ON A FIleT PASSAGE PROBLEM IN GENEL QUEUEING SYSTEMS WITH MULTIPLE VACATIONS

The author studies a generalized sngle-server queueng system wth bulk arrivals and batch service, where the server takes vacations each time the queue level falls below r (>_ I) n accordance wth the multiple vacation dscpIine. The nput to the system is assumed to be a compound Poisson process modulated by the system and the service s assumed to be state dependent. One of the essential part n the analyss of the system s the employment of new techniques related to the first excess level processes. A prefiminary analyss of such processes and recent results of the author on modulated processes enabled the author to obtain all major characteristics for the queueng process expficitly. Various examples and applications are dscussed.


I. INTRODUCTION
The class of queueing systems, where a server (or servers) periodically goes on "vacation" was offered in many works published in seventies because of its connection with congestion phenomena in local area networks.The interest in such systems has been further enhanced in the eighties and the beginning of the nineties due its applicability in communications, computer and production systems.Such models also apply to manufacturing processes that exhibit uninterruptible mainte- nance tasks, for instance, tool changes or alterations of a flexible manufacturing system.Vacation models have been studied in many queueing systems of types M/G/1, GI/G/1 and closed systems (see Dshalalow IS]).tReceived: September 1991, Revised: January 1992  Since the first publications on vacation models over twenty years ago the number of works on this topic has increased tremendously and as a result several survey papers appeared (see for ins- tance Doshi [10]) classifying various systems and their modifications.One of the most frequently studied types of vacation models are those with so-called exhaustive service and general nonexhaus- live service.The exhaustive service refers to the systems, where the server takes a vacation only when the queue of primary customers is empty.General nonexhaustive service refers to many other systems, where the server may go on vacation even if there are some customers waiting for service.
For example, if the server takes batches of customers of some minimum size (say r), he prefers not to wait until the queue accumulates so many and it goes on vacation.Upon his return the server checks if the queue length is of an appropriate size, and if not, leaves the system again, and so on.
Such systems are also classified as having multiple vacations.Clearly, this particular nonexhaustive system generalizes a class of exhaustive systems with r = 1.But there are also some other modifi- cations of general nonexhaustive models, such as those with Bernoulli schedules (see Pamaswamy  and Servi [16]).It is assumed in such a system that after each return to the system after a vacation the server may interrupt his vacation cycle regardless of how many customers are in the system, in the general case, with a probability dependent on the number of vacations taken (see Kella [13]).
Many papers in the literature on queueing theory dealt with such models under different assump- tions to the input stream, service discipline and the waiting room capacity.Queueing processes, busy period processes, and other processes were analyzed in those papers.The methods applied in- eluded decomposition (see Footman and Cooper [11] and Shanthikumar [17]) and supplementary variables techniques (see Takagi [19]).
In the present paper the author considers a class of queueing systems of type MX/Gr/1 with general nonexhaustive service and multiple vacations.It is assumed that if after completion of a service there are at least r customers in the waiting room the server takes a batch of customers of size r and begin their service.Otherwise, the server leaves the system for a vacation and upon its return the server checks if during its absence more customers arrived at the system and there are enough customers to pack the batch.If this is still not the case the server again leaves the system and so on.It is assumed that the input stream is compound Poisson.Under these assumptions, the model studied in this article generalizes a class of multiple vacations systems treated by various authors in the recent past (Abolnikov, Dshalalow and Dukhovny [3], Harris and Marchal [12], Lee  and Srinivasan [14], Minch [15] and Shanthikumar [17]).
One of the central problems in the analysis of such systems is the behavior the queueing process at the time the server begins processing groups of customers.Excluding trivial cases, it is clear that at those instants of time, the queueing process is more likely to exceed level r (first excess level) than just to reach it, and thus the results on "level crossing analysis" of processes [4,6,18] are inappropriate.Fortunately, as the author will show it, the paper [1] by Abolnikov and Dshalalow contains relevant results for the analysis of excess level processes needed for multiple va- cations systems.
