ON A MULTILEVEL CONTR . OLLED BULK QUEUEING SYSTEM MX / Gr ’ R / 1 LEV ABOLNIKOV

The authors introduce and study a class of bulk queueing systems with a compound Poisson input modulated by a semi-Markov process, multilevel control service time and a queue length dependent service delay discipline. According to this discipline, the server immediately starts the next service act if the queue length is not less than r; in this case all available units, or R (capacity of the server) of them, whichever is less, are taken for service. Otherwise, the server delays the service act until the number of units in the queue reaches or exceeds level r.


INTRODUCTION AND SUMMARY OF PREVIOUS WORK
A multilevel control strategy in a bulk queueing system is based on the utilization of cer- tain feedback relationships between parameters of both bulk arrival and bulk service processes and a current number of units in the system (or in the queue).Using this control strategy it is possible, for example, to respond to an excessively long (or, conversely, too short) queue by changing the rates of the arrival and service processes, or by changing the sizes of arriving groups of units or groups taken for service.
Another possibility to control the queueing process in bulk queueing systems is an assump- tion that the server of capacity R can delay a new service act until the number of units in the queue reaches a certain control level r (where r <_ R).A "service delay discipline" of this type may be useful in reducing start-up costs and, in combination with a multilevel control strategy, offers a considerable scope for improvements and optimization of the queueing process.
Special cases of e queueing system with service delay discipline appeared in Chaudhry and   Templeton [7] and Chaudhry, Madill and Brire [8] without bulk input.The authors called them queueing systems with a quorum and denoted M/Ga'b/1.A more general system was introduced and studied even earlier by Neuts [23], where, in the version of M/Ga'b/1 treated in IS], the author of [23] additionally assumed that service times may depend on groups sizes (between a and b) taken for service.
A queueing system M/Ga'b/1 becomes more attractive (both theoretically and practically) if, in addition to the assumptions about quorum systems accepted in [7,8,23], the input stream is allowed to be bulk.In contrast with models of type M/Ga'b/1, in which the queue length, being less than a, will always (with probability 1) reach a, in systems MX/Ga'b/1 (with general bulk input) the probability of reaching exactly level a in similar situations is less than 1 (in most cases even very small).Or more precisely, the queue length can exceed this level with a positive probability by any conceivable integer value.Behaviors of such processes about some critical level generally differs from those of processes with continuous state space which is treated by various methods called "level crossing analysis" [see 5,10,24].Consequently, this fact makes a preliminary analysis of a corresponding first passage problem (of the queueing process over some fixed level) a necessary and essential part of the treatment of the system.As a separate kind of the first passage problem, this was recently studied in Abolnikov and Dshalalow [3].
Another considerable generalization of the M/Ga'b/1 system, employed in this article, is the assumption that interarrival times of the input stream, the sizes of arriving batches of customers, and service time distributions of groups of customers taken for service, all depend upon the queue length (multilevel control policy).This essentially enlarges a class of real-world systems to which the results obtained are applicable.
It is necessary to mention that unlike a model studied in [8] the authors of this paper do not assume the dependence of service on group sizes.[The latter option, however, was included and extensively studied in Dshalalow [14] and [15], and Dshalalow and Tadj [17-19], and it may be combined with all options set in the present paper.]Other special cases of this model were considered in the literature on queueing theory.An idea to employ a multilevel control policy in bulk queueing systems, presumably, belongs to Bachary and Kolesar [6].However, methods used   there suffer from insufficient analytical justification.Some special cases of a multilevel control strategy with respect to more general queueing systems and inventory and dam models were considered in [1].An example of an application of a multilevel control strategy to a general bulk queueing system (but with no service delay discipline) is contained in [12].
The main purpose of this article is to develop a general mathematical model which would take into account mentioned above features of a queueing system with a bulk input, batch service, multilevel control strategy and a queue length dependent service delay discipline.
In the present article a general bulk queueing system with a multilevel control and a queue length dependent service delay discipline is studied.The results obtained in the paper generalize, compliment or refine similar results existing in the literature [2,[6][7][8]11,12,21,23].
