Multi-server preemptive priority queue with general arrivals and service times

Transcription

1 Mult-server preeptve prorty queue wth geeral arrvals ad servce tes Alexadre Bradwaj Bask School of Egeerg Uversty of Calfora Sata Cruz USA Thoas Beg LIP UMR CNRS - ENS Lyo - UCB Lyo 1 - INRIA 5668 Frace thoas.beg@es-lyo.fr ABSTRACT We preset a sple approxate soluto for preeptve-resue queues wth ultple servers geeral phase-type servce ad geeral phase-type terarrval te dstrbutos. I our soluto prorty levels are solved oe at a te the order of decreasg prortes. Each prorty level s solved approxately usg a reduced state descrpto. The coplexty of our approxate soluto ters of the uber of equatos solved grows learly wth the uber of servers ad prorty levels. We studed a large uber of uercal exaples wth a rage of values for ea servce tes ad offered loads across prorty levels varyg the uber of servers fro 8 to 48. Dscrete-evet sulato was used to assess the accuracy of our approxate soluto. Overall the case of Posso ad quas-posso arrvals expected relatve error for the ea uber of custoers the syste was below 2% whle the correspodg eda relatve error was below 0.25%. The good accuracy of our approxato appears to exted to the case of phase-type tes betwee arrvals wth expected relatve errors for the ea uber syste below 5% eve for a Pareto-lke dstrbuto of terarrval tes wth a large coeffcet of varato. Our uercal results dcate that the proposed approxato provdes a relatvely sple ad geerally accurate approach to preeptve-resue queues wth larger ubers of servers ad geeral dstrbutos of servce ad terarrval tes. Keywords: Multple servers prorty preeptve-resue geeral servce geeral arrvals Ph/Ph/c/N queue reduced-state approxato lear coplexty.

2 1. INTRODUCTION Systes wth ultple servers whch custoers are served accordg to dfferet prortes ca be foud ay areas such as custoer servce ceters [GAN03] arport securty checkpots [DEL13] hosptal eergecy roos [LIN14] cloud coputg systes [ELL12] or processor aageet certa coputer Operatg Systes [STA04]. Despte the large uber of systes that ca be vewed as staces of a prorty queue wth ultple servers the lterature devoted to ther theoretcal aalyss appears rather oderate as the heret coplexty of these queues hders ther aalyss. I fact eve the accurate approxate aalyss of a queue wth ultple servers ad geeral servce tes wthout prortes reaed a ope ssue for several decades cf. Gupta et al. [GUP07]. Wth the preeptve-resue servce dscple whch a hgher prorty custoer ca terrupt the servce of a lower prorty custoer ad the terrupted servce resues fro the pot of terrupto whe a server becoes avalable a sple exact aalytcal soluto s kow the partcular case whe there s oly oe server for all custoers [TAK91 ALL90]. To the best of our kowledge few exact results exst the case of prorty queues wth ultple servers eve uder the splest assupto of expoetally dstrbuted servce tes e.g. servce rates for all prorty levels ust be detcal the solutos proposed by Davs [DAV66] ad Kella et. al [KEL85]. However oteworthy progress has bee ade the aalyss of prorty queue wth ultple servers over the last decade or so. Recetly Wag et al [WAN15] proposed a ovel exact aalytcal soluto for the partcular case of two prorty levels ad expoetal servce tes M/M/c. They obta the geeratg fucto of the uber of custoers at the lower prorty level ad ther soluto becoes cubersoe whe the uber of servers exceeds 2. A drect uercal soluto of the balace equatos of a M/M/c queue wth prortes quckly becoes uaageable as the uber of prorty levels creases. Clearly geeral servce tes ca oly copoud the proble especally wth hgher ubers of servers as llustrated by the dffculty of solvg eve the uch spler top prorty level whch s just a M/G/c queue [BRA14]. Ths drves the developet of effcet approxate solutos. I 2004 Zelty et al. [ZEL04] studed the M/M/c queue wth L xed prorty classes soe preeptve others o-preeptve. They were able to derve exact ad approxate solutos uder the costrat that the servers have expoetal servce tes detcal for all prorty classes. Sce the several terestg approaches have bee proposed ag to lft the restrcto of detcal ad expoetally dstrbuted servce tes the soluto of prorty M/M/c queues. More recetly Al Habal et al. [ALH15] reoved the costrat of expoetal servce tes by proposg a approxate soluto to evaluate the frst two oets of the watg te a o-preeptve M/G/c prorty queue wth detcal servce te dstrbutos over all the classes. I the sae year as etoed above Wag et al. [WAN15] relaxed the costrat of havg detcal servce rate over all the classes by provdg the exact aalyss of a preeptve M/M/c queue wth two prorty classes havg dfferet

