Analysis of Alternating-priority Queueing Models with (Cross) Correlated Switchover Times

Transcription

1 IEEE INFOCOM 005 Analyss of Alternatng-prorty Queueng Models wth Cross Correlated Swtchover Tmes Robn Groenevelt, Etan Altman INRIA Sopha Antpols, 0690 Sopha Antpols, France Emal: {robngroenevelt, Abstract Ths paper analyzes a sngle server queueng system n whch servce s alternated between two queues and the server requres a fnte swtchover tme to swtch from one queue to the other The dstncton from classcal results s that the sequence of swtchover tmes from each of the queues need not be d nor ndependent from each other; each sequence s merely requred to form a statonary ergodc sequence Wth the help of stochastc recursve equatons eplct epressons are derved for a number of performance measures, most notably for the average delay of a customer and the average queue lengths under dfferent servce dscplnes Wth these epressons a comparson s made between the servce dscplnes and the nfluence of correlaton s studed Fnally, through a number of eamples t s shown that the correlaton can sgnfcantly ncrease the mean delay and the average queue lengths ndcatng that the correlaton between swtchover tmes should not be gnored Ths has mportant mplcatons for communcaton systems n whch a common communcaton channel s shared amongst varous users and where the tme between consecutve data transfers s correlated for eample n ad-hoc networks Inde Terms Stochastc processes/queueng theory I INTRODUCTION AND MOTIVATION SO far only few eplct results have been known n queueng theory for systems whose evoluton s descrbed by general statonary ergodc processes One lne of research that allows one to handle statonary ergodc sequences s based on dentfyng measures that are nsenstve to correlatons For eample, the probablty of fndng a G/G/ queue non-empty s just the rato between the epected servce tme and the epected nterarrval tme of customers whch follows drectly from Lttle s Law The epected cycle duraton n a pollng system under farly general condtons too, depends on the nterarrval, servce and vacaton tmes only through ther epectatons under general statonary ergodc assumptons see eg An eample of performance measures that depend on the whole dstrbuton of servce tmes but s nsenstve to correlatons s the growth rate of the number of customers or of the sojourn tme n a dscrmnatory processor sharng queue n overload, Other nsenstvty results on bandwdth sharng n a network can be found n 4, 5 The pollng models we study do not ehbt nsenstvty Appromatng correlated vacatons by ndependent ones can result n large errors n the performance, see eg 6 n the contet of Bluetooth To study these systems we make use of stochastc recursve equatons SRE ntroduced n 7 whch Ths work was partly sponsored by the EURONGI network of ecellence etend branchng processes wth mgraton on one hand, and lnear stochastc recursve equatons on the other It has already been shown n 7 that vector valued SRE can be used to descrbe some embedded processes appearng n pollng models In ths paper we dentfy one dmensonal SRE whch we use n order to compute the epected watng tmes and queue lengths n a system wth two queues where a sngle server alternates between two queues and requres swtch-over tmes modeled as vacatons to move from one queue to the other We consder the ehaustve servce dscplne where the server serves a queue untl t emptes before swtchng to the net queue as well as the gated dscplne where only customers present upon the arrval of the server are served Two systems are studed: one n whch both queues are served ehaustvely and one n whch one queue s served ehaustvely and the other accordng to the gated dscplne Our analytcal results are then used to study numercally the mpact of correlated swtchover tmes on the performance, as well as the dfference n performance due to the servce dscplne used The pollng system studed n ths paper, but wthout the correlaton, has been used n the past 0 to model communcaton systems n whch transmsson between two statons can take place only n one drecton at a tme The poston of the server then corresponds to the drecton data s travelng n A smlar stuaton arses n ad-hoc network; there s a common channel whch needs to be shared amongst varous users The more users there are, the longer one has to wat before beng able to capture the channel necessary to retransmt data In partcular, f one has to wat a long tme before beng able to transmt data, then t s very lkely that there are many users around and that the net tme one has to wat once agan for a long perod of tme For ths reason the correlaton of the number of users over tme n an ad-hoc network nherently ntroduces correlaton between the swtchover tmes, and ths n turn leads to an ncrease n the mean delay and queue lengths The remander of ths document s structured as follows In secton II the pollng system s descrbed n more detal, notaton and some formulas are establshed, and the one dmensonal SRE s dentfed Followng that, n secton III a number of performance measures are derved, most notably the Lnear SRE have been used to study the mpact of correlaton of the loss process on TCP throughput 8 SRE have also been used recently to study the nfnte server queue wth correlated arrvals 9