In the first section the authors give a formal definition of a multilevel control bulk queueing system MX/G"'R/1 (with a queue length dependent service delay discipline).In order to describe rigorously the input to the system, the authors use the general notion of a modulated random measure recently introduced and studied in Dshalalow [13].The authors establish necessary and sufficient conditions for the ergodicity of the queueing process with discrete and continuous time parameters and study its steady state distributions in both cases.A recently developed new analysis of a class of general Markov chains in Abolnikov and Dukhovny [4] is used.Due to the queue length dependent service delay discipline assumed in the article, an auxiliary random process describing the value of the first excess of the queue length above a certain control level appears to be one of the kernel components in the analysis of the queueing process (section 3).Using this random process, the authors derive the invariant probability measure of an embedded process in terms of generating functions and the roots of a certain associated function in the unit disc of the complex plane (section 4).The stationary distribution of the queueing process with continuous time parameter is obtained by using semi-regenerative techniques (section 6).The results of this section together with [13] enable the authors to intro- duce and study some functionals of the input and output processes via ergodic theorems.A number of different examples (including an optimization problem) illustrate the general methods developed in the article.

FOIMAL DESCttIPTION OF THE SYSTEM
We begin this section with the definition of the modulated random measure introduced and studied in Dshalalow [11] (formulated there for a more general case).All stochastic processes below will be considered on a common probability space {Q, , (Px)xe }, with @ = {0,1,...}.
2.1 Detion.Let E = R + with its natural topology and let be the space of all Radon measures on the Borel (r-algebra (E).Denote CK the space of all continuous functions on E with compact support and denote the Baire (r-algebra in 1 generated by all maps p-fdp, fee K. (i) Let {12,aY, P,(t), e E} @ be a stochastic process on Z and let denote the w-sec- tion of .Then for T' C_ @ and B (E) we define Yr = B gl , l(r) and call it the holding time of in F on set B. For each f'Lxed w, Y F is a measurable subset of E which can be measured by any Radon measure on (E).In general, Yr is a mapping from f into (E) which can be made a random set after we define the (r-algebra {A C'_ 2(E): (ii) Consider for each j a marked random measure Zj = .Sijeri j (where ex denotes the Dirac point mass) with mark space {0,1,...} and introduce Z = Z(',) = E Zj(r ).

j =o {j
The random measure Z: (f,) (.AI,E) is called a marked random measure modulated by the process .The marked random measure Z can be more vividly represented in the form Z= Let {Q(t); t _> 0} ---.@ = {0,1,...} be a stochastic process describing the number of units at time t in a single-server queueing system with an infinite waiting room.Following the introduction, {Tn;n N0,T 0 = 0} is the sequence of successive completions of service and Qn = Q(Tn + 0).
Input Process.Let C = a 0 eT Define (t) Q(Te([o,t]) + 0), t > 0. Then the input is a compound Poisson process modulated by according to definition 2.1, from which it follows that customers arrive at random instants of time rn, n--1,2, ..., that form a point process mo- dulated by with Si(t -Si_l,(t} = Xi(t as the ith batch size of the input flow.Thus, in our case {X} = {Xi(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 identically dist- ributed.Denote a(t)(z ) = E['i(t)], the generating function of ith component of the process {X}, with a(t) = E[Xi(t)] < c, = 1,2, Service Time and Service Discipline.If at time T n + 0 the queue length, Qn, is at least r (a positive integer number less than or equal to R), the server takes a batch of units of size R (a positive integer denoting the capacity of the server) from the queue and then serves it during a random length of time a n + 1" [We assume that r n + has a probability distribution function BQn E {B 0, B 1 ,...}, where the latter is a given sequence of arbitrary distribution functions with finite means {b0,b1,...}.]Otherwise, the server idles until the queue length for the first time reaches or exceeds the level r.Let 7n = inf{k N: r k >_ Tn} n N O Then the size of the first group after T n (which arrives at instant of time r.rn) is XTn, n.
For more convenience in notation, we reset the first index-counter of the process {X} on 1 after time hits T n.Therefore, in the light of the new notation, X1Qn, X2Qn,... are the sizes of successive groups of units arriving at the system after T n.Let St:n = Xo n + Xln+ + XkQ n, where XOQ n = Qn.Then, given Qn, {Skn k NO} is an integer-valued delayed renewal process.Denote v n = inf{k >_ O" Skn >_ r} the random index when the process {Skn} first reaches or exceeds level r given that the queue length is Qn.Thus, r,n is the instant of the first excess of level r by the queueing process after time T n (more accurate but less friendly notation for this instant would be r.rn + n_l,).
