}
1 x
0
b 1 u
bx
b
c = Joo xz/ b e-e(X)[ M(x,O) _ ~ rrO(z)] dx.
o
bx
b
Thus
Evidently, lim MO(s,z)=z-l;
00.
s~
But lim s-z/b ee(s)=oo.
that the terms inside the braces in (8) go to zero.
following expression for rro(z):
Then we must have
s~
This yields the
114
(9)
rro(z) = ~oo xz/ b- 1 e-e(x) M(x,o)dx
a ~ xz/ b e-e(x) dx
o
The value of rro(z,AO'W 1) is dependent on the initial distribution
W1(x)=W(X)*B(X). Thus we need to substitute M(u)B*(s) for M(x,o)=
M(x,0,Ao'W 1) in equation (9), yielding:
(10)
5.2 A Special Case: Computational Aspects of rro
We now have all the tools to evaluate n(T,A(·)).
is to designate values for a, b, B(x) and ¢(t).
All that is needed
We will specialize here
to the case ¢(t)=cos2~t, a=l, b=A O and B(X)=81e-x/81. We assume that
1+b81<e to ensure that a limit exists. This is so, since
AO
A
AoEY(U)= b E[ln(l+bU)] ~ ~ In(1+b8 1)<1 by Jensen's inequality.
x AO
that e(x)=b - lb In(1+8 1x).
Then
We -note
115
Also
Computationally, there are still a few considerations, as the
evaluation of rro(z) at z= ~ 2ni can present difficulty.
We determine
the inner integral of the numerator in (12) as
Let N stand for the numerator in (12); then
N is easily evaluated as proportional to a gamma function, using the
1
simple change of variable u=by; indeed
(13)
= bz/b+1 r(z/b)
116
The integral r(+ .2:gi) cannot readily be evaluated, but r(+ 2:gi + 1)
can, and we then can use the identity r(x+1)=xr(x). We have
r(+ 2~i + 1) = foo e-y y+ 2~i/b dy
- """l)
0
= foo e-y
o
~ 2~i/b
= ~ e _y cos [2~
0
00
log y dy
]
Y
]
log y dy + i ~ e- sin [2~
b log y dy
00
The two functions on the right are easily computable.
The evaluation of N2 cannot be done straightforwardly.
We note that the first integral on the right can be written,
z=.:!:.2~i,
as
If we also write D for the denominator in (12) we have
We write
for
117
With the above representations of N1 , N2 and D we can compute rro(z) and
th us IT (T , A( . ) ) •
5.3 Limiting Value of the Mean Store Content; r(x)=a+bx
The mean of the storage process is also of particular importance.
To derive the approximation we must determine the mean for the homogeneous
store.
Clearly, from (7)
lim aMo(s,z) = -~ rro(z) + ~ lim MO(s,z) + lim [M(S,O)-(Z+A o (l-B*(S)))M(S,Z)]
S-roo
as
S-roo
S-roo
bs
We note:
(1) lim MO(s,z) = 1 ;
s-roo
z
(2) By 1IHopital IS rule
lim M(s,o)-(z+Ao(l-B*(s)))M(s,z) = -].1(0)+Z].1°(z)-A081/z
s-roo
bs
b
This yields
118
whi ch imp 1i es
t
t
~(t) = ~(O)e-bt+(A Sl- a )J e-budu+a J IT(u)e-b(t-u)du
000
Having performed this preliminary calculation of
~(t),
(17) into equation 3.17 to obtain the desired approximation.
1
= ~aIT(oo,Ao)-a+AoS1)-£Ao
we substitute
Note that
~IT(OO,AO)~(t)
b
(18)
+£Ao[Err(oo'AO)+S1]~t¢(u)e-b(t-U)du+£Aoa~t¢(t-U)6uIT(X'Ao'W1)e-b(~~~~+O(E2)
5.4 A Special Case: p(u)=cos2nu
In the case ¢(u)=cos2nu, ~(t)_sin~;nt)
t
{
J cos2nue -b ( t-u )du=e -bt
o
and
.
e bu [bcos2nu+2nsln2nu]
b2+4n2
}t
0
119
= bcos2nt+2nsin2nt-be -bt
b2+4n 2
(19)
To determine
~(T,A(·))
we need to study, from (18),
u
t
~ cos2n(t-u) ~ IT(x,A o 'W 1) e-b(u-x)dxdu = J(t), say. Since
Icos2n(t-u)e-b(u-x)IT(X,AO'W ) I~ e-b(u-x), integrable in [o,t]x[o,u], we
1
can interchange integrals to get
{
bcoS2n(t-X)+2nsin2n(t-X)-be-b(t-X)}dX
b2+4n 2
To determine 0(T,A(·)) we need to determine lim J(n+T).
As we saw
n-7<lO
in the previous chapter
1 2TIlT
. ~.
. J n+T ITA( X,A ,WI ) cos2n (
) dX=-r
11m
t-x
ft ~2nl
n-7<lO
and
0
0
) 1 - 2·
nl TAO
. ,Ao,W )
IT (
-2TIl
I
,AO'~~ +~
120
From Lemma 3.2.2, since n(t,A o 'W ) is bounded and converges to a
1
· . t , an d e -bx £Ll' we have th at
11m1
Thus
+
£Aoan(oo'Ao)[bSi~2rrT
b2+4rr2
rr
- COS2rrT+1] - £Aoan(oo,Ao)
b2+4rr2
= an(oo,A o )-a+Ao81 + £Ao81 [bcoS2rrT+2rrSin2rrT ]
b
b2+4rr2
(20)
+ £A a
°
Equation (20) can alternatively be written without using complex numbers.
First we note that fio(z)=no(z)-n(oo)/z.
If we let
nO(2rri ,A o ,W )=x+iy, then after some algebraic manipulation we can write
1
(20) as
121
i1(T,A(·)) =
(21)
As before, we need to establish that lim
~(n+T,A(·),W)
is the expec-
n~
tation of the limiting value of store content.
store with r(x)=a+bx and A1=A o(1+S»A(t).
