Response time in a tandem queue with blocking, Markovian arrivals and phase-type services

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1 Operatons Research Letters 33 (25) Operatons Research Letters wwwelsevercom/locate/dsw Response tme n a tandem queue wth blockng, Markovan arrvals and phase-type servces B Van Houdt a,,1, Attahru Sule Alfa b a Department of Mathematcs and Computer Scence, Unversty of Antwerp, Performance Analyss of Telecommuncaton Systems Research Group, Mddelhemlaan 1, B-22 Antwerp, Belgum b Department of Electrcal and Computer Engneerng, Unversty of Mantoba, Wnnpeg, Mantoba, R3T 5V6, Canada Receved 22 January 24; accepted 2 August 24 Avalable onlne 6 October 24 Abstract A novel approach for obtanng the response tme n a dscrete-tme tandem-queue wth blockng s presented The approach constructs a Markov chan based on the age of the leadng customer n the frst queue We also provde a stablty condton and carry out several numercal examples 24 Elsever BV All rghts reserved Keywords: Tandem queue; Blockng; Markovan arrvals; Phase-type servces; Response tme; Matrx analytc method 1 Introducton In most communcaton networks we are nterested n obtanng the response tme of jobs as well as the number of jobs n the system It has been common practce to frst obtan the queue length and then use that to obtan the response tme The two part procedure could be cumbersome n many stuatons, Correspondng author Tel: ; fax: E-mal address: bennyvanhoudt@uaacbe (B Van Houdt) 1 B Van Houdt s a post-doctoral Fellow of the FWO-Flanders Ths work was performed whle Houdt was vstng the Unversty of Mantoba and TR Labs n Wnnpeg, Mantoba We would lke to thank the Unversty and TR Labs for ther hosptalty especally when dealng wth tandem queues Ths s because n order to obtan the queue length we have to set up the assocated Markov chan, obtan ts statonary vector and then use that n a process that nvolves a consderable amount of effort to obtan the response tme In ths paper we present a dfferent and novel approach n whch we set up the Markov chan, based on the age of the leadng job n the frst queue and other auxlary varables The statonary dstrbuton of ths Markov chan wll lead us to the response tme easly and the queue length can also be computed from t usng a smple procedure We beleve that the effort requred to compute the two quanttes s less usng ths age process approach we are presentng, when compared to the tradtonal approach of usng the Markov chan of queue length More mportantly, snce the /$ - see front matter 24 Elsever BV All rghts reserved do:1116/jorl2484

2 374 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) response tme s sometmes the only key measure of nterest, our approach s very favorable n such cases Tandem queues wth blockng have receved consderable attenton n the queueng, communcatons and manufacturng lterature because of ther pervasveness and sgnfcance n real lfe A number of survey papers have been publshed durng the last two decades [7,2,1] Some of the earler works on ths nclude those of Hunt [12], who frst studed the blockng effects n a sequence of watng lnes Later Av-Itzhak [1] studed the system wth arbtrary nput and regular servce tmes There has been a plethora of studes on ths subject and an addtonal lterature survey can be found n [21] A contnuous-tme tandem queue wth blockng, Markovan arrvals (MAP) and no ntermedate buffer was consdered by Gómez-Corral [6] Phasetype servce was assumed at the frst nfnte queue whereas the other queue s assumed to have a general servce tme Results for the jont queue length dstrbuton were provded and the stablty ssues were partally addressed Gómez-Corral used matrx analytcal methods (MAMs), as we do, to obtan hs results, the dfference n methodology beng that we propose a novel approach that keeps track of the age of the leadng customer n the frst queue as opposed to the number of customers, as ths more easly leads to the response-tme dstrbuton MAMs have been used on a varety of occasons when studyng tandem queues wth blockng, gong back to Latouche and Neuts [15] The bulk of the work done n ths area of tandem queues focussed on contnuous-tme models As ponted out by Daduna [4], the ntroducton of