For the next constructions we need a more universal notation for the instant of the first excess of level r, appropriate for the situations when this instant occurs after T n or at T n. in other words we define At that instant of time the server is supposed to take a batch of the size min{Q(On),R} for service.In other words, if Qn >r, T n + 1 Tn coincides with length of service time a n + of the n+lst batch.If Qn < r the interval (Tn,Tn+l] contains the waiting time for X1Q n + + X,nQ n units to arrive and the actual service time a n + 1" Finally, denoting V n = Z(crn) we can abbreviate the definition of the servicing process through the following relation for {Qn}" (2.3) n + = 3. FIRST PASSAGE PROBLEM In the following sections we will be using some basic results on the first passage problem stated and developed in Abolnikov and nshalalow [3].Some of the results of [3] (which the authors will highlight in this section) were obtained by Dynkin [20] and Takcs [25] more general processes.However, for convenience in notation and with the purpose of a more specific terminology only results of [3] will be mentioned.
First we treat the process {SVn } without any connection to the queueing system.For this reason we will temporarily simplify the notation introduced in section 2 by suspending the second subscript in the sequence {X} and the corresponding index in the probability measure pi and ex- pectation Ei.
Therefore, in this section we will discuss the critical behavior" of a compound Poisson process Z determined by a Poisson process 7"={r n=t0+t l+...+tn;n>_O, o=O) on R+ marked by a discrete-valued delayed renewal process S = {S n = X O + X + + X n n >_ 0} on @.
As mentioned above, we assume that the processes 7" and S are independent.We also assume that inter-renewal times tn = rn-rn-1, are described in terms of its common Laplace-Stieltjes e(O) = E[e-ot,] = ,X::O' n = 1,2, transform For a fixed integer r >_ 1 we will be interested in the behavior of the process S and some related processes about level r.
The following terminology is introduced and will be used throughout the paper.
(i) For each n the random variable Vn= inf{k >_ 0:S k >_ r} (defined in the previous section) is called the index of the the first excess (above level r-1).
(ii) The random variable Svn is called the level of the first excess (above r-1).
(iii) The random variable rvn is known as the first passage time of S of level r.
(iv) The random variable 1 n = SVn-S 0 is called the increment of the input process over the time interval [Tn, gn] or shortly, the total increment.Let 0r r, 0 z 0], (3.1a) 7(')(0,z) = E i[e o z 0], j(i)(0 z) = E i[e-or, s Gi(O,z = EjoEi[e-rJz 8j Iur_i(Sj)], where Up = {0,1,...,p} and I A is the indicator function of a set A. We call Gi(O,z the generator of the first excess level.We will also use the following functionals of marginal process: It is readily seen that Gi(z is a polynomial of (r-1)th degree.
We formulate the main theorems from Abolnikov and Dshalalow [3] and give formul for the joint distributions of the first psage time and the random variables listed in 3.1 (i-iii).
.2 Whrem.The functional 7(i}(0, z) (of the first passage time and of the index of the first excess leve 0 satisfies the following foula: (3.2c) Specifically, the Laplace-Stieltjes transform of the first passage time, 7(i)(0,1), is as From formula (3.2a) we immediately obtain that the mean value of the index of the first excess equals (3.4) From (3.2a) we also obtain the mean value of the first passage time: E [r0 = (i).
3.5 Theorem.The generator Gi(O,z) of the first excess level can be determined from the following formula: The rationale behind the use of the term "generator of the first excess level" comes from the following main result.
3.6 Theorem.The functional (i)(O,z) (of the first passage time and of the first excess level) can be determined from the formula z) = -[1 3.7 Remark.To obtain the functionals of the marginal processes defined in (3.1b-3.1d)we set e(9)= 1 in formulas (3.2a), (3.5a) and (3.6a).
3.8 Corollary.The generating function (i)(z) of the first excess level is determined by the following formula: (3.Sb) By using change of variables in (3.8a) we can transform it into an equivalent expression zr_ { a(z) a(: 3.9 Corollary. (3.9a) Specifically, the mean value ](i)= E [1o] of the total increment is then (3.9b) 3(i) ((i).
3.10 Remark.Now we notice that the above results can be applied to our queueing system, where in formulas (3.2a)-(3.9a)we supply a(z) with subscript i.

