The Erlag Model wth o-posso Call Arrvals Thomas Boald Frace Telecom R&D 38-40 rue du geeral Leclerc 92794 Issy-les-Mouleaux, Frace thomasboald@fracetelecomcom ABSTRACT The Erlag formula s kow to be sestve to the holdg tme dstrbuto beyod the mea Whle calls are geerally assumed to arrve as a Posso process, we prove that t s fact suffcet that users geerate sessos accordg to a Posso process, each sesso beg composed of a radom, fte umber of calls ad dle perods A key role s played by the retral behavor case of call blockg We llustrate the results by a umber of examples Categores ad Subject Descrptors G3 [Probablty ad statstcs]: Queueg theory Geeral Terms Performace Keywords Erlag formula, loss etworks, sestvty INTRODUCTION Sce ts publcato 97, the Erlag formula has proved very useful for dmesog telephoe etworks [8] It determes the requred umber of telephoe les gve a predcto of expected demad ad a target blockg probablty A key property of the Erlag formula s ts sestvty: the blockg probablty does ot deped o the holdg tme dstrbuto beyod the mea [8] Traffc s fact characterzed by a uque parameter, the traffc testy, whch s defed as the product of the call arrval rate ad the mea holdg tme Ths makes the Erlag formula both smple to apply ad robust to chages fe traffc characterstcs, ad explas ts edurg success The oly assumpto requred by the Erlag model s that calls arrve as a Posso process Ths property aturally Also afflated wth École Normale Supéreure, Frace Permsso to make dgtal or hard copes of all or part of ths work for persoal or classroom use s grated wthout fee provded that copes are ot made or dstrbuted for proft or commercal advatage ad that copes bear ths otce ad the full ctato o the frst page To copy otherwse, to republsh, to post o servers or to redstrbute to lsts, requres pror specfc permsso ad/or a fee SIGMetrcs/Performace 06, Jue 26 30, 2006, Sat Malo, Frace Copyrght 2006 ACM -59593-320-4/06/0006 $500 arses whe calls are geerated depedetly by a large umber of users Bascally, f each user geerates at most oe call a perod of tme where the system s steady state (eg, the busy hour), the arrval tme of each call s uformly dstrbuted over that perod so that call arrvals deed form a Posso process It s frequet, however, that users geerate a sequece of calls stead of a sgle, solated call The call arrval process s the ot a Posso process Defg a sesso as the sequece of calls geerated by the same user, t oly holds that the sesso arrval process s a Posso process We prove ths paper that the Erlag formula stll holds ths case, uder some mld assumptos o the retral behavor case of call blockg Specfcally, we cosder two types of retral behavor case of call blockg: ether the sesso goes o as f the call were accepted ad termated stataeously, or the call s reattempted wth a fxed probablty r after a dle perod of radom durato, the sesso gog o wth probablty r I the latter case, the durato of the dle perod betwee two attempts has the same dstrbuto as the dle perod betwee the ed of the prevous call ad the frst attempt of the cosdered call If the cosdered call s the frst of a sesso, there s o retral ad the sesso s lost wth probablty r ad goes o wth probablty r Uder these assumptos, the Erlag formula s sestve to all traffc characterstcs beyod the traffc testy The umber of calls per sesso, the call duratos ad the dle duratos may have arbtrary dstrbutos There may be arbtrary correlato betwee these radom varables The blockg probablty of a call s gve by the Erlag formula, depedetly of the characterstcs of the sesso t belogs to ad of ts locato ths sesso (eg, the frst, secod or last call of the sesso) The oly requremet s that sessos are mutually depedet ad arrve as a Posso process Sessos may ot arrve as a Posso process, however Ths s the case whe the umber of users s relatvely low so that the sesso arrval testy s ot depedet of the umber of ogog sessos (the hgher the umber of ogog sessos, the less lkely the arrval of ew sessos) All sessos must the be cosdered as permaet, the user actvty beg determed by the rato of mea call durato to mea dle durato Ths s the well-kow Egset model [6, 2] For equal traffc testes, the Egset formula gves a lower blockg rate tha the Erlag formula ad teds to the Erlag formula whe the umber of users teds to fty The sestvty property of the Erlag formula s sats-