the asynchronous transfer mode as a multplexng technque for broadband ntegrated servces dgtal networks has ncreased the studes n the areas of dscrete-tme queueng models Even though carryng out dscrete-tme analyss of tandem queues may be done, to some extent, n a smlar manner as ther contnuous-tme counterparts ther analyss often ntroduce some addtonal challenges Settng up ther transton matrces s more complex, and obtanng the response tmes of such a system after that requres a more consderable effort Gün and Makowsky [8,9] consdered a dscrete-tme tandem queue wth blockng (and falures), Bernoull arrvals and phase-type servces They used the MAM approach also and assumed both watng rooms to be fnte Daduna [4] consdered the case wth an nfnte watng room for the frst queue, but restrcted hmself to Bernoull servce processes Desert and Daduna [5] focused on dscrete-tme tandem queues wth state-dependent Bernoull servce rates and a state-dependent Bernoull arrval stream at the frst node They obtaned the jont sojourn tme dstrbuton for a customer traversng the tandem system under consderaton In our current paper we consder an nfnte watng room n front of the frst server and allow Markovan arrvals (D-MAP), enablng us to model arrval processes that have some elements of correlatons, whch s more common n the telecommuncaton feld where arrvals are usually bursty Moreover, whle Gün and Makowsky focus on the jont queue length dstrbuton, we provde an algorthm for the total response tme of a customer and address the stablty ssues rased by the nfnte queue Other related works, n the sense that they focus on the response tme as opposed to the jont queue length, are those by van der Me et al [22] and Knessl and Ter [13], who studed the frst two moments of the response tme n an open two-node queueng network wth feedback for the case wth an exponental processor sharng (PS) node and a FIFO node (whle the arrvals at the PS node are Posson) Chao and Pnedo [3] consdered the case of two tandem queues wth batch Posson arrvals and no buffer space n the second queue They allowed the servce tmes to be general and obtaned the expected tme n system We start wth a descrpton of the model under consderaton n Secton 2, whle the GI/M/1-type Markov chan constructed to obtan the performance measures s gven n Secton 3 An effcent method to compute the response tme and the jont queue length dstrbuton from the steady-state vector s presented n Secton 4, whereas Secton 5 addresses the stablty ssues surroundng our model We end by demonstratng the strength of our model through a varety of numercal examples 2 Model descrpton Consder two queues n tandem, where the frst queue has an nfnte watng lne and the second has a fnte one wth capacty B Customers arrve (to the frst queue) accordng to a dscrete-tme Markovan arrval process (D-MAP), characterzed by the

3 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) l l sub-stochastc matrces D and D 1 The matrx D = D + D 1 s the stochastc matrx of the underlyng Markov chan that governs the arrval process The element (D k ),j, 1, j l, k=, 1, represents the probablty of makng a transton from state to j wth k arrvals Let γ be the statonary dstrbuton assocated wth D, then the arrval rate s gven by λ = γd 1 1 l, where 1 l s a l 1 column vector of ones For more detals on MAP see [2,16] The servce requred by a customer n the th queue s phase-type (PH) dstrbuted wth matrx representaton (m, α,t ), for = 1, 2 It s well known that PH dstrbutons are very good for representng most of the types of servces encountered n communcaton systems [14] The mean servce tme of a PH s gven as =α (I m T ) 1 1 m, where I x s an x-dmensonal dentty matrx The matrx T s sub-stochastc and s of order m, whle t s defned as 1 m T 1 m The elements (α ) s of the stochastc vector α represent the probablty that a customer starts hs servce μ 1 n phase s Let r (k) be the probablty that the servce tme at node lasts for k or more unts of tme, then r (k) = α T k 1 1 m,k 1 Notce, the mnmum servce tme at node s at least 1 and the probablty that the servce tme equals exactly k s found as α T k 1 t For more detals on the phase-type dstrbuton see [19] Whenever a customer fnshes servce n the frst queue