Global and local asymptotics for the busy period of an M/G/1 queue

Transcription

1 Queueig Sys : DOI /s Global ad local asympoics for he busy period of a M/G/1 queue Deis Deisov Seva Sheer Received: 5 Jauary 2007 / Revised: 18 Jauary 2010 / Published olie: 17 February 2010 The Auhors This aricle is published wih ope access a Sprigerlik.com Absrac We cosider a M/G/1 queue wih subexpoeial service imes. We give a simple derivaio of he global ad local asympoics for he busy period. Our aalysis relies o he explici formula for he joi disribuio for he umber of cusomers ad he legh of he busy period of a M/G/1 queue. Keywords Busy period Busy cycle Heavy-ailed disribuios Mahemaics Subjec Classificaio K25 60G50 60F10 1 Iroducio Le A 1,A 2,... ad B 1,B 2,... be idepede sequeces of idepede ad ideically disribued radom variables. We call {A i } ier-arrival imes ad {B i } service imes. I is assumed hroughou ha ρ := E{B 1 }/E{A 1 } < 1, so ha he sysem is sable. Deoe also ξ i = B i A i ad S = i=1 ξ i. We shall deoe by A, B ad ξ radom variables wih he same disribuios as A 1, B 1 ad ξ 1, respecively. We use he sadard oaio: we deoe by M/G/1 he sysem wih expoeial ier-arrival imes A i ad by GI/G/1 he sysem wih arbirarily disribued A i. The mai ieres of our paper is he global ad local asympoics for he disribuio of he legh of he busy period τ = B 1 + +B ν, D. Deisov Deparme of AMS, Herio-Wa Uiversiy, Ediburgh EH14 4AS, UK S. Sheer Eidhove Uiversiy of Techology ad Euradom, P.O. Box 513, 5600 MB Eidhove, The Neherlads sheer@euradom.ue.l

2 384 Queueig Sys : where ν = if{ : S < 0} is he umber of cusomers which arrive ad ge served durig he busy period. I [10], i was show ha i he M/G/1 case, if B has a regularly varyig disribuio, he P{τ >} E{ν}P { B>1 ρ } as. 1 This resul was geeralised i [11] o he case of a GI/G/1 queue ad uder he assumpio ha he ail P{B >} is of iermediae regular variaio see he ex secio for a defiiio. Laer o, i was show i [3] ad [9] ha asympoics 1 hold for he GI/G/1 model i he case whe he service-ime disribuio belogs o aoher subclass of heavy-ailed disribuios. This class icludes, i paricular, Weibull disribuios wih parameer α<1/2. The ails of he disribuios cosidered i [3] ad [9] are heavier ha e.asisshowi[2], he laer codiio is crucial for asympoics 1 o hold. I was show i [6] ha i a M/G/1 queue Pτ > C PX > 0 as, 2 where C is a cosa, X = N i=0 B i ad N is a Poisso process wih iesiy λ = 1/. I was also show i [6] ha i he case whe he ail of B is heavier ha e ad uder some furher codiios he asympoic equivalece 2 reduces o 1. I his oe, we cocerae o he M/G/1 case ad have wo mai goals. The firs ad he more impora cosiss i providig ew resuls o local asympoics for τ. These resuls are o surprisigly similar o 2: P τ, + T ] PX 0,T] λ as, wihafixedt 0,. Aoher goal of his paper is o give a simple proof of 1 forhem/g/1 case for some classes of disribuios of service imes wih ails heavier ha e. However, our coribuio here does o cosis oly i providig a shorer proof; we believe ha our assumpios o he disribuio of he service imes are close o miimal. I paricular, hey are saisfied by all he disribuios for which relaio 1 has bee proved earlier. Boh aims are achieved by usig he explici formula for he joi disribuio of τ ad ν see e.g. [5, 4.63]: Pν =, τ > = λu 1 e λu PB 1 + +B du. 3! The paper is orgaised as follows. I Sec. 2, we give resuls o local asympoics for τ, ad i Sec. 3 we prese a simple proof of 1 forhem/g/1 queue.