2 IEEE INFOCOM 005 epected watng tmes and the average queue lengths These performance measures are then used n the eamples of secton IV to show the effect of correlated swtchover tmes Fnally, conclusons are gven n secton V To help the reader a lst of the notaton used s gven n Append C II MODEL DESCRIPTION We eamne the pollng of two queues, e one queue s served after whch the other queue s served No lmt s specfed for the length of ether queue After servng queue =, for the n-th tme, the server requres a swtchover tme of duraton V n, Assume all V n, have the same dstrbuton as V n,, where s assumed to form a general dstrbuton wth frst and second moment v and v, and wth varance δ := v v, =, Let R := v +v and := δ +δ The sequences of swtchover tmes are assumed to be statonary ergodc nstead of the usual d, and possbly dependent on each other Ths mples that there can be a correlaton between the swtchover tmes of the two queues and/or wthn the sequence of swtchover tmes for each queue The arrval of customers at queue s Posson wth rate λ and the servce tmes are nonnegatve, d random varables wth fnte frst and second moments for queue gven by, respectvely, b and b The load at queue s ρ := λ b and the system s stable, page 80 f and only f the overall load ρ := ρ + ρ <, whch we assume throughout Furthermore, we wll contnuously assume that the queues are operatng under statonary regme Introduce the covarance functons =, c n =IEV 0, V n, IEV 0, IEV n,, n N, c n =IEV 0, V n, IEV 0, IEV n,, n Z Note that c n s defned for n Z Wth ths conventon t s not necessary to work wth c n := IEV 0, V n, IEV 0, IEV n,, snce under statonary regme c n = IEV 0, V n, IEV 0, IEV n, = IEV n, V 0, IEV n, IEV 0, = c n In partcular, f for each queue the sequence of swtchover tmes s uncorrelated, then c 0 = δ and c n = 0, for n N If there s no correlaton between the swtchover tmes of the two queues, then c n = 0, for n Z Because of the assumpton of the queues operatng under statonary regme IEV 0, V n, = v + c n, =,, n N, a IEV 0, V n, = v v + c n, n Z b In order to establsh the SRE, let D n, N be the duraton of the busy perod n the th queue, ntated by N customers watng n that queue when the server arrves at that queue for the n th tme Smlarly, let N n, T be the number of customers arrvng at queue durng a perod of tme T durng the server s n th vst to queue We start by eamnng the ehaustve pollng of two queues, e one queue s served untl t s empty after whch the other queue s served untl empted Consder the system at the moment the server starts servng the frst queue for the n th tme wth L n, customers watng n the queue From here on the followng steps take place see Fgure for a vsual representaton of ths decomposton: Ehaustng the frst queue The L n, customers n the frst queue requre a busy perod duraton of D n, := D n, L n, to ehaust Swtchng to the second queue After servng the frst queue the server requres a swtchover tme of V n, unts of tme Ehaustng the second queue In the tme needed to swtch from the second to the frst queue V n,, to ehaust the frst queue D n,, and to swtch back to the second queue V n,, there have been L n, := N n, V n, + N n, D n, L n, + V n, customers arrvng at the second queue It requres D n, := D n, L n, unts of tme to empty ths queue Swtchng back to the frst queue After servng the second queue the server requres a swtchover tme of V n, unts of tme Busy perod Swtchover Busy perod Swtchover of duraton D n, tme Vn, of duraton Dn, tme Vn, Fg Decomposton of the n th cycle nto busy perods D n, and D n, and swtchover tmes V n, and V n, After ths the process starts over agan and a new cycle begns Hence the n th cycle s made up of C n =D n, + V n, + D n, + V n, The tme between the server fnshng work at queue and returnng to queue n the net cycle s the ntervst tme I n, and s gven by I n, = V n, + D n, + V n,, a I n, = V n, + D n+, + V n+, b A SRE wll be establshed for ths quantty and we wll see that t plays a central role for the dervaton of the epected watng tmes and queue lengths The tme D n+, spent at queue n the n+ th cycle s related to the ntervst tme accordng to D n+, = D n+, N n, I n,, a D n+, = D n+, Nn, V n, +N n+, Vn+, +D n+, N n, I n, b The epectaton s the sum of the epectaton of ndependent sub-busy perods, p7 and thus IED n, = b IEN n, I n, ρ = ρ IEI n, ρ 4a The star s added to L n, to dstngush t from the average queue length L n,

3 IEEE INFOCOM 005 Usng the statonarty and the dvsblty 4 of the arrval process t can be shown that IED n, = ρ IEI n, ρ 4b Snce the busy perods are sums of servce tmes, the dvsblty property also holds for D n, Ths means that from substtute b nto a I n+, =V n+, + V n+, + D n+, N n, V n, + N n+, V n+, + D n+, N n, I n,, n N, we see a SRE as presented and solved for statonary ergodc sequences n 7 arsng Although the system s two dmensonal as there are two queues, the reducton to a one dmensonal SRE s a key element n obtanng eplct formulas for the performance measures Theorem : SRE for ehaustve/ehaustve system The ntervst tme of the frst queue allows tself to be wrtten as a one-dmensonal SRE, wth I n+, =A n I n, + B n 5 A n := D n+, N n+, D n+, N n,, B n := V n+, + V n+, + D n+, N n, V n, + N n+, V n+, 6 Note that from 4a and 4b we have IEA n I n, = αiei n, where α := ρ ρ ρ ρ Now let the frst queue be served ehaustvely and the second be served n a gated manner The tme needed to serve N customers n the second queue n the n th cycle s denoted by S n, N Naturally, IES n, N = b IEN The servce tme of the second queue, S n,, satsfes the followng recursve relatonshp S n+, =S n+, Nn, Sn, + V n, + N n+, Dn+, + V n+, At the same tme, the tme the server works per cycle at the frst queue, D n,, satsfes D n+, =D n+, N n, Vn, + S n, + V n, By combnng these two epresson we obtan the followng theorems Theorem : SRE for the ehaustve/gated system The SRE for the servce tme at the gated queue s gven by S n+, =X n S n, + Y n, 7 4 The dvsblty property mples that N n, a+b = N n, a+n n, b, where N n, and N n, are stochastc processes ndependent of each other and each wth the same dstrbuton as N n, wth X n : = S n+, N n, +Nn+, Dn+, Nn,, Y n := S n+, N n, Vn, + Nn+, Dn+, Nn, V n, + V n, + V n+, 8 Note that IEX n S n, = γies n, where γ := ρ ρ Remark: In systems wth both queues served n a gated manner or wth more than two queues the SRE can not be wrtten n a one-dmensonal verson Although stll solvable, the analyss s more nvolved and wll be addressed n the future Before presentng a number of performance measures, a number of formulas need to be establshed Frst recall that f D s a random sequence wth IED = d and IED = d, ndependent of a random varable N, and then IEτ N = Smlarly, τn := n nie D n= = N D, 9 = n D N = n P N =n = =d IEN + d d IEN 0 IEN n,t = 0 IEN n,t T = tdt t =λ IET + λ IET Net we proceed n a smlar manner to obtan the second moment of the busy perod generated by Y customers ntally n the system Frst recall that D n, s a sngle busy perod ntated by a sngle customer n an M/G/ queue wth Posson arrvals wth rate λ and general servce tme wth frst and second moments b and b respectvely The frst two moments of a sngle busy perod ntated by a sngle customer are gven by, equatons 54 and 54 d := IED n, = b ρ, a := IEDn, = b, =, b ρ d The busy perod, D n, Y, generated by Y customers s the sum of Y ndependent sngle busy perods, each wth dstrbuton D n,,k D n, Hence the second moment s Y Y IEDn,Y =IE D n,,k = IE D n, k= =d IEY + By takng Y = N n, T we obtan =, k= d d IEY IED n,n n, T =λ d IET + d λ IET 4

4 IEEE INFOCOM III PERFORMANCE MEASURES Startng wth a system n whch both of the queues are served ehaustvely, an eplct epresson wll be gven for the frst two moments of the ntervst tme n the presence of correlated swtchover tmes Based on ths a number of performance measures follow, n partcular, the epected watng tmes and the average queue lengths After ths the epected watng tme and average queue length for the ehaustve/gated servce system wll be gven A Ehaustve/Ehaustve Servce Dscplne Central to the dervaton of a number of performance measures s Theorem Ths allows the dervaton of the followng theorem of whch the proof s gven n Append A Because of the symmetry for ths servce dscplne the results wll be presented for only the frst queue Theorem : Intervst tme n ehaustve/ehaustve system Under the statonary regme the epected ntervst tme of the frst queue s gven by EI n, = R ρ, ρ := ρ + ρ, 5 ρ The second moment s gven by βiein, = R λ ρ b ρ ρ + λ b + where and α := ρ ρ ρ K := δ + ρ + ρ ρ ρ + δ + K R 6 ρ ρ ρ ρ, β := ρ ρ + ρ ρ ρ j= c j+c j+ ρ ρ ρ c j 7 + ρ ρ c j+ ρ α c j α s the addton to the ntervst tme due to the correlaton between the swtchover tmes On the bass of ths theorem the frst two moments of a number of performance measures quckly follow Number of Customers Watng The ehaustve nature of the server mples that the number of customers buldng up at the frst queue s eactly the number of customers that arrved at that queue durng ts ntervst tme Thus, L n+, = N n, I n, From ths we mmedately obtan IEL n+, =λ IEI n, = Rλ ρ ρ α j as the epected length of the queue, under statonary regme, at the moment the server arrves at the frst queue The second moment follows through squarng, IEL n+, =IEN n,i n, = λ IEI n, + λ IEI n,, whch leads to IEL n, :=λ IEIn, + Rλ ρ 8 ρ Duraton of Busy Perods The epected tme per cycle, n steady state, for the server to work on the frst queue s gven by IED n, = Rρ ρ Ths follows drectly from IED n, = IED n, L n, = b IEL n, ρ Snce IEDn, = IEDn,N n, I n,, the second moment follows wth 4 and s gven by IEDn, = ρ IEIn, ρ + Rλ b ρ ρ Number Served per Cycle To derve the frst and second moments of the number of customers served per cycle, consder an M/G/ queue wth arrval rate λ, average servce tme b, and the second moment of the servce tme b Then the epectaton and the varance of the number of customers served n a sngle busy perod are known to be, equatons 55 and 554 IEΓ =, VarΓ = ρ ρ + λ b ρ ρ Let T n, N be the number of customers served at the frst queue durng the n th cycle f there are N customers n the queue at the moment of pollng Note that T n, N s the number of customers served startng wth N customers, whereas Γ s the number of customers served startng wth just one customer A dfferent notaton s used for these two quanttes to reflect the dependency on the n-th cycle and the queue number Snce the number served s the sum of the number served durng N busy perods, 9 tells us that the epected number of customers served, per cycle, at the frst queue s IET n, =IET n, L n, = IEL n, ρ = λ R ρ 9 To derve the second moment note that IETn, =IETn,L n, =IEΓ IEL n, + VarΓ n, IEL n, = IEL n, ρ + Rλ ρ ρ + λ b ρ ρ Usng equaton 8 gves IETn, = ρ λ IEIn,+ Rλ ρ +λ b ρ