EMBEDDED PROCESS
4.1 Definition.Let T be a stopping time for a stochastic process {fL*5, (PX)=eo, Q(t); t> 0} (, !8()).{Q(t)} is said to have the locally strong Markov property at T if for each bounded random variable (: ---, @r and for each Baire function f: r---,R, r = 1,2,..., it holds true that EZ[f o o 0 T Ir] = EZT[f o ] PX-a.s. on {T < where O u is the shift operator. From relation (2.3) and the nature of the input it follows that the process {f2,, Q(t); t>_ 0} = {0,1,...} possesses a locally strong Markov property at Tn, where T n is a stopping time relative to the canonic filtering a(Q(y);y <_ t), n = 1,2, Thus the embedded process {f,,(px)=eq, Qn;nENo} is a homogeneous Markov chain with transition probability matrix denoted by A = (aij).
4.2 Lemma.The generating function Ai(z of ith row of matrix A can be determined from the following formulas: /3i(0), Re(O) >_ 0, is the Laplace-gtieltjes transform of the probability distribution function Bi, and (i) satisfies one of the formulas (3.8a) or (3.8b), taking into account remark 3.10.Proo1'.Since Ai(z)= E i[zQ1], formulas (4.2a) and (4.2b) follow from (2.3) and probability arguments similar to those in the proofs of section 3. Observe that since (i)(z) = zi, > r, in (3.8a), formula (4.2b) reduces to (4.2d) /r, R) = z(i-R) + i> r, which also agrees with the result that could have been obtain directly from (2.3) for _> r.

El
For analytical convenience and without considerable loss of generality we can drop the modulation of the input process and service control when the queue length exceeds a fixed (perhaps very large) level N.In other words, we assume that Given assumption (AS), it can be shown that the transition probability matrix A is re- duced to a form of the AR, N-matrix introduced and studied in [4].There the stochastic matrix A = (aij; i,j E qt = {0,1,...}) is called a AR, N-matrix if it is of the form A= (aij" i,j t" aij = kj_i + r > N, j >_ i-R aij = O > N, j < i-R), where o ki is an atomic probability measure.The following two theorems are necessary to j=0 obtain all main results in this section.(Abolnikov and Dukhovny [4]).Let {Qn} be an irreducible aperiodic Markov chain with the transition probability matrix.A in the form of a At, N-matrix.Qn is recurrent- positive if and only if (4.3a) and (4.3b) dz z=l < x), = 0,1,...,N, <R.dz z=l 4.4 Theorem (Abolnikov and Dukhovny [41).Under the condition of (4.3b) the function z r-K(z) has exactly r roots that belong to the closed unit ball [(0, 1).Those of the roots lying on the boundary OB(O, 1) are simple.
Therefore, given that p < R, the Markov chain {Qn} is ergodic.Let P = (Pz;Z ) be the invariant probability measure of operator A and let P(z) be the generating function of the components of vector P. Now we formulate the main result of this section.
12 Below we give another version of theorem 4.6, where the corresponding formulas will be analytically less elegant but numerically are of greater advantage, especially for large N.
i=O Taking into account these relations and proceeding as in the proof of formula (4.6d), we obtain (4.8a).The rest of the statement can be proved similar to theorem 4.6.
Observe that, unlike formula (4.6b), that gives equations for finding N unknown probabilities, the right-hand side of (4.7a) contains only R unknown probabilities.This may be advantageous in computations, especially when N >> R. (i) Let j = EJ[T] and fl = (flj;j )T.Then we will call the value Pfl the mean service cycle of the system, where P denotes the stationary probability distribution vector of the embedded queueing process (it) Let a-(a ;z e @)T, , = (Ax ;z )T and let p a, fl.A be the Hadamard (entry- wise) product of vectors a, fl and A. We call the scalar product Pp the intensity of the system.
Observe that the notion of the "intensity of the system" (frequently called the offered load in queueing theory) goes back to the classical M/G/1 system, when Pp reduces to p = Ab.
5.2 Proposition.Given the equilibrium condition p < R, the mean service cycle can be determined from the following expression: )) (5.2a) Pfl b + E 2= o Pj(bj b + Proof.Obviously, flj = bj+ /(J)/Aj.The statement follows after elementary algebraic transformations.Using (5.2a) we similarly get r--I i).