The author will also show that the analysis is well suited for generalized versions with mo- dulated input and state dependent service times.In the present article queuing processes with con- tinuous and discrete time parameters are treated.The author establishes necessary and sufficient conditions for the ergodicity of the processes, and by the use of the semi-regenerative analysis, the author derives a simple and explicit relation between the stationary distributions for the both pro- cesses.The embedded Markov process is analyzed by methods developed by Abolnikov and   Dukhovny [2] and Abolnikov, Dshalalow and Dukhovny [3].The invariant probability measure of the transition probability matrix of the embedded process is obtained in terms of the probability generating function in an explicit form.Various examples and applications are discussed.

MODULATED INPUT PR.OCESS
2:1 Definition (Dshalalow [9]).Let E be a locally compact and (r-compact space with a countable base.Let be the space of all Padon measures on the Borel (r-algebra (E), and let be the Borel (r-algebra in alg generated by the vague topology ff.Denote by (K the space of all continuous functions on E with compact support and denote by the Baire (r-algebra in atg generated by all maps/ J" f dp, f E g" (i) Let {f2,r,P,(t), t E E} be a stochastic process on E and let o denote the -sec- tion of .T hen for F C_ I, and B @ (E) we define }/ = B Cl (F) and call it the holding time of in F on set B. For each fixed w, )/. is a measurable subset of E which can be measured by any Pdon measure on (E).In general, OF is a mapping from f2 into (E) which can be made a random sei with respect to the (r-algebra {A C_ %(E): /7 I(A) }.

= = o
The random measure Zb:(f,ff) (.Ag,iff/) is called a marked random measure modulated by the process .T he marked random measure Z can be more vividly represented in the form Z= S n n (iii) Consider he following special case.Let Z j be a compound random measure (with mark space {0,1,...}) (2.1b) Zj = E Si=ij' obtained from the underlying counting measure Ni = E irij by independene marking, i.e. for each {j}, {Xii = Sii-Si_l,j;i = 1,2,...} is a sequence of independen and identically distributed random variables with common mean denoted aj.Assume that N is a Poisson counting measure with mean measure #5 = AiL art,, where denotes the Lebesgue measure and A 5 is a positive con- san.Then, we call such a marked random measure modulated by a compound Poisson random measure modulated by the process Assume that E=IR+with its usual topology.Let {f2,,(PZ)z,,Q(t); tE} = {0,1,...} be a stochastic process describing the evolution of the number of units in a single- server queueing system, and le {Tn;n No, T O = 0} be the sequence of successive completions of service and Qn = Q(Tn + 0).
2.2 Notation.Denote C = n 0 T and (t) = Q(Tc([o,t]) + 0), E. Then the input is a compound Poisson process modulated by according to definition 2.1 (iii), from which it fol- lows that customers arrive at random instants of time 7"n, n = 1,2,..., (with arrival intensity A(t)), that form a point process modulated by with Xn(t as the ith batch size of the input flow.Thus, in our case X = {Xn(t)} is an integer-valued doubly stochastic sequence describing the sizes of groups of entering units.We assume that, given (t), all terms of X are independent and X identically distributed.Denote a(t)(z E[z n(t)], n = 1,2,..., the generating function of the nth component of the process X.
3. CONTROL OF THE SERVICING PROCESS At time T n + 0 the server can carry a group of units of size r and it takes that many for service if available.Otherwise (that is if Qn < r), the server leaves for vacation for a random period of time with probability distribution function VQn {Vo, V,...,Vr_.},after which the server returns to the system and leaves again if the total number of customers in the system is still less than r.The server will immediately start processing a group of customers if upon its return the queue has accumulated at least r customers.[Since the server leaves the system for a random time and it does not interrupt its vacation even if the queue length has accumulated the desired level, and because customers arrive in batches, it is more likely in general that the queue level by that time exceeds the level r rather than it will equal r.]The service time of that group of customers is supposed to last a random time (r n + distributed BQn i8 = {B0,B,...}, where !B is a given set of arbitrary probability distribution functions with finite means {b0,b,...}.By pi or E we denote the conditional expectation induced by the initial measure Q.We denote The capacity of the waiting room is assumed to be unlimited.