5 IEEE INFOCOM Theorem 4: Epected watng tme and queue length for the ehaustve/ehaustve servce dscplne The epected watng tme total tme n system mnus servce tme of a customer gong through the frst queue s decomposed of two parts, namely Ths gves IEW q, = λ b ρ + ρiei n, 0 R ρ IEW q, = λ b + ρ R ρ + ρ + λ ρ b +λ ρ b ρ ρ ρ+ρ ρ ρ ρ δ +K ψ, ρ ρ where ψ := R ρ+ρ ρ and K defned n 7 s the ncrease n the epected watng tme due to correlated swtchover tmes The average number of customers at the frst queue n servce and n the queue follows drectly from Lttle and s IEL s, =λ IEW q, + ρ The proof can be found n Append B In the uncorrelated case K = 0 ths s n correspondence wth, formula 5 B Ehaustve/Gated Servce Dscplne Now let the second queue be served wth a gated dscplne nstead of an ehaustve one We obtan the followng theorem Theorem 5: Epected watng tme and queue length for the ehaustve/gated servce dscplne The epected tme a customer wats n queue =, untl beng served s gven by where K and K are the ncreases n the epected watng tme due to correlated swtchover tmes The average queue lengths, IEL s, =λ IEW q, + ρ, =, follow mmedately because of Lttle If all of the swtchover tmes are ndependent of each other, then K = 0 = K and we obtan the results gven n, formulas 5 and 8 or, formula 6 4 Proof: The proof, although slghtly more nvolved, runs along the same lnes as the proofs of Theorems and 4 and s omtted due to space constrants The reader s referred to 4 for the full proof Although not shown here, t can be verfed that for any choce of parameters K K From we can see that f δ s suffcently large, f ρ + ρ > 0, and f K = 0 = K, then t may very well be possble that the 5 The last term n, formula s mssng a factor two and there s a m up between δ and δ The epresson should read + ρ ρ ρδ + ρ δ R ρ+ρ ρ 6 The formula presented n ths reference s coped ncorrectly from The frst term for the watng tme for customers arrvng at the second queue should contan + ρ nstead of + ρ epected watng tme at the gated queue s smaller than the epected watng tme at the ehaustve queue! However, the range of parameter settngs for whch ths s the case s farly small In partcular, t can be shown 4 that ths does not happen when the parameters for both queues are equal, f the swtchover tmes or equal to zero, or f the system s heavly loaded IV EXAMPLES In the followng paragraphs a number of eamples wll be consdered n whch the sequences of swtchover tmes are correlated The covarance functons wll be calculated eplctly after whch the mpact of the correlaton on the watng tmes wll be studed In all of the eamples the epected watng tme of a customer arrvng at the frst queue of an ehaustve/ehaustve served system s gven by whereas n the ehaustve/gated served system the epected watng tmes are gven by The dfference between each of the eamples s that K, K, and K take on dfferent values The frst eample studes a sngle server queue wth correlated vacatons by turnng off one of the queues, whereas n the subsequent eamples eplct epressons are derved for the epected watng tmes and these are compared to the epected watng tmes f there would be no correlaton A Sngle Server Queue wth Correlated Vacatons By turnng off one of the queues one obtans an M/G/ queue wth multple correlated vacatons Let us start by turnng off the second queue by settng λ = 0, ρ = 0, v = 0, and v = 0, whch leads to c j = 0, c j = 0, and γ = 0 to end up wth an ehaustvely served M/G/ queue where the epected watng tme IEW q, = λ b ρ + v v Ehaustve M/G/ s ndependent of the correlaton between the vacatons! Ths result was prevously ponted out n 7, paragraph 6 whch causes t to correspond to the epresson for the epected watng tme but wth d vacaton tmes 5, page On the other hand, by turnng off the frst queue n the ehaustve/gated system, we are left wth an M/G/ queue wth a gated servce dscplne After settng the approprate parameters to zero we obtan IEW q, = λ b ρ + v + ρ v + v ρ v c jρ j j= Gated M/G/ as the watng tme of a customer arrvng at a gated M/G/ queue wth correlated vacatons If there s no correlaton then ths epresson s n agreement wth the result prevously obtaned n 7, Theorem 5 and 5, equaton 54a It s nterestng to compare the dfference between these two watng tmes due to the server behavng dfferently Assumng queues wth dentcal parameters by droppng the ndces of

6 IEEE INFOCOM where γ := IEW q, = ρ R ρ + ρ ρ + ρ δ + R ρ + ρ IEW q, = + ρ R ρ ρ ρ + ρ δ + R ρ + ρ ρ ρ and K := K := j= j= ρ ρ + ρ + ρ ρ + ρ λ b + λ b ρ + + K R λ b + λ b + + K ρ R a b c j+ρc j + + ρ ρ c j+ρc j γ j + ρ ρ ρ ρ ρ c 0 a c j + ρc j + ρ ρ c j + ρc j γ j + ρc 0 ρ b ones and twos and settng ˆρ := ρ = ρ we see that IEW gated M/G/ IEW ehaustve M/G/ = ˆρv ˆρ + cjˆρ j, 4 v j= where the frst term on the rght hand sde s the mean length of a servce perod whch s the same for the ehaustve and the gated servce systems If there s no correlaton, then t s well known that the epected watng tme n an ehaustvely served queue s less than that n a gated servced queue In the presence of correlated vacaton tmes ths dfference s larger but remans a surprsngly smple epresson B Correlated Swtchover Tmes Consder a sequence of swtchover tmes where there s no correlaton between the swtchover tmes of the two queues ths gves c j = 0, for j Z Let the ndvdual sequence of swtchover tmes per queue satsfy V n+, = V n, + ε n,, =,, 5 where 0, s a constant and ε n, are postve d varables wth fnte epectaton IEε n, =: ε and second moment IEε n, =: ε The parameter