A NOTE ON THE JOINT DISTRIBUTION OF THE DURATION
OF A BUSY PERIOD AND THE TOTAL QUEUEING TIME*
by
Wa1ter·L. Smith
Depa1'tment o.f Statistics
University of North CaroZina at ChapeZ Hill
Institute of Statistics
June~
r4i~eo
1976
* Research supported by the Office of Naval
N00014-76-C-0550.
Series No. 1072
Research under Grant No.
ANOTE ON THE JOINT DISTRIBUTION OF THE DURATION
OF ABUSY PERIOD AND THE TOTAL QUEUEING TIME.*
by
Walter L. Smith
University of North Carolina
§1.
IntrOduction
Co~ter
simulation of queueing models is becoming the parammmt
method for their investigation; complexities of the model which present
fonnidable difficulties to any analytical attack can often be dealt with
in a trivial way in the computer program.
Nonetheless the existence of
exact mathematical formulae is useful, even when they refer to rather
specific llDdels.
For one thing they can indicate t in a rough'. way t
the effect of varying relevant parameters.
For another thing, they can
provide valuable checks on the correctness of a computer simulation for
\'lhich, maybe, a subtle tmdetected flow in the program is yielding results
which are wrong, but not so conspiciously so as to arouse suspicions.
For a further thing, the exact fonnulae can possibly provide indication
of the sanpling errors inherent in a proposed MJnte Carlo study of a
queueing system.
The present note is part of an attack on a number of analytical
problems which arose from work being done by Andrew Seila of the Curriculum
of Operations Research at the University of North Carolina, Chapel Hill.
* Research supported by the Office of Naval
NOOOI4-76-C-0550.
Research tmder Grant No.
2
He is investigating 1 mainly by computer methods 1 the estimation of quantiles
of the waiting time distribution in various queues; his methods focus on the
busy period as the basic sampling mit.
One check on the correct operation
of his methods is through a study of the joint distribution of the total
time lost
length
(U)
(2)
by customers in queueing during a busy period and the
of that period.
This note tackles this problem for the
M/r'1/l queue and shows that product moments of 2 and U are obtainable
in analytic form.
They are given later in this note up to order three.
§2 Derivation of a Joint Generating Function
Let a busy period begin by a customer Co ' say, with service-time
X1
arriving to find the server free.
period thus L'rlitiated and let
Let U be the length of the busy
Z be the stun of the queueing-times of
all customers who are served in that busy period.
distribution of U and Z depends upon x.
Plainly the joint
Let a > 0 and 13 > 0
be dur:1Il1Y transform-variables and set
M(a 1 13Ix) = E{e- aZ - 13U
(2.1)
I x},
where the conditional expectation has the obvious meaning.
we shall write M(a 113Ix)
For ease
simply as M(x).
The distribution of the ti.rne from the arrival of Co
to the next
customer Cl ' saY1 has a p.d.f. of exponential form and this leads us
to the integral equation
(2.2)
M(x)
= e- Ax - 13x
+
f:
Ae- AU-(3u-a(x-u)
In this equation the constants
and service, respectively.
J: M(x+z-u)~e-~zdzdu.
A and ~ are the intensities of arrival
3
If we provisionally set
then a straightforward computation yields the fact that the Laplace
transfonn~
denoted by the notation:
and defined for real s > 0, is given by
oO(s)
(2.3)
~
= Mo(~)-M0 (s+~)
s+a-~
If this result is used in (2.2) we obtain after a little computation
rrP( ) =
(2.4)
.
s
1
S+A+S
+
1
s+A+13
{t,P (~)s+a-~
+1° (s+aJ}
.
Since we are principally concerned with the unconditional expectation
E{e- aZ - SU}
=
J: ~~(x)e-~
=
w.P(~) ~
dx
it follows that MO(~) is of special interest to us.
obtain by putting s
From (2.4) we
= ~~
and hence
(2.5)
~bre
generally, if n is any positive integer, then putting s = ~ + na
4
in (2.4) yields
MO(~+na) = ~+A+~+na {I + a(~~l) [MO(~) - MO(~+n+la)]}.
(2.6)
Let us set
~
= A~/a,
K(~)
(2.7)
and
=1 -
~
~2
Ai~~8 + 2(A+~+S)(A+v+B+a)
~2
+
etc.
It is not difficult to see that this infinite series is always convergent
and that
K(~)
is actually an entire fu"1ction.
Somewhat tedious and repetitive use of (2.6) in (2.5) will then
show that
(2.8)
In obtaining this result it is helpful to note that
n -+
I'P (na)
-+ 0
as
co.
It is possible to express (2.7) as a certain Bessel fUnction, and,
indeed, if we temporarily set
then
(2.9)
This result does not seem particularly tractable. The most we can hope
5
e
to achieve is an expansion of (2.8) as a double Taylor expansion in powers
of ex and 13 from which the joint moments (unconditional) of Z and U
can be extracted.
§3 Expansion of Joint Generating
Func~ion
From a wel1-1mown property of the Bessel function
(Watson~
1958)
we have that ~ for any argument y ~ say,
J e-l ()
y + J 8+1 ()
Y
=
2(a~1) JeeY).,
from which one can derive
and thus obtain a continued- fraction expansion of the ratio of Bessel
functions in (2.9).
Indeed if we now set
a
b = ~
~
we find that
(3.1)
1
= ------:;-----
1
a - -----:;:.-...,...1-(a+b) - la+2b)-etc.