fed by the Egset formula uder the same assumptos o the retral behavor I partcular, whle call duratos ad dle duratos are geerally assumed to be depedet ad detcally dstrbuted, there may fact be arbtrary correlato betwee these radom varables Note that the two types of retral behavor descrbed above are equvalet for the Egset model uder d assumptos but dffer the presece of correlato The combato of the Erlag model ad the Egset model s suffcetly geeral to represet ay traffc codtos I both cases, the blockg probablty depeds o the traffc testy oly, whch makes these models eve more robust tha geerally beleved The sestvty property exteds to geeral loss etworks [2, 6] Such models aturally arse the performace aalyss of wred ad wreless etworks whe physcal or vrtual costat bt rate crcuts are used to trasmt varous traffc streams lke data, voce, audo ad vdeo flows Whle such a traffc mx may result a bursty flow arrval process, ths does ot affect the blockg probablty provded flows are geerated wth depedet sessos We start wth prelmary results o queueg etworks that are used Secto 3 to prove the sestvty of the Erlag formula The two types of retral behavor metoed above, referred to as jump-over blockg ad radom retrals, are preseted Secto 4 Secto 5 s devoted to some extesos of the ma result, cludg the sestvty of the Egset formula The results are llustrated by a umber of examples Secto 6 Secto 7 cocludes the paper 2 PRELIMINARY RESULTS 2 A queueg etwork Cosder a etwork of N fte server queues Customers arrve at queue as a Posso process of testy ν, wth ν def = ν > 0 Servce tmes at queue are depedet, expoetally dstrbuted of mea /µ, for some µ > 0 Oce served at queue, customers move to queue j wth probablty p j ad leave the etwork wth probablty: p def = j= p j All customers evetually leave the etwork The arrval rate λ at queue cludg arrvals from other queues s the uquely defed by the traffc equatos: λ = ν + λ jp j () j= Summg these equatos, we get the traffc coservato equato: ν = λ p (2) The traffc testy at queue s defed by ρ = λ /µ Let (t) be the N-dmesoal vector whose -th compoet (t) s the umber of customers queue at tme t It s well-kow that the statoary dstrbuto of the Markov process (t) has a product form We gve a proof of ths result stated as Theorem to show the smlarty wth the proof of Theorem 2 below I the rest of the paper, we deote by e the N-dmesoal vector whose -th compoet s equal to ad other compoets are equal to 0 Theorem The statoary dstrbuto of the Markov process (t) s gve by: wth NY ρ x π(x) = π(0) x π(0) = NY e ρ Proof It follows from the traffc equatos () ad (2) that for all queues ad all states x N N : π(x + e )µ (x + ) = π(x)ν + ad π(x + e j)µ j(x j + )p j j= π(x) ν = π(x + e )µ (x + )p The equlbrum equatos follow by summato: π(x) (ν + µ x ) = π(x e )ν + π(x + e j e )µ j(x j + )p j,,j= where we use the coveto that π(x) = 0 f x N N 22 Restrcto of the state space Cosder ow a set of admssble states, A N N We assume that A s coordate covex the sese that f x A the y A for all vectors y such that y x compoet-wse Customers arrve at queue as a Posso process of testy ν (x) state x, wth ν (x) = 0 f x+e A Oce served at queue, customers move to queue j wth probablty p j(x) state x, wth p j(x) = 0 f x e + e j A, ad leave the etwork wth probablty p (x), wth p (x) = p j(x) j= Let λ (x) be the arrval rate of customers to queue state x The correspodg traffc equatos are: λ (x) = ν (x) + λ j(x)p j(x + e j) (3) j= Summg these equatos, we get the correspodg traffc coservato equato: ν (x) = λ (x)p (x + e ) (4)