3 Queueig Sys : Local asympoics for he busy period I his secio, we shall prese our mai resuls o local asympoics for τ.le = 0,T] ad recall ha S = i=1 B i A i ad X = N i=1 B i. We will sar wih he followig simple resul. Theorem 2.1 Assume ha he radom variable B is absoluely coiuous. The ad P{ν = }= f S 0 λ 4 f τ = f X 0. 5 λ Here f ξ x deoes he desiy of a radom variable ξ a a poi x. The ex resul holds for radom variables ha are o ecessarily absoluely coiuous. Theorem 2.2 Fix a posiive ad fiie value of T. Assume ha P{B 1 + }e δ, as, for ay δ>0. The P{τ + } P{X } λ P{X [] } λ[] as, where [] deoes he ieger par of. These asympoics are o explici. However, he heorem above reduces he origial problem o a paricular case of he large-deviaios problem for radom walks, which is exesively sudied i he lieraure. I urs ou ha some kow largedeviaios resuls ogeher wih he heorem above yield explici asympoics for he disribuio of he busy period. Some of hese cases will be preseed below. Firs, we discuss corollaries of he above heorems ad he give he proofs. We sar by defiig wo raher large classes of disribuios. Defiiio 2.1 We say ha a fucio fxis iermediae regularly varyig if lim lim sup sup fy+ κ 1 fx+ 1 = 0. 6 x x y κx I paricular, 6 holds whe fxis regularly varyig a ifiiy. Le Fx = PB x be he disribuio fucio of B ad le Fx + = Fx+ T Fx. We will use he followig codiios: A EB < ad Fx+ is iermediae regularly varyig. B EB 2 <, sup y x Fx y + Fx+ 1 0

4 386 Queueig Sys : ad P{B 1 >, B 2 >, B 1 + B 2 x + } ε sup x 2 = o1/ as. PB x + Sufficie codiios for B o hold may be derived usig, for example, Lemma B.1 ad Lemma B.2 of [7]. I paricular, oe ca see ha Fx+ e xβ,β <1/2, as well as Fx+ e lβ x,β >1, saisfy Codiio B. The followig proposiio direcly follows from Corollary 2.1 of [7]. Proposiio 2.1 Le eiher Codiio A or B hold. The P{S } P { ξ Eξ + } as ad hece sice N is a Poisso process P{X [] } []P { X 1 [] EX 1 + } as. For desiies, Codiios A ad B ca be reformulaed as follows: A EB < ad f B x is iermediae regularly varyig. B EB 2 <, sup fx y 1 fx 0 ad ε = sup x 2 y x x fx yfydy fx = o1/ as. Proposiio 2.1, Theorems 2.1 ad 2.2 immediaely allow us o obai he local asympoics. Corollary 2.1 Le eiher Codiio A or B hold for some T>0. The P{τ + } PB ρ + as. A similar resul holds for desiies. Corollary 2.2 Assume ha B is absoluely coiuous ad le oe of Codiios A or B hold. The f τ f ρ. Now we give proofs of he saemes preseed i his secio.

5 Queueig Sys : Proof of Theorem 2.1 Pu a = λ 1 =,b = EB. The λu 1 P{ν = }= e λu fb u du! 0 = λ 1 0 f A u f B This implies he firs saeme of he heorem. We proceed o prove he secod saeme: f τ = λ 1 e λ fb! = 1 λ =1 = 1 λ = 1 λ =1 u du = λ 1 f =1 B A 0 P { N= } f B P { N= } P{B 1 + +B + du} du P{ N i=1 B i + du} du = f X 0. λ Proof of Theorem 2.2 We sar by showig ha he assumpios of he heorem imply ha P{X }e δ for ay δ>0. Ideed, fix a value of δ>0 ad assume ha P{X 1 + } e δ for sufficiely large which is guaraeed by he assumpios of he heorem. Le a = EX 1 > 0. Fix also ε>0 ad wrie P{X } P { X 1 a ε, a + ε ],X 1 + X X 1 } = a+ε a ε a+ε a ε P{X 1 du}p{x 1 u + } P{X 1 du}e δu P { X 1 a ε, a + ε ]} e δa+ε 1 εe δa+ε, where he laer iequaliy follows from he Law of Large Numbers. Sice δ>0is arbirary, his implies ha P{X }e δ for ay δ>0as. We will oly prove he asympoic equivalece P{τ + } P{X }/ as. The proof of he equivalece P{τ + } P{X [] }/[] may be give followig he same lies. Fix ay ε>0. Accordig o Formula 3,. Pτ + = =1 = =1 +T +T λu 1 e λu PB 1 + +B du! PNu = PB 1 + +B du λu