t advances to the second queue (at no swtchng cost), unless the watng lne of the second queue s already fully occuped In ths case, the customer remans wthn the servce faclty of the frst queue untl there s a servce completon n the second queue Thereby, preventng any other customers watng n the watng lne of queue 1 from enterng the server (meanng, we adopt the blockng-after-servce mechansm, see [2, p 6]) Both queues serve ther customers n a FCFS order All events such as arrvals, transfers from a watng lne to the server and servce completons are assumed to occur at nstants mmedately after the dscrete tme epochs Ths mples, amongst others, that the age of a customer n servce at some tme epoch n s at least 1 3 The GI/M/1-type Markov Chan A Markov chan (MC) that allows us to effcently obtan the response-tme dstrbuton of an arbtrary customer, s constructed next The state space of ths MC wll be subdvded nto an nfnte number of groups, called levels Level zero wll contan all the states that correspond to a stuaton n whch the frst server s dle Whereas level, for >, reflects the fact that the frst server s occuped by a customer of age (ether because he s beng served or because he s blocked by the second queue) To be more specfc, the states of level are dvded nto 2 sets: BI ={j 1 j l}: The MC s sad to be n state j of the set BI at tme n, f both servers are dle and the arrval process s n state j at tme n FI ={(b, s 2,j) b B,1 s 2 m 2, 1 j l}: If the frst server s dle, the second server s occuped by a customer whose servce s n phase s 2 and b customers are watng to be served by the second server, whle the state of the D-MAP s j at tme n, then the MC s sad to be n state (b, s 2,j) of the set FI at tme n The states of level, for >, are further subdvded nto three sets: SI={(s 1,j) 1 s 1 m 1, 1 j l}: The MC s sad to be n state (s 1,j)of the set SI, at tme n, n case the second server s dle, an age customer s served by server 1, the phase of hs servce equalng s 1, whle the arrval process s, at tme n + 1, n state j BS = {(b, s 1,s 2,j) b B,1 s v m v,v = 1, 2; 1 j l}: The stuaton n whch, at tme n, a customer of age s beng served by server 1, there are b customers watng for servce n the second watng lne, the phase of servce n the vth server equals s v, for v = 1, 2, and the state of the D-MAP at tme n + 1 equals j, wll correspond to the state (b, s 1,s 2,j) of level BL ={(s 2,j) 1 s 2 m 2, 1 j l}: The scenaro where, at tme n, there s an age customer blocked n server 1, a customer s served by server 2, whose current phase s s 2, and the state of the D-MAP at tme n +1 equals j s represented by state (s 2,j) of the set BL Let S denote the number of elements n a set S Defne d t and d b as SI + BS + BL =(B+1)m 1 m 2 l+(m 1 + m 2 )l and BI + FI =(B + 1)m 2 l + l, respectvely As we shall explan later on, the transton matrx P

4 376 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) of ths MC has the followng form: B 1 B B 2 A 1 A P = B 3 A 2 A 1 A B 4 A 3 A 2 A 1 A, (31) In case (), the number of customers n the second queue wll ether reman the same or decrease by one, dependng on whether there s a servce completon n server 2 In cases () and (), the number of customers n the second watng lne has to reman equal to B, mplyng that there can be no servce completon As a result, we fnd A = K I l, where T 1 T 1 t 2 T 1 T 2 T 1 t 2 α 2 T 1 T 2 K =, (32) T1 T 2 T 1 t 2 α 2 T 1 T 2 t 1 T 2 T 2 where the matrces A k are d t d t matrces, B k, for k 2, s a d t d b matrx, B a d b d t and B 1 a square matrx of dmenson d b Notce, the matrx A k represents the transton probabltes of gong from level >tolevel k + 1, for k, whereas the matrces B k are related to transtons from and/or to level Let us now dscuss the matrces A k and B k n detal, the structure of P wll be apparent from ths dscusson Assume the MC s n some state of level > at tme n, meanng that an age customer, referred to as customer c, s occupyng server 1 In order to get a transton to level + 1, customer c has to reman n server 1 Ths s because customers are served n a FCFS order and there are no batch arrvals; hence, the age of the very next customer who arrves after c cannot be larger than at tme n + 1 There are three scenaros