Time-dependent analysis of an M / M / c preemptive priority system with two priority classes Selen, J.; Fralix, B.H.

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1 Time-dependent anaysis of an M / M / c preemptive priority system with two priority casses Seen, J.; Fraix, B.H. Pubished in: Queueing Systems DOI: 1.17/s Pubished: 1/12/217 Document Version Accepted manuscript incuding changes made at the peer-review stage Pease chec the document version of this pubication: A submitted manuscript is the author's version of the artice upon submission and before peer-review. There can be important differences between the submitted version and the officia pubished version of record. Peope interested in the research are advised to contact the author for the fina version of the pubication, or visit the DOI to the pubisher's website. The fina author version and the gaey proof are versions of the pubication after peer review. The fina pubished version features the fina ayout of the paper incuding the voume, issue and page numbers. Lin to pubication Citation for pubished version APA: Seen, J., & Fraix, B Time-dependent anaysis of an M / M / c preemptive priority system with two priority casses. Queueing Systems, 873-4, DOI: 1.17/s Genera rights Copyright and mora rights for the pubications made accessibe in the pubic porta are retained by the authors and/or other copyright owners and it is a condition of accessing pubications that users recognise and abide by the ega requirements associated with these rights. Users may downoad and print one copy of any pubication from the pubic porta for the purpose of private study or research. You may not further distribute the materia or use it for any profit-maing activity or commercia gain You may freey distribute the URL identifying the pubication in the pubic porta? Tae down poicy If you beieve that this document breaches copyright pease contact us providing detais, and we wi remove access to the wor immediatey and investigate your caim. Downoad date: 2. Jan. 218

2 Time-dependent anaysis of an M/M/c preemptive priority system with two priority casses Jori Seen and Brian Fraix March 16, 217 Abstract We anayze the time-dependent behavior of an M/M/c priority queue having two customer casses, cass-dependent service rates, and preemptive priority between casses. More particuary, we deveop a method that determines the Lapace transforms of the transition functions when the system is initiay empty. The Lapace transforms corresponding to states with at east c high-priority customers are expressed expicity in terms of the Lapace transforms corresponding to states with at most c 1 high-priority customers. We then show how to compute the remaining Lapace transforms recursivey, by maing use of a variant of Ramaswami s formua from the theory of M/G/1-type Marov processes. Whie the primary focus of our wor is on deriving Lapace transforms of transition functions, anaogous resuts can be derived for the stationary distribution: these resuts seem to yied the most expicit expressions nown to date. 1 Introduction Priority modes with mutipe servers constitute an important cass of queueing systems, having appications in areas as diverse as manufacturing, wireess communication and the service industry. Studies of these modes date bac to at east the 195 s see, e.g., Cobham [7, Davis [8, and Jaiswa [15, 16 yet many properties of these systems sti do not appear to be we understood: recent wor addressing priority modes incude Septcheno et a. [27, and Wang et a. [29. We refer the reader to [29 for more specific exampes of appications of priority queueing modes. Our contribution to this stream of iterature is an anaysis of the time-dependent behavior of a Marovian muti-server queue with two customer casses, cass-dependent service rates and preemptive priority between casses. To the best of our nowedge, the joint stationary distribution of the M/M/1 2-cass preemptive priority system was first studied in Mier [24, who maes use of matrix-geometric methods to study the joint stationary distribution of the number of high- and ow-priority customers in the system. More particuary, in [24 this queueing system is modeed as a quasi-birth and death QBD process having infinitey many eves, with each eve containing infinitey many phases. Mier then shows how to recursivey compute the eements of the rate matrix of this QBD process: once enough eements of this rate Department of Mathematics and Computer Science, Eindhoven University of Technoogy Department of Mechanica Engineering, Eindhoven University of Technoogy Department of Mathematica Sciences, Cemson University E-mai address: j.seen@tue.n 1

3 matrix have been found, the joint stationary distribution can be approximated by appropriatey truncating this matrix. This singe-server mode is featured in many wors that have recenty appeared in the iterature. In Septcheno et a. [27 an exact, recursive procedure is given for computing the joint stationary distribution of an M/M/1 preemptive priority queue that serves an arbitrary finite number of customer casses. The M/M/1 2-cass priority mode is briefy discussed in Katehais et a. [2, where they expain how the Successive Lumping technique can be used to study M/M/1 2-cass priority modes when both customer casses experience the same service rate. Interesting asymptotic properties of the stationary distribution of the M/M/1 2-cass preemptive priority mode can be found in the wor of Li and Zhao [22. Muti-server preemptive priority systems with two customer casses have aso received some attention in the iterature. One of the earier references aowing for different service requirements between customer casses is Gai et a. [13, see aso the references therein. In [13, the authors derive the generating function of the joint stationary probabiities by expressing it in terms of the stationary probabiities associated with states where there is no queue. A combination of a generating function approach and the matrix-geometric approach is used in Septcheno et a. [26 to compute the joint stationary distribution of an M/M/c 2-cass preemptive priority queue. The M/P H/c queue with an arbitrary number of preemptive priority casses is studied in Harcho-Bater et a. [14 using a Recursive Dimensionaity Reduction technique that eads to an accurate approximation of the mean sojourn time per customer cass. Furthermore, in Wang et a. [29 the authors present a procedure for finding, for an M/M/c 2-cass priority mode, the generating function of the distribution of the number of ow-priority customers present in the system in stationarity. Our wor deviates from a of the above approaches, in that we construct a procedure for computing the Lapace transforms of the transition functions of the M/M/c 2-cass preemptive priority mode. Our method first maes use of a sight twea of the cearing anaysis on phases CAP method featured in Doroudi et a. [1, in that we show how CAP can be modified to study Lapace transforms of transition functions. The specific dynamics of our priority mode aow us to tae the anaysis a few steps further, by showing each Lapace transform can be expressed expicity in terms of transforms corresponding to states contained within a strip of states that is infinite in ony one direction. Finay, we show how to compute these remaining transforms recursivey, by maing use of a sight modification of Ramaswami s formua [25. Whie the focus of our wor is on Lapace transforms of transition functions, anaogous resuts can be derived for the stationary distribution of the M/M/c 2-cass preemptive priority mode as we. We are not aware of any studies that obtain expicit expressions for the Lapace transforms of the transition functions, or even the stationary distribution, as we do here: these resuts seem to yied the most expicit expressions nown to date. The Lapace transforms we derive can easiy be numericay inverted to retrieve the transition functions with the hep of the agorithms of Abate and Whitt [3, 4 or den Iseger [9. These transition functions can be used to study as a function of time ey performance measures such as the mean number of customers of each priority cass in the system; the mean tota number of customers in the system; or the probabiity that an arriving customer has to wait in the queue. The time-dependent performance measures can, for exampe, be used to anayze and dimension priority systems when one is interested in the behavior of such systems over a finite time horizon. Using the equiibrium distribution as an approximation of the time-dependent behavior to dimension the system can resut in either over- or underdimensioning, which can ead to poor performance. So, our method yieds a way of understanding the time-dependent behavior of muti-server priority queues. This type of behavior cannot be anayzed using any methods found in previous wor pertaining to this system: unti now, one woud have to resort 2

