••
t Phe l'esearoh in th~ pepon UJas partiaZly suppo:vted by the NationaZ
Science Poundation unde:v Gztant No. GU-20fJ9 and the Ai:v Po:vee Office of
Scientifio Reaearooh under-Contl'act No. APOSR-68-141fJ.
THE INFINITELY f4ANY SERVER QUEUE WITH SEMI-f1ARKOVIAN ARRIVALS
AND CUSTOMER DEPENDENT EXPONENTIAL SEVICE TIMES
by
Woollcott Smitht
Depa:vtment of Statistics
Unive:vsity ofNor-th CaroZina at Chapel BiZZ
Institute of Statistics Mimeo Series No. 746
May~ 1971
•
Abstract:
An
infinitely many
se~er
q-ueue with N types of customers arri"ing_
according to a semi-Markov process is studied.
pendent exponential random variables with means
type of customer being served.
The service times are indel/~i
where i
is the
The stationary distribution of the embedded
liarkov chain and the limiting distribution of the contiuuoustime process
are obtained•
••
1.
'e-
INTRODUCTION
In this note"we consider an infinitely many server queue where the
arrivals of customers form a semi-Markov process,
S - MP,
and the service
times are independent exponential random variables with parameter Pi which
depends on the type of customer being served.
This note is an extension of
a recent paper by Neutsand Chen (1970) which considers the same process
except that the
exponentia~
service times did not depend on the type of
customer being served.
Single server queues with semi-Markovian arrivals have been investigated
by Cinlar (1967a, 1967b), Neuts (1966) and others.
The queueing system
discussed in this note is a generalization of the GI/M/oo queue which has
been fully studied by Takacs (1962).
The results in this note follow from a
straightforward extension of TakAcs' arguments.
We will restrict ourselves
to finding some stationary and limiting properties of the queueing system.
However the time dependent properties of the queueing system can also be
obtained by a suitable extension of the GI/M/oo queueing results.
Consider a queueing system with finitely many types of customers,
Let the random variable X(t)
before time t.
N.
be the type of the last customer to arrive
We assume that X('t)
is a finite state S - MP.
Let
be the arrival, times of the customers and let Yn " '[,n - T·n-1
for n ~ 1. Let X be the type of the n-th customer, xn .. X(t) for
n
Tn s: t s: T n+l' The S - MP X(t) is defined by the set of defective distribution functions
••
Let
N
(1)
Fi(t) ..
2
j
Gij(t).
2
In this note we assume that
- -(2)--- ----
m
We also assume that Fi(t)
Let qij(s)
Fi(OO) = 1 and that
for
i == 1,2, ••• N is not a lattice distribution.
denote the Lap1ace-Stieltjes transform of Gij ,
(3)
and let Q(s) == [qij(s)]
P • Q(O)
denote the matrix of Laplace-Stieltjes transforms.
is the one-step transition matrix for the embedded Markov chain,
X. Throughout this note it is assumed that P is an irreducible stochastic
n
matrix. Let n == (nl' n2 ••• ~) be the vector of stationary probabilities,
nP
(4)
•
n•
A customer starts service immediately upon arrival.
The service times
-ll X
are independent random variables with distribution function
for
X ~ 0,
where i
1 - e~ i
,
is the type of customer being served.
denote the number of customers of type
i
Let n (t)
i
in service at time t, and let
• • • ~ (t) )
be the vector process denoting the. ;number
1
2
of customers of each type in the queue at time t.
n (t) == (n (t), n (t) J
Although this model is described in tenns of queueing theory terminology,
it was suggested by a generalization of a stochastic model for an enzyme
reaction in an open system (Smith (1971».
This application is too
specialized to develop in detail here.
For another application of this model consider the following simplified
model of a large multi-purpose computer communicating with many remote
3
the entire output.
'-.'
Assume that the transmission tilnes of the oUtput are
independent exponential random variables with parameter
-
-
\.1
i
where
i
is
-
-
the type of job being transmitted.