determnes the amount of correlaton n the sequence; wth = 0 the sequence s d, whereas when tends to one the correlaton s mamal Notce that there ests a statonary ergodc sequence of swtchover tmes whch satsfes 5 By takng the epectaton t follows that IEV n+, = IEV n, + n ε Due to the statonarty of the process IEV 0, = IEV n, = v s ndependent of, and therefore v = ε A smlar relatonshp can be derved for the second moments by takng the epectaton over the square of 5 to gve IEV n+, = IEV n, + IEε n, + IEε n, IEV n, Due to the statonarty IEVn+, = IEV n, = v ths mples that = ε + ε v, + v whch gves a second relatonshp snce ε = v, δ = + V arε n, Thus we see that for 0, there ests a ε n, such that any desred values of v and δ can be obtaned Now we wll derve the covarance functons and the epected watng tme By teratng 5 a number of tmes t s quckly seen that n V n, = n V 0, + ε n k, k 6 From ths we obtan k=0 IEV 0, V j, = j IEV 0, + j ε IEV 0, = j v + j ε v Ths means that the covarance functons, c j = IEV 0, V j, IEV 0, IEV j,, are gven by c j = j v + j ε v v j v + j ε = j v v = j δ 7 Snce c jα j =δ j= α j = α δ, 8 α j= we have from Theorem 4 that the epected watng tme n the ehaustve/ehaustve system s gven by wth K := α δ + α δ ρ + α α ρ ρ ρ ρ ρ ρ and α = Equvalently, n the ehaustve/gated system we have from equaton 8 and from Theorem 5 that the epected watng tmes are gven by where γ = ρ ρ and K = γ δ + ρ ρ ρ + γ δ γ ρ γ K = γ δ + ρ ρ + γ δ γ ρ γ Numercal eamples of the nfluence of the correlaton on the epected watng tmes can be found n Fgure Shown

7 IEEE INFOCOM n each of the fgures s the epected watng tme dvded by the epected watng tme for uncorrelated sequences of swtchover tmes There are Posson arrvals wth λ = 04 The frst two moments of the swtchover tmes are always kept fed frst moment for each of the swtchover tme dstrbutons s fed at v = and the servce tmes are taken to be eponental wth b = 04 or b = Based on the fgures and equatons above the followng mportant conclusons can be made: If = = 0 then there s no correlaton between the sequences of swtchover tmes and K = 0, K = 0, and K = 0; The ncrease n epected watng tmes due to correlated swtchover tmes can be up to several tmes 5 tmes n the eample the epected watng tmes f there would be uncorrelated swtchover tmes The ncrease n the epected watng tmes due to correlaton grows lnearly wth the varance δ of the swtchover tmes; Under lght traffc α and γ are small and so the ncrease n the epected watng tme s appromately lnear n Under heavy traffc α and γ are close to one and, due to the factor α or γ n the denomnators, the ncrease n watng tme due to correlated swtchover tmes can be sgnfcant Hence the presence of correlaton has the bggest mpact on the watng tme f the system has a heavy load and the swtchng tmes have a hgh varance Ths can be seen clearly n Fgure In can be shown that, under dentcal parameter settng, n the ehaustve/gated system the epected watng tme at the ehaustve queue s always larger than at the gated queue In addton to ths, we see from Fgure that n lghtly loaded systems the gated queue Q suffers most from correlated swtchover tmes whereas n heavly loaded traffc both queues are effected relatvely equally by the correlated swtchover tmes C Identcal Swtchover Tmes Set V n, = V n, Ths ntroduces cross-correlaton between the two sequences of swtchover tmes and t gves v = v and δ = δ In addton to ths let V n+, = V n, + ε n, just as n the prevous eample From 7 we have c j = j δ after whch c j =IEV 0, V n, IEV 0, IEV n, =IEV 0, V n, IEV 0, IEV n, = c j = j δ c j =IEV 0, V n, IEV 0, IEV n, =IEV 0, V n, IEV 0, IEV n, = c j = j δ c j =IEV 0, V n, IEV 0, IEV n, =IEV 0, V n, IEV 0, IEV n, = c j = j δ mmedately follow Ths means that =, c jα j = c jα j = c jα j = αδ α j= j= j= can all be plugged nto Theorem 4 so that the epected watng tme n the ehaustve/ehaustve system s gven by wth K = αδ ρ α ρ + + α α ρ ρ + ρ, ρ ψ := ρ v ρ+ρ ρ, and α = ρ ρ ρ ρ Equvalently, the epected watng tmes n the ehaustve/gated system are gven by wth K = γδ + ρ ρ ρ+ ρ ρ ρ γ ρ ρ δ, K = γδ + ρ ρ ρ + ρδ γ ρ To get a feelng of the mpact of the cross correlaton, the epected watng tmes are plotted n Fgure for varous swtchover tme dstrbutons and traffc loads Shown n each of the fgures s the epected watng tme dvded by the epected watng tme for uncorrelated sequences of swtchover tmes There are Posson arrvals wth λ = 04 The frst two moments of the swtchover tmes are always kept fed frst moment for each of the swtchover tme dstrbutons s fed at v = and the servce tmes are taken to be eponental wth b = 04 or b = Strkng s the mpact of the cross correlaton on the watng tmes For eample, f there s no correlaton wthn each sequence of swtchover tmes = 0, then there s stll an ncrease n the epected watng tme due to the crosscorrelaton For the ehaustve/ehaustve system ths ncrease s ρ αψδ and for the ehaustve/gated system ths ncrease s gven by ρ ρ ρ ρ δ and ρδ for, respectvely, the ehaustve and the gated queue For eponentally dstrbuted swtchover tmes ths can mean an ncrease of tens of percents n the epected watng tme Besdes ths, all of the conclusons made n the frst eample also hold here, wth the ecepton that the ncrease n epected watng tme can up to a factor 5 D Stochastc Recursve Swtchover Tmes Consder a sequence of swtchover tmes whch satsfy the followng stochastc recursve relatonshp V n+, =F n, V n, + E n,, 9 where F n, are ndependent, nfntely dvsble stochastc processes wth IEF n, T = IET and IEFn, T = IET + y IET Here 0,, y 0 and The sequence E n, s a sequence of ndependent varables wth IEE n, = ε and IEEn, = ε n V n, = F k, V 0, + and so k=0 k=0 Iteratng gves n n k=0 l=k+ F l, E n k,, n IEV n, = n v + ε k = n v + n ε