3 servce rates. However oe of these two approaches hadles the case of a prorty queue havg both o-detcal servce rates ad geeral servce tes. I 2005 Harchol-Balter et al. [HAR05] ade a sgfcat step the aalyss of the geeral case of a prorty ult-server queue by cosderg servers that cobe both o-expoetal servce tes ad o-detcal servce rates over all the classes. Ther approach reles o reducg the desoalty of the uderlyg Markov cha to a oe-deso Markov cha wthout trucatos. Ther results spag a rage of loads ad varablty of the servce tes show good accuracy. Ther soluto sees best applcable to systes wth a oderate uber of servers ad classes. Ideed although theory ther ethod ca hadle systes wth ay uber of servers ad ay uber of prorty classes the authors develop aother approxato whe the uber of classes L s large by approxatg the L -prorty syste wth a two-class prorty syste. Besdes all uercal results preseted ther paper perta to systes wth oly two servers. We have etoed ths troducto oly pror work that appears ost relevat to ths paper. The terested reader ay refer to the paper of Harchol-Balter et al. [HAR05] for a thorough ad sghtful revew of the lterature pror to I ths paper we focus o preeptve-resue ult-server queues ad we propose a coceptually sple approxate soluto for such a preeptve prorty syste wth geeral terarrval ad servce tes. I our soluto prorty levels are solved oe at a te the order of decreasg prortes. Ths akes the coputatoal coplexty of our soluto lear the uber of levels. Each prorty level s solved approxately usg a reduced state descrpto so that the coplexty of the soluto grows learly wth the uber of servers. Thus the cotrbuto of ths paper s to troduce a sple approxate soluto for preeptve queues wth ultple servers geeral servce ad terarrval tes that s coputatoally scalable both the uber of servers ad the uber of prorty levels. Our approach ca accoodate tes betwee arrvals ad servce tes that deped o the uber of custoers at a gve prorty level. Addtoally although we focus ths work o preeptve-resue prortes the proposed approach ca be readly appled to ult-server queues wth preeptve-restart prorty. Ths paper s orgazed as follows. We start by the case of eoryless.e. Posso or quas-posso arrvals. I Secto 2 we descrbe detal the syste cosdered ad defe the a sybols used the sequel. Secto 3 outles the proposed approxate soluto ad clearly detfes the approxatos ade. Secto 4 s devoted to the uercal results llustratg the accuracy of our approxato the case of eoryless arrvals. Secto 5 presets the exteso of our ethod to geeral arrvals. Fally Secto 6 cocludes ths paper. 2. SYSTEM CONSIDERED - CASE OF MEMORYLESS TIMES BETWEEN ARRIVALS Wth eoryless arrvals the prorty queueg syste cosdered s show Fgure 1. It coprses C hoogeeous servers agets ad arrvg custoers are dvded to L prorty classes levels ubered 1... L where level 1 s the hghest. Custoers of class arrve accordg to a quas-posso

4 process wth rate λ l where l s the uber of level custoers curretly the syste. A arrvg custoer who fds ts level queue epty terrupts preepts the servce of a lower-prorty custoer f ay. The terrupted custoer ay resue ts servce at the pot of terrupto preeptve-resue or restart a whole ew servce perod preeptve-restart. We do ot cosder the case whch the terrupted custoer repeats a detcal servce perod preeptve-repeat detcal. The uber of custoers at each prorty level s lted to N l. Custoers arrvg to fd ther respectve queue at capacty are sply lost. 1 1 Hgh-prorty queue C servers 2 2 L L Arrvals Low-prorty queue Departures Fgure 1. The prorty queueg syste wth L prorty levels ad C servers. The servce te dstrbuto at each prorty level ca be dfferet. Custoers at a gve level are assued to be statstcally detcal ad the queueg dscple s assued to be FCFS wth each class. The servce tes at level l l =1... L are dstrbuted accordg to a phase-type dstrbuto see Fgure 2 wth a total of b l phases. Referrg to level we deote by σ l the probablty that servce starts phase ad by µ l the testy of the correspodg phase. The probablty that the servce proceeds phase j followg the copleto of phase s gve by q jl =1...b l j =1...b l ad the probablty that the servce eds wth the copleto of phase s deoted by ˆq l. The prcpal otato used ths paper s suarzed Table 1. Note that ay dstrbuto ca be represeted arbtrarly closely by a phase-type dstrbuto [BOL05]. If oly the frst two oets of a dstrbuto are kow ad f the dstrbuto s squared coeffcet of varato s greater tha 0.5 a phase-type dstrbuto wth oly two phases b l = 2 suffces to atch the kow frst two oets. If ore oets are kow or oe s atchg a whole theoretcal or eprcal dstrbuto a good ft wll typcally requre ay ore tha two phases. Also dstrbutos wth a squared coeffcet of varato below 0.5 requre ore tha 2 phases. Recall that the coeffcet of varato of a dstrbuto s defed as the rato of the stadard devato to the ea. Readly avalable tools exst to effect such dstrbuto fttg e.g. [BOB05 OSO06].