----
---
This system corresponds to the queueing
system described above, where the number of messages in the buffer from
jobs of type
of type
Let
i
i
at time
t
corresponds to
in service at time
nn
ni(t),
the number of customers
t.
be the vector denoting the number of customers in service just
before the n-th customer arrives,
n
n
= n(n-0).
In Section 2 we investi-
gate the stationary distribution of the embedded Markov chain
{(Xn , nn)}
and in Section 3 we find the limiting distribution of the process
2.
{(X{t),n(t»}.
STATIONARY DISTRIBUTION OF THE EMBEDDED ~KOV CHAIN
Since the service times for this queue are Markov. the imbedded process,
{ (X , nn)} ,
n
is a homogeneous Markov chain.
To simplify the computations
for this vector-valued process we introduce the following notation:
i, m,
and IL be integer valued row vectors of length N,
rr ~ tm ]
N
j;=l.
(5)
i
let
define
if
r'i
o
otherwise
and
(6)
Let
p(k,
i;
j, m)
denote the one-step transition matrix for the em-
bedded chain
(7)
p(k, t; j, m)
= Pr(Xn+l
D
k, nn+l • t
I
Xn • j, nn • m).
4
'['n a new customer of type Xn starts service. Thus at time '['n + 0
the number of customers in the queue is nn + o(Xn ), where 0(1) is a row
vector of length N with a 1 in the i-th place and zeros everywhere else.
At time
Given that nn
and no new customers arrive in the interval
R.
D
(L, '[' +t) ,
n
n
the number of customers of each type are independent binomial random variables,
Pr(n('['n+t) • tlxn .. j. nn • m, '['n+l"- '['.n
> t)
• b (R.; m + 0 (j), lJ t).
Thus the one-step transition probability is
p{k, R.; j, m).
(8)
Let n(k, l)
chain,
n(-k, l)
~
b(R.; m + 6(j),
lJ t)
d Gjk(t).
denote the stationary distribution for the embedded Markov
0
t
n(k, l)
(9)
f:
=
N
r
j=l
I
n(j, m)p(k. R.; j, m)
m
and
nk
(10)
=
I
n(k. R.).
l
where nk'
k
chain
Throughout this note
X •
n
a
1,2, ••• N,
is the stationary distribution of the Markov
I
m
will denote the sum overall ~~n-ne8ative
interger-va1ued vectors of length N.
We define a generalized binomial moment Bi (IL),
the stationary distribution,
••
(11)
=
{ :
[~]
n(k,t)
if
rj
i" 1,2, ... N,
~
otherwise
for
0, for j = 1,2, ••• N.
5
and let
B(r)
denote the vector
We can now state a result which gives an algorithm for obtaining the binomial
moments of the stationary distribution of the Markov chain
(X , nn).
n
THEOREM 1:
The stationary binomial moments of the embedded Markov chain, {(X , nn)}
n
satisfy
(13)
where
B(~)
(14)
III
(Bl(~
- ~(l», B2(~ - 0(2», ••• , Bj(~. o(j», ••• ,
~(~ - o(N»
and
B(o).
n
PROOF:
(~)
Multiplying (9) by
(15)
Bi(~)
III
D
III
I I I (:)
l j
J:{
IT· (j, m) (m+O(j)]
j
I
j
~
one obtains
p(i, l; j, m) IT(j, m)
II
m
l
' ..
n
· I
III
In
I~
j
".
and summing over all
(j. m)
[~]
~
I t (j, m)
(B (~)
j
f
00
0
~
e -(~·~)t d.G
I[~J + [4 _m6(j)f qj1(~'4)
-
+
bet; ....6(j).
-
B (~ - o(j» qj1 (~.~) •
j
tld Gj1 (t)
(t)
j1
6
'-.'
Line two of (15) is obtained by substituting (8).
Line three follows
from a standard result for the binomial moments of a binomial random variable.
Lines four and five follow from the definitions of qji
and B (It).
i
In
matrix form (15) becomes
(16) .
8(1t) •
which yields
[B(It)
+
B(Jt).] Q(Jj.It),
(13).
