Sojourn times in the M/G/1 FB queue with light-tailed service times

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1 Sojour times i the M/G/ FB queue with light-tailed service times M. Madjes M. Nuyes arxiv:math/032444v2 [math.pr] 23 Feb 2004 September 22, 2008 Abstract The asymptotic decay rate of the sojour time of a customer i the statioary M/G/ queue uder the Foregroud Backgroud (FB) service disciplie is studied. The FB disciplie gives service to those customers that have received the least service so far. We prove that for light-tailed service times the decay rate of the sojour time is equal to the decay rate of the busy period. It is show that FB miimises the decay rate i the class of work-coservig disciplies. Keywords: decay rate, sojour time, Foregroud Backgroud (FB), service disciplie, light tails, busy period AMS 2000 Subject Classificatio: Primary 60K25, Secodary 68M20; 90B22 Itroductio The sojour time of a customer, i.e. the time betwee his arrival ad departure, is a ofte used performace measure for queues. I this ote we study the asymptotic decay rate of the tail of the sojour-time distributio of the statioary M/G/ queue with the Foregroud Backgroud (FB) disciplie. The FB disciplie gives service to those customers who have received the least amout of service so far. If there are such customers, each of them is served at rate /. thus, whe the age of a customer is the CWI, 090 GB Amsterdam, The Netherlads, ad, Uiversity of Twete, Faculty of Mathematical Scieces, 7500 AE Eschede, The Netherlads KdV Istitute for Mathematics, Uiversity of Amsterdam, Platage Muidergracht 24, 08 TV, Amsterdam, The Netherlads

2 amout of service a customer has received, the FB disciplie gives priority to the yougest customers. Let V deote the sojour time of a customer i the statioary M/G/ FB queue, i.e. the time betwee his arrival ad departure. Núñez Queija [6] showed that for servicetime distributios with regularly varyig tails of ide η (, 2), the distributio of V satisfies P(V > ) P(B > ( ρ)),, () where ρ is the load of the system ad B is the geeric service time. Usig Núñez Queija s method, Nuyes [7] obtaied () uder weaker assumptios. I case of regularly varyig service times the tail behaviour of V uder other disciplies, like FIFO, LIFO, PS ad SRPT, has bee foud to be the same or worse, see Borst et al. []. Additioal support for the optimality of FB uder heavy tails is give by Righter ad Shathikumar [8, 9, 0]. They show that for certai classes of service times the FB disciplie miimises the queue legth, measured i umber of customers. For light-tailed service times, however, the FB disciplie does ot perform so well. For epoetial service times (ad also for a subclass of Gamma-distributed service times ) FB still miimises the queue legth, but for service times with a log-cocave desity the queue shows opposite behaviour ad the queue legth is maimal, see Righter ad Shathikumar [8], [9], [0]. This udesirable behaviour of the FB disciplie is very proouced for determiistic service times. I this etreme case i the FB queue all customers stay till the ed of the busy period. Hece the sojour time is maimal i the class of all work-coservig disciplies. I this ote we cosider the (asymptotic) decay rate of the sojour time, defied as follows. Furthermore, let the (asymptotic) decay rate of a radom variable X be defied by lim log P(X > ). The it may be see that i the M/D/ FB queue the sojour time ad the busy-period legth have the same decay rate. It turs out that this decay-rate property holds true for all service-time distributios with a epoetially fast decreasig tail, i the followig sese. Assume that the service times have a fiite epoetial momet, or equivaletly, the Laplace trasform is aalytic i a eighbourhood of zero. The mai theorem of this ote is the followig. Theorem Let V be the sojour time of a customer i the statioary M/G/ FB queue, ad let L be the legth of a busy period. If the service-time distributio has a fiite epoetial momet, the the followig limits eist ad lim log P(V > ) = lim log P(L > ). (2) 2