5 µ 1` 1` b` 2` q 12` q 1b` µ 2` ˆq 1` ˆq 2` ˆq b` µ b` Fgure 2. The phase-type dstrbuto wth b phases for servce tes of prorty level. The perforace dces of terest clude custoary perforace etrcs such as the ea uber of custoers at each level the ea respose sojour te loss probabltes the server utlzato or custoer throughput for each class etc. I the ext secto we outle a effcet approxate soluto to copute the perforace etrcs of terest. 3. EFFICIENT APPROXIMATE SOLUTION A classcal state descrpto for the prorty syste cosdered s the jot probablty of the ubers of custoers at each prorty level ad the uber of custoers each phase of servce cludg custoers whose servce has bee suspeded due to preepto by hgher prorty classes. It allows oe to geerate drectly the full balace equatos of the syste. However the uber of states ths full descrpto grows cobatorally wth the uber of prorty classes ad servers the systes akg such a drect descrpto uaageable. Sce wth the preeptve prorty cosdered a gve prorty level s oly affected by hgher prorty levels we elect to look at a sgle level at a te startg fro the top level. The top level s sply a stace of the M/Ph/c/N queue ad we solve t approxately usg the reduced state descrpto 1 1 where 1 s the curret uber of custoers at ths level ad 1 descrbes the curret servce phase of a selected servce posto see [BRA14]. For a syste wth preeptve-resue prorty we descrbe the state of a prorty level l l = 2... L by the trple l l where l s the curret uber of custoers at ths level l s the uber of servers agets curretly uavalable at ths level busy servg hgher prorty custoers ad descrbes the progress of the servce at the curret level. Followg the dea of reduced state descrpto [BRA14] we descrbe explctly the progress phase uber of the servce of oly oe arbtrarly selected servce posto so that = 1... b f the servce posto s curretly actve at ths level.e. a aget s servg

6 the level custoer at ths posto l = 1... b l f the level custoer at ths posto s curretly suspeded.e. the level custoer at ths posto s preepted by a hgher level custoer ad we use the value l = 0 to descrbe a servce posto wthout a level custoer. The possble values for the uber of uavalable servers are l = 0...C ad for the curret uber of level custoers the syste l = 0... N l. Thus the uber of states our state descrpto for level s at ost N l 12b l 1C 1 sce ot all values of are feasble for all sets of l ad l. The set of values for l correspods to the case of preeptve-resue prorty dscple. If the servce dscple s preeptve-restart there s o eed to keep track of the servce phase whch the custoer was preepted so that oe suspeded state suffces ad the total uber of possble values for l s lted to b l 2. As a exaple a syste wth C = 6 agets servers cosderg lower prorty level > 1 the state = 4 l = 4 l > 0 ples that there are a total of 4 custoers at level the uber of servers avalable for custoers at ths level s C l = 2 ad the selected servce posto s curretly actve the custoer at ths servce posto beg phase l of ts servce. If l = 0 ths ples that the selected servce posto s oe of the axc l 0 = 2 uoccuped servce postos. If l < 0 ths eas our exaple that the custoer at the selected servce posto s oe of the C l l = 2 custoers curretly suspeded due to uavalablty of servers preepted by custoers at hgher prorty levels. Note that our descrpto each prorty level has a total of C servce postos ad ay avalable server aget ay serve ay servce posto occuped by a custoer. Sce there s o affty betwee servce postos ad agets a custoer terrupted by hgher prorty custoers wll geeral resue ts servce wth a dfferet server tha before preepto. At the top level all servers are always avalable 1 = 0 ad thus we ust have 1 0. Fgure 3 shows the savg ters of the uber of states wth the proposed reduced-state descrpto as copared to full-state descrpto for a preeptve-resue queue. I Fgure 3a wth a syste wth C = 8 N l = 64 for l =1... L ad b l = 4 for l =1... L we let the uber of prorty levels vary betwee L = 2 ad 10. We observe that the reduced-state descrpto leads to several hudreds of states whle the full-state descrpto results several tes of thousads. I Fgure 3b the syste uder cosderato has a fxed uber of prorty levels of L = 4 ad N l =128 for l =1... L ad b l = 4 for l =1... L. Depedg o the specfc uber of servers C we observe that the dfferece betwee the total uber of states cosdered the reduced-state descrpto ad the full-state descrpto aouts to roughly two ad three orders of agtude.

7 Nuber of prorty levels L Nuber of states logarthc scale Full state descrpto Reduced state descrpto Nuber of servers C Nuber of states logarthc scale Full state descrpto Reduced state descrpto Fgure 3a. Savg uber of states usg the reduced state descrpto wth C = 8 N l = 64 ad b l = 4. Fgure 3b. Savg uber of states usg the reduced state descrpto wth L = 4 N l =128 ad b l = 4. Let p l l be the steady-state probablty of the retaed state descrpto at level. Fro the perspectve of the gve prorty level the fluece of hgher prorty levels ca be vewed sply as servers agets dsappearg ad reappearg wth soe rates correspodg to preeptos ad hgher prorty levels becog dle. It s a straghtforward albet soewhat tedous atter to derve the balace equatos for p l l. As a exaple for the case where l > C ad l < C wth l =1...b l we have ˆ ] [ 1 1 p C C p p p q j p q j p p p b j j j b j j j a b b µ µ l b a µ l = å å = = 1 Equato 1 volves the kow paraeters for level custoers as wells as ew quattes vz. ν l l l α l l l ad β l l l. The frst quatty ν l l l represets the rate of copletos of custoers at servce postos other tha the chose oe the servce progress of the latter s descrbed by l gve the curret state l l. α l l l deotes the rate wth whch servers dsappear fro level gve the curret state ad β l l l s the rate wth whch servers reappear gve that the curret state s l l.