1
~(oo,A1)=E-<aTI(OO,A1)-a+A1B1)
We look at the homogeneous
We know from (6) that
as long as a+bx2:.AoS'1.
We also see from (17)
that
Since we know that a limiting distribution of store content exists,
1im
t~
~(t,A(·) ,W)=~(oo,A
), and all random variables are nonnegative, we
0
know that the store content is uniformly integrable. In addition, since
by Lemma 3.2.1 we have
~(t,A(.),W)~(t,A1'W),
the content of the nonhomo-
geneous process must be uniformly integrable, which gives us what we seek-that for each
~i1(T,A(·))
is the approximation to the expectation of the
quasi-limiting value of store content.
5.5 Some Graphs
Figures 5.1-5.6 are examples of graphs of the probability of an empty
122
store and mean store content of the nonhomogeneous store with r(x)=l+bx,
A(t)=Ao(1+£cos2nt) and Ao=b.
9.5.
We have 81=.1 for all figures and Ao=l, 5,
In each figure, £ takes on the four values .1, .25, .5 and .9.
The probability of an empty store when r(x)=l+bx has form similar
to the earlier examples.
The mean store content, while similar in form,
varies considerably more than other stores.
Tand£, i1(T,A(')) is less than O.
In fact, at some values of
This is possible, since i1(T,A(')) is
only an approximation, and many of the remaining terms in the series
expansion of
~(T,A('))
may be positive.
Of course, if
~(T,A('))<O
we
take it to be zero.
Tables 5.1 and 5.2 give the extreme values, percentage deviation of
the extrema from the average, and response time for n(T,A(')) and
i1(T,A(')) respectively. While not so indicated, the values of i1 are
assumed to be nonnegative.
It should be noted that the difference in
the figures reflect not only the differences in AO ' but also the release
rules, since we have assumed that Ao=b.
Thus we cannot directly compare
either the graphs or the entries in the table.
Table 5.1
AO
Extreme
Values
% Deviation
from II(OO,A O)
Response
Time
1
5
9.5
.91+.08£
.67+.27£
.51+.37£
9£
41£
72£
.0775
.0777
.0659
123
Table 5.2
AO
Extreme
Values
1
5
9.5
.009+.007E:
.033+.044£
.049+.038£
Response
Time
% Deviation
from ll(OO,A )
o
78£
133£
78£
.0399
.4497
.4249
5.6 A Type B Process
Up to now we have spent a good deal of time examining the emptiness
of a store.
But there are some stores which cannot empty in a finite
amount of time.
These are Type B processes.
The content of a Type B
process can get arbitrarily close to zero, but
interest here is the mean store content.
Type B process is r(x)=cx.
IT(t,A(·),W)~O.
What is of
A typical release rule for a
For this particular store we will see that
we can get more than just an approximation to the mean content.
We begin with the basic storage equation
t
Z(t)=Z(O) + A(t) - c
6Z(u)du
where A(t) is the total input in (O,t].
Assuming the input is Compound
Poisson, A(t)=A (l+£¢(t)), we take expectations of both sides of (22) to
o
get
t
1l(t,A(·),W)=ll(O,A(.),W)+B1N(t)-c ~ ll(X,A(·),W)dx
t
(23)
=1l(0,A(·),W)+B 1(A o+£A o¢(t))-c ~ ll(x,A(·),W)dx
We take Laplace-Transforms with respect to t, of both sides of (23) to
get
llO(S,A(O),W)=ll(O,A(.),W)+B1Ao+£81Ao¢O(S)-CllO(S,A(O),W)
s
s
s
e
124
FIGURE 5.1
Probability of an Empty Store
r(x)=l+x
A =1.0 8 =.10
1
O
A(t)=A (1+Ecos2nt)
o
1.
0.9 Ir ======:::::::::::::=:::::::::~"..A~;:;;:;;;================~~"""'~~~~
0.8
0.7
~
I-
......
--l
......
0.6
0
O.S
ca
c::(
ca
~
0-
0.lJ-
0.3
0.2
O. 1
0
0
0.1
0.2
0.3
0.lJ-
O.S
0.6
TAU
0.7
0.8
0.9
1.
e:=. 10
e:=.25
e:=.50
e:=.90
125
FIGURE 5.2
Probability of an Empty store
r(x)=1+5x
1. =5.0
0
S(.10
A(t)=A o (1+£cos2TIt)
l.
r~-r-T-
0.9
0.8
0.7
>-
I-
......
£=.10
£=.25
0.6
£=.50
--J
......
co
~
co
0
0.5
£=.90
0::
c..
0.4
0.3
0.2
O. 1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
TAU
0.7
0.8
0.9
1.
126
FIGURE 5.3
Probability of an Empty Store
r(x}=1+9.5x
Ao=9.5
81=.10
A(t}=AO(1+£cos2~t}
l.
0.9
0.8
0.7
>-
I-
......
......
0.6
...J
co
~
co
0
0::
0.5
£=.10
£=.25
Q..
O.lf£=.50
0.3
£=.90
0.2
O. 1
L----.l-.._..I..-_JL-._.L.--I_I_---l.-I_..1-1_-1..I__
0
0
0.1
0.2
0.3
O.lf-
0.5
0.6
TAU
0.7
0.8
L.----J
0.9
1.
127
FIGURE 5.4
Mean Store Content
r(x)=1+x
AO=1.0
61=·10
A(t)=Ao(1+Ecos2nt)
0.020
0.018
0.016
0.01~
0.012
z
c:r:
l.LJ
::E
0.010
0.008
0.006
,
0.00lJ0.002 Q
o.
0.1
0.2
0.3
O.~
0.5
0.6
TAU
0.7
0.8
0.9
1.
