A QUEUEING SYSTEM WITH A FIXED ACCUMULATION LEVEL , RANDOM SERVER CAPACITY AND CAPACITY DEPENDENT SERVICE TIME

This paper introduces a bulk queucing system with a single server processing groups of customers of a variable size. If upon completion of service the queueing level is at least r the server takes a batch of size r and processes it a random time arbitrarily distributed. If the qucueing level is less than r the server idles until the queue accumulates r customers in total. Then the server capacity is generated by a random number equals the batch size taken for service which lasts an arbitrarily distributed time dependent on the batch size. The objective of the paper is the stationary distribution of queueing process which is studied via semi-regenerative techniques. An ergodicity criterion for the process is established and an explicit formula for the generating function of the distribution is obtained.


INTRODUCTION.
In many queueing systems with bulk service a server does not start service unless the number of waiting customers is at a certain fixed level. In this case the server is waiting for more arriving customers until the desired level is reached. A typical situation arises in computer network service, where every job to be done must go through a chain of computers (or parallel processors). The job can not get started until all necessary computer components are free. So the job (which now plays the role of a server) waits until the queue of waiting computers (in this case customers) accumulates a necessary group to run the job. A version of such a queue was modeled in Dshalalow and Russel [4]. A relevant modification of this model occurs when during waiting time a task can be reset up or being on a preliminary service insofar requiring a different (generally smaller) number of computer components by the time a group of the initially desired size becomes available. Such situations are common whenever a server, resting due to a queue accumulating more customers, lends a part of its capacity which perhaps may not be restored by the time the queue has reached the desired level. So by then the server begins to pro-190 J.H. DSHALALOW AND L. TADJ cess a group of custo, nors in accordance with the available capacity.
In the present paper the authors introduce and study a queueing model with an orderly Poisson input flow of customers and a single server of a variable capacity. The server usually takes a group of fixed size r if such a group is available and processes it a random time with a given general distribution. Otherwise, the server idles until the level of the queue reaches level r. By then however capacity is a random number less than or equal to r and it takes the corresponding batch for service which lasts a random time with a general distribution dependent on the batch size. The authors target the queueing process {Q(t)} with continuous time parameter.
They establish a steady state condition and obtain the stationary distribution for the process by using tools for semi-regenerative processes. An imbedded Markov chain is also given a detailed treatment.

DESCRIPTION OF THE MODEL.
We consider a servicing system with an infinite waiting room and a single channel processing a stream of customers described by an orderly stationary Poisson point process {rk;k E Ihl} with intensity A. Denote N(-) the associated counting measure. Let Q(t) denote the total number of customers in the system a.t time _> 0 and let o 0, tl, t2, be the sequence of the successive completions of service of groups of customers. Defining Q(t) as a right continuous process we introduce the imbedded process Qn Q(t,), n 1, 2, Let a denote the service time of nth group of customers. If Q, >_ r then the server takes a group of size r for service and immediately begins processing this group completing the service by t,, + 1-In this case a,, + ,, + 1-t,, and it is distributed according to a probability distribution function B (B(0) 0) with a finite mean b. If Q, < r then the server waits r-Q, exponentially distributed phases, i.e. until r-Q,, more customers arrive at the system reaching exactly level r and only then the server is ready to begin service. But its capacity now becomes a random number 3',,+1: fl --{1, 2, r} generated by the begin of n + 1st service. We assume that 3'0, 3'1, 3'2, are independent identically distributed random variables with the common probability mass function (91, 92, 9r). Now given the server capacity 3',+ a group of the same size will be processed during a random time distributed according to B3',, E {BI, B2, Br}, where the latter is a tuple of arbitrary probability distribution functions with finite means {b, b2, b,}. In this case t, + t, is the sum of server waiting time and the actual service time a,, + l-With the above formalism, the terms of the sequence {Q,} therefore satisfy the following recursive relation  where qkJ fc'Au (jr + k)/ q =0 j =0, r-k-l, and the lower block matrix which is upr tril matrix (with Ml sitive elements on the main diagonal and above the mn diagonM d zero elements low the mn diagonM.) Thus A is a A.-matrix, a speciM ce of a cls of ,N-matrices studied by Alnikov d Dukhovny [2]. According to the Alnikov/Dukhovny criterion, the equilibrium of the prs (Q,) is bica.lly up to a certain quity of the generating function of the rth row of matrix A. Let A,(z) denote the generating function of the ith row of A. (0)  a.
(a.) Formula (a.a) foos fom the relation P(z) The determination of the unknown probabiliti P0, P-is subjt to the following THEOREM 3. The unknown probabilities is a unique lution P0, P-of the following system of line equations: ,: 0 , A()-] 0, 0,..., . , ,..., s, (3.6) where z, e the rts of the function z -( z) that long to the closed unit bMl (0,1) in C with their multiplicity k, such that s,= xk, r-1. (ii) Let C denote the (stationary.) capacity of the system, defined C APf (equals the ratio of the mean "service cycle" Pfl and the mean interarrival time l/A). Observe that the notion of the capacity of a system goes back to the classical model M/G/l, where C is reduced to C --p.
(3.8) One remarkable property of the system in the equilibrium is that the capacities of the server and the system coincide. DEFINITION 9. Let {fl,, (P)z,E, g(t); 0} (E, (E)) a semi-regenerative cess relative to the sequence {t.} of stopping times. Intruce the probability g(t) e{Z() ,, > }, j, .
We will call the functional matrix K(t) (Kid(t) j,kE) the semi-regenerative keel.
Before stating the main result of this section, we will recMl the mn convergence threm d its corMlary.
THEOREM 10. (The Main Convergence Threm, cf. qinl [3], p. 347). Let {fl,ff, 0} (E, (E)) be a semi-regenerative sth=tic press relative to the sequence {t.} of stopping time and let K(t) the corresnding semi-regenerative kernel. Suppose that the sociated Mkov renewal press is ergic d that the semi-regenerative kernel is emn integrable over R +. Then the stationary distribution r (x,; z E) of the press (g(t)) exists d it is determined from the formula: E,P, f o K(t)dt, ke E.  (ii) Let g(t)= (Kit(t); j,k e E) denote the semi-regenerative kernel (definitions 8 d 9), where gi(t) {Q(t) k, > t}. By element probability guments we deduce that is C=r e'-i= P' due to proposition 4.
2) The expected length of an idle period of the server in the equilibrium satisfies the below formula (due to straightforward probability arguments): The probability that the server is idle in the stationary mode is r-I i= 0r 7 + where B denotes the mean busy period which thus can be expressed in terms of known values: l?= 'r, . 0 s