I
= a.:
1
I
I
(a+b)- (a+2b)- (a+3b)- ...
Let us call this continued fraction C(a,b).
(3.2)
1
C(a,b)
=a
- C(a+b,b).
Then~
evidently,
6
e
In particular we see
This quadratic equation gives us the equivocal result
(3.4)
C(a,O)
= A+~+a
± I[(A-~)2+a2+2(A+~)a]
2/fll
However, the substitution
Ct
.
21(XiiJ
=
° and
S
=
° must make
l-lf'ti (~)
0
= 1.
Thus, from (2.9), we need
= _A_+_~ ± _1-,(~A_-~=)_2_
2/fii
2/fil
and so we may conclude the plus sign to be correct, and hence
= a+~
C(a,O)
•
This result will be found to agree with the well-known formula
(Cox and Smith, 1961) for the transfonn of the distribution of the duration
of a busy period, for it yields from (2.9):
(3.5)
E(e-~) = (~~)~.{A~ + )(A+~~~)' - 4}
= ~A {(A+~+B)
Let us adopt the nota-c.ion
C for
C•• (a,b) =
1J
+ IrA+~+a)2-4A~ .
ai +j
•
C(a,b)
.
• C(a,b)
aa1 abJ
and be prepared to contract Cij (a,b)
and
,
even further to
Cij .
Then
7
(3.2) gives
-cz- =
{- :z- = CIO
(3.6)
1 - C10 (a+b s b)
COl
C10(a+bsb) - COI(a+b,b)
.
C
If we set y
= yea) = C(a,O)
and
y .. =y .. (a)
1J
1J
= C··CasO),
1)
then (3.6) gives
(3.7)
and
Y01
(3.8)
= - (..1:-).
y2_ l ~ .
Since the function y2 J(yZ_l) occurs frequently we shall denote
it by cp.
Thus
YlO =
_ 4> Z
{ YOI - -4> •
(3.9)
If we return to (3.6) and perform further partial differentiations
then we find
2
zelO
-3- -
(3.10)
CZo
T = - CZO(a+bsb)
C
C
ZC C
C
10 01 _ 11
CZ
C3
Z
ZC OI
CO2
C2
---r- - ._-
C'
= -C
ZO
(a+b b)
~
= -CZO(a+b,b)
Computation then yields the equations:
8
YZO
=-
Z
~ ¢
3
Y
_ 4
(3.11)
Yll - 3" ¢4
Y
_ 2 <b4 10 A,5
Y02 - ~. - ~ 'I'
Y
Y
It is plain that, at the expense of greater and greater complication,
one
can
continue to obtain equations of higher order.
We merely list
the following results:
_ 6 ¢4 + 12 ¢5
(3.12)
Y30
-:jf
Y21
=-
y6
6
y4
_
1J
Y and (j).
20
6
. __ 6 A,5 + 12 A,6 + 4 A,6 + 56 A,7
'4."" 'I'
""4 'I'
-0 'I'
6' 'I'
Y12 -
Y
Y
6 A,3
Y03 - - - 4
y
Let y..
5
¢ - y6 ¢
'I'
Y
Y
6 A,4 + 36 A,5
'I'
""4 'I'
-"4
y
y
-
36 A,6
'4."" 'I'
y
12 A,5 + 60 A,6
-
6'
y
'I'
denote the value of y.. when we set S = o.
1J
0"
y
'I'
228 A,7
-
-0
y
'I'
Similarly for
Then it is easy to see that y = (Vj1)~ and ep = :>,,/(A-j1).
(Note that we assume 11 >:>.. for stability of the queue, so (j) < 0 .)
Then we have the expansion
and so
(3.13)
Wi\11)
= (~) ~
C(a ~b)
= C(a,b)
y
= 1 + 1 {(1..)YIO
Y
irdJ
+
We are now in a position to extract the requisite moments, for we see that
9
ruiz j = (_l)i+j
(3.14)
;tj
'J
(Al1)~ 1+J Y
•
Thus the equations (3.9), (3.11), (3.12) will yield product moments
up to those of the third order.
We list our results in the next section,
and spare the reader the detailed calculations.
§4 The Product Moments
The
following results are obtained as outlined in the previous sections.
=~
h~p).
(4.1)
Ell
(4.2)
EZ=l.(P)2
A FpJ
(4.3)
Var U = 1 p (l+p) .
~ (1_p)3
(4.4)
Var Z = 1-. p (2+7p+p ) •
.
2
.
3
1.2
2
(1_p)5
Cov (U,Z) - 1 p3(3+P)
(4.5)
- i2 (1_p)4
•
From the last three results we obtain for the correlation coefficient
PU,z ' say,
(3+p){P"
(4.6)
Thus, as p
-)0
1,
Pu,z
and, as p + 0,
+
~ = .894427,
10
Puz
. . ,L;p·
~
IZ
Actually ~ the correlation of U and Z is high for lOOderately low values
of
p~
as this table displays: -Traffic Intensity
-+-
p
Pu
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
.5678
.7044
1.0
~
Z
.7745
.8160
.8427
.8607
.8734
.8825
.8893
-+-
.8944
(The arrows in the bottom line indicate limit values.)
For product moments of order three we find:
11
References
Cox~
D.R. and Smith, W.L. (1961), Queues, Methuen & Co. Ltd., London,
England.
Watson, G.N. (1958), A Treatise on the Theory of BesseZ Functions
ed. Cambridge University Press, England.
3
2d.