4 2 15 E[X1t 1 5 ρ 2.5 ρ 2.55 ρ t Figure 1: Mean number of ow-priority customers in the system as a function of time for increasing oad of the high-priority customers. Parameter settings are c 1, ρ 1 1/3, and ρ 2 varies. number of servers stationary probabiities Lapace transforms Tabe 1: Computation time in seconds required to cacuate, for a states in S, the stationary probabiities or the Lapace transforms for a specific α 1/2 1/2i. We use the numerica impementation outined in Section 5.5 with accuracies ɛ 1 8. Parameter settings are ρ 1 1/3, ρ 2 1/2 and c varies. to simuation in order to study the time-dependent behavior. Having expicit expressions for the Lapace transforms of the transition functions greaty simpifies the computation of some performance measures: for instance, these transforms yied expicit expressions for the Lapace transforms of the distribution of the number of ow-priority customers in the system at time t. We now present some numerica exampes of the time-dependent performance measures, where we wi mae use of the notation introduced in Section 2. In Figure 1, we pot the mean number of ow-priority customers in the system as a function of time. Simiary, in Figure 2 we pot the time-dependent deay probabiities for each priority cass. The Lapace transforms used to obtain Figures 1 and 2 can be computed numericay using the approach discussed in Section 5.5: here we used an error toerance of ɛ 1 8. Once these transforms have been found, numerica inversion can be done via the Euer summation agorithm of [3 where we again used an error toerance of 1 8. From Figure 1 we can aso informay derive the mixing times of each scenario. It seems that the mixing time vasty increases with an increase in the oad. As expected, in Figure 2 we see that the deay probabiity of a high-priority customer is much ower than the deay probabiity of a ow-priority customer. Furthermore, as time passes, the deay probabiity of the high-priority customer tends to the deay probabiity in an M/M/c queue with ony high-priority customers. Finay, in Tabe 1 we show the computation times of the agorithm, which was impemented in Matab and run on a 64-bit destop with an Inte Core i7-377 processor. The computation time scaes reasonaby we with the number of servers and therefore the agorithm can be used to evauate any practica instance. This paper is organized as foows. Section 2 describes both the M/M/c 2-cass preemptive 3

5 .6 Deay probabiity.4.2 high priority ow priority t Figure 2: Probabiity that an arriving customer has to wait in the queue as a function of time for the two priority casses. Parameter settings are c 1, ρ 1 1/3, and ρ 2 1/2. priority queueing system, as we as the two-dimensiona Marov process used to mode the dynamics of this system. In the same section we introduce reevant notation and terminoogy, and detai the outine of the approach. In Sections 3 5 we describe this approach for cacuating the Lapace transforms of the transition functions. We discuss the simpifications in the singe-server case in Section 6. In Section 7 we summarize our contributions and comment on the derivation of the stationary distribution. The appendices provide supporting resuts on combinatoria identities and singe-server queues used in deriving the expressions for the Lapace transforms. 2 Mode description and outine of approach We consider a queueing system consisting of c servers, where each server processes wor at unit rate. This system serves customers from two different customer casses, referred to here as cass- 1 and cass-2 customers. The cass index indicates the priority ran, meaning that among the servers, cass-2 customers have preemptive priority over cass-1 customers in service. Reca that the term preemptive priority means that whenever a cass-2 customer arrives to the system, one of the servers currenty serving a cass-1 customer immediatey drops that customer and begins serving the new cass-2 arriva, and the dropped cass-1 customer waits in the system unti a server is again avaiabe to receive further processing, i.e., the priority rue is preemptive resume. Therefore, if there are currenty i cass-1 customers and j cass-2 customers in the system, the number of cass-2 customers in service is minc, j, whie the number of cass-1 customers in service is maxmini, c j,. Cass-n customers arrive in a Poisson manner with rate λ n, n 1, 2, and the Poisson arriva processes of the two popuations are assumed to be independent. Each cass-n arriva brings an exponentiay distributed amount of wor with rate µ n, independenty of everything ese. We denote the tota arriva rate by λ : λ 1 λ 2, the oad induced by cass-n customers as ρ n : λ n /cµ n, and the oad induced by both customer casses as ρ : ρ 1 ρ 2. The dynamics of this queueing system can be described with a continuous-time Marov chain CTMC. For each t, et X n t represent the number of cass-n customers in the system at time t, and define Xt : X 1 t, X 2 t. Then, X : {Xt} t is a CTMC on 4

6 X 2 t λ 2 λ 2 j 2 λ 1 λ 1 cµ 2 cµ 2 c 1 λ 2 λ 2 λ 2 j 1 λ 1 mini 1, c j 1µ 1 λ 1 c j 1µ 1 λ 1 j 1µ 2 j 1µ 2 j 1µ 2 λ 2 λ 2 λ 2 λ 1 i 1µ 1 λ 1 cµ 1 i 1 c 1 i 2 λ 1 X 1 t Figure 3: Transition rate diagram of the Marov process X. the state space S N 2. Given any two distinct eements x, y S, the eement qx, y of the transition rate matrix Q associated with X denotes the transition rate from state x to state y. The row sums of Q are, meaning for each x S, qx, x y x qx, y : qx, where qx represents the rate of each exponentia sojourn time in state x. For our queueing system, the non-zero transition rates of Q are given by qi, j, i 1, j λ 1, i, j, qi, j, i, j 1 λ 2, i, j, qi, j, i 1, j maxmini, c j, µ 1, i 1, j, qi, j, i, j 1 minc, jµ 2, i, j 1. Figure 3 dispays the transition rate diagram. We further associate with the Marov process X the coection of transition functions {p x,y } x,y S, where for each x, y S with possiby x y the function p x,y : [, [, 1 is defined as p x,y t : PXt y X x, t. 2.1 Each transition function p x,y has a Lapace transform π x,y that is we-defined on the subset of compex numbers C : {α C : Reα > } as π x,y α : e αt p x,y t dt, α C. 2.2 We restrict our interest to transition functions of X when X, with probabiity one w.p.1, and so we drop the first subscript on both transition functions and Lapace transforms, 5