3 It is show below that the decay rate of V for ay work-coservig disciplie is bouded by the decay rate of the residual life of a busy period. For service times with a epoetial momet the latter decay rate is equal to that of a ormal busy period. Hece (2) is the lowest possible decay rate for a work-coservig disciplie. Usig the decay rate of V as a criterio to measure the performace of a service disciplie the leads to the followig coclusio. For service times with a epoetial momet, the FB disciplie is the worst disciplie i the class of work-coservig disciplies w.r.t. the decay rate of the sojour time. The chapter is orgaised as follows. I Sectio 2 we preset the otatio, some prelimiaries, ad show that the decay rate of the sojour time uder FB is pessimal, as described above. I Sectio 3 Theorem is proved. Sectio 4 discusses the result ad the decay rate of the sojour time i queues operatig uder other service disciplies. 2 Prelimiaries Throughout this ote we assume that the geeric service time B with distributio fuctio F i the M/G/ queue satisfies the followig assumptio. Assumptio 2 The geeric service time B has a epoetial momet, i.e. E ep(γb) < for some γ > 0. Let i additio the stability coditio ρ = λeb < hold, where λ is the rate of the Poisso arrival process. The proofs i this ote rely o some properties of the busyperiod legth L ad related radom variables, which we derive i this sectio. Uder assumptio 2, Co ad Smith [3] have show that P(L > ) b 3/2 e c for certai costats b, c > 0. I particular, L has decay rate c. I fact, by epressio (46) o page 54, c = λ ζ λg(ζ), where g is the Laplace trasform of the service-time distributio, ad ζ < 0 is such that g (ζ) = λ. Hece ζ is the root of the derivative of the fuctio m() = λ λg(). Sice m() attais its maimum i the poit ζ, we may write c i terms of the Legedre trasform of B, c = lim log P(L > ) = sup{θ λ(ee θb )}. (3) θ Remark Cosider a Poisso stream, with itesity λ, of i.i.d. jobs, where every job is distributed accordig to the radom variable B. Let A() deote the amout of work geerated i a arbitrary time widow of legth. It is a easy corollary of Cramér s 3

4 theorem that lim Notig that Ee θa() = log P(A() > ) = sup{θ log Ee θa() }. (4) θ k=0 ( ) k ( ) e λλk Ee θb = ep λ(ee θb ), k! we observe that P(L > ) ad P(A() > ) have the same decay rate. This is somewhat surprisig, as the evet {L > } relates to the arrival patter withi a iterval of legth (ad, i fact, also the jobs already preset at time 0), rather tha just A(). I reewal theory the otio of residual life, also kow as ecess or forward-recurrece time, is stadard. Let L be the residual life of a busy period. The P( L > ) = (EL) P(L > y)dy, see for istace Co [2]. Usig stadard calculus we fid lim log P( L > ) = lim log Hece L has the same decay rate as L. y 3/2 e cy dy = c. (5) Aother igrediet used i the proofs below is the M/G/ queue with trucated geeric service time B τ, τ > 0. Call this the τ-queue ad let L(τ) deote the legth of a busy period (a τ-busy period) i this queue. Let L(τ) be its the residual life ad defie L (τ) to be the legth of a τ-busy period that starts with a customer with service time at least τ, i.e. L (τ) = [L(τ) B τ]. These radom variables satisfy the followig relatio. Lemma 3 The radom variables L(τ), L(τ) ad L (τ) have the same decay rate c(τ), where c(τ) = lim log P(L(τ) > ). Proof Let B deote the first service time i the busy period, hece B d = B. The for τ such that P(B τ) > 0, P(B τ)p(l(τ) > B τ) = P(L(τ) >, B τ) P(L(τ) > ) P(L(τ) > B τ) = P(L (τ) > ), (6) where the iequality follows from the observatio that the value of B is maimal i the τ- queue, sice all service times are bouded by τ. The for every τ such that P(B τ) > 0, 4