Note that these equatos may ot have a uque soluto If ν (x) = 0 for all some state x for stace, the arrval rates are defed up to a multplcatve costat (the etwork s the locally a closed queueg etwork) Let (t) be the N-dmesoal vector whose -th compoet (t) s the umber of customers queue at tme t We have the followg key result: Theorem 2 If the arrval rates defed by: λ (x) = λ f x + e A, λ (x) = 0 otherwse are a soluto of the traffc equatos (3) for all queues ad all states x N N, the the statoary dstrbuto π of the Markov process (t) s the restrcto of π to the set of admssble states A, that s: wth NY π (x) = π ρ x (0) x π (0) = f x A, NY x A π (x) = 0 otherwse, ρ x x Proof It follows from the traffc equatos (3) ad (4) that for all queues ad all states x N N, π (x + e )µ (x + ) = π (x)ν (x) + π (x + e j)µ j(x j + )p j(x + e j),j= ad π (x) ν (x) = π (x + e )µ (x + )p (x + e ) The equlbrum equatos follow by summato: π (x) + (ν (x) + µ x ) = π (x e )ν (x e ) π (x + e j e )µ j(x j + )p j(x + e j e ),,j= where we use the coveto that π(x) = 0 f x N N The followg Departures See Tme Average property s a drect cosequece of Theorem 2 A queue s sad to be free f x + e A for all admssble states x A Corollary Uder the codtos of Theorem 2, customers leavg ay free queue (ether movg to aother queue or leavg the etwork) see the etwork steady state mmedately after ther departure Proof The probablty π (x) that customers leavg free queue see the etwork state x mmedately after ther departure s proportoal to π (x + e )µ (x + ), so that: π (x) = π (x + e )µ (x + ) π (y + e )µ (y + ) y A Now t follows from Theorem 2 that for all x A, π (x + e )µ (x + ) = π (x)λ We deduce: π (x) = π (x)λ = π (x) π (y)λ y A 3 INSENSITIVITY OF THE ERLANG FORMULA The above results are ow used to prove the sestvty of the Erlag formula We frst show that the queueg etwork troduced 2 ca represet ay traffc characterstcs, wth arbtrary dstrbutos of the umber of calls per sesso, call duratos ad dle duratos There may be arbtrary correlato betwee these radom varables The oly requred assumpto s that sessos are depedet ad arrve as a Posso process 3 Traffc characterstcs We decompose each call ad each dle perod a fte, radom umber of expoetal phases, as llustrated by Fgure The correspodg dstrbutos, geerally referred to as phase-type dstrbutos or Cox dstrbutos, are kow to form a dese subset wth the set of all dstrbutos wth real, o-egatve support 2 3 4 Fgure : A Cox dstrbuto wth four expoetal phases: after the completo of phase, =, 2, 3, the servce eters phase + wth probablty q ad eds wth probablty q Each fte-server queue of the etwork descrbed 2 represets the phase of a call or a dle perod We deote by C the set of queues correspodg to call phases, by I the set of queues correspodg to dle phases These sets form a partto of the set of queues {,, N} By coveto, all sessos start ad ed wth a call (ad ot a dle perod) We deote by S C the set of call phases correspodg to the startg phase of a sesso These are the oly queues for whch ν > 0 We assume that queues are umbered such a way that phases of successve calls ad dle perods of the same sesso correspod to creasg dces Specfcally, for ay call phase C, we let p j = 0 for all j except f j s the followg phase of the same call, that s j = + wth j C \ S, or f j s the startg phase of the followg dle perod, that s { +,, j } C \ S ad j I Smlarly, for ay dle phase I, we let p j = 0 for all j except f j s the followg phase of the same dle perod, that s j = + wth j I, or f j s the startg phase of the followg call, that s { +,, j } I ad j C \ S We have p = 0 for all except f s a phase of the last call of a sesso, that s C ad { +,, j } C \ S for some j S correspodg to the startg phase of aother sesso For ay call phase C, we refer to the smallest dex j such that j C ad {j +,, } C \ S as the startg phase of the call Smlarly, for ay dle phase I, we refer