128
FIGURE
5.5
Mean Store Content
r(x)=1+5x
A =5.0
=.10
1
A(t)=A ( 1+Eeos 27ft)
o
13
O
0.09
0.08
e
0.07
0.06
. 0.05
z
L5 O.Oif
~
0.03
0.02
0.01
0
e
0
0.1
0.2
0.3
O.if
0.5
0.6
TAU
0.7
0.8
0.9 "1.
129
FIGURE 5.6
Mean Store Content
r(x)=1+9.5x
Ao=9.58 1=·10
A(t)=Ao(l+£cos2TIt)
£=.10
0.09
0.08
£=.25
0.07
£=.50
0.06
£=.90
:z:
~
::E:
0.05
O.o't
0.03
0.02
Q.01
...
Oi.-
0.1
0.2
0.3
O.'t
0.5
0.6
TAU
0.7
0.8
0.9
1.
130
Then
~(t,A(·),W) = ~(O,A(·),W)e-
(24)
=
ct
+S1A
t
o
t
()
f e-cudu+EA S1 f ~(u)e-c t-u du
0
0
[~(o,A(.),W)-l]e-ct+SlAO
+EA
O
0
le-ct~t~(U)eCUdU
c
We take note here of a few important points.
First, equation (24)
t
only requires that ~ ~(u)e-c(t-u)du exist for all t.
shown that the entire expansion of
terms.
~
Second, we have
in E is contained in the first two
We have no need to rely on an approximation.
Third, we see that
the mean store content depends on the input size distribution only
through the mean of the distribution.
We now restrict ourselves to the case of periodic
~(t).
The last
term in (24) becomes
t t
-ct M
t 2 ·k
EA OS1 e- c ~ ~(u)ecudu = EA S1e k=~M a ~ e nl uecudu
k
o
=
Q
EAO~l
1
[
~
k=-M
~
a e2nikt _ -ct
a
]
k
e
k=-M k
2nik+c
=2n-i~k-+-c
If we go to the limit as n goes to infinity we get
(25)
(
()) =n::
1· ~ ( n+T,A ( . ) , W) =SlA + EAOSl k=-M
~ a ke2nikT
(2nik+c)
O
c
~ T,A·
We would like to be able to say that
~(T,A(·))
does indeed equal the
mean of the quasi-limiting distribution of store content.
First, we show
131
that a quasi-limiting distribution exists, as long as A081<c, by virtue
of Theorem 2.4.
For
dx = -1 [ lnU-ln£ ]
y(u,s) = J u -cx
£
c
'
and
The function f(x)=lnx is concave, so by Jensen's inequality
Thus, for any £>0
II
< 1.. A0[8 1-1 n£) -c
< 1.. A013 1 < 1,
£-c
and the assumptions of Theorem 2.4 are satisfied.
Let us now consider the homogeneous store with A1=A o (1+£M), where
¢(t)~M.
The Laplace-Stieltjes Transform of store content can be determined from the partial differential equation
(26) aM(s,t,A ,W)+A (1-B*(s))M(s,t,A ,W)= - scaM(s,t,A ,W)
at
1
1
1
as
1
Let M* (s,A 1 ) denote the Laplace Stieltjes Transform of the limiting value
of store content. We know this exists, since r(x»A1S1 for x>A P>1' so
1
Theorem 2.5 applies. Then M*(s,A 1) satisfies
132
(27) d M*(s,A 1) = -A 1(l-B*(S))M*(s,A )
1
ds
cs
To find ~(oo,A1) we take the limit of d~: as s~, in (27).
We get
But we already know that
~(t,A1'W) = ~(O,A1'W) e- ct+ SIAl (l_e- ct )
c
so
Since the store content and limiting value of store content in the
homogeneous store are both nonnegative random variables, we have that the
content is uniformly integrable.
that
~(t,A(·),W) ~ ~(t'A1'W),
By Lemma 3.2.2, since
A(t)~A1
we have
from which we can infer that the content
in the nonhomogeneous store is uniformly integrable.
conclude that for the nonhomogeneous store
5.7 An Example: p(t)=cos2nt
When p(t)=cos2nt equation (25) becomes
With this we can
133
= B1Ao + EA oB1 [c cos27T't+27Tsin27T't]
-c- c2+47T2
We now look at a specific example, the store with r(x) =
A(t) = Ao(l+Ecos27Tt).
j- x,
Figures 5.7-5.9 are graphs of the mean content for
this store.
We let B =.1 and A. =l, 5, 9.5 for the three figures, while E
1
o
takes on the values .1, .25, .5 and .9 in each figure.
The graphs of the mean store content are symmetric about
~(oo,Ao).
The extrema occur at points 'to such that
tan27T't 0
=27T
.
C
We note that 'to does not depend on either the input rate or input size
distribution.
Thus the response time is only a function of c.
The
actual value of the extrema points, though, do depend on the mean input
size and the average intensity
11. •
0
Table 5.3 gives the extrema, percent deviation of extrema from
~(oo,Ao)
and the response time for each graph.
The response time, of'
course, is the same for all Ao , while the percent deviations of the
extrema from ~(oo,A ) remain virtually constant. Thus, except for a scale
o
factor, for a given E the graphs are essentially identical.
Table 5.3
11.
0
1
5
9.5
Extreme
Values
.3 +.02E
1. 5 +.1£
2.85+.2£
% Deviation
from ~(oo,Ao)
Response
Time
6.6E
6.6E
.2416
.2416
.2416
7E
e
134
FIGURE 5.7
Mean Store Content
r(x)-~x
~'o =1.0
S(.lO
A(t)=Ao(1+£cos2TIt)
1.
0.9
0.8
0.7
0.6
z
0.5
a::
w
L
O.if
0.2
0.1
o
o
0.1
0.2
0.3
O.if
0.5
TAU
0.6
0.7
0.8
0.9
1.