6 388 Queueig Sys : = For he firs sum we have: S 1 1 λ 1 λ ε 1 λ ε +T 1 λ P N λ ε + λ ε< λ+ε + >λ+ε P Nu= PB 1 + +B du 1 λ ε S 1 + S 2 + S 3. P{B 1 + +B + }=o e δ wih some δ>0as, sice N is Poisso disribued wih parameer λ. Ideed, i follows immediaely from he Cherov boud ha for some δ>0 P { N λ ε } = o e δ as. I ca be show i a similar way ha S 3 = oe δ as. Le us ow ivesigae he asympoic behaviour as of he remaiig sum: S 2 = 1 + o1 1 λ +T We ow cosider e λu 1 + u = e u λu +log1+ λ ε< λ+ε e λu 1 + u e λ λ! PB 1 + +B du. { exp λu + u } exp { λu + λ + εu } = exp { εu } exp{εt }. Here we used he facs ha for he sum uder cosideraio λ1 + ε ad u + T, ad he iequaliy log1 +. I a similar way, wih he help of he iequaliy log1 + /2, 1, oe ca prove ha e λu 1 + u exp{ εt/2}. Therefore, ad S o1 e εt λ P X,λ ε N λ + ε S o1 e εt/2 P X,λ ε N λ + ε. λ Noe ow ha P X, N λ ε P N λ ε = o e δ as.

7 Queueig Sys : Hece, we have he followig upper ad lower bouds: ad P{τ + } 1 + o1 e εt λ PX + o e δ P{τ + } 1 + o1 e εt/2 PX + o e δ λ as as. Usig he codiio e δ P{X } for ay δ>0 ad he leig ε 0, we arrive a he saeme of he heorem. Proof of Corollary 2.1 I follows from Proposiio 2.1 ad Theorem 2.2 ha P{τ + } λ 1 P { X 1 [] EX 1 + } = λ 1 P { X 1 [] ρ[]+ } as, sice EX 1 = EN1EB 1 = ρ 1. Now we should oe ha P { X 1 [] ρ[]+ } { N1 } P B i [] ρ[]+ i=1 EN1P B [] ρ[]+ EN1PB ρ + as, by he heorem o local behaviour for radomly sopped sums from [1]. Ideed, eiher Codiio A or B implies ha F S,see[1] for his heorem ad defiiios. Proof of Corollary 2.2 As is o difficul o see, Codiio A implies Codiio A ad Codiio B implies Codiio B for ay fixed T. Therefore, Corollary 2.1 implies ha Pτ + PB ρ + Tf B ρ. 7 The laer equivalece holds sice by boh Codiios A ad B f B is log-ailed, ha is, f B + u f B as uiformly i u [0,T]. Nex, i follows from he explici formula 3 ha f τ is logailed as well ad, herefore, Pτ + Tf τ. 8 Combiig 7 ad 8, we arrive a he coclusio. Noe ha he resuls of his secio hold oly for some classes of disribuios such ha Fx + e x. We believe ha i is also possible o obai asympoics for

8 390 Queueig Sys : he disribuios wih ails ligher ha e x, bu oe should use resuls for large deviaios of radom walks oher ha Proposiio 2.1. There are a lo of kow resuls i ha direcio; we refer he reader o a rece paper of Borovkov ad Mogulskii [4] for some ew resuls ad a review. 3 Global asympoics for he busy period I his secio, we give a shor proof of he kow resul 1. I oher words, his secio deals wih he case T =. Pu Fx = P{B >x}. We cosider he same codiios o he disribuio of B as hose used i he previous secio. I he case T =, hey rasform io: A EB < ad F is iermediae regularly varyig, i.e. B EB 2 <, Fx x Fxad Fx lim lim sup κ 1 x Fκx = 1. P{ξ 1 >, ξ 2 >, S 2 >x} ε sup x 2 = o Fx The followig proposiio follows from Corollary 2.1 of [7]. 1,. Proposiio 3.1 Le eiher Codiio A or B hold. The P{S > 0} F Eξ as. Noe ha he so-called square-roo isesiiviy codiio Fx Fx x implies ha l Fx = o x as x ; hece, he proposiio above deals wih disribuios whose ails are heavier ha e x. We ow sae he mai resul of his secio. Theorem 3.1 Le eiher Codiio A or B hold. The P{τ >} E{ν}P { B>1 ρ } as. We also eed he followig resul. Proposiio 3.2 Le eiher Codiio A or B hold. The P{ν >} E{ν}P { B> E{ξ} } as. 9 The proof of his proposiio is raher sadard see, for isace, [8] ad is hus omied here.