8 Aalogous equatos ca be obtaed for all other values of l l ad l. We ust have N C b å å å p = 1. The uber of equatos the resultg set of equatos s oderate = 0 = 0 = -b depedg o the value of N l ad grows oly learly as the uber of servers C creases. Of course we eed to kow the rates ν l l l α l l l ad β l l l to solve ths set of equatos. If we had the exact values for these rates the soluto of our set of equatos would gve us the exact probabltes p l l. We are ot able to obta the exact values for the ukow rates but we ca obta good approxatos by assug that certa varables each of these rates are ore portat tha others. We start by α l l l the rate at whch servers dsappear gve l l. It sees logcal to assue that the curret uber of users at the gve level ad the servce progress at the selected servce posto have uch less fluece o the rate α l tha the uber of servers already uavalable so that α l l l α l l l = 0...C 1. 2 We ake slar assuptos for the rate at whch servers reappear at level gve l l β l l l β l l l =1...C. 3 Clearly we have α l C = β l 0 = 0. Note that the soluto of the top prorty level produces the steady-state probablty p 1 1 sce there are o uavalable servers at level 1. The probablty that there are a total of 1 custoers at level 1 s b 1 gve by p 1 1 = p 1 1. Sce the servers ad custoers at a gve level are assued to be 1 =0 statstcally detcal we readly obta the overall rate of copletos gve that the curret uber of custoers s 1 deoted by u 1 1 as b 1 u 1 1 = C p 1. 1 µ 1 ˆq 1 / p =1 The rate of server dsappearace for the followg level α 2 2 s sply α 2 2 = λ 1 1 = 2 for 2 = 0...C 1. 5 The rate wth whch servers reappear at level 2 s gve by

9 u 1 1 = 2 2 =1...C 1 β 2 2 = u 1 1 = C p 1 1 = C / p = C 1 C 6 At level 2 the rate of copletos of the selected servce posto whe there are 2 custoers ad 2 servers agets uavalable gve that the posto s ot dle ca be expressed as ξ = b 2 b 2 p µ 2 ˆq 2 p =1... N 2 ; 2 = 0...C =1 2 =1 The rate of copletos by servce postos other tha the selected oe ca be approxated ters of ξ ν C 2 ξ [ 2 C 2 1]ξ > 0 8 At ths stage we ca solve the balace equatos for level 2 to obta the steady-state probabltes p Note that because the rates ν are effectvely expressed ters of p the syste of equatos to solve becoes o-lear. The steady-state probabltes that there are 2 C b 2 custoers at ths level s gve by p 2 2 = p =0 2 = b 2 Havg solved level 2 we use the probabltes p to assess the rates of server dsappearace ad reappearace for the edately followg prorty level. level 3 ca be obtaed as The rate of server dsappearace for 3 b 2 α 3 3 = p 2 = [α 2 2 λ 2 2 ] / P = 0...C =0 2 = b 2 where P 2 3 deotes the probablty that a total of 3 servers are uavalable for the followg level.e. busy wth custoers at level 1 ad 2. The rate of server reappearace at level 3 ca be expressed as β 3 3 = 3 b 2 p 2 = [β ξ ] / P =1...C =0 2 = b 2 The probablty P 2 3 s gve by

10 3 P 2 3 = C 2 =0 2 = b 2 b 2 p 2 = N 2 2 =0 2 =C 2 2 = b 2 b 2 p = 0...C 1 3 = C. 11 The rates of copletos by servce postos other tha the selected oe ν are coputed usg forulas aalogous to 7 ad 8 ad the soluto of level 3 becoes the possble. We proceed ths way level by level. The rates of server dsappearace ad reappearace are evaluated at each prorty level except the lowest level for the soluto of the level edately below t usg forulas drectly aalogous to forulas 9 ad 10. The aalyss of prorty level l =1... L yelds p l the steady-state probablty that there are l custoers at ths level. The ea uber of custoers at level s gve by l = l p the attaed throughput of custoers at ths level ca be expressed as θ l = λ l p ad the loss probablty ca be wrtte as ξ l = λ l N l pn l / λ l p. Kowg the ea uber of custoers ad the throughput t s easy to obta the correspodg ea sojour te ad server utlzato. Algorth 1 suarzes our approach. Algorth 1. Solvg preeptve-resue queues wth ultple servers geeral phase-type servce tes ad Posso or quas-posso arrvals. Step 1. Cosder level 1 Solve the top level to obta p = 0... N 1 1 = 0...b 1 ad p 1 1. N l l =0 N l 1 l =0 N l l =1 Evaluate perforace dces of terest pertag to level 1. Copute α = 0...C 1 ad β =1...C for use the soluto of level 2 forulas 5 ad 6. Step 2. Cosder levels l = 2... L the order of decreasg prorty. At level Solve the balace equatos for the gve level usg approxato forula 8 to obta p l l ad p l. Evaluate perforace dces of terest pertag to level.