135
FIGURE 5.8
Mean Store Content
r(x)-~x
~\=5.0
Sl=.lO
A(t)=A (1+€cos2TIt)
•
0
2.0
1.8
1.6
a:z==:c:
~
1. it
£=.10
£=.25
"£=.50
£=.90
~
e
1.2
z
1.
cr:
W
L
0.8
0.6
O.lf
0.2
0
_I
a
0.1
0.2
0.3
O.lf
0.5
TAU
L_--l
0.6
0.7
I
0.8
I_0.9
l.
e
136
FIGURE 5.9
Mean Store Content
1
r(x)=3 x
>':0=9.5
13 =.10
1
A(t)=Ao(1+£cos2nt)
if. 0 , - - - - - r - - - , - - - - - r - - - - , - - . . , - - - , - - - - , - - - - . , . - - r - - - - - ,
3.6
3.2
2.8
e:=.25
e:=.10
~~~~~~~~~~-_~::::::::::::======:::=;~~ e:=.50
e:=.90
2.if
z
a:
w
L:
2.0
1.6
1.2
0.8
o
o
0.1
0.2
0.3
O.~
0.5
TAU
0.6
0.7
0.8
0.9
1.
CHAPTER 6
A MATRIX APPROACH TO THE NONHOMOGENEOUS QUEUE
6.1 A Step Function Approximation to A(t)
In this chapter we depart from our previous perturbation approach
to the nonhomogeneous store and look at an alternative method for dealing
with the M(t)/M/1 queue which relies heavily on the computer.
was discussed in a paper by Leese (1967).
The method
Basically, the method involves
using a step function approximation to the arrival intensity to generate
transition matrices for each time period during which the arrival intensity is constant.
More specifically, suppose we assume that the nonnega-
tive real line [0,00) is divided into intervals
I n=[t n_1 ,t n) where to=O,
00
k~l Ik=[O,oo) , and on each interval the arrival intensity is approximated
by a constant, i.e. A(t):Ak for t£I k. If this is done, then on each
interval I k we essentially have an M/M/1 queue with arrival intensity Ak.
We can compute the probability transition matrix tk(t), which has
entries
where Qk(t) is the number of customers in the system at time t£I k, using
the well known formula for the transient probability:
138
(l )
+ (~ k)
+
(N-n+l)/2
In+N+1 ( ~ ~\II t)
(1- >r~)" 1=n!N+2 uS/\ (~Ak~ t) ]
Since the system is Markovian, by repeated multiplication of the
Ek
we can find the system size probabilities.
For example, suppose td 2
and we want to know the probability there are n in the system at time t
if we started at time 0 with N.
Looking at t} we see that there might
have been j customers in the system at that time with probability
P~,j(tl). The probability of going from j to n in the system in the
2
remaining t-t 1 time is P.J,n
(t-t 1).
This gives
This is just the (N,n) entry in the matrix product E1(t 1)E2(t-t 1).
follows in general that if tsl k then P{Q(t)=n !Q(O)=N} is just the
It
(N,n) entry in the matrix product.
In his paper, Leese complained that even with a judicious choice of
step functi on and form of the formul a to compute (l), the amount of
cOO1putation required would be formidable and perhaps excessive.
Thus
he did not recommend this technique as a practical approach to the nonhomogeneous queue.
Since Leese wrote his paper more than 10 years ago,
139
much progress has been made in computer technology.
It is likely that
in many instances the amount of computation required would now be considered routine.
In the particular case of periodic arrival intensity, this matrix
method seems well suited, even perhaps in Leese's time.
For periodic
A(t) the step function need only be fitted to the first period,
n
I=[o,w).
to find
If
~(w).
I=k~lIl
and A(t)=A i for tEl i then we only need N matrices
To find the limiting value of the size of the system, we
continue squaring
P=~(w)
until the limit is reached.
This should not
require many multiplications, as we have:
~(2w)
= P • P =
p2
~(4w) = ~(2w) • ~(2w) = p4
(3)
Thus to get ~(2kw) required k multiplications once ~(w) is obtained.
6.2 Construction of the Transition Matrices
Let us look at this method in greater detail.
We shall first con-
sider the construction of the basic transition matrix fk(t).
mula for the entries in this matrix was given by (1).
The for-
This form is not
well suited for numerical work, as it is not clear how many terms of the
infinite series of Bessel functions are needed for any particular level
of accuracy.
Leese suggests rewriting the formula as a power series in
A, but leaving the factor e-(l+A)t.
This gives the much simpler formula:
140
(4)
P (t) =
N,n
m~-o
m
[(At)me-At] F (t)
m!
N,n
where
F~,n(t) = Sm+N_n(t) + Tm_n(t)-Tm_n_1(t) . M(m+N+1,t)
T (t)
m
T .(t)
J
=f~;)j
J.
Q
j>O
j>O
(5)
S. (t) = e- tj.lT . (t)
J
J
j-1
(J()
M(j ,t) =
L:.
U=J
S (t) = 1- L: Su (t) .
u
u=o
It is interesting and helpful to note that the terms in (4) can be
interpreted probabilistically.
The factor
(;>..t)me-At is the probability
m!
of exactly m arrivals in [O,tJ, while the factor Fm
N,n (t) is the probability
of there being n in the system at time t, given that there were N in the
system at time 0, and there were exactly m arrivals in [O,t].
Since F is
a probabil ity we can make use of the fact that: it is bounded above by 1 to
determine the number of terms we need to use in (4).
The error term
We have that
141
k
00
= 1-e-).t E
m=o
E
m=k+1
can easily be calculated, and an appropriate value of k chosen for any
given level of accuracy.
Another nice aspect of (4) is that, if we can
choose a step function with equal-sized time intervals then only one set
of F values is needed.
The transition matrix
~
is, theoretically, doubly infinite.
course, computationally that is not feasible.
this particular problem in the method.
matrix.
nxn.
Of
Leese did not deal with
Naturally, we must use a finite
What we have done here is to truncate the matrix, making it
To retain the stochastic nature of the matrix, we replace the last
entry in each row by
n-2
P.. (t).
J=o lJ
Thus the last entry is in a
1 - .E
sense the probability that, starting with i customers, the queue will
have n-1 or more customers at time t.