5 we have by (5) ad (6) c(τ) = lim log P(L(τ) > ) = lim log P(L (τ) > ) = lim log P( L(τ) > ), (7) which was to be show. I this ote we eed the followig lemma about the decay rate of the sum of two idepedet radom variables. Lemma 4 Let X ad Y be o-egative, idepedet radom variables such that lim log P(X > ) = lim log P(Y > ) = c for some c > 0. The also lim log P(X + Y > ) = c. Proof The lower boud for the limif is obvious. For the upper boud let N be fied. Clearly, P(X + Y > ) i=0 ( P X i ) ( P Y ( i ) ). Fi ε > 0. For sufficietly large, for all i {0,..., }, Hece, ( P X i ) ( P Y lim sup ( i ) ) ( ep ( = ep (c ε) i (c ε) i ) (c ε) ( ) ). ( log P(X + Y > ) (c ε) ). (8) Sice (8) holds for every N ad ε > 0, we may take the limits ad ε 0, ad the result follows. Propositio 5 Whe a customer eters a statioary work-coservig queue, the time D till the system is empty agai satisfies D d = A L+L, where L is distributed like the residual life of a busy period, L is a ormal busy period legth, P(A = ) = ρ = P(A = 0) ad A, L ad L are idepedet. 5

6 Proof The radom variable D is idepedet of the service disciplie, ad ca be decomposed as follows. The customer fids the system empty with probability ρ. I that case D is just the legth L of the busy period started by the customer. If our customer eters a busy system, the the server may first fiish all the work i the system apart from the work of our tagged customer, see the Figure below. The momet this remaiig busy period of legth L is fiished, our customer starts a sub-busy period which is distributed like a ormal busy period L ad is idepedet of L. After this sub-busy period the system is empty agai. workload 0 L L + L time Figure A realisatio of the workload process Hece D = d A L + L, where L is the residual life of a busy period, L is a ormal busy period, P(A = ) = ρ = P(A = 0) ad A, L ad L are idepedet. For the τ-queue we have the followig corollary. Corollary 6 I the statioary τ-queue, the time D from the etrace of a customer till the system is empty agai satisfies D d = A L(τ) + L(τ). If the customer has service time τ, the D d = A L(τ) + L (τ). Sice the system is work-coservig, the sojour time of a customer is ot loger tha D, where D is the time till the system is empty agai. Hece V st D for every service disciplie. Sice A L ad L satisfy the coditios of Lemma 4, the followig corollary holds. Corollary 7 For every work-coservig service disciplie the sojour time V of a customer i the statioary queue satisfies lim sup log P(V > ) lim log P(A L + L > ) = c 6

7 Corollary 8 For service times with a epoetial momet, the FB disciplie miimises the decay rate of the sojour time i the class of work-coservig disciplies. I Sectio 4 we show that there are service disciplies with a strictly larger decay rate, e.g. FIFO. Iterestigly, for a subclass of Gamma-distributed service times the FB disciplie miimises the queue legth, but the sojour time has the smallest decay rate. This shows that optimisig oe characteristic i a queue may have a ill effect o other characteristics. The eistece of a fiite epoetial momet i the corollary is crucial: for heavytailed service times the tail of V caot be bouded by that of L. For eample, i the M/G/ FIFO queue with service times satisfyig P(B > ) = ν L(), where L() is a slowly varyig fuctio at ad ν >, De Meyer ad Teugels [4] showed that P(L > ) ( ρ) ν ν L(). It may be see that i this case the tail of L is oe degree heavier tha that of L. Now ote that for the FIFO disciplie we have V A L. Hece the tail of V is at least oe degree heavier tha that of L, see also Borst et al. [] for further refereces. I the lighttailed case this pheomeo is abset sice the tails of L ad L have the same decay rate. 3 Results The results i this sectio rely o the followig decompositio of V. Let V (τ) is the sojour time i the statioary queue of a customer with service time τ. The sojour time V of a arbitrary customer i the statioary queue satisfies P(V > ) = P(V (τ) > )df(τ). (9) Here F is the service-time distributio. Hece we may write P(V > ) = E B P(V (B) > ), where B is a geeric service time idepedet of V (τ), ad E B deotes the epectatio w.r.t. B. Theorem is proved usig this represetatio of V. I the et lemma we compute the decay rate of V (τ). Propositio 9 Let V (τ) be the sojour time of a customer with service time τ i the statioary queue. If the service-time distributio satisfies Assumptio 2, the for τ > 0, lim log P(V (τ) > ) = c(τ), 7