11 If l < L copute α l1 l1 ad β l1 l1 usg forulas 9 ad 10. Note that the proposed approxate soluto replaces the soluto of a sgle syste of balace equatos whose coplexty grows cobatorally wth the uber of prorty levels ad the uber of servers by the soluto of L systes of equatos whose coplexty ters of the uber of equatos grows oly learly the uber of servers Fgure 3 copares the coplexty of the two approaches. We solve the equatos for each prorty level uercally. The ext secto presets uercal results to llustrate the accuracy of the proposed approach the case of eoryless arrvals. C L l λ l Nuber of servers agets Nuber of prorty levels classes Nuber of level custoers curretly the syste Arrval rate of level custoers gve there are l custoers the syste N l Maxu uber of custoers at prorty level b l Nuber of phases for the servce te dstrbuto of custoer of prorty level σ l Probablty that servce starts phase for custoer of prorty level µ l Itesty of the phase for custoer of prorty level q jl Probablty that the servce proceeds phase j followg the copleto of phase for custoer of prorty level ˆq l Probablty that the servce eds wth the copleto of phase for custoer of prorty level l Nuber of servers agets curretly uavalable at level l Curret phase of the servce o the selected servce posto for level p l l Steady-state probablty at level ν l l l Rate of copletos at servce postos other that the selected oe gve the curret state α l l l Rate wth whch servers dsappear fro level gve the curret state Table 1. Prcpal otato used the paper the case of eoryless arrvals. β l l l Rate wth whch servers reappear for level gve the curret state p l u l ξ l l Probablty that there are a total of l custoers at level l Overall rate of copletos gve that the curret uber of custoers s l Rate of copletos by servce postos other tha the selected oe gve there are l custoers ad l servers agets uavalable P l l1 Probablty that a total of l1 servers are uavalable for level l 1

12 4. NUMERICAL RESULTS Wth eoryless arrvals the approxato proposed ths paper cotas two possble sources of errors. Frst eve wth expoetally dstrbuted servce tes there are possble accuraces due to the assupto that the rates of server dsappearace ad reappearace at a lower prorty level deped oly o the uber of servers occuped at hgher prorty levels cf. equatos 2 ad 3 Secto 3. Secod eve at the hghest prorty level where there are o servce terruptos the reduced state descrpto used to accout for o-expoetal servce te dstrbutos troduces possble errors. As dcated by a study of the reduced state descrpto [BRA14] the errors attrbutable to ths approxato are geerally sall ad ted to decrease as the uber of servers grows. To assess the overall accuracy of our approxate level-by-level soluto approach we studed a farly large set of uercal exaples of preeptve-resue queues usg the results of dscrete-evet sulato as coparso bass. For the latter we used 7 depedet replcatos [MAC89] of betwee ad copletos each. These sulato paraeters were chose a attept to ze war-up effects. The resultg estated cofdece tervals at 95% cofdece levels ted to be suffcetly sall so that we used oly the d-pot value. We studed two dfferet eoryless arrval patters. I the frst oe arrvals to each prorty level coe fro a separate Posso source wth rate λ l for level l =1... L. The uber of custoers at level l s lted to N l resultg possble lost custoers. I the secod arrval patter as a exaple of quas-posso arrvals custoers at each prorty level coe fro a separate fte set of eoryless sources wth K l sources for level l. A custoer s ether at the source or the prorty queue watg or beg served ad o custoers are lost so that we have N = K. The rate of custoer arrvals to level l wth l custoers already preset s K l l φ l where 1/ φ l s the ea te a custoer speds at the source o each pass through the syste. We start by the case of Posso arrvals. The ubers of servers cosdered were C = ad 48. The buffer sze for each prorty level was set to N = 3C. The uber of prorty levels was kept at L = 4 ad we used a set of 4 values for the ea servce tes at dfferet levels. The ea servce te for level 1 hghest prorty was set to 1 for level 2 to 1/2 or 2 for level 3 to 1/4 or 4 ad for the last level to 1/8 or 8. We cosdered 4 values for the coeffcet of varato of the servce tes at dfferet levels: ad 4. The arrval rates for dfferet prorty levels raged fro 0.1 to 1.5 per te ut per server. These arrvals rates l were explored so that our results spa cases whch dfferet prorty classes doate the syste. The above cobatos of paraeter values aouted to a total of 960 exaple pots for each of the 4 prorty levels. For each prorty level we used the ea uber of custoers the attaed throughput ad the loss probablty as perforace dces. Tables 2 3 ad 4 suarze the relatve errors versus sulato results obtaed for the ea uber syste attaed throughput ad loss probablty respectvely.

13 These tables clude the ea expected ad eda relatve errors as well as the dstrbuto of relatve errors. Table 2a shows how the accuracy of our approxato geerally proves as the uber of servers grows. We ote that already for 16 servers the ea error s below 2% ad the percetage of exaple pots whch the relatve error exceeds 10% s less tha 5%. I Table 2b we have cluded results for each prorty level separately ad for all levels cobed. We observe Table 2b that although ot surprsgly the accuracy of the approxato degrades for lower prorty levels t reas geerally qute good. Eve for the lowest prorty level our exaples the expected relatve error for the ea uber of custoers the syste reas below 3% ad the percetage of cases whch the relatve error was below 10% s over 90. Overall the exaple pots cosdered our study the ea error for the ea uber of custoers was less tha 1.5% whle the eda error dd ot exceed 0.03%. Table 3 yelds slar observatos for the attaed throughput. Table 4 suarzes the results obtaed for the loss probablty. Note that order to avod potetally large relatve errors we have cluded Table 4 oly exaple pots whch the loss probablty the sulato was above Here the ea relatve error reas below 1%. The fact that the relatve error sees to decrease for lower prorty levels appears to be due to larger values of loss probabltes at lower levels leadg to saller relatve errors. Nuber servers Mea % Meda % <1% 1-5% 5-10% >10% < < All Table 2a. Dstrbuto of the relatve errors for the ea uber syste wth Posso arrvals. Class Mea % Meda % <1% 1-5% 5-10% >10% % < < All Table 2b. Dstrbuto of the relatve errors for the ea uber syste wth Posso arrvals. Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 3. Dstrbuto of the relatve errors for the attaed throughput wth Posso arrvals.