9 Queueig Sys : Proof of Theorem 3.1 For Codiio B, he esimae from below is a corollary of he CLT ad he codiio Fx x Fx,asx, a proof may be foud i [11]. For Codiio A, he esimae from below follows similarly from he Law of Large Numbers. Therefore, we will cocerae o he esimae from above. We will cosider Codiio B. The proof for Codiio A is similar. I follows from Proposiio 3.2 ad Codiio B ha Pν > P ν> as. Therefore, here exiss a fucio l such ha Pν > P ν> l as. By he oal probabiliy formula, Pτ > = P 1 + P 2 + P 3 = P τ>,ν 1 ε + P τ>,1 ε <ν l + P τ>,ν> l. Firs, P 1 = P B 1 + +B ν >,ν 1 ε P P 1 1 ε 1 B > = o1/p 1 B >, P B i > 1 1 ε 1 1 ε,, A > A > where ha laer equivalece follows from he fac ha, whe 2 <, P A i i >ε = o1/ i=1

10 392 Queueig Sys : as. The, P 2 = 1 ε < l PNu = 1 PB 1 + +B du. For u 1, probabiliy PNu = 1 is o-icreasig i u. Therefore, PNu = 1 PN = 1, ad his fac implies ha P 2 1 ε < l P 1 B > 1 ε PN = 1 PB 1 + +B > P 1 ε N By he Ceral Limi Theorem for reewal processes, P 1 ε N 1/2 l = o1 as ad, herefore, P 2 = o1/p As a resul, we have P 1 + P 2 = o1/p 1 1 B i >,. 1/2 l. B > = o PB > ρ,, where he laer follows from Corollary 2.1 of [7]. For he hird erm, P o1 P ν> = 1 + o1 E{ν}PB > ρ as see 9. Fially, we have Pτ > 1 + o1 E{ν}PB > ρ,. Ackowledgemes The auhors would like o hak Serguei Foss for drawig heir aeio o his problem ad Oo Boxma for a umber of useful discussios ad for may suggesios ha helped improvig he mauscrip. The auhors are also graeful o he aoymous referees for heir horough readig of he paper ad may valuable remarks. The research of boh auhors was suppored by he Duch BSIK projec BRICKS ad he EURO-NGI projec. Ope Access This aricle is disribued uder he erms of he Creaive Commos Aribuio Nocommercial Licese which permis ay ocommercial use, disribuio, ad reproducio i ay medium, provided he origial auhors ad source are credied.

11 Queueig Sys : Refereces 1. Asmusse, S., Foss, S., Korshuov, D.: Asympoics for sums of radom variables wih local subexpoeial behaviour. J. Theor. Probab. 16, Asmusse, S., Klüppelberg, C., Sigma, K.: Samplig a sub-expoeial imes, wih queueig applicaios. Soch. Process. Appl. 79, Balrūas, A., Daley, D.J., Klüppelberg, C.: Tail behaviour of he busy period of a GI/GI/1 queue wih subexpoeial service imes. Soch. Process. Appl. 111, Borovkov, A.A., Mogulskii, A.A.: Iegro-local ad iegral heorems for sums of radom variables wih semiexpoeial disribuios. Sib. Mah. J 476, Cohe, J.W.: The Sigle Server Queue, 2d ed. Norh-Hollad, Amserdam/New York Deisov, D., Sheer, V.: Asympoics for firs passage imes of Levy processes ad radom walks. EURANDOM Repor Deisov, D., Dieker, T., Sheer, V.: Large deviaios for radom walks uder subexpoeialiy: he big-jump domai. A. Probab. 365, Doey, R.A.: O he asympoic behaviour of firs passage imes for rasie radom walks. Probab. Theory Rela. Fields 81, Jeleković, P.R., Momčilović, P.: Large deviaios of square roo isesiive radom sums. Mah. Oper. Res. 292, De Meyer, A., Teugels, J.L.: O he asympoic behaviour of he disribuios of he busy period ad service ime i M/G/1. J. Appl. Probab. 173, Zwar, A.P.: Tail asympoics for he busy period i he GI/G/1 queue. Mah. Oper. Res. 263,