8 IEEE INFOCOM EW / EW no correlaton 4 Normalzed Epected Watng Tme at Q n Eh/Eh LogNorm0,ln ep/ Posson Gamma5,/5 Lght traffc Normalzed Epected Watng Tme at Q n Eh/Gated 4 V LogNorm0,ln ep/ Lght Posson traffc V Gamma5,/5 EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated 4 V LogNorm0,ln ep/ Lght Posson traffc V Gamma5,/5 EW / EW no correlaton EW / EW no correlaton Normalzed Epected Watng Tme at Q LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc Fg Eample: Correlated Swtchover Tmes The epected watng tme dvded by the epected watng tme wth uncorrelated swtchover tmes The dfferent lnes correspond to dfferent swtchover tme dstrbutons, all wth mean v = Here := = determnes the level of correlaton, there s no cross correlaton, the servce tmes are eponental, and λ = 04 The top fgures are wth mean servce tmes b = 04 ρ = 0 whereas the second row of fgures are under heavy traffc wth b = ρ = 096 The frst column of fgures correspond to the ehaustve/ehaustve system, whereas the second and thrd column of fgures show the normalzed watng tmes for, respectvely, the ehaustve queue Q and the gated queue Q n the ehaustve/gated system EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Eh LogNorm0,ln ep/ Posson Gamma5,/5 Lght traffc Normalzed Epected Watng Tme at Q n Eh/Gated 4 V LogNorm0,ln V ep/ Lght V Posson traffc V Gamma5,/5 EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated 4 V LogNorm0,ln V ep/ Lght V Posson traffc V Gamma5,/5 EW / EW no correlaton EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Eh LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc EW / EW no correlaton Normalzed Epected Watng Tme at Q n Eh/Gated LogNorm0,ln ep/ Posson Gamma5,/5 Heavy traffc Fg Eample: Identcal Swtchover Tmes The epected watng tme dvded by the epected watng tme f there would be no correlaton between the swtchover tmes The dfferent lnes correspond to dfferent swtchover tme dstrbutons Here determnes the level of correlaton, cross correlaton s ntroduced by settng V n, = V n,, the servce tmes are eponental, and λ = 04 The fgures n the frst row are wth mean servce tme b = 04 ρ = 0 and the fgures on the bottom row are under heavy traffc wth b = ρ = 096 The frst column of fgures correspond to the ehaustve/ehaustve system, whereas the second and thrd column of fgures show the normalzed watng tmes for, respectvely, the ehaustve queue Q and the gated queue Q n the ehaustve/gated system

9 IEEE INFOCOM Ths leads to the condton v = ε The second moment of the swtchover tmes s found by takng the epectaton over the square of 9 Dong ths produces IEV n+, =IEF n,v n, +IEE n,+ief n, V n, E n, = v + y v + ε + v ε Snce, by defnton, IEV n+, v = ε also equals v, we have + y + ε v Thus we see that there ests a E n, whch generates a sequence V n, wth arbtrary correlaton and any desred values of v and Smlarly, t s easy to show that v IEV 0, V n, = n v Ths gves the covarance functons c j =IEV 0, V j, IEV 0, IEV j, + n ε v = j v + j ε v v j v + j ε = j δ whch are completely dentcal to 7! Ths means that f the frst two moments of the swtchover tmes are kept fed whle varyng 0,, that the epected watng tmes are once agan gven by and wth K, K, and K as gven n the frst eample Furthermore, the conclusons of the frst eample also hold here As a specal case of 9 we can take V n+, = V n + ε n, where 0, s a constant and ε n, s a postve sequence of d varables V CONCLUSIONS We have studed the performance of alternatng-prorty queues wth very weak assumptons on the swtchover tme sequences; all we assume s that these sequences are statonary ergodc In spte of ths generalty we were able to derve eplct epressons for the epected watng tmes and number of customers n each queue The epressons obtaned nvolve the weghted sum of all correlatons where the weghts decrease eponentally fast to zero Wth the help of our eplct epressons, we studed numercally the role of correlaton and gave eamples where they add up to 400% to the epected watng tmes Ths has mportant mplcatons for ad-hoc networks where a common communcaton channel s shared amongst a number of users and the number of users between consecutve data transfers are correlated REFERENCES Etan Altman, Panagots Konstantopoulos, and Zhen Lu, Stablty, monotcty and nvarant quanttes n general pollng systems, Queung Systems, vol, pp 5 57, 99, Specal ssue on Pollng Systems Etan Altman, Tanja Jmenez, and Danel Kofman, DPS queues wth statonary ergodc servce tmes and the performance of TCP n overload, n Proceedngs of IEEE Infocom, Hong-Kong, March 004 Alan Jean-Mare and Phlppe Robert, On the transent behavour of the processor sharng queue, QUESTA, vol 7, no -, pp 9 6, Thomas Bonald, A Proutère, G Régné, and JW Roberts, Insenstvty results n statstcal bandwdth sharng, n ITC, Salvador, Brazl, 00, number 7 5 Thomas Bonald, M Jonckheere, and A Proutère, Insenstve load balancng, n ACM Sgmetrcs Performance Evaluaton Revew, June 004, vol, pp Gl Zussman, Ur Yechal, and Adran Segall, Eact probablstc analyss of the lmted schedulng algorthm for symmetrcal