We need more details and formalism of how exactly vacations and service are functioning.
Suppose that the server leaves the system at time T n and returns at time T n + q[' reaching the system with Q(T n + q(') < r.Then the server leaves the system again and it is coming back and forth until upon its arrival at T n + "i' + + f the queue length for the first time reaches or ex- ceeds level r, where is an integer-valued random variable counting the total number of server va- cations prior to the service begin.[Of course, (u) = also depends upon n.] Let Yn,Yn2,... be the increments of the arrival process over time intervals (T n T n + ], (T n T n + "('], Then, nk = Yno + Ynl "t" + Ynk where Yno = Qn is a modulated integer-valued delayed renewal process with the increments Ynl,Yn2,..., distributed in accordance with a common generating function qQn(Z) which can easily be computed as (3.1) qQn(Z) = Vn(Qn-QnaQn(Z)), where V(a) denotes the Laplace-Stieltjes transform the distribution function V i.Thus v(n) = inf{k: Snk >_ r) and the instant 5r n = T n + q[' + + q[' is the first passage time (after Tn) of the queueing process to reach or to exceed level r.In a situation when Qn >-r, we may denote ff'n to be T n to extend the notion of the first passage time for all cases.
Finally, figuring that Z((rn) gives the total number of customers that arrive during the period of nth service, an, we obtain the following relation for process {Qn}: Qn r+ Z((ir n + 1)' Qn -> r.

FIRST EXCESS LEVEL PROCESSES
Throughout sections 4-8, we will use some basic results on a first passage problem stated and developed in Abolnikov and Dshalalow [2].
Let {f2,,P} be a probability space.Consider on this probability space a delayed renewal process {S n = o + 1 +''' + n n >-O} valued in a set __N.Let o =So = a.s. (for some E hi o) and let 1 be distributed in accordance with the generating function q(z) = E[zl], analy- tic inside the open unit ball B(0,1) centered at the origin and continuous on its boundary 0B(0,1), and with finite mean q = E[I].
For a fixed integer s >_ 1 we will be interested in the behavior of the process {Sn} and some related processes about level s.
The following terminology from [2] will be used throughout the paper.
(i) Denote t/= inf{k >_.O: S k >_ r} and call it the in&a: of the the first excess of level r.
(ii) The random variable S, is called the level of the first excess of r.
(iii) The random variable ff is known as the first passage time of {Sk} of level r.where Up = {0,1,...,p} and I A is the indicator function of set A. We call G(O,z) the generator of the first excess level.We will also use the following functionals of marginal processes: (4.1b) 7i)(z)-7i)(0,z), = = It is readily seen that Gis(z is a polynomial of (r-1)th degree.
We formulate the main theorems from Abolnikov and Dshalalow [2] and give formul for the joint distributions of the first psage time and the random variables listed in 4.1 (i-ill).To adopt these results for the processes to be treated in the next sections we additionally sume that the parameters ( Ai, ai(z), ai and qi(z)) of the point, compound and vacation processes intr duced sections 2 and 3 may depend on index (in agreement with the modulation).Consequently, these parameters will be indexed by i.The random variable ff will then become the first term 1 of the point process {n} of first psage times.
4.2 Threm.The functional 7i)(fl, z) (of the firsf passage fime 1 and of fhe index 1 of Let the first excess level) satisfies the following formula: (4.4) Specifically, the conditional mean value of the first passage time is Ei[I] = // 4.5 Theorem.The generator Gir(O,z) of he first ezcess level can be determined from the following formula: The use of the term "generator of the first excess level" is due to the following theorem.
4.6 Theorem.The functional i)(O, z) (of the first passage time and of the first excess level) can be determined from the formula (4.6a 4.7 Remark.To obtain the functionals of the marginal processes defined in (4.1b-4.1d)we set ei(O) = 1 in formulas (4.2a), (4.5a) and (4.6a).