Since we also limit the number of
rows, we are actually restricting the system to be no bigger than n-1,
and hence the value of IT will be somewhat overestimated, and
~
under-
estimated.
At what size should
to answer.
~
be truncated? This is a difficult question
Here we only begin to explore the answer.
Since we are
essentially limiting the system size by truncating, we should look at
the loss involved.
For the homogeneous queue in steady state
P{Q~}= pn where P=).Sl is the traffic intensity.
If, say, p=.95 then
P{Q>45}=.099; i.e., if we use a 45x45 transition matrix, allowing at
most 44 customers in the system, we "turn away" or ignore 1 in 10 customers.
Even if n=100,
P{Q~100}=.006.
It seems that in some cases the
142
size of the matrix required to give reasonable results will surpass the
1imits of many computers.
Cons i der further th,at to model a nonhomo-
geneous i ntens i ty wi 11 in general requi re a number of these matri ces.
The following table shows the 1I10s s
involved in truncating a queue at
ll
size n-1, for a few values of p and n.
It can be used as an aid in
determining the size of matrix needed to be within a particular level
of accuracy.
Table 6.1
"P"\n
10
20
40
50
75
100
140
.95
. 90
.75
.50
.30
.599
.349
.056
.001
<.001
.358
.122
.003
<.001
.129
.015
<.001
.077
.005
.021
<.001
.066
<.001
e
6.3 An Example: A(t) a Step Function
It is interesting to note an additional use of the matrix approach.
For an M(t)/M/1 queue where A(t) is a step function, the matrix approach
should yield values of IT and
~
which are very nearly exact.
If, in
addition, A(t) is a periodic function, we can compare the values of
IT(T,A(·)) and
~(T,A(·))
with the approximations ff(T,A(·)) and O(T,A(·))
presented earlier.
We now turn our attention to a specific e'xample.
a step function on each cycle.
Suppose A(t) is
The simplest example is where A(t) takes
on only two values: A(t)=A 1 for tE[k,k+w ) and A(t)=A 2 for tE[k+w 1 ,k+1).
1
Here we will set A1=9, A2=3, p=lO and w1=.1. This could be a crude model
for a situation where for a short amount of time we have a very high
arrival rate (busy or peak period) while the rest of the day the arrival
143
rate is moderate.
We find from Table 6.1 that a 40x40 matrix should be
sufficient for our purpose.
Only 5 multiplications of
~(1)
were necessary
To find the mean waiting time, we can use the
to reach convergence.
relationship
(6)
~(T,A(·))
= IIQ(T)]
~
where Q(T) is the quasi-limiting value of system size at time T which was
shown by Luchak (1956).
Note that rr(T,A(·))=P[Q(T)=O].
We will compare these two quantities, rr(T,A(·)) and
the approximations yielded by the perturbation approach.
write A(t)=A o (l+£¢(t)), where ¢(t) is periodic.
k~t<K+wl
k=O,l ,2, ...
we can choose AO' £ and ¢i' i=1,2 such that
k=O,1,2, ..•
k+w1~t<1
and A(t) = Ao(l+£¢(t)).
We would like to be able to write ¢(t) in the form
(7)
~ a e27fikt
¢(t) = k=-M
k
or, in the more traditional form of the Fourier series
00
(8)
¢(x) = Co + k~1(ckcOs27fkx+dksin27fkx)
with
We need to
If we have that
k+w1~t<1
k~t<k+w1
~(T,A(·))
144
Term by term integration of (8) gives
1
6ep(x)dx
=
Co
61ep(x)cos2rrnxdx = k~1 Ck 61cos2rrkxcos2rrnxdx
00
00
+ k~1 d k
1
bsin2rrkxcos2rrnxdx
Now
1
~ cos2.kxcos2.nxdx
={;
k=n
and
1
6cos2rrkxsin2rrnxdx
= o.
Thus we have that for each n
1
2
b ep(x)cos2rrnx
=
cn·
Similarly, we can show
1
2
b ep(x)sin2rrnx
= dn•
For our particular ep(t),
wI
en
=
2
epl
= -TIn
=
1
b cos2TInxdx + 2 ~lep2cOS2TInXdX
.
S 1 n2TInw
¢2 .
--1 1m s 1 n2-,rnw 1
(ep -ep )
;n 2 sin2rrnw 1
n>O
145
and
w1
1
dn = 2 0J ¢1sin2TInxdx + 2 w
J ¢2sin2TInxdx
1
¢1
= - -TIn
cos27fnw
¢1 ¢2 ¢2
- cos2rrnw 1
1 + -TIn - -TIn + TIn
=
n>O
w1
1
J ¢2 dx
c o = J0 ¢1dx + w
1
We can now write ¢(t) as in (7) by noting that
= k=_oo
~ a k e 2TIikx
where
a k=
k>O
k=O
k<O
But our method requires that the series be finite.
such that
We can choose an N
146
~
k-N+1
(a e2rrikx + a e-2rrikx)
k
-k
is small.
1
To use the perturbation method, we also need of ¢(x)dx=O .
requires a transformation of A(t) in the following way.
that f 1¢(u)du=a =c. Let ¢(t)=¢(t)-a o and
This
We have now
000
where
With the revised values of AO '
£
and ¢ we can now use the formulae 4.8
and 4.12 to determine IT(T,A(')) and ~(T,A('))'
quantities graphically with rr(T,A(·)) and
We can compute these two
~(T,A(·)).
The question arises, though, whether the results will depend on the
particular choice of
~
AO'
By definition
~
= A (l+£a )
000
£,
Ao and ¢i' i=1,2.
Let us look at the value of
147
Recall that AO' £ and ~i were chosen such that
which means also that A1-A2~£AO(~1-~2). Thus
A1=Ao(1+£~0)'
i=1,2,
So whatever the choice of parameters, ~o will be the same.
