ON SOME QUEUE LENGTH CONTROLLED STOCHASTIC PROCESSES

The authors establ}sh an ergod}c}ty criterion for both the queueing process with continuous time parameter and the embedded process, study their transient and steady state behavior and prove ergodic theorems for some functionals of the nput, output and queue}ng processes. The following results are obtained: Invariant probability measure of the embedded process, stationary dstr}but}on of the process w}th continuous tme parameter, expected value of a busy period, rates of input and output processes and the relative speed of convergence of the expected queue length. Various examples (including an optimization problem) llustrate methods developed }n the paper.

1. INTRODUCTION Let (t) denote the number of units in a single-server queueing system at time t > 0 and let n = (tn + 0), n = 1,2,..., where t n are successive moments of service completions.We assume here that the server capacity, service time and the input process depend upon the number of units in the system as follows.If at time n + 0 the number of units n is j then: within the random time interval (tn, t n + ) the input flow of units is a compound Poisson 1Received: May 1990 Revised: October 1990  process with interarrival rate ,j and with batch size distributed according to {a!J);i = 1,2,...} and with mean a [j].
the server takes for service the nth group of units of size rain{j, m(j)} (where re(k) is the ser- ver capacity, k = 0,1, ...), provided that j > 0. If j = 0 the server begins its idle period which ends as soon as a next group of units arrives at the system.This group is of random size a (0) and the server takes min{c! ),m(0)} units for the next service.
the nth group is being served a random time with the distribution function B j E {B o B 1, ...} of general type with the finite first moment bj.
Such a system supplements and generalizes the established class of single-server bulk queues with state dependent parameters, therefore, including a larger variety of real situations subject to sto- chastic control.Many authors, who treated queueing processes in special models of this class (pioneer- ed by Finch [12]) (el. [3-5,7-9,12-16]) analytically, were facing real problems to justify results obtain- ed and to reduce a solution to an explicit form.Some other authors (cf.[11]) instead developed nu- merical algorithms.While there is no doubt about the need for queueing systems with broad stochas- tic control, it is also desirable to improve the analysis of existing results.
In the present paper the authors consider the queueing, input, output processes and some functionals of them in the above queueing system.After a formal description of all participating pro- cesses, the authors treat the embedded process {n} and establish a necessary and sufficient criterion of its ergodicity.The transient and stationary distribution of {n} and the stationary distribution of {(t)} are found .interms of their generating functions.In the last section, besides the queueing pro- cess, the authors study the input and output processes obtaining explicit formulas for expected rates of these processes and establishing several ergodic theorems.Various examples, including an optimization problem, demonstrate applications of the general results obtained in this article.

FORMAL DESCPAPTION OF THE SYSTEM
The stochastic process {f, if, (PX)zcQ (t) t >_ 0} ---, Q = {0,1,...} or shortly {(t)} de- scribes the number of units in a single-server queueing system at time t.Other processes are related to {(t)} as follows.Denote by a) {f2, , (PX)xeQ n nEN0=NU{0} (t0:=0)}---,(IR+,B+) (where!B+ --.!B(IR+) is the Borel it-algebra) the point process of successive moments of service completions; b) {f2, , (PZ)zeQ, Cn: = (tn +) n No} Q the embedded process over the moments of time {tn}; c) {r e s N} a stationary Poisson point process with rate A describing the flow of groups of arriving customers; d) {N(t); t>_ 0} the associated compound Poisson counting process with as: = N(rs+l) N(rs) as the size of the sth arriving batch.{as} is assumed to be a lattice renewal process indepen- dent of {(t)} and {tn} and such that a(z): = E[zas] = > 1 ai zi, z E B(0,1) U { 1,1}, (where B(zo,R is an open ball in C (with Euclidean norm) centered at z 0 and with radius R), 0 < a: = E[as] < c, s = 1, 2, e) {vJ) s N} the jth partial input process that will govern the flow of group of units on those intervals (tn, t n + ) when n = J, i.e. let {J); s e N} be a stationary Poisson point process with rate Aj then .o x (t., t. + ) o (., e )), where I A denotes the indicator function of a set A. The corresponding controlled arrival point pro- cess denoted by {f2, , (PX)xcQ rs s E N } --, (R +, iB +) is then the superposition of partial input processes {rJ)}, j = 0,1,... f) {Nj(/); l>_ 0} the compound Poisson counting process associated with the above point process {r!J)}.It should count only those units that arrive during the intervals (t,,t,+ 1) when n = J" More formally, let {.j(/)} be a compound Poisson process with rate j, the batch size distri- buted according to {a!J);i = 1,2,...} and the batch size mean a[J]).Then With the formalism like in d) we have: J)'=.N(-'(j).+l)-N(rj))denotes the size of sth arrival batch of units on those intervals (tn,ln+ 1) when n-J" Unlike {as}, the process (a) depends upon {n} and {tn} But we assume that given n the random variables a (n) O n + 1, are conditionally independent (on (tn, n + 1)), where O n = inf{s e N: rs >_ tn}.