(5.3) PP = P + E iN= 0 Pi(Pi P) + E 0 Pi'J(   5.4 Theorem.Given the equilibrium condition p < R, the intensity of the system Pp and the mean server load coincide. Proof.Because of (2.2) we have (5.4a) S 0] + q-'i=oP i=RPi" Formula (4.6a) can be rewritten in the form Then (P, 1) = 1 yields The last expression allows modification of (5.3) into ,-x o:U ]+ Z: n-x oo a Rp {So -0 pi which, because of (5.4a), obviously equals I. N R)Zu, o S.0 + E = a (-R) + E = o P(.-+ R-0 = R . Therefore, for the mean server load we can use formula (5.3) which requires the know- ledge of PO,"',PN" The following formula for is less friendly but it requires just Po,'",PR-I" 5.5 Proposition.The mean server load can alternatively be obtained from the following formula: (5.5a) Proof.The statement directly follows from (5.3) and (4.6c).I:l 6. GENEILL QUEUEING PROCESS In this section our main objective is the stationary distribution of the queueing process with continuous time parameter.Although this section is developed in a similar way as section 5 in Dshalalow [16] for the sake of consistency and better readability we include all necessary details.We need the following 6.1 Definitions.
(ii) Let (Qn, Tn) be an irreducible aperiodic Markov renewal process with a discrete state space @.Denote / = E[T1] as the mean sojourn time of the Markov renewal process in state Let, K(t)-(Kjk(t) ;j, kE P) be the semi-regenerative kernel (see definition 6.1 (iv)).The following statement holds true.
6.3 Lemma.The semi-regenerative kernel satisfies the following equations: where 6jk is as defined in (6.2a) or (6.2b) and j denotes the density of the joint probability dist- ribution function of the random variable SUo and the instant 0 o of the first passage time of level r by the queueing process { Q(/)} (defined by (2.2)).
Proof.The above assertion follows from direct probability arguments.Now we are ready to apply the Main Convergence Theorem to the semi-regenerative kernel in the form of corollary 6.5.
6.4 Theorem (The Main Convergence Theorem, cf. C. inlar [O]).Let {f,51:,(PZ)z,, Q(t); t>_ O} (@, (@)) be a semi.regenerativestochastic process relative to the sequence {tn} of stopping times and let K(t) be the corresponding semi-regenerative kernel.Suppose that the associated Markov renewal process is ergodic and that the semi.regenerativekernel is Riemann integrable over R +.Then the stationa distribution x = (rx x ) of the process {Q(t)} exists and it is determined from the formula: (6.4a) k = Z j pl f o KJ k(l)dt' k e .
6.5 roH.Denote g= (h;j, k e )= f o K(t)dt as the integrated semi-regene- rative keel, hi(z) the generaling funclion of jlh row of malx H, z)= (hi" j @)T and as lhe generating function of vector .Then the following foula holds = z) PH Finally, Proof.From (6.4a) we get an equivalent formula in matrix form, =.
formula (6.5a) is the result of elementary algebraic transformations.
6.6 Theorem.Given the equilibrium condition p < R for the embedded process {Qn}, the stationary distribution x = (rx;z t) of the queueing process {Q(t)} exists; it is independent of any initial distribution and is expressed in terms of the generating function r(z) of x by the following formulas: = + E o 1 Kj(z) (6.6b) dj(z) = AX-i where P(z) is the generating function of P, Pfl is determined in proposition 5.2, Gi(z is deter.mined in (3.5a) (taking into account Remark 3.7), and d(z) is defined as dj(z) with all subscripts dropped.
Proo[.Recall that the Markov renewal process {Qn, Tn} is ergodic if p < R. By corollary 6.5 the semi-regenerative process {Q(t)} has a unique stationary distribution r provided that p < R. From (6.3a) we can see that the semi-regenerative kernel is Riemann integrable over R +.. Thus, following corollary 6.5 we need to find the integrated semi-regenerative kernel H (which is done with routine calculus) and then generating functions hi(z) of all rows of H. First we find that (6.6c) E o o p zp fOOo 6i, p i(u)[1 Bi(u)] du = .,idi(zThen it follows that (6.6d) hi(z)=zil r-i-l{ 1 } di(z)!r)(z)O<i<r //x (I x)(l'-ai(z)) "[- where !r)(z) denotes the tail of the generating function (i)(z) summing its terms from r to However, it is easy to show that (]!r)(z) and (i)(z) coincide.Then it appears that (6.6e) hi(z = zidi(z), >_ r,   where the index can be dropped for all exceeding N, in accordance with assumption (AS) made in section 4. Formula (6.6a) now follows from corollary 6.5, equations (6.6c-6.6e), (3.5a), (3.6a)   and some algebraic transformations.