bluetooth pconets, n Personal Wreless Communcatons PWC, Vence, Italy, Sept Etan Altman, Stochastc recursve equatons wth applcatons to queues wth dependent vacatons, Annals of Operatons Research, vol, no, pp 4 6, 00 8 Eten Altman, Chad Barakat, and Konstantn Avratchenkov, A stochastc model of TCP/IP wth statonary ergodc random losses, n ACM- Sgcomm, Aug 8 - Sept 000, See also INRIA Research Report RR-84 9 Etan Altman, On stochastc recursve equatons and nfnte server queues, n Proc of IEEE Infocom 005, Mam, FL, March Jack S Sykes, Smplfed analyss of an alternatng-prorty queueng model wth setup tmes, Operatons Research, vol 8, no 6, pp 8 9, November-December 970 Hdeak Takag, Queueng analyss of pollng models: An update, Stochastc Analyss of Computer and Communcaton Systems, pp 67 8, 990 Leonard Klenrock, Queueng Systems, Volume I: Theory, vol I, John Wley and Sons, 976 T Ozawa, An analyss for mult-queueng systems wth cyclc-servce dscplne - models wth ehaustve and gated servces, Tech Rep 4, The Insttute of Electroncs, Informaton and Communcaton Engneers IEICE n Japanese, Robn Groenevelt and Etan Altman, Analyss of alternatng-prorty queueng models wth cross correlated swtchover tmes, Tech Rep RR-568, INRIA, Sopha-Antpols, November Hdeak Takag, Queueng Analyss, Vacatons and Prorty Systems, Part, vol, Elsever Scence Publshers BV, The Netherlands, 99 6 Davd M Lucanton, New results on the sngel server queue wth a batch markovan arrval process, Commun Statst - Stoch Models, vol 7, pp 46, 99 7 Davd M Lucanton, Kathleen S Meer-Hellstern, and Marcel F Neuts, A sngle server queue wth server vacatons and a class of non-renewal arrval processes, Adv Appl Prob, vol, pp , Martn Esenberg, Queues wth perodc servce and changeover tme, Operatons Research, vol 0, no, pp , March-Aprl 97 9 JDCLttle, A proof for the queueng formula: L = λw, Operatons Research, vol 9, pp 8 87, May-June 96 0 F Baccell and P Brémaud, Elements of Queueng Theory, Sprnger- Verlag, 994 Robert B Cooper, Shun-Chen Nu, and Mandyam M Srnvasan, A decomposton theorem for pollng models: The swtchover tmes are effectvely addtve, Operatons Researchs, vol 44, no 4, pp 69 6, Jul-Aug 996 S W Fuhrmann and Robert B Cooper, Stochastc decompostons n the M/G/ queue wth generalzed vacatons, Operatons Research, vol, no 5, pp 7 9, 985 Hdeak Takag, Analyss of Pollng Systems, MIT Press, Cambrdge, Massachusetts, 986 A Proof of Theorem : APPENDIX Takng the epectaton on both sdes of equaton 5 gves IEI n+, =R + ρ R + ρ IEI n, ρ ρ Under the statonary regme IEI n+, = IEI n, whch mmedately leads to the frst moment EI n, = R ρ 0 ρ

10 IEEE INFOCOM To obtan the second moment we take the epectaton of the square of 5 to gve IEI n+, =IEA n I n, + IEB n + IEA n I n, B n Ths epresson s central to proof n ths secton and each of the terms wll be derved pece by pece Frst of all, IEA ni n, = =IE D n+, N n+, D n+, N n, I n, =λ d IED n+,n n, I n, + λ d IED n+,n n, I n, =λ d λ d IEIn, + λ d IEI n, + λ d λ d IEI n, Pluggng equaton and 5 nto ths results n IEA ρ ρ IEIn, ni n, = ρ ρ + R ρ ρ λ ρ b ρ + λ ρ b Net we proceed wth the second unknown of epresson, IEBn, where we recall that B n s defned n equaton 6 We have IEBn = IE V n+, +V n+, +D n+, N n, V n, +N n+, V n+, =v + v + v v + c 0 + λ d v +v +v v +c +c +c 0 + λ d v + v + v v + c + λ d R = + R ρ ρ δ + Rλ b ρ ρ + ρ c + c 0 ρ + ρ c ρ To solve the last part frst notce that the processes N n,, N n,, N n+,, N n+,, D n+,, and D n+, are all ndependent of each other, and each of them s ndependent of I n,, V n,, V n+,, and V n+, Ths means that IEA n I n, B n =αiei n, B n, 4 ρ ρ wth once agan α := ρ ρ The last pece of the puzzle can be derved wth the help of Theorem n 7 whch states that n I n, = A n j B n j, n Z, j=0 =n j where for each nteger, {A j } j are ndependent of each other and have the same dstrbuton as A To apply the theorem t s suffcent to have α <, whch turns out to be equvalent to ρ <, and that IEB n < see Lemma n 7 The latter ndeed holds as IEB n = R ρ Applyng the theorem gves IEI n, B n = n = IE D +, N +, D+, N, B n j B n j=0 j=0 =n j = α j IE B n j B n = α j IE B 0 B j+ 5 j=0 because of the ndependence of the processes D,, D n,, N,, and N,, for all Z Wrtng out the last term yelds IE B 0 B j+ = =IE V, + V, + D, V j+, + V j+, N 0, V 0, + N, V, + D j+, N j+, V j+, + N j+, V j+, = R ρ + c j+ + c j+ ρ 6 + ρ c j c j+ + c j+ ρ + c j + c j+ ρ + ρ c j + c j ρ Puttng 4-6 together and re-ndeng the summaton for eample, j=0 αj+ c j = j= αj c j = αc 0 + j= αj αc j produces IEA n I n, B n = = j= R + c j + c j ρ α + α c j ρ + ρ + c j ρ ρ + α ρ + c j + αρ ρ ρ + ρ αc 0 c ρ + ρ c + αc 0 ρ All of the terms wth v and v can be pulled out of the summaton and under statonary regme c 0 = δ Ths gves IEA n I n, B n = α j ρ ρ R ρ ρ + ρ ρ δ ρ ρ ρ c + ρ c + αc 0 ρ ρ + ρ α ρ j= + c j + ρ c j ρ ρ c jα j + ρ + ρ c j ρ j= c j α j 7 Puttng equatons,, and 7 nto and collectng