The Poisson processes introduced in c) and d) will be considered as special cases of the cor- responding controlled processes described in e)g).In this event, the associated superscripts will be dropped.
Units arrive at the system in accordance with processes introduced in e)-g) and they are placed in a line in a waiting room of an infinite capacity.Arriving groups of units obey the FIFO dis- cipline.Within any group the order is arbitrary.At nth moment of servicing time following tn, the server takes for service the nth group of units of size M(n) defined as min{o(rOn), m(0)}, n = 0 (2.1) M(n)-min{n, m(n)}, n > 0 where m(-) denotes the capacity of the server.Observe that the server takes units for service at time t n + O, on a busy period, and at time r ) + 0, at the end of a corresponding idle period.The service time n of the nth batch is distributed according to Bn picked from a given sequence {B0, B 1 ,...} of distribution functions each of which is concentrated on R + and with the properties Bi(0 + = 0, Bi(+cx) = 1, hi: = E[r n In-1 = i] e (0,), i= 0,1, The service times n are supposed to be conditionally independent given n, n = 1,2, We introduce another notion.Let {C(t)} be the counting process associated with the point process {tn}.Then, the first group of units that arrives in interval (tc(t) tc(t)+ 1) will be of size where tgt:-inf{s E N" r s _> to(t) }.Then, as in (2.1), we can si- N(T$t), and it arrives at time TOt milarly define the size of group which the server takes for service in interval (tc(t where (t) denotes the value of the queueing process (t) at time tC(t).
a .ote,i.iti of the {(t)} we u,e ro .the u.i mass e= to emphasize that the process started from a state {z}.The corresponding conditional proba- bilities and expectations are denoted by px and E=.

EMBEDDED PROCESS {n}
Let u n be the number of units arriving at the system during the service in (tn, n + 1)" Obvi- ously, for every n = 0,1,..., the elements of the process {n} are connected by the following relation: (where (u) +: = sup{u,O}).Due to the nature of the arrival process, it follows then that {tn} is a se- quence of stopping times relative to the canonic filtering cr((u); u _< t) (t-past r-algebra of the pro- cess (t)) and that {,Y, (PZ)zeQ (n, tn): n 0,1,...) --, (Q x R +, !8(Q R +)) is a Markov rene- wal process (that will appear again in section 6).{n} is a homogeneous Markov chain whose transi- tion probability matrix is denoted by A = (aij i,j E o)" We assume that for some N (which may be arbitrarily large) and Aj = , aj(z) = a(z), j = N + 1, N +2, ..., m(N+l)=m(N+2)= =:m(<_N+l) BN+ 1 = BN+2=...='B Under the above restrictions, the transition probability matrix A of the Markov chain {n} is reduced to a so-called Am, N.matrix (3.1)A = (aij" i,jE Q; aij-kj_i+m, i> N, j> i-m; aij=O i> N, j< i-m) studied earlier by Abolnikov and Dukhovny [2], where the values kj_i+ m of the corresponding entries aij can be determined from formula (3.3) given below.
Let K(z)" = E j > o kj zj and Zi(z E j >_ o aij zj" The evaluation of A results (3.2) The following two theorems were established by Abolnikov and Dukhovny [2]" 3.1 Theorem.Let {n} be an irreducible aperiodic Markov chain with the transition probabili- ty matrix A in the form of a Am, N-matrix (3.1).{n} is recurrent-positive if and only if (3.4) tim d Zi(z < , i-0,1, ..,N, z-*l: zeB(0,1) dz and The condition (3.5) is equivalent to (3.5a) p:-ab < m 3.2 Theorem.Under the condition of (3.5) the function z m-K(z) has exactly m roots that belong to the closed unit ball (0,1)-{z e C: I I z ll _< 1}.The roots lying on the boundary 0B(0,1) of B(0,1) are simple and all of them are rlh roots of 1 for some r.
