ON A SINGLE-SERVER QUEUE WITH FIXED ACCUMULATION LEVEL , STATE DEPENDENT SERVICE , AND SEMI-MARKOV MODULATEO INPUT FLOW

The authors study the queueing process in a single-server queueing system with state dependent service and with the input modulated by a semi-Markov process embedded in the queueing process. It is also assumed that the server capacity is r > 1 and that any service act will not begin until the queue accumulates at least r units. In this model, therefore, idle periods also depend upon the queue length. The authors establish an ergodicity criterion for the queueing process and evaluate explicitly its stationary distribution and other characteristics of the system, such as the mean service cycle, intensity of the system, intensity of the input stream, distribution of the idle period, and the mean busy period. Various special cases are treated.

However, for the class of systems under consideration, we assume that r (_> 1) units are necessary for service, and if fewer are available the server waits for more units to arrive.The different possible values of r, as well as the formation of the idle period (which in our case does not necessarily end with the arrival of the next unit), allows our system to encompass more situations.Applications in transportation seem probable.
Other authors, such as Neuts, Sumita, and Takahashi (see [4]), have studied a Poisson process modulated by a Markov process.In this article we consider the more general case of a Poisson process modulated by a semi-Markov process.We also assume the service process is state dependent.After a formal description we study the embedded queueing process construc- ted over the sequence of instants of service completion (under no restriction to service time dis- tributions).We extend the result for the continuous time parameter process by using tech- niques appropriate for semi-regenerative processes.We establish a necessary and sufficient cri- terion for ergodicity and give explicit formulas for the limiting distributions of the processes.We derive the mean service cycle, intensity of the system, intensity of the input stream, distri- bution of the idle period, and the mean busy period.Examples are presented throughout the paper.