7 X 2t vertica boundary interior c c 1 horizonta boundary X 1t c 1 Figure 4: Terminoogy of the various sets of states. i.e., p x t : p,,x t for each t and π x α : π,,x α for each α C. Our goa is to derive efficient numerica methods for cacuating each Lapace transform π x α, x S. We often refer to the Lapace transform π x α associated with the state x as the Lapace transform for state x. 2.1 Notation and terminoogy It heps to decompose the state space S into a countabe number of eves, where for each integer i, the i-th eve is the set {i,, i, 1,...}. We further decompose the i-th eve into an upper eve and a ower eve: upper eve i is defined as U i : {i, c, i, c 1,...}, whie ower eve i is simpy L i : {i,, i, 1,..., i, c 1} and the union of ower eves L, L 1,..., L i is denoted by C i i L. The set of a states in phase j is denoted by P j : {, j, 1, j,...}. We sometimes refer to upper eve U as the vertica boundary. The union of upper eves is caed the interior of the state space. Finay, the union U 1 U L L is caed the horizonta boundary or the horizonta strip of boundary states. Figure 4 depicts these sets. The indicator function 1{A} equas 1 if A is true and otherwise. Given an arbitrary CTMC Z, we et E z [fz represent the expectation of a functiona of Z, conditiona on Z z, and P z denotes the conditiona probabiity associated with E z [. In our anaysis it shoud be cear from the context what is being conditioned on when we write P z or E z [. We wi aso need to mae use of hitting-time random variabes. We define for each set A S, τ A : inf{t > : im Xs Xt A} 2.5 s t 6

8 as the first time X maes a transition into the set A so note X A does not impy τ A and τ x shoud be understood to mean τ {x}. 2.2 Notation for M/M/1 queues Most of the formuas we derive contain quantities associated with an ordinary M/M/1 queue. Given an M/M/1 queueing system with arriva rate λ and service rate µ, et Q λ,µ t denote the tota number of customers in the system at time t. Under the measure P n, which, in this case, represents conditioning on Q λ,µ n, et B λ,µ denote the busy period duration induced by these customers. Under P 1, the Lapace-Stietjes transform of B λ,µ is given by φ λ,µ α : E 1 [e αb λ,µ λ µ α λ µ α 2 4λµ λ Reca that under P n, B λ,µ is equa in distribution to the sum of n i.i.d. copies of B λ,µ under the measure P 1, see, e.g., [28, p. 32. Thus, for each integer n 1 we have E n [e αb λ,µ φ λ,µ α n. 2.7 We wi aso need to mae use of the foowing quantities in Sections 5 and 6. Suppose {Λ θ t} t is a homogeneous Poisson process with rate θ that is independent of {Q λ,µ t} t. For each integer i, define w λ,µ,θ i α : E 1 [e αb λ,µ 1{Λ θ B λ,µ i}. 2.8 Lemma C.3 of Appendix C deveops a recursion for the w λ,µ,θ i α term and in Lemma C.4 of Appendix C we give expicit expressions for these terms by soving the recursion. The foowing quantities associated with M/M/1 queues wi appear at many paces of the anaysis. To increase readabiity, we adopt the notation used in [1 and define the quantities and φ 2 : φ λ2,cµ 2 λ 1 α, r 2 : ρ 2 φ λ2,cµ 2 λ 1 α, 2.9 Ω 2 : ρ 2 φ λ2,cµ 2 λ 1 α λ 2 1 ρ 2 φ λ2,cµ 2 λ 1 α Further resuts for M/M/1 queues are presented in Appendix C. 2.3 Outine of our approach Our approach for computing the Lapace transforms of the transition functions of X when X, w.p.1 is divided in three parts. 1. For each integer i, we use a sight modification of the CAP method [1 to write each Lapace transform for each state in U i, i.e., π i,c 1j α, j 1, in terms of the Lapace transforms π,c 1 α, i as we as additiona coefficients {v, } i that satisfy a recursion. 2. In Section 4 we obtain an expicit expression for the coefficients {v, }. This in turn shows that for each i, each Lapace transform for each state in U i, i.e., π i,c 1j α, j 1, can be expicity expressed in terms of π,c 1 α, i. 7

9 3. In Section 5 we derive a recursion with which we can determine the Lapace transforms for the states in the horizonta boundary. Specificay, we derive a modification of Ramaswami s formua [25 to recursivey compute the remaining Lapace transforms π i,j α, i, j c 1. The techniques we use to derive this recursion are exacty the same as the techniques recenty used in [17 to study boc-structured Marov processes. Ony the Ramaswami-ie recursion is needed to compute a Lapace transforms: once the vaues for the Lapace transforms of the states in the horizonta boundary are nown, a other transforms can be stated expicity without using additiona recursions. 3 A sight modification of the CAP method The foowing theorem is used in mutipe ways throughout our anaysis. It appears in [17, Theorem 2.1 and can be derived by taing the Lapace transform of both sides of the equation at the top of page 124 of [21. Equation 3.2 is the Lapace transform version of [1, Theorem 1. Theorem 3.1. Suppose A and B are disjoint subsets of S with x A. Then for each y B, or, equivaenty, π x,y α z A π x,y α z A π x,z αqz αe z [ τ A π x,z α z A c qz, z E z [ τ A e αt 1{Xt y} dt, 3.1 e αt 1{Xt y} dt Lapace transforms for states aong the vertica boundary In this subsection we empoy Theorem 3.1 to express each Lapace transform π,c 1j α, j 1 in terms of π,c 1 α. Using Theorem 3.1 with A U c we obtain, for j 1, π,c 1j α π z α [ qz, z τ U c E z e αt 1{Xt, c 1 j} dt z U c z U [ τ U c π,c 1 αλ 2 E,c e αt 1{Xt, c 1 j} dt. 3.3 From the transition rate diagram in Figure 3, we find that the expectation in 3.3 can be interpreted as an expectation associated with an M/M/1 queue having arriva rate λ 2 and service rate cµ 2. Indeed, τ U c is equa in distribution to the minimum of the busy period initiaized by one customer of this M/M/1 queue and an exponentia random variabe with rate λ 1 that is independent of the queue. Aternativey, τ U c can be thought of as being equa in distribution to the busy period duration of an M/M/1 cearing mode, with arriva rate λ 2, service rate cµ 2, and cearings that occur in a Poisson manner with rate λ 1. Appying Lemma C.2 of Appendix C shows that λ 2 E,c [ τ U c Substituting 3.4 into 3.3 then yieds e αt 1{Xt, c 1 j} dt r j π,c 1j α π,c 1 αr j 2, j

