A SINGLE-SERVER QUEUE WITH RANDOM ACCUMULATION LEVEL

The author studies the queueing process in a single-server bulk queueing system. Upon completion of a previous service, the server can take a group of random size from customers that are available. Or, the server can wait until the queue attains a desired level. The author establishes an ergodicity criterion for both the queueing process with continuous time parameter and the imbedded process. Under this criterion, the author obtains explicit formulas for the stationary distributions of both processes by using semi-regenerative techniques.

In a large class of bulk queueing models, the server takes groups of a fixed size for service if enough group members are available; otherwise, it waits until the queue reaches a desired (fixed) level.Several versions of such systems are considered in Dshalalow/Russel [4] and DshMa- low/Tadj [5].We call such systems queues with fixed accumulation level.Practically more attrac- tive and versatile, but analytically more complicated, is a system with a random accumulation level.In such a system, the server capacity is a random number generated by the completion of previous service and this number is the desired group size to be taken for service.The server will therefore rest until the queue accumulates that many customers if that group size is unavailable by the time the server becomes free.1Received: March 1991 Revised: June 1991  For instance, for shipment of certain goods not only are transportation units of different capacity used, but arriving units can also be partially occupied.Units can take some of the load and move that quantity farther, or wait until the load reaches a specified level.Although such situations are most common in air and surface transportation, postal delivery, inventory-transpor- tation systems and assembly lines, there are other real systems of the same nature that can be modeled by queues with random accumulation levels.For example, a computer user needs a specif- ic task to be performed on several parallel or networked computers or processors.The job can only be started when all necessary computer components become free.So in this case the job to be done will be regarded as a server and computers will play the role of the customers.Again each particular job needs a different (random) number of computers.Thus the situation can be de- scribed in terms of a model with a random accumulation level.
In the present paper, the author introduces a class of stochastic models with a random ac- cumulation level and studies the queueing process with discrete and continuous time parameter.
In both cases the author establishes the ergodicity criterion and derives explicit formulas for the limiting distribution of the processes.

DESCRIPTION OF THE SYSTEM AND NOTATION
Let Q(t) denote the number of customers in a single-server queueing system at time >_ 0 and let Qn = Q(tn + 0), n = 1,2,..., where t n is the moment of time when the server completes the processing of the nth group of customers.At time n+O the server can carry a group of customers of size c n + 1 and it takes that many for service if available.If not available, that is if Qn < Cn + 1' the server prefers to rest as long as necessary for the queue to accumulate to the level of c n + 1" Only then does it begin to process a group of the appropriate size, with the pure service time lasting a n + .We assume that each of the sequences {Cn} and {n} are families of independ- ent identically distributed random variables, independent of each other and of the input stream.
The probability distribution of c is given by gk = P{c = k}, k = 1,...,r.The random variable r I has an arbitrary probability distribution function B, with B(x)= 0 for x < 0, and with a finite mean b.We denote The input stream is formed by an orderly stationary Poisson point process {rn) with intensity ,; and the capacity of the waiting room is assumed to be unlimited.

IMBEDDED PROCESS
Let N(-) denote the counting measure associated with the point process {rn}.Denote v n = N((rn).Then the terms of the sequence {Qn} satisfy the following recursive relation: (3.1) Clearly the process {,, (PX)eE Q(t); t >_ 0} ---.E = {0,1,...} possesses a locally strong Markov property at t n (see Definition A.1 in Appendix), where t n is a stopping time relative to the canonic filtering r(Q(y);y _< t), n = 1,2, Thus the imbedded process {Qn} is a homogeneous Markov chain with the transition probability matrix A = (Pij; i,j E).Due to (3.1) the upper block (Pij; 0,1,...,r-1, j @ E) of A consists of purely positive elements, and the lower block of A is an upper triangular matrix (with all positive entries on the main diagonal and above it and all zero elements below the main diagonal).Clearly the Markov chain {Qn} is irreducible and aperiodic.According to Abolnikov and Dukhovny [2], A is a At, r-matrix and the ergodicity of {Qn} is given by the following criterion.
3.2 Proposition.The generating function Ai(z of ith row of the transition probability ma- trix A satisfies the following formula: Proof.Formulas (3.4) and (3.5) follow from (3.1) by use of standard probability alcu- Now we turn to Lemma 3.1.While condition (3.2) is obviously satisfied, formula (3.3) applied to (3.4) and (3.5) leads to the following.
3.3 Theorem.The imbedded Markov chain { Qn} is irreducible and aperiodic.It is recur- rent positive if and only if (3.6) p<, where p = Ab and = E[Cl] is the mean server capacity.