14 Class Mea % Meda % <1% 1-5% 5-10% >10% < < < < All 0.35 < Table 4. Dstrbuto of the relatve errors for the loss probablty wth Posso arrvals. Fgure 4 shows a exaple of the behavor of dfferet prorty levels a preeptve-resue queue wth Posso arrvals C =16 servers ad the axu uber of custoers at each prorty level lted to N = 3C. The ea servce tes are 1 ½ ¼ ad 1/8 for prorty levels ad 4 respectvely whle the coeffcet of varato of the servce te s kept at 2 for all custoer classes. The rates of custoer arrvals are gve by λ 1 = λ λ 2 = λ / 2 λ 3 = λ / 4 ad λ 4 = λ / 8 for levels 123 ad 4 respectvely ad we vary the factor λ to study the perforace of custoers at dfferet prorty levels as a fucto of overall offered load. We observe the close agreeet betwee sulato ad our approxate results ea uber syste Class =1 Appx Class =1 Su Class =2 Appx Class =2 Su Class =3 Appx Class =3 Su Class =4 Appx Class =4 Su offered rate factor λ Fgure 4a. Mea uber syste as a fucto of the offered rate wth Posso arrvals. Grr=16

15 16 14 attaed throughput Class =1 Appx Class =1 Su Class =2 Appx Class =2 Su Class =3 Appx Class =3 Su Class =4 Appx Class =4 Su offered rate factor λ Fgure 4b. Attaed throughput as a fucto of the offered rate wth Posso arrvals.grr=161 We ow cosder the secod arrval patter wth K l sources for level. We used 3 values for the uber of sources: K = ad 50. The values of the utary source rate j raged fro 0.1/ K l to 0.9/ K per te ut. We used the sae set of ea servce tes as the case of Posso arrvals. The coeffcet of varato of the servce tes was set to 2. Here the total uber of exaple pots explored was over Tables 5 ad 6 suarze the relatve accuracy of our approxato for the ea uber of custoers the syste ad the custoer throughput respectvely. As was the case for Posso arrvals we show the ea eda ad dstrbuto of relatve errors for each prorty level as well as for all levels cobed. We observe that whle the relatve errors crease for lower prorty levels the ea error for class 4 reas below 3% our study ad over 90% of exaple pots cosdered the relatve errors rea below 10% for ths prorty class. The eda errors are qute sall less tha 0.5%. It has bee our experece that the frequet larger relatve error ted to occur whe the ea servce tes at hgher prorty levels are loger tha at lower prorty levels ad whe the uber of sources at the latter s sall. Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 5. Dstrbuto of the relatve errors for the ea uber syste wth dscrete sources.

16 Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 6. Dstrbuto of the relatve errors for the attaed throughput wth dscrete sources. Fgure 5 shows a exaple of the behavor of a preeptve-resue prorty syste wth a fte set of eoryless sources the partcular case where the uber of custoer sources K l s the sae for each of the 4 prorty levels cosdered. I ths exaple there are C =16 servers the utary source rate s set to j = 0.5 / K l the ea servce tes are ad 8 for prorty classes ad 4 respectvely whle the coeffcet of varato of the servce te s kept at 2 for all classes ea uber syste Class =1 Appx Class =1 Su Class =2 Appx Class =2 Su Class =3 Appx Class =3 Su Class =4 Appx Class =4 Su uber of sources per class Fgure 5a. Mea uber syste as a fucto of the uber of sources.grr=321

17 attaed throughput uber of sources per class Class =1 Appx Class =1 Su Class =2 Appx Class =2 Su Class =3 Appx Class =3 Su Class =4 Appx Class =4 Su Fgure 5b. Attaed throughput as a fucto of the uber of sources. Grr=322 Overall the exaple cosdered the results of our approxate soluto closely atch sulato results. I the ext secto we exted our ethod to clude geeral terarrval te dstrbutos. 5. EXTENSION TO GENERAL ARRIVALS We ow cosder a prorty syste slar to the oe descrbed Secto 2 but whch the tes betwee cosecutve custoer arrvals at each prorty level are dstrbuted accordg to a phase-type dstrbuto possbly dfferet for each custoer class. Specfcally the tes betwee custoer arrvals at level l l =1... L are dstrbuted accordg to a phase-type dstrbuto see Fgure 6 wth a total of a l phases. Referrg to level l we deote by τ l the probablty that the te betwee arrvals starts phase ad by λ l the testy of the correspodg phase. The probablty that the servce proceeds phase j followg the copleto of phase s gve by r jl =1...a l j =1...a l ad the probablty that the servce eds wth the copleto of phase s deoted by ˆr l. To exted our approxate soluto of Secto 3 to such phase-type dstrbutos of te betwee arrvals we ote that our level-by-level approach we are essece dealg wth a ult-server queue whch wth the excepto of the hghest prorty level servers dsappear ad reappear wth rates depedet o the uber of servers curretly uavalable to the level cosdered. Fgure 6 shows a sgle level wth phase-type tes betwee arrvals ad phase-type servce tes. Therefore we propose to exted the approach used recetly by the authors for Ph/Ph/C/N queues [ATM16]. I ths approach such systes are solved by teratg betwee two spler odels: a odel wth eoryless state-depedet arrvals ad phase-type servce M/Ph/c/N queue ad a odel wth phase-type tes betwee arrvals ad eoryless state-depedet servce Ph/M/c/N queue. I our case as show