10 3.2 Lapace transforms for states within the interior We next deveop a recursion for the Lapace transforms for the states within the interior. First, we express the transforms in upper eve U i in terms of the transforms in upper eve U i 1 and in state i, c 1. Second, we use this resut to express the transforms in upper eve U i in terms of the transforms for the states, c 1, 1, c 1,..., i, c 1 and some additiona coefficients. Empoying again Theorem 3.1, now with A Ui c, yieds for i, j 1, π i,c 1j α π z α [ qz, z τ U c i E z e αt 1{Xt i, c 1 j} dt z U i The expectation z U c i [ τ U c i π i 1,c 1 αλ 1 E i,c 1 1 π i,c 1 αλ 2 E i,c [ τ U c i E i,c 1 [ τ U c i e αt 1{Xt i, c 1 j} dt e αt 1{Xt i, c 1 j} dt. 3.6 e αt 1{Xt i, c 1 j} dt, j, has the same interpretation as the expectation in 3.3, except now the M/M/1 queue starts with customers at time. Using Lemma C.2 of Appendix C, we obtain [ τ U c i λ 1 E i,c 1 e αt 1{Xt i, c 1 j} dt Υj,, j, 1, 3.8 where, for j, 1, Υj, : { λ 1 Ω 2 r j 2 1 r 2 φ 2, 1 j 1, λ 1 Ω 2 φ j 2 1 r 2 φ 2 j, j. Substituting 3.8 into 3.6 and simpifying yieds a recursion. Specificay, for i and j 1, π i1,c 1j α r j 2 π i1,c 1α 3.9 Υj, π i,c 1 α, 3.1 with initia conditions π,c 1j α π,c 1 αr j 2, j 1. The recursion 3.1 can be soved, i.e., π i,c 1j α can be expressed in terms of the transforms π,c 1 α, π 1,c 1 α,..., π i,c 1 α. Theorem 3.2. Interior For i, j 1, π i,c 1j α i 1 j 1 v i, 1 V 2 r j 2, 3.11 where the quantities {v i,j } i j satisfy the foowing recursive scheme: for i, v i1, π i1,c 1 α, v i1,j V 1 v i,j 1 V a i v i,, 1 j i 1, 3.12b with initia condition v, π,c 1 α. Here V 1 λ 1Ω 2 1 r 2 φ 2, and V 2 r 2 φ 2. j 9

11 Throughout we foow the convention that a empty sums, such as number zero. 1, represent the Proof. Ceary, when i, 3.11 agrees with 3.5. Proceeding by induction, assume 3.11 hods among upper eves U, U 1,..., U i for some i. Substituting 3.11 into 3.1 yieds π i1,c 1j α r j 2 π i1,c 1α Υj, 1 i 1 v i, 1 V 2 r Next, interchange the order of the two summations and appy Lemma A.1 of Appendix A to get 3.13 r j 2 π i1,c 1α i i v i, 1 V 2 V 1 V 2 j 1 1 v i, 1 V 2 λ 1 Ω 2 r j 2 1 m1 1 1 V 2 m j 1 m m r j To further simpify the right-hand side of 3.14, increase the summation index of the first summation by one by setting 1, mutipy its summands by 1 r 2φ 2 1 r 2 φ 2, and change the order of the doube summation. This yieds 3.14 r j 2 π i1 j 1 i1,c 1α V 1 v i, 1 1 V 2 r j 2 i V 1 V 2 1 i 1 v i, 1 V 2 j 1 r j 2 π i1 i1,c 1α V 1 v i, 1 V 2 1 i r j 2 j 1 v i, 1 V 2 r j 2, 3.15 which shows π i1,c 1j α satisfies 3.11, competing the induction step. 4 Deriving an expicit expression for v i,j Theorem 3.2 suggests that the Lapace transforms for the states within the interior can be computed recursivey. 1. Initiaization step: Determine π,c 1 α, which yieds each transform π,c 1j α, j Recursive step on i: Given π,c 1 α for i and the coefficients {v, } i a. Compute π i1,c 1 α. b. Compute {v i1, } i1. c. Once steps 2a. and 2b. are competed, a transform vaues π i1,c 1j α, j 1 are nown. 1

12 Our next resut, i.e., Theorem 4.1, shows that for each i, the {v i, } i terms can be expressed expicity in terms of π,c 1 α, i. If our goa is to ony compute π i,c 1j α for some arge i, then Theorem 4.1 aows us to avoid computing a intermediate {v, } i 1 terms, which means we can avoid computing an additiona Oi 2 terms. Not ony that, nowing exacty how these v i,j coefficients oo coud aid in future questions ased by researchers interested in the M/M/c 2-cass priority queue. Readers shoud eep in mind that the expressions we have derived for the v i,j coefficients do contain binomia coefficients, and one shoud be carefu to avoid roundoff errors whie computing these expressions. Theorem 4.1. Coefficients The coefficients {v i,j } i j from Theorem 3.2 are as foows: for i j, i v i,j V j 1 π j i j,c 1α V1 j j 1 π i,c 1 α V j j1 Proof. From 3.12 we find, for each i, that v i, π i,c 1 α and v i,i V1 iπ,c 1α; these expressions agree with 4.1. Next, assume for some integer i that v i,j satisfies 4.1 for j i. Our aim is to show v i1,j aso satisfies 4.1 for j i 1: we do this by substituting 4.1 into 3.12b and simpifying. There are three cases to consider: i j 1; ii 2 j i 1; and iii j i. We focus on case ii, with cases i and iii foowing simiary. We first examine the V 1 v i,j 1 term in 3.12b by substituting 4.1. Here, V 1 v i,j 1 V j 1 π i1 j,c 1α V j 1 π i1 j,c 1α i j i1 j1 1 1 j V1 1 j 1 π i,c 1 α 1 j V 1 j j 1 1 π i1,c 1 α j 1 j 1 j 1 V2 1 2 V Next, write 4.2 in a form where the binomia coefficients match the ones in 4.1: 4.2 V j 1 π i1 j i1 j,c 1α V1 j 1 j π i1,c 1 α j 1 j V The remaining terms on the right-hand side of 3.12b can be further simpified by substituting 4.1. Doing so reveas that V 1 V 2 i v i, j i j i 1 V 1 1 π i,c 1 αv 2 i j 1 V 1 1 π i,c 1 α m1 Swapping the order of the tripe summation in 4.4 gives i 4.4 V1 1 π i,c 1 αv 2 j i j1 j V1 1 π i,c 1 α m m1 j m m 1 m 1 1 m 1 V m V m

13 The inner-most summation over of 4.5 can be evauated using Lemma A.3 of Appendix A: m j m m jm m 1 j 1 j. 4.6 m 1 Next, substitute 4.6 bac into 4.5 and focus on the inner-most doube summation of 4.5. This gives j m m1 j m 1 m 1 V m1 2 j m1 j m1 1 j m2 m jm m 1 j m jm 1 j 1 jm j m 1 j 1 j 1 V2 m1 m 1 m 1 1 j V2 m1 m 1 m 1 j Substituting 4.7 into 4.5 and changing the two outer summation indices shows 4.5 i1 j1 i1 j2 V 1π i1,c 1 αv 2 j V1π i1,c 1 α m2 jm j m j 1 j m m V2 m. m V2 m. 4.8 m 1 Furthermore, since V 2 j 1 j 1 j 1 j 1 1 V2 1, we can merge the singe summation with the doube summation in 4.8. In other words, 4.8 i1 j1 j V1π i1,c 1 α m1 jm j m j 1 Finay, summing 4.3 and 4.1 produces 4.1, as V 1 v i,j 1 V 1 V 2 i j v i, V j 1 π i1 j,c 1α i1 j1 i1 j1 V j m j j 1 1 π i1,c 1 α j 1 1 j j j 1 π i1,c 1 α j 1 V 1 1 V2 m. 4.1 m 1 j j 1 V2 1 1 V Summing the coefficients in front of the binomia coefficient terms proves case ii. Cases i and iii foow simiary. 12