11 IEEE INFOCOM 005 terms gves IEIn+, ρ ρ IEIn, = ρ ρ R λ ρ + b ρ ρ ρ + λ b + ρ ρ ρδ ρ+ρ ρ + R ρ ρ + ρ α ρ c jα j + ρ j=0 c j j= + c j + ρ c j + ρ c j α j ρ ρ ρ Under statonary regme IEIn+, = IEIn, The Theorem follows by puttng these terms on the same sde and by makng use of the dentty ρ ρ = ρ ρ + ρ ρ ρ ρ ρ ρ B Proof of Theorem 4: For a customer arrvng at the frst queue the system behaves as an M/G/ queue where the server goes on vacaton as soon as the queue s empty The random varable for the n th vacaton from the frst queue s eactly I n, Condtonng the watng tme n the queue on whether or not a customer arrves when the server s busy or on vacaton produces IEW q, = IEI n, IEC n, IEW q, vac + IED n, IEC n, IEW q, busy 8 A tagged customer that arrves durng a vacaton has to wat for the vacaton to fnsh plus the tme needed to serve the customers that arrved before hm/her n the vacaton The epected remanng vacaton tme s IEI n,/iei n, and the epected number of customer that arrved before the tagged customer s λ IEI n,/iei n, Ths means that IEW q, vac = IEI n, IEI n, + λ b 9 A tagged customer that arrves when the server s busy has to wat for the current customer n servce to fnsh plus the epected tme needed to serve the L q, customers that arrved at and stll are n the queue before the tagged customer dd Ths gves IEW q, busy = b + b IEL q, busy 40 b To obtan the number of customers n the queue, frst realze that the epected watng tme of a customer n the system s IEW s, = b + IEW q, Lttle 9 tells us that the epected number of customers, L s,, at the frst queue n servce and n the queue s IEL s, := λ IEW s, = ρ + λ IEW q, On the other hand, IEL s, = IED n, IEC n, IEL s, busy + IEI n, IEC n, IEL s, vac =ρ + IEL q, busy + ρ λ IEI n, IEI n, Combnng these last two equatons gves IEL q, busy = b IEW q, ρ IEIn, IEI n, Puttng together equatons 8-4 gves IEW q, = ρ IEI n, IEI n, + λ b 4 b + ρ + IEW q, ρ IEIn, b IEI n, = λ b ρ + IEI n, IEI n, 4 The theorem follows by pluggng n the values of IEI n, and IEI n, Notce that not once have we assumed the swtchover tmes and the busy perods to be uncorrelated! C Lst of Notatons A n = A nested combnaton of stochastc processes defned n 6 B n = A nested combnaton of stochastc processes defned n 6 D n, N = Total busy perod generated by N customers n queue wth arrval rate λ and frst and second moment of the servce tme b and b, respectvely D n,,k = Sngle busy perod generated by the k-th customer n queue wth arrval rate λ and frst and second moment of the servce tme b and b, respectvely S n, N = Servce tme of N customers at queue wth frst and second moment of the servce tme b and b, respectvely N n, T = Number of arrvals at queue n tme T n the n th cycle cycle startng from the pollng nstant of the frst queue T n, N = The number of customers served at queue durng the n th cycle f there are N customers n the queue at the moment of pollng X n = A nested combnaton of stochastc processes defned n 8 Y n = A nested combnaton of stochastc processes defned n 8

12 IEEE INFOCOM 005 C n, = Duraton of the n th cycle startng from the pollng nstant of the th queue D n, = Duraton of the busy perod at queue n the n th cycle I n, = Intervst of the th queue n the n th cycle Ths s the tme between the server swtchng away from queue untl the tme that the server comes back to queue I n, = V n, + D n, + V n, for ehaustve/ehaustve queues and I n, = V n, + S n, + V n, for ehaustve/gated queues K = Some constant used n the epressons for the ntervst tme and the watng tmes Sometmes K or K s used f there s a dfference between the two queues L n, = Number of customers n queue n the nth cycle at the moment the queue s polled L q, = Average number of customers n queue Ths s also the number of customers that arrved at queue durng a vacaton, and are stll n the queue, before a tag customer arrved n that same vacaton L s, = Average number of customers at queue ncludng the customer n servce Ths s also the number of customers that arrved at queue durng a vacaton and are n the queue or n servce before a tag customer arrved n that same vacaton S n, = Servce tme at queue n the n th cycle =smlar to the duraton D n, of the busy perod but then wth no arrvals T n, = The number of customers served, per cycle, at queue V n, = Swtchng tme from queue to the other queue n the n th cycle W q, = Random varable for the watng tme of a customer n queue not ncludng servce W s, = Random varable for the total sojourn tme of a customer at queue watng tme plus servce tme Γ = The number of customers served durng a busy perod, where the arrval rate s λ, average servce tme s b, and the second moment of the servce tme s b ρ ρ ρ ρ α = = Central quantty n the ehaustve/ehaustve queueng system Comes forth from IEA n I = αiei, wth A n defned n 6 b = IEB = Epected servce tme at queue b = IEB = Second moment of the servce tme at queue c n = IEV 0, V n, IEV 0, IEV n, = Covarance functon for the vacaton sequences at queue c n = IEV 0, V n, IEV 0, IEV n, = Covarance functon for the vacaton sequences between the two queues d = ρ / ρ = The epected duraton of a sngle busy perod, where the arrval rate s λ and the average servce tme s b d = b / ρ = Second moment of the duraton δ of a sngle busy perod, where the arrval rate s λ, the average servce tme s b, and the second moment of the servce tme s b = v v = Varance of the swtchng tme from queue to the other queue γ = ρ ρ = Central quantty n the ehaustve/gated queueng system Comes forth from IEX n I = γiei, wth X n defned n 8 λ = Posson arrval rate at queue ρ = λ d = load at queue ρ = ρ + ρ = load of the system The assumpton s made throughout that ρ < / ˆρ = ρ = ρ f the parameter settngs for the two queues are equal R = v + v v = EV n, = Epected swtchng tme from queue to the other queue v = EV n, = Epected second moment of the swtchng tme from queue to the other queue