The first and second moments for the enbedded Markov chain can easily
be computed from Theorem 2.1.
The first moments are
For the second moments we have for
B(6 (~)
and for
+
i
~ j
IS (j»
i. j
With a little effort it can be shown that for the special case
i
= 1,2
••• N,
lJ
i
= lJ
for
these results correspond to the results given in Neuts and
Chen (1970).
Because of the complexity of the equations for the binomial moments, in
practice one would usually be satisfied with calculating the first few
binomial moments.
However, in theory, given all the multivariate binomial
moments, one can use an extension of the univariate inversion formula for
binomial moments to find the stationary probabilities
••
TI(i, l) •
7
'.
CoROLLARY:
The stationary probabilities for the· {(Xn , nn)} process are
N
N
1 +
r
(17)
. n (1. t) (_l)j-l j j-l j
B (It) •
I
r
r
[1)
1
PROOF:
For N = 1
(17) is the binomial inversion formula for the univariate
case, (Riordan 1968).
The proof for the multivariate case follows from a
straight forward inductive argument on N.
SECTION
3.
In this section the limiting distribution of the continuous time process
{(X(t), n(t»},
IT *
(i, k)
(18)
= Lim
Pr(X(t)
t-+co
= i,
net)
= k),
and its binomial moments
*(i,
(19)
k)
are investigated by using the results for S - MP with auxiliary paths
(Pyke and Schaufele, 1966).
element is Bl(lt) ,
Let B*(It)
denote the row vector whose i-th
then:
THEOR~ 2.
The binomial moments for the limiting distribution of the
'.
(X(t), n(t»
process are,
(20)
8*(It)
=«~·It)a)
-1 -8(1t)[1 - Q(It·~)] -1 [1
-
V(It·~)],
8
'.
where·
(22)
(23)
and
S(IL)
18 defined in (14).
PRooF:
Let
n-(t)
be the value of the
transition in the
(X( t) ,
X('t)
process~
n- (t) ) process 18 a
n- (t)
S - MP,
auxiliary path of this process.
net)
III
and
process just before the last
if '['.n+1 > t ;;;: '["n.
nn
n (t)
can be considered as an
Using a known result for the limiting dis-
(Pyke and Schaufele (1966, page 1459»
tribution of auxiliary paths
The
we have
(24)
Xn
From the properties of the
. n
(25)
where
<!)
(26)
b
and
F
i
*(1~J)
n (t)
=;
III
i,
nn .. k.) d' U •
process we have
~
n
(1,k.)
I:
b(j; I<.+o(i), Hl t)(l-Fi(t»dt
are defined in (6) and (1) respectively_
and summing over all
j
one obtains
Multiplying by
9
,-e
Using-matrixunotationand equation (13) we obtaUlu (20).
Again the limiting probabilities,
*
IT (i,h.),
can be found by applying
the multivariate inversion formula, equation (17).
IffiRENCES
Cinlar, Erhan
365-379.
(1967~).
Cinlar, Erhan (l967b).
' Queues with semi-Marko""ian arrivals ... .. J .. AppZ. Prob.
4,
Time dependence of queues with semi-Markovian services.
J. AppZ. Prob. 4, 356-364.
Neuts, M. F. (1966). The single server queue with Poisson input and semiMarkov service times. J. Appl.. Prob. 3, 202-230.
Neuts, M. F. and Chen, Shun-Zer (1970). The infinite server queue with semiMarkovian arrivals and negative exponential services. Purdue University,
Depa!"tment of Statistics MimeogPaph Senes No. 233.
Pyke, Ronald and Schaufele, Ronald (1966). The existence and uniqueness of
stationary measures for Markov renewal processes. Ann. Math. Stat. 31 t
1439-1462.
Riordon, John (1968).
Combinatorial.
Identities~
John Wiley, New York.
Smith, Woollcott (1971). Stochastic models for an enzyme reaction in an open
linear system. BuZZ. Math. Biophys. 33, 97-115.
Takacs, L. (1962). Introduction to the Theory of Queues.
Press, New York•
••
Oxford University