that would cause customer c to reman n server 1: () hs servce (n server 1) dd not fnsh at tme n, () hs servce fnshed, but he s blocked by the second queue, or () customer c remans blocked I k represents the dentty matrx of dmenson k, t = 1 m T 1 m, for = 1, 2; and 1 k sa1 k vector wth each entry set to 1 Notce, ths matrx does not depend on the age of customer c Next, we consder the transtons from level to k + 1, for k 1 Thus, as before we have a customer, called c, occupyng server 1 at tme n To get a transton to level 1 k + 1, customer c has to leave server 1 Moreover, a new customer, whose age should equal k + 1 at tme n + 1, has to enter the server at tme n, call hm customer c Meanng, the nterarrval tme between c and c has to equal k The fact that customer c leaves server 1 mples that the MC cannot make a transton to one of the states SI BL of level k + 1 Also, gven that the watng lne of server 2 was fully occuped at tme n, there should have been a servce completon n server 2 (otherwse c would become/reman blocked) Fnally, f there stll was a vacancy n the watng lne of the second queue at tme n, the number of watng customers there ether ncreases by one or remans the same, dependng on whether there s a servce completon n server 2 Hence, transtons from level to k + 1 are governed by the matrx A k = K 1 D k 1 D 1, where t 1 α 1 α 2 t 1 α 1 t 2 α 2 t 1 α 1 T 2 t 1 α 1 t 2 α 2 K 1 = (33) t1 α 1 t 2 α 2 t 1 α 1 T 2 t 1 α 1 t 2 α 2 α 1 t 2 α 2

5 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) The matrces B k, for k, descrbng the transtons to and from level can be obtaned usng smlar arguments (see [11] for detals) B k, for k>1, equals K1 c(α 1 1) D k 1, where K1 c(α 1 1) s obtaned from K 1 by replacng all vectors α 1 by the scalar 1 and removng the last m 2 columns The matrx B 1 can be wrtten as K r(t 1 α 1,t 1 ) D 1 To obtan K r(t 1 α 1,t 1 ) we replace all matrces T 1 appearng n the expresson for K nto the vector α 1, the vector t 1 nto the scalar and remove the last m 2 rows of K Fnally, the matrx B 1 obeys the followng equaton: 1 t 2 T 2 t B 1 = 2 α 2 T 2 D T2 t 2 α 2 T 2 (34) Let π =[π, π 1, π 2,] be the steady-state vector of P, where π sa1 d b vector and π, for >, a1 d t vector Snce, the MC characterzed by P s a GI/M/1-type MC, π exsts f and only f θβ > 1, where θ k A k =θ, θ1 dt =1 and β= k 1 ka k1 dt The stablty condton θβ > 1 s dscussed n Secton 5 The steady-state vector of a GI/M/1-type MC can be found by teratvely solvng for the mnmal nonnegatve R n the non-lnear equaton R = k Rk A k (see [19]) Havng solved ths equaton we fnd π as [π, π 1 ] =[π, π 1 ] [ B 1 B k 1 Rk 1 B k+1 k 1 Rk 1 A k ], (35) π = π 1 R, (36) where >1, π and π 1 are normalzed as π 1 db + π 1 (I R) 1 1 dt = 1 However, the computaton of π can be done much more effcently (both n terms of the tme and memory complexty) by constructng a quas-brth Death (QBD) Markov chan and applyng the cyclc reducton algorthm [17] to solve ths QBD We refer to [11] for detals on the QBD reducton 4 Performance measures In ths secton we demonstrate how to get the response-tme dstrbuton from the steady-state vector π We start by ntroducng the followng set of random varables: T W : The amount of tme a tagged customer has to wat n the th watng lne, for = 1, 2 T S : The servce tme duraton of a tagged customer n server, for = 1, 2 T B : The tme that elapses whle a tagged customer s blocked n server 1 Havng defned these varables the total response tme T R of a tagged customer s defned as T W1 + T S1 + T B + T W2 + T S2, whle the response tme n queue 1, denoted as T R1, equals T W1 + T S1 + T B We need two more varables before we can proceed: F N : The number of customers stll requrng full servce by server 2 before a tagged customer who just left server 1 can start hs servce F T : The remanng servce tme of the customer occupyng server 2 when a tagged customer leaves server 1 Wrte π as [π SI, π BS, π BL ] n accordance wth the three sets of states of level, then P [T R1 = r, F N = b, F T = h] = (t 1 ) s1 λ 1 {b=&h=} s {b<b h=} s2,j π BS r + 1 {b=b&h=} λ j r (s 1,j) π SI (b, s 1,s 2, j)(t2 h t 2) s2 π BL r (s 2, j)(t 2 ) s2, s2,j where (x) denotes the th component of the vector x From ths t s straghtforward to compute the probabltes P [T R1 + F T = r, F N = b] The total