The other terms in IT(T,A(.)) and ~(T,A(·)) which involve our parameters are of the form £~oan' £~obn and £~o~(t).
and
~
Similarly we can show
We have that
148
£>.. o a n = (>"1->"2) sin2TInw 1
TIn
and
~>..
E:
0 bn = (>"2->"1) (cos2TInw 1-1)
TIn
Thus we see that our results will be independent of the choice of
E,
>"0' and </J.1 , i=1,2.
In Figure 6.1 we present the graphs of rr(T,>..(·)) and n(T,>..(·)) for
the parameters >"1=9, >"2=3,
~=10.
The solid 1i ne is the graph produced
by the matrix approach, while the triangles form the graph of the approximation.
Figure 6.2 displays the two graphs of the mean waiting time.
Again, the solid line is the graph produced by the matrix approach, while
the triangles form the graph of the approximation.
To quantify the com-
parison we have calculated the average, maximum and minimum values of the
vertical distance between the two graphs in each figure.
These numbers
are based on the values at T=k/50, k=1,2, ... 50 and are found in Table
6.1, below.
Table 6.1
Avera e
Maximum
Minimum
.00317
.00106
.02622
.00219
.00005
.00060
It is clear that in this example, at least, the approximations n(T,>..(·))
and
~(T,>..(·))
are very good.
149
FIGURE 6.1
Probability of an Empty Queue
A1=9 A2=3
~=10
1.
0.9
0.8
e
0.7
0.6
>t-
I--f
0.5
...J
A
I--f
co
a:
co
O.If
0
a:::
CL
0.3
0.2
0.1
0
0
e
0.1
£-Approximation
- Matrix Approach
~
0.2
0.3
O.If
0.5
TAU
0.6
0.7
0.8
0.9
1.
150
FIGURE 6.2
Mean Waiting Time
A1=9
A2=3
~=10
r
I
0.09
0.08
e
0.07
0.06
0.05
z
w
a:::
O.OIt
L:
0.03
0.02
0.01
0
0
O. 1
6 £-Approximation
- Matrix Approach
0.2
0.3
O.It
0.5
TAU
0.6
0.7
0.8
0.9
1.
e
151
6.4 Another Example: Periodic A(t)
We would like to be able to judge the adequacy of our approximations
for situations where A(t) is not a step function.
In such cases the
matrix approach to the M(t)/M/1 queue can only give an approximation
itself.
Suppose we were to approximate A(t) above and below by step
functions AL(t) and AU(t), where AL(t)~A(t)~ AU(t).
We know from Lemma
3.2.1 that
and
The matrix approach, at the least, can provide upper and lower bounds to
rr
and~.
We also know that the closer we approximate the intensity with
a step function, the closer that approximation will be to the "true"
value.
So as we let AU(t)_AL(t) become small, rr(T,A(.)) and ~(T,A(·))
will be squeezed in between their upper and lower bounds, and we can
evaluate the approximations n(T,A(.)) and ~(T,A(·)).
We shall look here at the example in Chapter 4 where
~(x)=cos2TIx.
As mentioned earlier, it is computationally efficient to choose equal
sized time intervals for the step function.
We let I k=[(k-1)/2N,k/2N )
where 2N is the number of intervals into which we divide the interval
[0,1).
Thus we choose for our step functions
AL( t)=A O(l +e:~~)
AU(t)=AO(l+e:~~)
te:l.1
152
where
<l>u = {
k
C:s(2~(k-l)/2n)
k=l, ... , N
<l>2N+1-k
k=N+1, ... , 2N
and
<l>L = { c:s (2~k/2N)
k
k=1,2,
<l>2N+1-k
k=N+1,
...
, N
, 2N
Figures '6.3 and 6.4 display the graphs of the probability of emptiness and mean waiting time, respectively, for Ao=5, ~=10, and €=.5. The
solid lines are the graphs of the upper and lower bounds, while the triangles form the graphs of IT(T,A(-)) and ~(T,A(-)).
We see from the
figures that, at least for the approximations, IT and
~
Here we have used 80 intervals for the step function.
are quite good.
Since the average
difference between the upper and lower bound is .0125 for IT and .005 for
~,
this seems to be a sufficiently fine step function approximation. '
We can make a few comparisons of interest.
First we look at the
deviation between the perturbation approximations and their bounds.
It
is also interesting to consider the mean waiting time and probability of
emptiness of a queue which has the homogeneous intensity AO- the average
intensity over a period.
This is the queue that is generally used as a
substitute for the periodic queue.
This can give us a basis of compari-
son to judge the error our approximation gives us.
In the following two
tables we present the average, maximum and minimum absolute differences
between the bounds of n(T,A(-)) and
from the two queueing models_
~(T,A(·))
We let
and the values obtained
153
01 = I~(T,AL('))-~(T,AO)I
02 = I~(T,AL('))-~(T,A('))I
03 = I~(T,AU('))-~(T,AO)I
04
where
= I~(T'AU('))-~(T'A(')) I.
~=rr
or
~,
as appropriate
Table 6.2 Probability of Emptiness
Average
Minimum
Maximum
e
°1
°2
°3
°4
.07334
.00076
.12728
.00687
.00025
.01415
.07219
.00286
.11394
.00625
.00064
.01243
Table 6.3 Mean Waiting Time
Average
~1inimum
Maximum
°1
°2
°3
°4
.02013
.00052
.03722
.00093
.00033
.00153
.01966
.00012
.03229
.00596
.00509
.00690
Since 01>02 and 03>04 it seems apparent that the approximations for
the periodic queue are better to use than the results for the homogeneous
queue.
It also seems that n(T,A(')) and ~(T,A(')) are sufficiently good
approximations that they should be the choice over the matrix results,
particularly when ease of computation is considered.
154
FIGURE 6.3
Probability of an Empty Queue
A =5.0
o
11=10.0
1.