SEMI-REGENETIVE ANALYSIS
In this section we will analyze the queueing process {Q(t)}.It will be shown that this pro- cess is semi-regenerative relative to the point process {Tn} and we will obtain its ergodicity condi- tions.
We need the following notion introduced in Dshalalow [7].
5.1 Definition.Let T be a stopping time for a stochastic process {,, (P)ze, Q(t); > 0} ---.(4, !B()).{Q(t)} is said to have the locally strong Markov property at T if for each bounded random variable : gt r and for each Baire function f" r , r = 1,2,..., it holds true that where O u is the shift operator.
From the nature of the input process and relation (3.2) it follows that {f,hr, (PZ)x, Q(t); t >_ 0} ---, = {0,1,...} possesses a locally strong Markov properly at Tn, where T n is a stopping time relative to the past of the process r(Q(y);y < t), for each n = 1,2, Thus the embedded process Qn is a homogeneous Markov chain with the transition probability matrix denoted A = (aij; i,j E t).In the next section we will show that Qn is irreducible and aperiodic, and that under a certain (necessary and sufficient) condition, it is recurrent-positive.We assume that this condition is met and denote by P the invariant probability measure of the operator A. Consequently, the two- dimensional process {Qn,Tn} is a Markov renewal process.Therefore, we conclude that the process {,zh,(PX)xe (t); t>_ 0) --+ 4, defined in notation 2.2, is the minimal semi-Markov process associated with Markov renewal process {Qn, Tn} and therefore, following definition 2.1, the input process {12, , (PZ)ze Z([O,t]);t >_ 0} is a compound Poisson process modulated by the semi- Markov process {(t)}.Then the value Zi = Zi[T1] is obviously the mean sojourn time of in state {i}.Denote fl= (flz;z E )T.In one of the next sections we show that the value Pfl (scalar product of the invariant probability measure and the vector of the mean sojourn times of the process ) which is called the mean inter-renewal time of the Markov renewal process, is finite.5.2 Notation.Let = p {ze([0, Tx > Then, given that (0)=z and because Z is not modulated by a new value input process takes on value Z z (introduced in definition 2.1 (it)).Therefore, we have of , the = t]) = Let {fl,, (PZ)xe Q(t); t >_ 0} (, !8()) be a semi-regenerative process relative to the sequence {Tn} of stopping times.Introduce the probability gik(t = Pi{Q(t) = k, T 1 > t}, i,k 4.
We will call the functional matrix K(t)--(Kik(t); i,ke) the semi-regenerative kernel.The follow- ing proposition holds true.[We set the value of any sum is zero whenever the lower index is greater than the upper index.] 5.3 Proposition.The semi-regenerative kernel satisfies the following equations: (5.3a) Kjk(t = where 6j, t is as defined in (5.2a) or (5.2b) and j denotes the joint probability density function of the first excess level S v and the first passage time fit" Proof.The above assertion follows from direct probability arguments.Now we are ready to apply the Main Convergence Theorem.5.4 Theorem.Let {f2,, (PX)x,, Q(t); t>_ 0} --(, ()) be a semi-regenerative process relative to { Tn} and let K(t) be the semi-regenerative kernel.Suppose that the associated Markov re- newal process is ergodic, the embedded Markov chain Qn is ergodic (and its invariant probability measure is P) and that K(t) is integrable over +.Then the stationary distribution x = (rz; x E ) of Q(t) exists and it can be determined in terms of its generating function 7r(z) as follows.Denote H = (hjk; j, k E )= fo K(t)dt, hi(z) the generating function ofjth row of matrix H and h(z)= (hi" j )Z.Then, (5.4a) a'(z) = Ph(z) where P is the stationary probability vector of Qn and fl is the vector of mean sojourn times of the semi-Markov process .
One of the main results of this section follows.
Therefore, given that p < r, the Markov chain {Qn} is ergodic.Let P = (Px ;x ) be the invariant probability measure of operator A and let P(z) be the generating function of the compo- nents of vector P. Denote (0,1)= {z e C: I I z I! <-1}.Now we formulate the main result of this section.