14 We now have an expicit expression for the coefficients. {v i,j } i j into 3.11 to obtain, for j 1, π i,c 1j α i 1 i 1 i π i,c 1 α V 1 1 V 2 j 1 r j 2 π i,c 1 αv 1 m1 Substitute the expressions for 1 j 1 V2 m 1 V 2 r j 2 m m Swapping the order of the doube summation and grouping coefficients in front of each Lapace transform reveas the dependence of π i,c 1j α on π,c 1 α, π 1,c 1 α,..., π i,c 1 α: π i,c 1j α r j 2 π i,c 1α 1 j 1 1 V 2 i 1 r j 2 m1 [ j 1 V1π i,c 1 α 1 V 2 r j 2 m 1 m 1 V m From this expression, we see that for each fixed i, as j, π i,c 1j α behaves in a manner anaogous to that found in Theorem 3.1 of [22, which addresses, when c 1, the asymptotic behavior of the stationary distribution as the number of high-priority customers approaches infinity, whie the number of ow-priority customers is fixed. The expicit expression 4.13 can be used to obtain an expression for the Lapace transforms of the number of cass-1 customers in the system. That is, e αt PX 1 t i X, dt where we can simpify the fina infinite sum as j1 via the identity j π i,c 1j α r 2 1 r 2 π i,c 1 α 1 1 V 2 r 2 1 r 2 1 j1 m1 e αt PXt i, j X, dt j c 1 π i,j α π i,j α π i,c 1j α, m i 1 1 m 1 j j1 [ 1 V1π V2 r 2 i,c 1 α 1 r 2 1 V m , 4.15 j 1 r j 2 r r Lapace transforms for states in the horizonta boundary In the previous section we showed how to express each Lapace transform for the states on the vertica boundary and within the interior expicity in terms of transforms for the states in the horizonta boundary. So, it remains to determine the transforms for the states in the 13

15 horizonta boundary. In this section, we show that the atter Lapace transforms satisfy a variant of Ramaswami s formua, which wi aow us to numericay compute these transforms recursivey. The approach we use to compute the above-mentioned variant of Ramaswami s formua maes, ie the CAP method, repeated use of Theorem 3.1. This approach is highy anaogous to the approach used in [17 to study boc-structured Marov processes, yet sighty modified since we are interested in recursivey computing Lapace transforms ony associated with states within the horizona boundary. This idea of restricting ourseves to a subset of the state space seems simiar in spirit to the censoring approach featured in the wor of Li and Zhao [23, but it is not currenty obvious to the authors if this approach is appicabe to our setting. We first introduce some reevant notation. Define the 1 c row vectors π i α as π i α : [ π i, α π i,1 α π i,c 1 α, i. 5.1 To propery state the Ramaswami-ie formua satisfied by these row vectors, we need to define additiona matrices. First, we define the c c transition rate submatrices corresponding to ower eves L i, i c as A 1 : λ 1 I, A 1 : diagcµ 1, c 1µ 1,..., µ 1 and λ 2 λ 2 A : µ 2 λ 2 µ 2 λ 2 2µ 2 λ 2 2µ 2 λ 2... c 1µ 2 λ 2 c 1µ 2 A 1 A 1, 5.2 where I is the c c identity matrix and diagx is a square matrix with the vector x aong its main diagona. We further define the c c eve-dependent transition rate submatrices associated with L i, 1 i c 1 as A i 1 : diagxi with x i j : mini, c jµ 1, j c 1, A i : A A 1 A i 1, and for L we have A : A A 1. Next, we define the coection of c c matrices {W m α} m. Each eement of W m α is equa to except for eement W m α c 1,c 1, which is defined as W m α c 1,c 1 : λ 2 w λ 2,cµ 2,λ 1 m α. We aso need the coection of c c matrices {G i,j α} i>j, where the, -th eement of G i,j α is defined as Gi,j α, : E i,[e ατ L j 1{Xτ Lj j, },, c Finay, we wi need the coection of c c matrices {N i α} i 1, whose eements are defined as foows: Ni α, i,[ τ : E Li 1 e αt 1{Xt i, } dt,, c Note that N i α N c α for i c, and we therefore denote Nα : N c α. 5.1 A Ramaswami-ie recursion The foowing theorem shows that the vectors of transforms {π i α} i satisfy a recursion anaogous to Ramaswami s formua [25. Theorem 5.1. Horizonta boundary For each integer i, we have π i1 α π i αa 1 N i1 α i π α 14 i1 W αg,i1 αn i1 α, 5.5

16 where we use the convention G i1,i1 α I. Proof. This resut can be proven by maing use of the approach found in [17. Using Theorem 3.1 with A C i, we see that for i and j c 1, π i1,j α z C i π z α z C c i qz, z E z [ τ Ci e αt 1{Xt i 1, j} dt. 5.6 Due to the structure of the transition rates, many terms in the summation of 5.6 are zero. In particuar, 5.6 can be stated more expicity as π i1,j α i [ τ Ci π,c 1 αλ 2 E,c c 1 [ τ Ci π i,m αλ 1 E i1,m m e αt 1{Xt i 1, j} dt e αt 1{Xt i 1, j} dt. 5.7 We now simpify each expectation appearing within the first sum on the right-hand side of 5.7. Summing over a ways in which the process reaches phase c 1 again yieds E,c [ τ Ci e αt 1{Xt i 1, j} dt [ τ Ci E,c e αt 1{Xt i 1, j} dt τ Pc 1 [ τci E,c 1{Xτ Pc 1, c 1}e ατ P c 1 e αt τ P c 1 1{Xt i 1, j} dt. τ Pc 1 i1 Appying the strong Marov property to each expectation appearing in 5.8 shows that 5.8 i1 i1 E,c [ 1{Xτ Pc 1, c 1}e ατ P c 1 E,c 1 [ τ Ci 5.8 e αt 1{Xt i 1, j} dt w λ 2,cµ 2,λ 1 α G,i1 αn i1 α c 1,j, 5.9 where the ast equaity foows from the definitions of G,i1 α and N i1 α, and Lemma C.4 of Appendix C. The expectations appearing within the second sum of 5.7 can easiy be simpified by recognizing that they are eements of N i1 α. Hence, we utimatey obtain π i1,j α i π,c 1 α c 1 m which, in matrix form, is 5.5. i1 λ 2 w λ 2,cµ 2,λ 1 α G,i1 αn i1 α c 1,j π i,m α A 1 m,m Ni1 α m,j, 5.1 It remains to derive computabe representations of {G i,j α} i>j, as we as the matrices {N i α} 1 i c. 15