18 Fgure 7 the odel wth eoryless state-depedet arrvals s fact the odel solved at each level as descrbed Secto 3. Thus o ew soluto approach eeds to be developed for ths part. The Ph/M/c/N queue the other odel ca be solved usg a sple uercally stable recurrece [BRA14]. The terato betwee these two odels for each prorty level stops whe the values for a perforace etrc such as the ea uber of custoers obtaed fro both odels becoe suffcetly close. As dscussed [ATM16] whle such a fxed-pot terato betwee odels yelds oly a approxate soluto the errors t troduces see geerally sall. Moreover typcally the terato betwee odels teds to coverge qute fast. ` uavalable servers 1` 1` 2` a` r 1a` r 12` 2` ˆr 1` ˆr 2`ˆra` a` The phase dstrbuto for terarrval tes Server appearace rate `` Server dsappearace rate `` Fgure 6. Model of a sgle level wth phase-type tes betwee arrvals ad phase-type servce tes. Copute the arrval rate at the queue at level ` gve the curret uber of custoers at ths level s ` Model solved at level of Secto 3 Ph/M/c/N queue ` Copute the departure rate fro the servers at level ` gve the curret uber of custoers at ths level s ` Fgure 7. Iterato betwee a odel of a sgle level wth state-depedet arrvals ad a Ph/M/c/N queue to solve the sgle level odel of Fgure 6. Algorth 2 suarzes our exteded approach.

19 Algorth 2. Solvg preeptve-resue queues wth ultple servers geeral phase-type servce ad geeral terarrval tes. Step 1. Cosder level 1 Use terato see Algorth 3 to obta approxate values for p = 0... N 1 1 = 0...b 1 ad p 1 1. Evaluate perforace dces of terest pertag to level 1. Copute α = 0...C 1 ad β =1...C for use the soluto of level 2 forulas 5 ad 6. Step 2. Cosder levels l = 2... L the order of decreasg prorty. At level Use terato see Algorth 3 to obta approxate values for p l l ad p l. Evaluate perforace dces of terest pertag to level l. If l < L copute α l1 l1 ad β l1 l1 usg forulas 9 ad 10. Algorth 3 suarzes the fxed-pot terato betwee odels at each level deotg by M the selected perforace etrc for covergece test. Algorth 3. Deterg state probabltes at a sgle level for Algorth 2. Step 1. Italze the arrval rate values λ l to the verse of the ea te betwee arrvals. Step 2. Solve the odel wth state-depedet eoryless arrvals usg the curret values of λ l. Obta curret values for p l l ad p l as well as the equvalet servce rate u l as u l = λ l 1 p l 1 / p l. Copute curret value of M fro ths odel. Step 3. Solve the Ph/M/c/N queue usg the curret values of u l fro Step 2 Obta curret values for p l ad λ l. Copute the curret value of M fro ths odel. Step 4. If the values of M fro Step 2 ad Step 3 devate by less tha ε > 0 the stop the terato otherwse go to Step 2. Step 5. Use the values of p l l ad p l fro last executo of Step 2 as the soluto of level.

20 To assess the fluece of the varablty the arrval process o the accuracy of the proposed approxate soluto we studed a prorty syste wth four prorty levels ad three dfferet values of the coeffcet of varato of the arrval process vz. 2 4 ad close to 15. For the latter we used a Pareto-lke dstrbuto wth 16 phases. I our study we used the sae set of values of the ea arrval rates ad ea servce tes as for the case of Posso arrvals Secto 4 but cosdered oly servce tes wth a coeffcet of varato of 2. Table 7 suarzes the relatve errors for the ea ubers of custoers at each prorty level a syste wth C = 16 servers ad the axu uber of custoers at each prorty level lted to N = 3C. Note that for each value of the coeffcet of varato of the te betwee arrvals we studed 60 exaple pots. Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 7a. Dstrbuto of the relatve errors for the ea uber syste wth a coeffcet of varato for terarrval tes of 2. Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 7b. Dstrbuto of the relatve errors for the ea uber syste wth a coeffcet of varato for terarrval tes of 4. Class Mea % Meda % <1% 1-5% 5-10% >10% All Table 7c. Dstrbuto of the relatve errors for the ea uber syste wth a Pareto-lke dstrbuto of terarrval tes coeffcet of varato of 15. We observe that the accuracy of the ethod reas good as the value of the coeffcet of varato of the terarrval te doubles fro 2 to 4. Eve wth the large coeffcet of varato of the Pareto-lke dstrbuto of 16 phases the ea relatve errors for the ea ubers of custoers the syste rea uder 5% for all four prorty levels. Iterestgly the exaple pots cosdered the degradato accuracy for lower prorty levels s qute oderate.