6 378 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) response-tme dstrbuton s then found by P [T R = ]= P [T R1 + F T = r, F N = b] b r b P [S ( b+1) 2 = r], (47) where S ( x) 2 s the x-fold convoluton of the PH servce tme dstrbuton of server 2 characterzed by (m 2, α 2,T 2 ) The blockng probablty p BL can be computed as p BL = 1 π BS r (B, s 1,s 2, j)(t 1 ) λ s1 (T 2 ) s2, r> s 1,s 2,j (48) and the blockng tme dstrbuton as P [T B = t]= 1 π BS r (B, s 1,s 2, j)(t 1 ) λ s1 r> s 1,s 2,j (T t 2 t 2) s2, (49) for t> and P [T B = ]=1 p BL Fnally, notce that the probablty of havng a jont queue contents equal to (q 1,q 2 ), equals the probablty of havng a customer of age a n the frst servce center, q 2 customers n the ntermedate queue and havng q 1 arrvals durng a tme nterval of length a (that starts mmedately after the customer n the frst servce center arrved) Hence, a smple procedure can be devsed to obtan the jont queue contents dstrbuton from the steady state probablty vector π (see [11]) 5 Stablty condton The GI/M/1-type Markov chan ntroduced n Secton 3 s ergodc f and only f θβ > 1, where θ k A k = θ, θ1 dt = 1 and β = k 1 ka k1 dt The followng theorem summarzes the stablty results (see [11] for detals) Theorem 51 The MC developed n Secton 3 s stable f and only f (1) λ/k<1, where λ s the D-MAP mean arrval rate and k = μ 1 (1 κ BL 1 m2 ) = μ 2 (1 κ SI 1 m1 ), (51) wth μ the servce rate of the th servce center and κ=[κ SI, κ, κ 1,,κ B, κ BL ] s the nvarant probablty vector of K = K + K 1 Thus, the arrval process only affects the stablty through ts mean arrval rate λ (2) the queue contents process of the nfnte watng lne of queue 1 has a steady state Remark 1 Eq (51) suffces to prove that nterchangng both servce-tme dstrbutons does not affect the stablty of the system Intutvely, when studyng the stablty, we may assume that there are always customers ready to be served n the nfnte watng lne Now, usng an argument by Melamed [18], we can regard the empty places (holes) n the ntermedate buffer as dual customers For each regular customer that moves through the system, a dual customer recevng dentcal servce moves n the opposte drecton Hence, the maxmum stable throughput of the regular customers and the dual customers must be dentcal Clearly, the dual system s dentcal to the nterchanged system Ths type of nterchangeablty result was already establshed long ago for exponental servers (and Posson arrvals) [19, Secton 52] Remark 2 In the specal case where the sze of the ntermedate watng lne B equals, one can prove that 1/k s nothng but the mean of the maxmum of both PH servce tme dstrbutons 6 Numercal results In ths secton we present some numercal examples that provde nsght on the system behavor 61 Influence of the capacty Band the correlaton of the D-MAP arrval process Consder an nterrupted Bernoull process (IBP), where the mean sojourn tme n both states equals x c and an arrval occurs n the on-state wth probablty x p Thus, [ ] 1 1/xc 1/x D = c, D 1 = (1 x p )/x c (1 x p )(1 1/x c ) [ x p /x c x p (1 1/x c ) ] (611)

7 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) (a) (b) Fg 1 Response-tme dstrbuton for varous capactes B, x p = 1 16 and (a) x c = 4, (b) x c = 4 The servce tme dstrbuton n server 1 s hypergeometrc wth parameters: [ 45 ] α 1 =[ 1 3, 2 3 ], T 1 = (612) 19 2 Thus, wth probablty 1 3 and 2 3 the servce tme s geometrcally dstrbuted wth a mean of 5 and 2 tme unts, respectvely The mean of ths dstrbuton s 15 tme unts The servce-tme dstrbuton of the second server s characterzed by 3 1 α 2 =[ 16 1, , 16 ], T 2 = , (613) x l where x l = s chosen such that the mean servce tme equals 15 We have chosen the mean servce tme dentcal n both servers as ths ought to create a strong couplng between both queues Fg 1 depcts the response-tme dstrbuton for varous capactes B, for x p = 16 1 (meanng that the arrval rate λ = 32 1 ), and x c ether 4 or 4 Clearly, the larger x c the more correlated the arrval process Obvously, stronger correlated arrvals gve rse to slower response tmes, whle addng more capacty B between both servers reduces the response tme However, at some