17 5.2 Computing the G i,j α matrices The next proposition shows that each G i,j α matrix can be expressed entirey in terms of the subset {G i1,i α} i c 1. Proposition 5.2. For each pair of integers i, j satisfying i > j, we have G i,j α G i,i 1 αg i 1,i 2 α G j1,j α Furthermore, for each integer we aso have G c,c 1 α G c,c 1 α Proof. Equation 5.11 can be derived by appying the strong Marov property in an iterative manner, whie 5.12 foows from the homogeneous structure of X aong a ower eves L i, i c. In ight of Proposition 5.2, our goa now is to determine matrices {G i1,i α} i c 1. We first focus on showing that Gα : G c,c 1 α is the soution to a fixed-point equation. Proposition 5.3. The matrix Gα satisfies Gα αi A W α 1 A 1 A 1 Gα 2 W αgα Proof. Observe that for i c 2 and j c 1, a one-step anaysis yieds Gα i,j E c,i[e ατ L c 1 1{XτLc 1 c 1, j} i,i1 1 Gα A 1 i,i 1{i j} A A i,i α A i,i α i1,j 1{i } A i,i 1 Gα A i,i α i 1,j A 1 i,i Gα 2 A i,i α i,j On the other hand, for j c 1, we aso have Gα c 1,j E c,c 1[e ατ L c 1 1{XτLc 1 c 1, j} A 1 c 1,c 1 A c 1,c 2 1{c 1 j} Gα A c 1,c 1 α A c 1,c 1 α c 2,j A 1 c 1,c 1 Gα 2 A c 1,c 1 α c 1,j λ [ 2 A c 1,c 1 α E c,c e ατ L c 1 1{XτLc 1 c 1, j} To simpify 5.15 further, notice that an appication of the strong Marov property at the time τ Pc 1 produces E c,c [e ατ L c 1 1{XτLc 1 c 1, j} E c,c [e ατ P c 1 1{XτPc 1 c, c 1}e ατ L c 1 τ Pc 1 1{Xτ Lc 1 c 1, j} [ E c,c [e ατ P c 1 1{XτPc 1 c, c 1} E c,c 1 e ατ L c 1 1{XτLc 1 c 1, j} w λ 2,cµ 2,λ 1 α Gα 1 c 1,j,

18 where the ast equaity foows from Lemma C.4 of Appendix C and Using 5.16, we can write more eeganty in matrix form as A 1 A αi Gα A 1 Gα 2 or, equivaenty, assuming αi A W α is invertibe, Gα αi A W α 1 A 1 A 1 Gα 2 W αgα 1, 5.17 W αgα It remains to show that the matrix αi A W α is indeed invertibe. Define a c c matrix Hα with eements Hα i,j : E c,i [ τ Lc Uc c 1 e αt 1{Xt c, j} dt, i, j c Again, using a one-step anaysis and the strong Marov property at the first transition time T 1 yieds, for i, j c 1, Hα [ τ i,j E Lc Uc c [ c,i e αt T 1 1{Xt c, j} dt E c,i e αt 1{Xt c, j} dt T 1 1{i } A i,i 1 Hα A i,i α i 1,j 1{i c 1} A i,i1 Hα A i,i α i1,j W c 1,c 1 1{i c 1} Hα A c 1,c 1 α c 1,j 1{i j} A i,i α. 5.2 This can be written in matrix form as αi A W α Hα I, 5.21 proving αi A W α is invertibe, with its inverse being Hα. The next proposition shows that through successive substitutions one can obtain Gα from Proposition 5.4. Suppose the sequence of matrices {Zn, α} n satisfies the recursion Zn 1, α αi A W α 1 A 1 A 1 Zn, α 2 with initia condition Z, α. Then, W αzn, α im Zn, α Gα n Proof. This proof maes use of Proposition 5.3, and is competey anaogous to the proofs of [18, Theorems 3.1 and 4.1 and [17, Theorem 3.4. It is therefore omitted. Now that we have a method for approximating Gα, it remains to find a method for computing G i1,i α, i c 2. The next proposition shows that these matrices can be computed recursivey. 17

19 Proposition 5.5. For each integer i satisfying i c 2, we have G i1,i α αi A i1 A 1 G i2,i1 α i1 1A i1 W i1 αg,i1 α Proof. This resut can be proven using a one-step anaysis. Fix an integer i, i c 2, and observe that for j c 2, c 1, Gi1,i α j, A i1 i1 1 j,j A j,j1 A i1 1{j } Gi1,i j,j α A i1 α j1, j,j α 1{j } Ai1 j,j 1 Gi1,i A i1 α j 1, j,j α A 1 j,j Gi2,i A i1 α j, j,j α Simiary, for c 1, Gi1,i α c 1, A i1 1 c 1,c 1 A i1 1{c 1 } c 1,c 1 α A i1 c 1,c 2 A i1 c 1,c 1 α Gi1,i α c 2, A 1 c 1,c 1 A i1 c 1,c 1 α Expressing 5.25 and 5.26 in matrix form yieds the equaity A i1 1 A i1 αi G i1,i α A 1 G i2,i α i1 W i1αg,i1 α i1 A i1 c 1, c 1,c 1 α Gi2,i α c 1, W i1 αg,i1 α Then, from the reation 5.11 we estabish The matrix A i1 1 is a diagona matrix whose diagona eements are a positive, so it has an inverse. This shows the inverse stated in 5.24 exists. We now have an iterative procedure for computing a {G i1,i α} i c 1 matrices: first compute Gα from Proposition 5.4, then use Proposition 5.5 to compute G c 1,c 2 α, then G c 2,c 3 α, and so on, stopping at G 1, α. 5.3 Computing the N i α matrices The matrices {N i α} 1 i c can be expressed in terms of {G i,j α} i j. Proposition 5.6. For each integer i satisfying 1 i c, we have N i α αi A i A 1G i1,i α where we use the convention A c A. 1, W i αg,i α 5.28 i 18

20 Proof. Observe that for each j, c 1, we can use a one-step anaysis and the strong Marov property at the first transition time T 1 to show that Ni α j, E i,j[ τ Li 1 T 1 [ e αt T 1 1{Xt i, } dt E i,j e αt 1{Xt i, } dt 1{j } Ai j,j 1 Ni A i α j 1, j,j α 1{j c 1} A 1 j,j A i j,j α 1{j c 1} Ai j,j1 A i j,j α 1 W i αg,i αn i α c 1,c 1 α c 1, i Gi1,i αn i α j, A i Expressing these equations in matrix form yieds proving Computing π α I A i αi A 1G i1,i α Ni α j1, 1{j } A i j,j α i W i αg,i α N i α, 5.3 It remains to devise a method for computing the vector π α so that the Ramaswami-ie recursion from Theorem 5.1 can be propery initiaized. The foowing is an adaptation of [17, Section 3.3. We define the c c matrix N α whose eements are given by N α i,j : E,i[ τ, e αt 1{Xt, j} dt, i, j c In the derivation to foow, we require the notation A [i,j which represents the matrix A with row i and coumn j removed meaning it is a c 1 c 1 matrix, whist eeping the indexing of entries exacty as in A. Simiary, A [i, has row i removed from A meaning it is a c 1 c matrix and A [,j has coumn j removed from A meaning it is a c c 1 matrix. Proposition 5.7. We have N α [, α I [, A [, [, A1 G1, α [, W α [, G, α [, Proof. We can restrict our attention to eements i, j with 1 i, j c 1. Simiar to the proof of Proposition 5.6, we perform a one-step anaysis and the strong Marov property at 19