4. TNSIENT AND STATIONARY PROBABILITIES OF PROCESS 4.1 Theorem.The generating function w(z,z) of the transient probabilities of the Markov chain {n} with the transition probability Am, N-matrix A = (aij) satisfies ihe following equalions -,,(=) 5-= o "" '"" where s(x) are the roots of function z N + 1-m(zm--xK(z)) inside B(0,1) with their multiplicities r s s such that s 1 re N+ 1.The system of equations (4.2) has a unique solution Uo, ,u N for all x such that 0 < I1 < 1.
It only remains to prove that the system of equations (4.2) has a unique solution.Assume that v= (v0,... VN) is another solution of (4.2).Consider w in (4.1) as an operator applied to a vector-function n = (Uo,...,UN) (in notation w(u)[x,z]), w(v)[z,z] is clearly analytic in z within the unit ball B(0,1) and continuous on the boundary c0B(0,1) for every xl < 1.As in Theorem 1, Dukhovny [s], it can be shown that the coefficients v(z) of the expansion of w(v) in Maclaurin series in powers of z form a sequence Y* which is an element of (l 1, I1" II) for every fixed z < 1.Now because of (4.1) Y*= (v), v,...)satisfies (3.8)and due to (4.2) v = v, i=O,...,N (but its components certainly differ from ui, = 1,..., N).Thus, we have two different solutions U and V* of (4.2) in 1 which yields the following contradiction II u-v" II II z(u-V*)A I I < II(U-V*)A II < II U-V* I! ]1A I! < !] U-V* II 0 4.2 Theorem.The invariant probability measure P of the Markov chain (n) with the transi- tion probability Am, N.matrix A exists if and only if p < m, where p:--Aab N + 1" Under this condi- tion, the generating function P(z) of the invariant probability measure P satisfies the following rela.tions: = O, k-0,..., r s-1, s = 1,...,S ZZ$ (4.5) E .Ni=o Pi[Ai'(1)-i+ m-p] = m-p where Ai(z satisfy formulas (3.2)-(3.3)and z s are the roots of the function s inside the ball B(0,1) not equal 1 with the multiplicities rs, such that s of equations (4.4)-(4.5)has a unique solution Po,'",PN" Proof.The necessary and sufficient condition for the ergodicity of (n) follows from Theo- rem 4.1, where the condition (3.4) is obviously satisfied and the condition p < rn follows directly from (3.7).Now taking into account (3.4) and (3.5a) from equation m(0))+ ,,, E[z(-.(.)) + = j > o Aj(z) PX{ n = j} we obtain formula (4.3).
Proceeding similarly as in Proof of Theorem 4.1 we get equations (4.4) and (4.5).As in Theorem 4.1, we can show that the non-uniqueness of the solution of the system of equations (4.4)-(4.5)would imply the non-uniqueness of the invariant probability measure P which is impossible after we meet the ergodicity condition.
4.3 Example.We did not discuss how to evaluate the roots of the functions in Theorems 4.1 and 4.2 assuming that in general they can be found by various numerical methods.However, for a wide class of special systems it is possible to find the generating function P(z) in an explicit form, avoiding numerical procedures.As an illustration to Theorems 4.1 and 4.2, consider a special case of the bulk queueing system under the assumption that the server takes for service all units available in the system provided it does not exceed N and a service is not initiated right after an idle period.In other vords, assume that re(j) = j, j = 1,...,N, re(j) = m, j = N / 1, N + 2, The capacity of the server at the beginning of a busy period m(0) remains under the general assumptions.In addition, assume that B = B1, )'i = 1 and ai(z = a:t(z), i= 1,...,N.Then, in (3.2a) Ai(z are reduced to Kl(Z), i= 1,...,N.