0.9
0.8
0.7
0.6
r
IH
-.l
0.5
H
CCl
cr::
CCl
e>
O.ll-
0:::
0-
0.3
0.2
0.1
0
a
O. 1
0.2
£-Approximation
- Upper and Lower Bounds
~
0.3
O.ll-
0.5
0.6
TAU
0.7
0.8
0.9
1.
155
FIGURE 6.4
Mean Waiting Time
A =5.0
o
11=10.0
0.18
0.16
O. 1lf
0.12
z
a:
~
O. 10
w
L:
0.08
0.06
O.Olf
0.02
o
o
0.1
0.2
£-Approximation
- Upper and Lower Bounds
~
0.3
O.lf
TAU
0.5
0.6
0.7
0.8
0.9
1.
CHAPTER 7
STORAGE SYSTEMS WITH STOCHASTIC INPUT INTENSITY
7.1 The Method
The input process of a storage system will not always be a deterministic function of time.
It might seem more realistic to assume that
the rate of input varies randomly over time.
This would be the case, for
example, if inputs to the system originate from a variety of different
sources and the number of sources is a random variable.
We would like
to be able to examine processes such as these in which the input rate is
generated by some stochastic mechanism.
In particular, it would be good
if the method employed previously on storage processes with nonhomegeneous Compound Poisson input could be adapted to allow for stochastic
A(t).
Fortunately, it is a relatively simple matter to do this.
As before, we model our arrival intensity as A(t)=A o (l+£¢(t)). We
assume now, though, that ¢w(t)=¢(t,w) is a measurable stochastic process
t
such that E{¢(t,w)} exists and J EI¢(x,w) Idx <00, t>O. If we have a
o
realization of the process, ¢w(t), then A(t)= Ao (l+£¢w(t)) is just as
before in the deterministic case. Thus the probability of an empty store,
given that realization, is just the conditional probability
O<s<t}
157
t
As long as E<pw
(u) exists
andw
J EI<p (u)ldu<oo, the integral in (1) will
o
exist. We can make this claims since IIT<PI<I<p1 implies that
A
E{<P w(U)II(t-u,A o 'W 1)} exists and
Jt EI<P (U)II(t-u,Ao'W )ldu<00. We note
w
1
A
0
that II¢ is itself now a random variable.
To obtain the unconditional probability we can take the expectation
of
II<p
with respect to the distribution of <p w.
II( t ,A ( . ) 'Wo ) =EP{Z (t) =0 I<p w( s),
Then
O~s<t}
= II(t,Ao,Wo)+CAoE[6t<pw(U)IT(t-u,AO,W1)dU]+0(c2)
By Fubinis Theorem we can move the expectation inside the integral to get
where m(·) is the mean function.
We can also find the mean content of the storage process by using
the relationship
and
EZ(t)=E[E(Z(t) l<P w)].
~\[ l<Pw(u)~(t-u,Ao,W1) I]dU < 00
If E{<P w(u)U(t-u,A o 'W 1)} exists
then
It appears that the complex behavior of the stochastic input process
is captured in the mean of the arrival rate.
2
II(t,A(.),Wo)=II(t,Ao'WO)+O(c ) and
If m(t)=O then
2
~(t,A(.),Wo)=IJ(t,Ao'WO)+O(c).
Unfor-
tunately, in most situations of interest this will not be the case.
We
note that the formulae in (2) and (3) are now the same as those in Chapter
4It
158
3, with m(.) in place of ¢(.).
tion of rr and
~
Thus to look at the limiting distribu-
we are now in the same situation as that Chapter.
It is unlikely that the mean function will be periodic, so much of
the results we have looked at will not apply.
A more likely event is
that m(t)=a+b(t) where b(t)£L 1• Then
t
rr(t,A(.),Wo)=rr(t,Ao,Wo)+£Aoab IT(t-u,A o ,W 1)du
Since fi is a bounded function which converges to a limit (which is 0) we
can use Lemma 3.2.2 to show that
t
lim J b(u)IT(t-u,A ,W )du=rr(oo,A )Joob(u)du=O
o 1
t-l<>O 0
0 0
This leads to the limit
(4)
lim rr(t,A(.),Wo)=rr(oo,Ao)+£Aoa6°Ofi(u,Ao,W1)dU+0(£2)
t-l<>O
7.2 An Example: A Queue with Two Phases
The most commonly studied process with stochastic arrival rates is
the queue in which the arrival rate depends on the
II
phase of the queue
ll
at the time of arrival, and the amount of time spent in a phase is a
negative exponential random variable.
This essentially yields an inten-
sity A(.) which is a step function which changes levels at random times.
This queueing process has been discussed by Yechiali and Naor, Eisen and
Tainiter and Neuts, to name a few.
While the simple queueing problem
159
with only two phases has been studied extensively with traditional
methods, it is informative to look at this example in the light of the
alternative method presented here.
Suppose we have a situation where the arrival rate fluctuates between two levels A1=A o(l+E¢1) and A2=A o (1+E¢2)' where the time spent at
level i has a negative exponential distribution with mean 1/n i . To find
IT and ~ we need to determine the mean of ¢. We can write m(t) as
Pi(t)=Pfsystem is in phase i at time t}
The phase process forms a simple two state Markov Chain.
We can use
the Kolmogorov forward equations to determine Pi(t), i=1,2, as follows.
Solving this differential equation we get
n2
( n2
P (t) =
1
n1+n2 - n1+n2
(6)
) - ("1+"2)t
PI (0) e
n1
( n2
) -(n +n2)t
P2(t) = n n + n n - P1(O) e 1
1+ 2
1+ 2
~
160
We can now write m(t) as
then we see from (4) that
(7)
~~rr(t'A(.)'Wo)=rr(OO'Ao)+£Ao[~2+(~1-~2)n2]&OOfi(U'Ao'W1)dU+O(E 2 )
n1+ n2
We recall from Chapter 4 that for an MIMl1 queue, if Wo=W then
= -(l-A oIJ1)
- - 1
J1
Substituting into (7) we have
We can use the relationship between the parameters Ai'
~
rewrite this as
~i
and £ to
161
(9)
1imrr(t,A(.),W)
t~
We notice that, as we would expect, the limit does not depend on the
choice of €, Ao , ¢ 1..