21 the first transition time T 1 to obtain N α i,j E,i[ τ, 1 i T 1 [ e αt T 1 1{Xt, j} dt E,i e αt 1{Xt, j} dt λ [ τ 2 1{i c 1} A i,i α E,,c e αt 1{Xt, j} dt c 1 A 1 [ τ i, A i,i α E, 1, e αt 1{Xt, j} dt c 1 A i, N A α,j i,i α 1{i j} A i,i α We now simpify the two expectations appearing in We first mae a sighty more genera statement that wi provide the second expectation, and wi be used in the derivation of the first expectation. Summing over a ways at which the process enters L yieds [ τ, E,i e αt 1{Xt, j} dt c 1 E,i [e ατ L 1{XτL, m} m1 c 1 τ, τ L e αt τ L 1{Xt, j} dt G, α N α i,m m,j m1 Now, the first expectation in 5.33 foows by the above and by conditioning on the state at which the process enters phase c 1 see aso the proof of Proposition 5.3: [ τ, E,c e αt 1{Xt, j} dt E,c [e ατ P c 1 1{XτPc 1, c 1} w λ 2,cµ 2,λ 1 τ, τ Pc 1 e αt τ P c 1 1{Xt, j} dt c 1 α G, α N α c 1,m m,j m1 Combining with 5.33 and writing it in matrix form proves the caim. We empoy 3.2 and Proposition 5.6 to determine the eements of the row vector π α. Set A {, } in Theorem 3.1: then, for 1 j c 1 and c 2, π,j α π z α [ qz, z τ A E z e αt 1{Xt, j} dt z A z A [ π, α λ τ, 1 E 1, e αt 1{Xt, j} dt π, α λ 2 E,1 [ τ, λ 1 c 1 1 e αt 1{Xt, j} dt G1, α N α,,j λ 2 N α 1,j

22 Hence, the transforms π,j α, 1 j c 1 can be expressed in terms of π, α. Section 5.5 we describe a numerica procedure to determine π, α. Remar 5.8. An aternative method for determining π α The Komogorov forward equations can aso be used to derive π α. The transition functions are nown to satisfy these equations, since sup x S qx <. Taing the Lapace transform of the Komogorov forward equations for the states in L yieds, after using 3.5, π α αi A π αw α π 1 αa 1 1 e, 5.37 where e i is a row vector with a eements equa to zero except for the i-th eement which is unity. Finay, using Theorem 5.1 to express π 1 α in terms of π α yieds π α αi A W α A 1 W αg,1 α N 1 αa 1 1 e If one is instead interested in the stationary distribution and in particuar the stationary probabiities of L, i.e., im α α π α, 5.38 resuts in a homogeneous system of equations, but it is not cear that this system sti has a unique soution. On the other hand, the approach outined earier for determining π α can straightforwardy be empoyed to obtain im α α π α. 5.5 Numerica impementation In order to compute the vectors {π i α} i, we first need to compute {G i1,i α} i c 1, {N i α} 1 i c, and N α [,. The first step is to compute Gα G c,c 1 α. Proposition 5.4 shows that this matrix can be approximated by using the recursion Using this recursion requires us to truncate the infinite sum appearing within the recursion. One way of appying this truncation is as foows: given a fixed toerance ɛ, pic an integer κ ɛ arge enough so that w λ 2,cµ 2,λ 1 α ɛ/λ κ ɛ1 Once κ ɛ has been found, we can use the approximation κ ɛ W αzn, α 1 W αzn, α 1, since the moduus of each eement of the matrix on the eft-hand side of 5.4 can be shown to be within ɛ of what is being approximated. Here we used that the matrices W α ony have one eement and the absoute vaue of each eement of Zn, α and Gα is ess than or equa to 1. Hence, we propose using the recursion 1 Zn 1, α αi A W α 1 κ ɛ A 1 A 1 Zn, α 2 W αzn, α to approximate Gα. Notice that we can determine κ ɛ satisfying 5.39 by writing the eft-hand side of 5.39 as κ ɛ κ ɛ1 w λ 2,cµ 2,λ 1 α 1 w λ 2,cµ 2,λ 1 α 1 1 In w λ 2,cµ 2,λ 1 α

23 An expicit expression for the infinite sum on the right-hand side of 5.42 can be derived with the hep of Lemma C.4 of Appendix C and the generating function of the Cataan numbers; the finite sum can be computed numericay. Once Gα has been found, we can use Proposition 5.5 to compute each G i1,i α matrix, for i c 2. For this computation, we use the same truncation procedure as outine above. The next step is to compute the matrices {N i α} 1 i c and N α [, using Propositions 5.6 and 5.7, respectivey. For both computations we again use the above truncation procedure. It remains to recursivey determine {π i α} i using Theorem 5.1, where we again use the truncation procedure for the infinite sum. As we have seen in Section 5.4, this recursion shoud be propery initiaized by the vaue of π, α. The random-product representation in Theorem B.1 of Appendix B shows that a Lapace transforms π x α satisfy, for each x S, π x α π, αψ x α, 5.43 where π, α is an unnown transform and ψ, α 1. It is cear that ψ x α can be computed using the exact same procedure as for π x α, and 5.36 shows that the transforms ψ, α, ψ,1 α,..., ψ,c 1 α are computabe expressions. We can cacuate π, α from the normaization condition which yieds x S π x α π, α x S π, α ψ x α 1 α, α x S ψ xα Since we cannot compute this infinite sum, we determine ψ i,j α for a i, j in a sufficienty arge bounding box S : {i, j S : i } for some. Notice that 4.15 aows S to be an infinitey arge rectange. The choice of in S ceary infuences the quaity of the approximation. A simpe procedure to choose is the foowing. Define Ψ : x S ψ x α Pic ɛ sma and positive and continue increasing unti Ψ 1 Ψ Ψ < ɛ. Then, set π, α 1/αΨ 1 to normaize the Lapace transforms π x α π, αψ x α. For c 1 we can normaize the soution as outined above, or we can expicity determine the vaue of π, α; see the next section. 6 The singe-server case We now turn our attention to the case where c 1, i.e., the case where the system consists of a singe server. In this case, the anaysis of the Lapace transforms π i, α, i simpifies consideraby. The expressions for the Lapace transforms for the states in the interior and on the vertica boundary are identica to the muti-server case. From [12, Coroary 2.1 see aso Theorem B.1 in Appendix B we have π, α 1 q, α1 E, [e ατ,