FORMAL DESCRIPTION OF THE SYSTEM
Let {ft, , (P=)feE, Q(t) t>_ 0} E {0,1,...} denote the number of units in a single-server queueing system at time t, let {T.n E N, To 0,} be the sequence of successive instants of service completion, let Q. Q(T.+ 0), let C(-) be the counting measure associated with the point process {T.}, and let (t)= Q(Tc(t)+ 0), >_ O. Then the input is a Poisson process modu2ated by {(t)} due to a definition to follow.
Let (t) be an integer-valued jump process (with successive jumps at T, n E N, noting that 0 is the increment of a jump in the case of Q.-1 Q.) and let {T; k N} be a non-sta- tionary orderly Poisson point process with its intensity function A(t).Then we call the doubly stochastic Poisson point process with intensity A((t)) the Poisson process modulated by the jmp process {(t)} and it is denoted by {,}.Let Ne( denote the associated counting measure.[For a formal definition of modulated processes see Dshalalow [3].]If the queue length Q. is at least r, then at time T. + 0 the server takes a batch of units of size r from the queue (according to the FIFO discipline) and serves it for a random time r.+ 1.Otherwise, the server idles until the queue length Q(t) first reaches level r after T. and then it begins to process a group of r units taken from the waiting room of infinite capacity (again, according to the FIFO discipline) with actual service time again equal to .+ 1.In both cases we assume that .+ has a probability distribution function B. {B0, B1 ,...}, Bi being an arbitrary distribution function with finite mean b i.
3. EMBEDDED PROCESS {Q.}.Let V. Ne(a.).Then the process {Q.} is defined recursively by Q.+I=(Q.-r)++ V.+, (3.1), where operator (f) + is defined as (f) + sap{f,0}.From relation (3.1) and the nature of the input process it follows that the process {ft,5,(P=)=g, Q(t);t >_ 0} ---, E has at Tn, n _ 1, the locally strong Markov property (see definition A.3 in Appendix) and that Q, ;n 6 No} --E is a homogeneous Markov chain with transition probability matrix T (Po)" Let A,(z) denote the generating function of ith row of matrix T. From A,(z)= E'[zQ*] and (3.1) we obtain Ai(z) z (' -")+ l,(A, A,z), e E (3.2), where (8), Re(8)>0, is the Laplace-Stieltjes transform of the probability distribution function B and A, A(i).For analytical advantage and with very little sacrifice of generality we drop the modulation and service control when the queue length exceeds a fixed (perhaps very large) level N.In other words, we assume that B,(z) S(z), fl,(O)= (0), b, b,A, A, i> N (AS).Without loss of generality we also assume that N > r-1.
Given assumption (AS), we can show that the transition probability matrix T is reduced to a form of the A,.N-matrix introduced and studied in [1].According to theorem A.1 (see Appendix), the Markov chain {Q.} is recurrent-positive if and only if z Ai(z) < oo, 0,1 ..,N, lira z--,l zeB(O, I) and (3.3) "---:(A Az) < r (3.4).
Given that p < r, the Mkov chin {Q} is ergc.Let P (p= ;x E) denote the invit probability meure of operator T d let P(z) be the generating function of vector P. Now we formulate the mMn result of this section.
(i) Let /j EJ[T], the mean sojourn time of the process {(t)} in state {j}, and let j (/;j E E) T. Then PJ is the mean service cycle of the system, where P denotes the stationary probability distribution vector of the embedded queueing process (ii) Let ,=(x;zEE) T and let p=.be the HaAamard (entry-wise) product of vectors/ and .We call the scalar product Pp the intensity of the system.
The concept of "intensity of the system goes back to the classical M/G/1 system, where Pp reduces to p Ab.It is worth noting that the intensity of the system and the capacity of the server (in our case r) coincide, as stated in proposition 4 and proved thereafter.PROPOSITION 3. Given the equilibrium condition p < r, the mean service cycle satisfies the following equation: PROOF.Obviously, /j b + (r j) +/A.The statement is now due to elementary algebraic transformations.
13 PROPOSITION 4. Given the ergodicity condition p < r, the intensity of the system and and the capacity of the server coincide.
PROOF.From definition of Pp and considerations as in propositi.on 3it follows that Pp= p+ N=op[(pj--p)+ (r-i) +] (4.2).The statement is due to relation (3.6c) and elementary algebraic transformations.(i) Consider a special case of our model with r 2, N 4 and with B as a negative ex- ponential distribution with parameter }.However, we retain all other assumptions about the modulation and service control having Bo,...,B 4 arbitrary.For this case we obtain/(-z) (1 + p-pz)-and it follows that the only root of the equation z -/(A-z) inside the ball B(0,1) is zx 2p Thus for equation (3.6b) we will be using z with multiplicity one and 0 with multiplicity three.This will give 4 of total 5 equations in the unknowns P0,.-.,P4: E' (z '-)+ The fifth equation will be (3.6c) with r 2 and N 4.This system can be solved by elementary algebraic methods.The solution of this system will then be put into equation (3.6a) to have the generating function P(z) in an explicit form.
(ii) By dropping the modulation and service control and setting r 1, we immediately arrive at the classical formula by Kendall established for the model M/G/1. 5. CONTINUOUS TIME PARAMETER PROCESS Q(t)}.
In this section our main objective will be the derivation of the stationary distribution of the queueing process with continuous time parameter.Prior to this, we will be concerned with some preliminaries.
From section 3 and definition A.4, it follows that {fl,J,(P=)=,E, Q(t); t>_ 0} ---, (E, (E)) is a semi-regenerative process with conditional regenerations at points T., n E N. Let f,, (P=)=,E, Q,, T,: n 0,1,... (E x R +, 9(E x R + )) be the associated Markov renewal process and let Y(t) be the corresponding semi-Markov kernel.With a very mild restriction to the probability distribution functions B, we can specify that the elements of Y(t) are not step functions and thus {Q., T,} is aperiodic.By proposition 3, the mean service cycle Pfl, which js also the mean inter-renewal time of the Markov renewal process, is obviously finite.There- fore, following definition A.5 and given that p < r, the Markov renewal process is ergodic.p It also follows that the jump process {f,,( )E, f(t); _> 0} --, E, defined in section 2, is the minimal semi-Markov process associated with the Markov renewal process {Q.,T.} and therefore, following the definition in section 2, the input process {f,,(P)=,E, Ne} -E is a Poisson process modulated by the semi-Markov process {f(t)}.
Let g(t)= (g,k(t);j, k e E) be the semi-regenerative kernel (see definition A.6).The following statement holds true.
PROOF.The above assertion follows from straightforward probability arguments. ['l Now we are ready to apply the Main Convergence Theorem to the semi-regenerative ker- nel in the form of corollary A.8, thereby arriving at the stationary distribution of the queueing process {Q(t)}.THEOREM 7. Given the equilibrium condition p < r for the embedded process {Q.}, the stationary distribution (r;x e E) of the queueing process {Q(t)} exists; it is indepen- dent of any initial distribution and is expressed in terms of the generating function r(z) of by the following formula: PflrCz)(1 z) i f(A Az)]P(z) + v=op,z/( where P(z) is the generating function of P, Pfl is determined in proposition 3, and A(z) is defined in (3.2).
PROOF.Recall that the Markov renewal process {Qn,Tn} is ergodic if p < r.By corollary A.8 the semi-regenerative process provided that p < r.We can see that the semi-regenerative kernel is Riemann integrable over [+.Thus, following corollary A.8, we need to find the integrated semi-regenerative kernel H (which is done with routine calculus) and then the generating function h(z) for each row of H.
First we find that zp h(z), i > r (5.2b).
(i) By dropping the modulation of the input process we obtain from proposition 4 that Pfl, -ands(z) rl_z)P(z).
(ii) By using obvious probability arguments we derive the probability density function of an idle period in the steady state: The mean idle period in the steady state is then s (s.3) 0 p, (iii) Formula (5.3a) and theorem 7 allow us to derive the mean busy period B in equilib- ,-I r is the probability that the server idles.On the other hand, it also rium.Clearly 0 equals j . .Thus we have -1 (iv) Now we turn to the special case in example 5 (i) applying its results for the process with continuous time parameter.We use probabilities P0,--.,P4,substituting them into formulas (5.2a) and (4.1) for r 2 and N 4, thereby reducing the generating function r(z) to an explicit form.
(v) If the input is a stationary Poisson process then its intensity is A, which is also the mean number of arrivals per unit time.In the case of a modulated input process its intensity is no longer a trivial fact.We define the intensity of any counting measure N by the formula where pt(z)= E=[N([0,])].We will apply ergodic theorem 3.9 established in Dshalalow [3] for more general Poisson process modulated by a semi-Markov process: : P/P/ where by proposition 4.3 P r and P satisfies equation (4.1) and thus we have: - (s.3).