It is interesting to compare the approximation (9) with the actual
value as found in the literature.
Yechia1i and Naor (1971) present the
relationship
where Pio=P{Z(t)=O and system is in phase i}, i=I,2 in steady state,
~=Pl.Al+P2.A2' ~=Pl.].11+P2.].12 and Pi.=~: Pi(t).
If ].11=].12=].1
then
(10) simplifies to
But Plo+ P20=P{Z(t)=O}.
e
Thus we have that
lim rr(t,A ,W) = 1 - ~/].1.
t~
0
Notice that from equation (6) we have that
If we substitute (12) into (9) and rearrange some terms, we find that
O(€2)=O.
Yechia1i and Naor (1971) state that Plo+P20=1-~/0 if and only if
1./].11=1. 2/].12.
Whil e it is true that A1/].11 = A2h 12 imp1 ies that
e
162
PIo+P2o=I-~/D, we have already shown in (11) that this is not the only
circumstance under which this relationship holds.
indeed hold true when
~If~2.
But when
~I=~2
one point they inadvertently divide by zero.
Their theorem does
the proof fails because at
APPENDIX
A REVIEW OF STANDARD RESULTS
The standard storage theory model is the following: Inputs to the
system occur at points to' t I , • . . where the time between inputs
Xi=ti-t i _I are i.i.d random variables. The size of each input, Ui , is
governed by a distribution B(x) with B(O+)=O.
Thus input to the store
can be described by the sequence {Xl' UI , X2 , U2 , . . . }. In many instances the Xi are assumed to be from a negative exponential distribution.
In such cases the total input into the system is Compound Poisson.
The content of the store, in standard theory, is released at a con•
stant rate c (r(x)=c)
as long as the store is not empty.
Thus in an
interval of length y, the output of the store will be at most c y, with
the maximum being achieved if the store has positive content throughout
the interval.
When the rate of release is 1, the storage process is
equivalent to a single server queue.
In this case the Xi are interpreted
as interarrival times and B(x) is the service time distribution.
A
simple change of time scale will convert a storage process with r(x)=c
into one with r(x)=I.
Thus without loss of generality we can focus our
attention on the single server queue, where much of the literature is
found.
We cite here results of the MIGII queue which concern us in this
research.
We look to the dissertation of Smith (1953) to summarize the
results for this appendix.
164
The following notation will be needed:
G(x) is the distribution function of the busy period
W(x,t) is the distribution function of the waiting time at t
Sj is the jth moment of service time
~(t)
is the mean of the waiting time at t
A is the arrival intensity.
This notation agrees with the notation used in later chapters and is
different from that used originally in Smith (1953).
The first theorem and its corollary provide information about the
event
€,
which occurs at t if Z(t)=O and Z(t-»O, through an examination
of the busy period.
Theorem 28:
If AS <1, G(x) possesses as many moments as B(x), and if the
1
jth moment of B(x) is Sj the moments Yj of G(x) are given by
Y2 = B2/(1-AS 1)
Y3 =
where
00
3
{B2+3AS~/(1-AS1)}/(1-AS1)4
Corollary 28.1:
(i)
•
If
Assuming B(+oo)=l we have that
AS1~1,
G(+oo)=l
is the root in R(s»O of the equation s-A(l-B*(s))=O.
We see from these theorems that the condition
to ensure that the event
€
is certain.
AS1~1
is sufficient
The next theorem demonstrates
~
165
conditions under which a limiting distribution of waiting time exists.
Theorem 30:
(i) If AS{l and W{oo,O) =1 then there is proper distribution
function W{x) such that
lim W{x,t) = W{x)
t~
where
6e00
W*{s) =
(i i)
If
A81~1
SX
dW{x) = s{1-A81)
s-AO-B*{s))
then for all x>O
lim W(x,t) = 0
t~
We can gain some additional information about the queue which we
will need from W*(s).
First, if we take the limit of W*(s) as s goes
to infinity we find that for the queue in steady state, the probability
of an empty queue is (1-AS 1).
Second, if we look at
lim -dW*(s) we find that the mean waiting time in steady state is
ds
s~
AS2/2{1-A81).
These are two well known and often used facts.
The last theorem we present is an estimate of I(t), the expected
idle time in (O,t) and
~(t).
I(t) = (1-AS 1)t + A8 2
20-A8 1)
and
~(t)
= AS 2
2(1-AS 1)
+ 0(1).
~(O)
+ 0(1)
,
BIBLIOGRAPHY
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Clarke, A. B. (1956). IIA Waiting Time Process of Markov Type", Annals
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----;:;---
ll
,
Koopman, B. O. (1972). IIAir Terminal Queues under Time Dependent
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_---=~_.
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.
Ph.D. Dissertation,
•
(1954). IIAsymptotic Renewal Theorems ll , Proceedings of the
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,
•
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Statistics & Probability Program
ONR, Arlington, VA 22217
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DISTRIBUTION STATEMENT (<>1 rh .. "h.,rrlt'/ ""t"r"d /" RI"ck 20, il dllierflnt Irom Report)
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SUPPLEMENTARY NOTES
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Ph.D. dissertation under the direction of Walter L Smith
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KEY WORDS (Continue on re':tof.,fJ "Id"
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Limit theorems, Storage Systems, Queueing Theory
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ABSTRACT (Contlntlt· on ,everse
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It is shown that under a variety of circumstances, the presence of a periodic
pattern of input is enough to insure that a quasi-limiting distribution of store
t-ontent exists. For stores with nonhomogeneous compound Poisson input and general
"'"elease rule a probabilistic approach is developed which allows equations to be
~itten down easily for such quantities as the mean store content and the probablility of an empty store. For at least some instances of the Met) /M/I queue it is
shawn that the approximations are very good.
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1473
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