SOME FINITE WAITING SPACE BULK QUEUEING ... - Springer Link

SOME FINITE WAITING SPACE BULK QUEUEING SYSTEMS Tapan P. Bagchi Imperial Oil Ltd., sarnia, canada, and J.G.C. Templeton University of Toronto

1.

SUMMARY. The imbedded Markov chain (Qn} of the single server bulk queue, with Poisson input and continuously busy server,

MX/GY/l

satisfies the recurrence relation given by Bhat [1964] (l) where Qn is the number of customers in the system at epoch (tn+O), , d'~a t e 1 y a ft er th e en d 0 f th e n th serv~ce , , d ' t he ~mme per~o, Yn ~s number of customers which can be served by the server in the service period commencing at (tn+O), and Xn+l is the number of customers arriving in (tn' t n + l ]. Finally, (.)+ is defined by z+ = max{Z,O). If (Xn } and (Yn } are sequences of independently distributed random variables, then the sequence (Qn} will be a Markov chain. For a finite waiting space MX/GY/1,K queue, in which the number of customers in the system must not exceed

K

at any time, equation

(l) for the continuously busy server must be replaced by Qn+l where

(.)

=

K + [(Q n -Yn )+ + Xn +1 - K]

is defined by

Z

=

(2)

min{Z,O).

Equation (2) is used, following Cohen [1969], to obtain and solve an integral equation (of the type used by Pollaczek [1957]) for the time-dependent state probabilities run state probabilities p[Q = j],

P[Qn j\Ql = z] and for the longfor j = O,l, ••• ,K. The formulas

obtained are not simple. The time-dependent and long-run state probabilities for the

M/G/1,K

queue given by Cohen [1969], can be obtained by specializa-

tion from our results for the

MX/GY/1,K

queue.

2. TIME-DEPENDENT SOLUTION. Applying the contour integral expansions given by Cohen [1969] for pX+ and pX- to equation (2) gives

A. B. Clarke (ed.), Mathematical Methods in Queueing Theory © Springer-Verlag Berlin · Heidelberg 1974

134

I

K --~ 2n i

Ds

(1 1 ) Xn + 1 -K+(Qn-Yn) s S -1- S-p

+

where DS is a circle in the complex s-plane with radius lsi and centre at the origin. Q The solution for the p.g.f. E[p n] follows the solution for the M/G/l,K queue in Chapter 111.6 of Cohen [1969], with necessary modifications to allow for bulk arrival and bulk service. Let the capacity Yn of the server have a maximum value m such that p[y =m] > o and P[Y>m] = 0 for all n, and let y(p) be the n n p.g.f. of Y. Then Cohen's formulas will apply if we replace (l/p) n -m sometimes by y(l/p) and sometimes by p Define Kn as the number of arrivals up to the departure at tn+l n

I

Xj + l

n

1,2,3, ••.

with

KO

o

j=l

and define

z] n=l X

Let ~(p) E[p n+l] and let ~i(r'P3)' i = 1,2, ••• , be the zeros of [l-ry(l/p) S(p P3)] where Irl < 1, and define a polynomial f(r,p,P3'z) of degree (m-l) in p, and a polynomial h(r,p,P3'z) of degree (n-l) in p, such that

o is satisfied for Then we have

p

1,2, •••

THEOREM 1. The function C(r,p,P3'z) I rl < 1, I P3 1 < 1, I pi < ex> and 0 s; z s; K C(r,p,P3'z)

=

1

defined above - - -for -

135

with

f(r,p,P3 ,z)

and

h(r,p,P3'z)

defined above, where

3.

STATIONARY SOLUTION.

as

n~,

The limiting distribution of queue length

First, by taking limits as

dependent solution given above, one obtains THEOREM 2.

f(p)

and

n~oo

in the time-

(see [2])

The stationary probability generating function of

Ipl