8 where c(τ) = lim log P(L(τ) > ). (0) Proof By the ature of the FB disciplie, the sojour time V (τ) of a customer with service time τ who eters a statioary queue is the time till the first epoch that o customers youger tha τ are preset. This is the time till the ed of the τ-busy period that he either fids i the τ-queue, or starts. By Corollary 6, V (τ) the satisfies V (τ) d = A(τ) L(τ) + L (τ), where L(τ) is the residual life of a τ-busy period, L (τ) is a τ busy period that starts with a customer with service time τ, P(A(τ) = ) = P(A(τ) = 0) = λe(b τ) ad A(τ), L(τ) ad L (τ) are idepedet. By (7) the radom variables A(τ) L(τ) ad L (τ) satisfy the coditio of Lemma 4. Hece lim log P(V (τ) > ) = lim log P(A(τ) L(τ) + L (τ) > ) = c(τ). () This fiishes the proof. The followig lemma provides the basis for fidig the eeded lower boud for the decay rate. The edpoit of the service-time distributio F is defied as F = if{u 0 : F(u) = }. Lemma 0 Let V be the sojour time of a customer i the statioary queue. Suppose the service-time distributio satisfies Assumptio 2. If τ 0 > 0 ad P(B τ 0 ) > 0, the lim if log P(V > ) P(B τ 0) F Here F is the distributio fuctio of the geeric service time B. τ 0 c(τ)df(τ). (2) Proof Let B 0 ad V deote the service time ad the sojour time of a customer i the statioary queue. Let τ 0 > 0 be such that P(B 0 τ 0 ) > 0. The P(V > ) P(V >, B 0 τ 0 ) = P(V > B 0 τ 0 )P(B 0 τ 0 ). (3) Usig the represetatio (9), we fid log P(V > B 0 τ 0 ) = log E B0 [P(V > ) B 0 τ 0 ]. (4) Sice log is a cocave fuctio, applyig Jese s iequality to the coditioal epectatio i (4) yields log E B0 [P(V > ) B 0 τ 0 ] E B0 [ log P(V > ) B 0 τ 0 ]. (5) 8

9 From (3), (4) ad (5) it follows that Θ := lim if log P(V > ) satisfies Θ lim if log F τ 0 log P(V (τ) > )df(τ)/( F(τ 0 )). (6) To iterchage limit ad itegral, we show that the itegrad is bouded ad the apply the Domiated covergece theorem. Sice lim log P(V (τ 0) > ) = c(τ 0 ) <, there is a γ R such that log P(V (τ 0) > ) < γ for all say. Sice log P(V (τ) > ) is decreasig i τ, the iequality log P(V (τ) > ) < γ holds for all τ τ 0 ad. From (6) it the follows Θ P(B τ 0 ) F which was to be show. τ 0 lim log P(V (τ) > )df(τ), Lemma Let c(τ) = lim log P(L(τ) > ). The c(τ) is decreasig i τ. Furthermore c(τ) c( F ) as τ F, where c( F ) = c. Proof The fuctio h τ (θ) = θ λ(ee θ(b τ) ) is cocave sice ay momet geeratig fuctio is cove. Furthermore lim θ h τ (θ) = lim θ h τ (θ) =. By defiitio of L(τ) ad (3), we may write c(τ) = sup θ {h τ (θ)}. The c(τ) is decreasig i τ, sice h τ (θ) is decreasig i τ. Sice c(τ) h τ (0) = 0 for all τ, ad c(τ) is decreasig, c(τ) coverges for τ F. Now ote that h τ (θ) is cotiuous i τ for all θ [0, sup{η : Ee ηb < }), eve if B has a discrete distributio. Sice the supremum of θ λ(ee θb ) is attaied i this iterval, we have lim τ F c(τ) = c( F ). Propositio 2 Let V be the sojour time of a customer i the statioary queue. If the service-time distributio satisfies Assumptio 2, the where lim if log P(V > ) c, (7) c = lim log P(L > ). Proof If P(B = F ) > 0, the choosig τ 0 = F i (2) yields lim if log P(V > ) c( F) = c ad (7) holds. Let ε > 0. If P(B = F ) = 0, the by Lemma there eists a (ε) < F such that c(u) c + ε for all u (ε). Choosig τ 0 = (ε) i (2) the yields lim if log P(V > ) P(B (ε)) F (ε) P(B (ε)) F 9 (ε) c(τ)df(τ) (c + ε)df(τ) = c ε.