INVAPdANT PROBABILITY MEASURE
Given the equilibrium condition (3.6), the invariant probability measure P = (Pi ;i E E) of the operator A exists and equals the stationary distribution of the Markov chain {Qn}" The following statement obviously holds true.
4.1 Lemma.Let P(z) denote the generating function of the invariant probability measure P of a transition probability matriz A of a homogeneous Markov chain {Qn}, and let Ai(z denote the generating function of ith row of A. Then (4.1) Using lemma 4.1 and the ideas of the last two sections, we obtain the following main result.
4.2 Theorem.Given the ergodicity condition in theorem 3.3, the generating function P(z) of the stationary distribution of the imbedded queueing process {Qn} satisfies the following formula: (4.2) where G is defined in (3.5).

P(z)
#(A Az) E o Pi z' = Although formula (4.2) contains r unknown probabilities, Po,"',Pr-1, they can be deter- mined from an additional condition which yields relatively simple equations.The latter can be solved numerically.tions: 4.3 Theorem.The probabilities Po,"',Pr-1 satisfy the following system of linear equa- where a isfies forml (a.g) and {; = 1,...,S} i he e 4 roo 4 he ncio z r-(A-Az)k= gzr inside the unit ball B(0,1) with their multiplicities ks, such that s k s r-1 The system of equations (4.3)-(4.4)has a unique solution, Po,"" Pr " s:l Proof.Formula (4.2) can be rewritten in the form E 0 <X> The rest of the proof is similar to that of theorem 4.2 [1].Let i = Ev[tl] This gives the expected length of the service cycle given that the initial queue length was equal to i. Let/ = (fli; E E) T. Then the scalar product P/ gives the value of the mean service cycle of the system in the stationary mode.We wish to call the ratio of the mean service cycle Pfl and the mean inter-arrival time the capacity of the system.Thus the capacity of the system is defined as AP.
Earlier we denoted the mean server capacity by .O bserve that for the classical M/G/1 queue the capacity of the system is Ab + P0 = 1, which coincides with server capacity.Below we show we have this remarkable property in our case also, when the system is in the equilibrium. 4.5 Proposition.Given the equilibrium condition, the capacity of the system AP3 and server capacity -ff are equal.
Proof.Evaluating fli we have I{0, r 1 (s i)g, where I D is the indicator function of a set D. The statement follows from the last equation and formula (4.4).

ANALYSIS OF THE CONTINUOUS TIME PARAMETER PROCESS
From the discussion in section 3 and from definition A.1, it follows that {f,ff, (PX)xeZ, Q(t); t>_ 0} (Z, S(E)) is a semi-regenerative process with conditional regenerations at points tn, n = 0,1,..., t o = 0. {f2,ff, (PX)xeE (Qn, tn): n = 0,1,...} (E x R +, IB(E x R +)) is the associated Markov renewal process.Let Y(t) denote the corresponding semi-Markov kernel.With a very mild restriction to the probability distribution function B, we can have that the elements of Y(t) are not step functions and thus we can have (Qn, tn) aperiodic.By proposition 4.5 the mean inter-renewal time Pfl of the Markov renewal process equals y/A < oo).Therefore, following de- finition A.3, the Markov renewal process is ergodic given the condition p < .
Let K(t) be the semi-regenerative kernel (see definition A.4).The following proposition holds true.[We will agree throughout the paper that the value of any sum is zero whenever the lower index is greater than the upper index.]