[21 7. EXAMPLES AND SPECIAL CASES In one of the first examples we drop the modulation of the input preserving all other special features of the system.
7.1 Proposition.In the bulk queueing system with no modulation of the input the generating function r(z) of x can be derived from the following formula: Proof.Formula (7.1a) follows from (6.6a) and (4.6a) after noticing that (7.1b) y (yif(y)) = 0 for all i> R, where f is any function analytic at the origin.Formula (7.1a) may look unfriendly but in the way it is presented it yields a number of elegant special cases.
The mean server load can be evaluated from formula (5.5a) that leads to (7.2f) = R E ( i) (R + ,).+. + i=r p2 i=oPi (iii) in the condition of (it) assume additionally that r= R-1.Then the upper expression in (7.2e) vanishes reducing (7.2d) to 7r(z) i-P!qz = P(z) 1 " +a z PrKr(z)zr" [:1 In the next situation we suppress bulk arrivals however preserving the modulation.
(i) In the condition of theorem 7.3 the mean server load is defined by l--Eie* Pi min{ R, max{r, i}} = R E ooi I:t, pi + "E 1 0 Pi + Z r F 1 -1 ip that also agrees with formula (5.5a) to which this special case is applied.
(iii) If the input is a stationary compound Poisson process (i.e.nonmodulated) then its in- tensity is c, which is also the mean number of arriving units 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 random measure Z by the formula x = lira t_oo{Ez[Z([O,t])].We will apply the formula from theorem 7.5 (Dshalalow [9]) for more general Poisson process modulated by a semi-Markov pro- tess: Pp P#' where by theorem 5.4 Pp = and PB satisfies (5.2a).Thus we have that: (7.4a) : = e--.
(iv) By virtue of obvious probability arguments we can derive the probability density function of an idle period in the steady state: v-., r-l i=oPi distributed if given on the trace w-algebra t gl I,.J Qff {j}) (where t is the canonic filtering induced by the process {Q(t))).The latter enables one to evaluate the functional E:[S([O,t])] by using the independence of Q(Oo(j) and I{j}x[o,t]o{Qn, Tn}.Applying the monotone convergence theorem and in the light of definition 6.1 (iii) we have E:[S([O,t])] = E j,, E J[inf{Q o Oo}]RX(j,t) E:[ E j, inf{ Q(Oc([o,,l) o (0), R}I (} o (t)].
Since E: [,,,inf{Q(ac([o,t])o(t)),R}I{j} oh(t)] < R for all t>0 which simplifies the output rate to (7 = lira E j,, E J[inf{Q o 0o} ,t) finally yielding from theorem A.1 (ii) (see Appendix) that o = Finally, by theorem 5.4 the mean server load and the intensity of the system coincide and this proves the theorem.

I-1
Since Z is the input process modulated by the semi-Markov process we can use formula (7.5a) in theorem 7.5 which gives the mean input rate of the modulated semi-Markov process Z .From (7.6a) and (7.6c) it therefore follows that -.T his is to be expected in most of the systems thereby proving valid one of the conservation laws: "In an ergodic stochastic system ihe inpu and output raes are equal". 7.7 Coroflary.For p < R the ezpected number of units in the system in equilibrium is either finite or diverges slower than with the unit speed.
Proof.Since the number of units in the system at Q(t) = Q(0)+ Z([0,t])-S([0,t]) the statement follows by theorems 7.5 and 7.6.time is 7.8 Example.As an application of the ergodic theorems 7.6 and A.1 (see appendix), we consider the following optimization problem.Let cl, c2, c 3, c4, w be real-valued Borel-measur- able functions that represent the following cost rates and functionals: c(k) denotes the total expenses due to the presence of k customers in the system per unit time.Then rl[Cl,Q](z,t)= EX[ o Cl(Q(u))du] gives the expected expenses due to the presence of all customers in the system in time interval [0,t] given that initially z units were present.By Fubini's theorem and theorem A.1 (ii) we have lira ri[ci Q](r,t) = j > 0cI(j)rrj = xe 1 as t---*oo the expected cost rate due to the presence of all units in the system, where c 1 = (ct(0), c1(1),...) r. c2(j) denotes the expenses for 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].
On the other hand, from Markov renewal theory it is known that Pj i = pand the statement (i) then follows.