Let k(z) and .0(z)be Nth degree interpolating polynomials of gl(z and Ao(z), respective- ly, taking on the values of K(z) and Ao(z and all its derivatives up to (r s-1)st order at z = zs, s = 1,...,S, and at z = 1.Then, denoting c = Pl + + PN we have N -' :i o pj[Aj(z) zj] = PoAo(z) + cKx(z E 7= o Pi z' and from (4.4) it follows that N (4.6) -, j = o Pj z: = Porto(z) + Cgl(Z) Substituting (4.6) into (4.3)we obtain (4.7) To find the unknown constants Po and c we set up a system of two equations following the condi- tions P(1) = 1 and P(0) = Po and taking into account (4.6) and (4.7): (4.8) The system (4.8)-(4.9)along with (4.6) is equivalent to the system (4.4)-(4.5) in Theorem 4.2, thus it has a unique solution when p < m.Observe that the form of the polynomials 0(z) and l(z) de- pends upon a relation between m(0), N and m.For instance, if a function f is analytic at zero and thus f(z)j >_ o fj zj' and if is its interpolating polynomial defined in the same way as k and fi o for K 1 and A 0, respectively, then can be represented in the form )(z) = + ?+ ?+ where j'*(z) is the interpolating polynomial of the series f*(z) j > N + 1 m fj zi + m-N-.It takes on the values of f* and all its derivatives up to (r s-1)st order at z = zs, s = 1,...,S, and at z = 1, where z s are the roots of z m-K(z) in (0,1).
where D is defined as d 1 lim (z).
(v) Under the condition of (iii) assume that m(0) = 1, i.e. after an idle period the server is in a "warm-up mode" by reducing its initial capacity.Then, from (5.8) we have M o = 1 and by Theorem 5.3 it follows that (5.12) P'fi = )aP = PVI-Po + = 1Pj min(j,m(j)) + m j >_ N + 1 Pj (vi) Assuming in addition re(j)= 1, j = 0, 1, 2, ..., (i.e.there is no batch service) we obtain from (5.12) 1 (5.13)Pfl = A' Equations (5.11), (5.13) and Theorem 5.3 lead to the following elegant formula (5 14) r(z) a(1 z)P(z) 1 a(z) Recall that the ergodicity condition in this case is p < 1.This is a formula of Pollaczek-Khintchine type.It reduces to the "eollaczek-Khintchine relation" when a(z) = z shown by Schgl [16] for a simi- lar system with state dependent service (note that Schgl did not give a formula for rr(z) = P(z) but just stated this relation).Formula (5.10) for the expected value of a busy period is then reduced to (5.15) = 1 ap..___q o aP o (vii) Assuming further that the group size is distributed geometrically, a = pqS-1, = 1, 2,..., we obtain from (5.14) qz r(z) = P(z) 1 V that enables us to obtain an explicit relation between r and P: k >_ 1 6.CONVERGENCE TIIEOREMS FOR SOME FUNCTIONALS OF INPUT, OUTPUT AND  QUEUEING PROCESSES   In this section we will treat the input, output and queueing processes that are special in cont- 6.4 Theorem.The expected rate of the input flow is (6 5) lira E[N(t)]-P'fi t Proof.From the definition of T j) it also follows that _--I o + o = I o z o du Applying Fubini's theorem in (6.6) we get another representation for T x (6.7) Tz(j, t) = Io PX{(u) = j} du.
6.5 Corollary.For p < ra the expected number of units in the system in equilibrium is either EXit(t)] finite or diverges slower than with the unit speed.In other words, lira = O.
Proof.Observing that the number of units in the system at time t is (t)= (0) -S(t) the statement follows from Theorems 6.3 and 6.4. 6.6 Examples.
(i) A trivial special case follows from Theorem 6.4 when the input stream is independent of the queue.The counting process {N(t)} is now the compound Poisson process {N(t)} with parame- ters (A,a) and the expected rate then should be ,a.On the other hand, from (5.12) we have P'fi = AaP which yield the same result Aa in the right-hand side of (6.5).Note that by Theorem 6.3, Aa is also the expected rate of the number of processed units.
(it) As an application of the above ergodic theorems, we consider the following optimization problem.Let Cl, c2, c3, r be real-valued Borel-measurable functions that represent the following cost rates: Cl(k) denotes the total expenses due to the presence of k units in the system per unit time; c2(j) denotes the expenses of the service act of type j per unit time [observe that the decision to "apply a certain distribution function Bj" when the system accumulated j units, will be affected by the cost function c 2 that is usually inverse proportionally to the service rates]; c 3 is the penalty for every idle period per unit time; r denotes the reward for each completely processed unit per unit time.
In connection with the above cost functions, we introduce the following functionals (whenever the explicit integral is given, the integration with respect to Lebesgue measure is understood): cl((u))du denotes the expected expenses due to the presence of all F 1 [c1,](x,t) EX[ 1 o customers in the system in time interval [O,t] given that initially x units were present; generating function of the jth row of the semi-regenerative kernel is