21 As a sple exaple of the applcato of the proposed soluto we copare Fgure 8 the ea ubers of custoers a preeptve-resue prorty syste wth two classes of custoers wth Posso arrvals versus o-posso arrvals. The ea servce tes are 1 ad ½ for prorty levels 1 ad 2 respectvely wth the coeffcet of varato of the servce tes kept at 2 for both custoer classes. The ea rates of custoers arrvals are λ ad λ / 2 for custoers at prorty levels 1 ad 2 respectvely. The coeffcet of varato of the te betwee arrvals the case of o-posso arrvals s kept at 4 for both prorty classes. The syste has C = 8 servers ad the uber of custoers at each prorty level s lted to N = 24 for = ea uber syste Posso Class =1 Appx Posso Class =1 Su Posso Class =2 Appx Posso Class =2 Su o Posso Class =1 Appx o Posso Class =1 Su o Posso Class =2 Appx o Posso Class =2 Su offered rate factor λ Fgure 8. Mea uber syste as a fucto of offered load factor. We observe that our approxate results wth o-posso arrvals closely atch sulato results. Thus the proposed approxato provdes a tool to study the fluece of the arrval process o the perforace of such a preeptve prorty syste. 6. CONCLUSIONS We have preseted a sple approxate soluto for preeptve-resue queues wth ultple servers geeral phase-type servce ad geeral phase-type terarrval te dstrbutos. I our approach prorty levels are solved oe at a te the order of decreasg prortes. We use a reduced state descrpto to deal wth geeral servce te dstrbutos each prorty level. Thus the coplexty of our approxate soluto ters of the uber of equatos solved grows learly wth the uber of prorty levels ad the uber of servers. We studed ay thousads of exaples to assess the accuracy of our approxato coparg ts results wth those of dscrete-evet sulatos. For Posso ad quas-posso arrvals we cluded systes wth fro 8 to 48 servers ad a rage of values for the ea servce tes as well as a large rage of values for the offered load at dfferet prorty levels. Overall for these types of arrval processes our exaples the ea relatve error for the ea uber of custoers the syste

22 was below 2% whle the correspodg eda relatve error was below 0.25%. Whe exaed for each prorty level separately as could be expected relatve errors ted to crease for lower prorty levels. However ths crease errors appears relatvely slow. As a exaple the case of Posso arrvals the ea error for the ea uber syste grows fro about 1.4% for level 2 to about 2.3% for level 4. Therefore oe ca reasoably expect errors to rea acceptable eve wth a larger uber of prorty levels. The accuracy of our approxato teds to prove as the uber of servers grows. Based o several hudred exaples the good accuracy of our approxato appears to exted to the case of phase-type tes betwee arrvals. As a exaple wth 16 servers ad 4 prorty levels the ea error for the ea uber syste grows fro 1.4% to about 1.7% ad about 3.2% as the coeffcet of varato of the tes betwee arrvals grows fro 2 to 4 ad 15 respectvely. Eve the latter case of a Pareto-lke dstrbuto wth 16 phases the ea error reas below 5% for each of the 4 prorty levels. For splcty of exposto we used classcal phase-type dstrbutos for the tes betwee arrvals ad the servce tes. It s a straghtforward atter to let the testy ad phase routg paraeters of these dstrbutos deped o the uber of custoers at the correspodg prorty level. Ths exteso ay be useful soe applcatos. Addtoally the proposed approach ca be readly appled to ult-server queues wth preeptve-restart prorty levels. 7. ACKNOWLEGEMENTS Ths work was supported part by the LABEX MILYON ANR-10-LABX-0070 of Uversté de Lyo wth the progra Ivestsseets d Aver ANR-11-IDEX-0007 operated by the Frech Natoal Research Agecy ANR ad by PALLAS Iteratoal Corporato of Sa Jose Calfora. 8. REFERENCES [ALH15] A. Al Habal E.M. Alvarez ad M.C. va der Hejde. "Approxatos for the watg-te dstrbuto a M/PH/c prorty queue." OR Spectru : [ALL90] A.O. Alle. Probablty statstcs ad queueg theory. Secod Edto. Elsever [ATM16] T. Ataca T. Beg A. Bradwaj ad H. Castel-Taleb. "Perforace evaluato of cloud coputg ceters wth geeral arrvals ad servce." IEEE Trasactos o Parallel ad Dstrbuted Systes : [BOB05] A. Bobbo A. Horváth ad M.Telek. "Matchg three oets wth al acyclc phase type dstrbutos." Stochastc odels : [BOL05] G. Bolch S. Greer H. de Meer ad K.S. Trved. Queueg etworks ad Markov chas: odelg ad perforace evaluato wth coputer scece applcatos. Joh Wley & Sos 2006.

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