pont there s lttle use n further augmentng the capacty as the response tme seems to converge for B large Ths s easly understood as the blockng probablty tends to decrease to zero whle ncreasng B The fgure further llustrates that the rate of convergence s affected by the correlaton of the arrval process: stronger correlated arrval processes more easly justfy ncreasng the capacty B 62 The maxmum arrval rate λ and the varaton of the servce tmes We consder the same IBP process as n the prevous secton The servce-tme dstrbuton s ether geometrc (Geo), Erlang-5 (Er5) 1 or hypergeometrc (HypGeo) characterzed by [ 45 ] α 1 =[ 1 9, 1 1 ], T 1 = (614) The mean of each of these servce-tme dstrbutons s 15 tme unts The HypGeo dstrbuton s the most varable of the three, followed by Geo and Er5 Fg 2 shows the maxmum stable load ρ max, defned as E[S 1 ]λ max = 15λ max, as a functon of B for dfferent server confguratons λ max s the maxmum arrval rate λ for whch the system s stable Recall, λ max =μ 1 (1 κ BL 1 m2 )=μ 2 (1 κ SI 1 m1 ), see Secton 5 The notaton (X, Y ) ndcates that the servce tme n servers 1 and 2 s dstrbuted as X and Y, respectvely 1 That s, the sum of 5 ndependent and dentcally dstrbuted geometrc random varables

8 38 B Van Houdt, AS Alfa / Operatons Research Letters 33 (25) Fg 2 Maxmum stable load ρ max as a functon of the capacty B for dfferent server confguratons Fg 3 Response-tme dstrbuton for dfferent server confguratons for B = 1 and IBP arrvals (x c = 4,x p = 1 16 ) Fg 2 clearly demonstrates that more varable servce tmes, that s, HypGeo, gve rse to a lower maxmum stable nput rate λ max Ths result stems from the fact that a more varable servce tme, whether n the frst or second server, causes a hgher degree of blockng n comparson wth a more determnstc servce tme dstrbuton As proved n Secton 5, nterchangng both servce tme dstrbutons does not alter the system stablty Notce, the actual nature of the D-MAP arrval process s rrelevant as the stablty s only affected by the arrval process through ts mean The response-tme dstrbuton for dfferent server confguratons s depcted n Fg 3 We assume that arrvals occur accordng to a IBP wth x c = 4 and x p = 16 1, see Secton 61 The capacty of the ntermedate watng lne B s assumed to be 1 Fg 3 confrms that the response of the system slows down as the servce tmes become more varable It further demonstrates that nterchangng the servce tmes generally causes a (lmted) change n the response tme (as opposed to the stablty) Placng the more varable server frst seems to result n a somewhat slower response Ths mght be explaned by notcng that the output process of server 1 s more bursty n such case, causng a hgher blockng probablty On the other hand, less varablty n server 2 decreases the blockng probablty, so when we nterchange both servce tme dstrbutons both these effects nfluence the degree of blockng n the system Varous numercal exper- ments, ncludng Fg 3, seem to ndcate that reducng the varaton of server 1 should be slghtly favored References [1] B Av-Itzhak, A sequence of servce statons wth arbtrary nput and regular servce tmes, Manage Sc 11 (5) (1965) [2] C Blonda, A dscrete-tme batch markovan arrval process as B-ISDN traffc model, Belg J Oper Res Statst Comput Sc 32 (3,4) (1993) [3] X Chao, M Pnedo, Batch arrvals to a tandem queue wthout an ntermedate buffer, Stochastc Models 6 (4) (199) [4] H Daduna, Queueng Networks wth Dscrete Tme Scale, Sprnger, Berln, 21 [5] B Desert, H Daduna, Dscrete tme tandem networks of queues: effects of dfferent regulaton schemes for smultaneous events, Performance Evaluaton 47 (22) [6] A Gómez-Corral, A tandem queue wth blockng and markovan arrval process, Queueng Systems 41 (22) [7] L Gün, Annotated bblography of blockng systems Techncal Report, Insttute for Systems Research ISR , 1987 [8] L Gün, M Makowsk, Matrx geometrc soluton for fnte capacty queues wth phase-type dstrbutons, n: Proceedngs of Performance 87, Brussels, 1987, (pp ) [9] L Gün, M Makowsk, Matrx geometrc soluton for two node tandem queueng systems wth phase-type servers subject to blockng and falures Techncal Report, Insttute for Systems Research, ISR 87-21, 1987

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