10 Sice ε > 0 was arbitrary, the lower boud (7) follows. Proof of Theorem The upper boud is established i Corollary 7 ad the lower boud i Propositio 2. 4 Discussio The decay rate of V i the M/G/ FB queue is the same as for the preemptive LIFO queue. Ideed, the sojour time of a customer i the statioary M/G/ queue uder the preemptive LIFO disciplie is just the legth of the sub-busy period started by that customer. The sojour time of a customer i the statioary queue uder FIFO satisfies V = B + W, where W is the statioary workload. The decay rate of W is the positive root θ 0 of h(θ) = θ λ(ee θb ). Sice h(0) = 0, h (0) = λeb < ad h is cocave, we have c FB := c = sup θ h(θ) < θ 0, see also Figure 4 below. Furthermore, θ 0 is strictly smaller tha the decay rate c B of the geeric service time B, which is give by c B = if{θ : h(θ) = }. A aalogue of Lemma 4 yields that the decay rate c FIFO of the sojour time i the FIFO system is strictly larger tha c. Madjes e Zwart [5] cosider the PS queue with light-tailed service requests. They show that the decay rate of P(V > ) is c, uder the additioal requiremet that, for ay positive costat k, lim log P(B > k log ) = 0. For determiistic requests, clearly this criterio is ot met. Ideed, i [5] it is show that the decay rate i the M/D/ queue with PS is larger tha c. h(θ) 0 c, c FB c FIFO c B θ Figure 2 The decay rates of the sojour time uder FB ad FIFO. 0

11 Refereces [] Borst, S., Boma, O. J., Núñez Queija, R., ad Zwart, A. P. The impact of the service disciplie o delay asymptotics. Performace Evaluatio 54 (2003), [2] Co, D. R. Reewal theory. Methue, 962. [3] Co, D. R., ad Smith, W. L. Queues. Methue, 96. [4] De Meyer, A., ad Teugels, J. L. O the asymptotic behaviour of the distributios of the busy period ad service time i M/G/. Joural of Applied Probability 7, 3 (980), [5] Madjes, M., ad Zwart, A. P. Large deviatios for waitig times i processor sharig queues. Submitted. [6] Núñez Queija, R. Processor-Sharig Models for Itegrated-Services Networks. PhD thesis, Eidhove Uiversity, [7] Nuyes, M. The Foregroud-Backgroud queue. PhD thesis, Uiversity of Amsterdam, to appear i [8] Righter, R. Schedulig. I Stochastic orders ad their applicatios, M. Shaked ad R. Shathikumar, Eds. Academic Press, 994. [9] Righter, R., ad Shathikumar, J. G. Schedulig multiclass sigle server queueig systems to stochastically maimize the umber of succesful departures. Probability i the Egieerig ad Iformatioal Scieces 3 (989), [0] Righter, R., Shathikumar, J. G., ad Yamazaki, G. O etremal service disciplies i sigle-stage queueig systems. Joural of Applied Probability 27 (990),

Sojourn times in the M/G/1 FB queue with. light-tailed service times

Sojourn times in the M/G/1 FB queue with. light-tailed service times Sojourn times in the M/G/ FB queue with arxiv:math/032444v3 [math.pr] 9 Jun 2004 light-tailed service times M. Mandjes M. Nuyens November 2, 208 Keywords: decay rate, sojourn time, Foreground-Background