24 In ight of 6.1, to evauate π, α the ony thing that needs to be determined is the expectation E, [e ατ,. This quantity is the Lapace-Stietjes transform of the sum of two independent exponentia random variabes: one is the exponentia random variabe E λ having rate λ, the other is the busy period B of an M/G/1 queue having arriva rate λ and hyperexponentia service times having cumuative distribution function F. More specificay, F t λ 1 λ 1 e µ 1t λ 2 λ 1 e µ 2t, t. 6.2 The Lapace-Stietjes transform ϕα of B is nown to satisfy the Kenda functiona equation ϕα λ 1 λ µ 1 µ 1 α λ1 ϕα λ 2 λ µ 2 µ 2 α λ1 ϕα. 6.3 Furthermore, ϕα can be determined numericay through successive substitutions of 6.3, starting with ϕα : see [1, Section 1 for detais. Using independence of E λ and B, E, [e ατ, E[e αe λb meaning that see aso [2, eq. 36 for c 1, π, α λ ϕα, 6.4 λ α 1 λ1 ϕα α. 6.5 We now turn our attention to the horizonta boundary. When c 1, the matrices Gα and Nα become scaars, which we denote as Gα and Nα, respectivey. More precisey, Gα : E i1, [e ατ L i, which are independent of i 1. Nα : E i, [ τ Li 1 Proposition 6.1. The scaar Gα is a soution to e αt 1{Xt i, } dt, 6.6 λ µ 1 αgα µ 1 λ 1 Gα 2 λ 2 Gαφ λ2,µ 2 λ 1 1 Gα α. 6.7 Proof. From Proposition 5.3, we easiy find that when c 1, λ µ 1 αgα µ 1 λ 1 Gα 2 λ 2 w λ 2,µ 2,λ 1 αgα The infinite series appearing in 6.8 can be simpified using Lemma C.5 of Appendix C; doing so yieds 6.7. Even though we cannot use 6.7 to write down an expicit expression for Gα, we can sti use it to devise an iterative scheme for computing Gα. The next resut shows that Nα can be expressed in terms of Gα. Proposition 6.2. We have Nα 1 α λ µ 1 λ 1 Gα λ 2 φ λ2,µ 2 λ 1 1 Gα α

25 Proof. Using Proposition 5.6, we observe that when c 1, Nα α λ µ 1 λ 1 Gα λ 2 w λ 2,µ 2,λ 1 αgα The proof is competed by appying Lemma C.5 of Appendix C to 6.1. We now focus on the recursion for the horizonta boundary. When c 1, Theorem 5.1 reduces to π i1, α λ 1 π i, αnα λ 2 For the inner-most sum over we have i1 w λ 2,µ 2,λ 1 i π, α αgα i1 Gα i1 i1 w λ 2,µ 2,λ 1 αgα i1 Nα mi1 w λ 2,µ 2,λ 1 m αgα m Ceary, i 1 1. So et us try to evauate the tai of the generating function of {w λ 2,µ 2,λ 1 m α} m evauated at the point Gα, i.e., mi1 w λ 2,µ 2,λ 1 m α Gα m φ λ2,µ 2 λ 1 1 Gα α i m w λ 2,µ 2,λ 1 m α Gα m, 6.13 which foows from an appication of Lemma C.5 of Appendix C. The remaining finite summation is easy to compute since each w λ 2,µ 2,λ 1 m α term, by Lemma C.4 of Appendix C, can be stated in terms of b K functions, and these satisfy the recursion found in Lemma A.4 of Appendix A. These observations aow us to state the foowing theorem, which yieds a practica method for recursivey computing Lapace transforms of the form π i, α. Theorem 6.3. Horizonta boundary, singe server When c 1, the Lapace transforms of the transition functions on the horizonta boundary satisfy the foowing recursion: for i, π i1, α λ 1 π i, αnα i λ 2 π, αgα i1 i φ λ2,µ 2 λ 1 1 Gα α w λ 2,µ 2,λ 1 αgα Nα. 7 Concusion 6.14 In this paper we anayzed an M/M/c priority system with two customer casses, cass-dependent service rates and a preemptive resume priority rue. This queueing system can be modeed as a two-dimensiona Marov process for which we anayzed the time-dependent behavior. More precisey, we obtained expressions for the Lapace transforms of the transition functions under the condition that the system is initiay empty. Using a sight modification of the CAP method, we showed that the Lapace transforms for the states with at east c high-priority customers can be expressed in terms of a finite sum of the Lapace transforms for the states with exacty c 1 high-priority customers. This expression contained coefficients that satisfy a recursion. We soved this recursion to obtain an expicit 24

26 expression for each coefficient. In doing so, each Lapace transform for the states on the vertica boundary and in the interior can easiy be cacuated from the vaues of the Lapace transforms for the states in the horizonta boundary. Next, we deveoped a Ramaswami-ie recursion for the Lapace transforms for the states in the horizonta boundary. The recursion required the coections of matrices {G i1,i } i c 1 and {N i α} 1 i c for which we showed that they can be determined iterativey. We demonstrated two ways in which the initia vaue of the recursion, i.e., the vector π α, can be cacuated. Finay, we discussed the numerica impementation of our approach for the horizonta boundary. In the singe-server case the expressions for the Lapace transforms for the states on the vertica boundary and in the interior were identica to the muti-server case. The expressions for the horizonta boundary, however, simpified consideraby. Specificay, the initia vaue π, of the recursion coud be determined by comparing the queueing system to an M/G/1 queue with hyperexponentiay distributed service times. Moreover, the cacuation of Gα and Nα, which are now scaars, simpified greaty. We now comment on how our expressions for the Lapace transforms of the transitions functions can be used to determine the stationary distribution. It is cear from the transition rate diagram in Figure 3 that the Marov process X is irreducibe. Moreover, it is we-nown that X is positive-recurrent if and ony if ρ < 1. In that case, X has a unique stationary distribution p : [p x x S. To compute each p x term from π x α, simpy note that p x im α α π x α. 7.1 Using this observation, we see that the procedure for finding p is highy anaogous to the one we presented for finding the Lapace transforms of the transition functions. A Combinatoria identities In this appendix we coect some combinatoria identities that are used throughout the paper. These emmas are iey nown, but we prove them here to mae the paper sef-contained. Lemma A.1. For j 1, and Υj, defined in 3.9, we have 1 Υj, r2 1 j 1 1 λ 1 Ω 2 r j 2 1 λ 1Ω 2 r 2 φ 2 1 r 2 φ 2 m1 1 1 r 2 φ 2 m j 1 m Proof. The resut is neary identica to [1, Lemma 1, so we omit its proof. Lemma A.2. We have the identity K K 2! 2m! 2!2m! K!! Proof. Rewrite the fractions as two binomia coefficients A.2 2!2m! m K 2 2m 1 2 2m 1 r j 2. A.1. A.2 K K 2 2m. A.3 K The rest of the proof foows from a direct appication of [11, Chapter 2, eq