to the smallest dex j such that {j, j +,, } I as the startg phase of the dle perod For ay call or dle phase C I, we refer to to the largest dex j such that j S as the startg phase of the correspodg sesso We deote ths dex by s() It s clear that ths queueg etwork ca represet ay type of sesso, as llustrated by Fgure 2 Sessos are mutually depedet ad the probablty that a ew sesso s of type S s gve by: 3 4 ν ν 5 6 7 2 where ρ = λ /µ deotes the traffc testy at queue The arrval rate λ at ay queue that correspods to the startg phase of a call or dle perod s equal to ν s, wth s = s() For ay other queue j, we have: λ j = ν sp,+p +,+2 p j,j, (6) where deotes the startg phase of the correspodg call or dle perod ad s = s() 32 Blockg probablty We ow evaluate the blockg probablty the presece of C telephoe les The set of admssble states s the: ( ) A = x N N : x C Assume the retral behavor s such that the codtos of Theorem 2 are satsfed By the Posso Arrvals See Tme Average property, the frst call of ay sesso sees the system steady state The probablty such a call s blocked s equal to the statoary probablty that all les are occuped Now vew of Corollary, customers leavg ay free queue see the etwork steady state Notg that queues correspodg to dle phases are free, we deduce that the probablty a arbtrary call s blocked s equal to the statoary probablty that all les are occuped For all = 0,,, C, the statoary probablty that les are occuped s gve by: π() = π (x) C x: P C x = I vew of Theorem 2, we get: 8 9 0 2 Fgure 2: Two types of two-call sessos represeted as sequeces of expoetal phases, wth startg phases S = {, 5}, call phases C = {, 2, 4, 5, 6, 7,, 2} ad dle phases I = {3, 8, 9, 0} (dashed crcles) Provded the umber of queues N s suffcetly large, we may represet ay traffc characterstcs, wth arbtrary dstrbutos for the umber of calls per sesso, call duratos ad dle duratos There may be arbtrary correlato betwee these radom varables I the example of Fgure 2 for stace, assumg all expoetal phases have the same mea, large dle duratos are typcally preceded ad followed by large call duratos The overall traffc testy, defed as the product of the call arrval rate ad the mea call durato ad usually expressed Erlags, s gve by: ρ = C ρ, (5) wth π() = π(0) ρ, π(0) = C ρ, where ρ deotes the overall traffc testy, gve by (5) We deduce the blockg probablty of a arbtrary call: B = ρ C C (7) C ρ Ths s the well-kow Erlag formula It does ot deped o ay traffc characterstcs descrbed 3 beyod the traffc testy ρ The blockg probablty s the same for all calls depedetly of the sesso they belog to ad of ther locato that sesso (eg, the frst, secod or last call of the sesso) 4 RETRIAL BEHAVIOR We ow descrbe two types of retral behavor for whch the codtos of Theorem 2 are satsfed We frst cosder jump-over blockg the radom retrals wth fxed retral probablty r Though the latter cotas the former as a specal case (for r = 0), we preset them as separate schemes for the sake of clarty
4 Jump-over blockg We frst assume that sessos go o case of blockg, as llustrated by Fgure 3 If a call s blocked, the followg dle perod starts mmedately If the call s the last of the sesso, the sesso eds We here assume that the retral behavor s the same for all sessos Ths assumpto wll be relaxed 42 where the retral probablty may deped o the cosdered sesso The durato of the dle perod betwee two attempts has the same dstrbuto as the dle perod that precedes the cosdered call If the cosdered call s the frst of a sesso, there s o retral ad the sesso s lost wth probablty r ad goes o wth probablty r r r r r Fgure 4: Blockg wth radom retral Fgure 3: Jump-over blockg Let x A be a admssble state For ay startg phase of a sesso S such that x + e A, we have ν (x) = 0 If the sesso cossts of more tha oe call, we also have ν j(x) = ν, where j deotes the startg phase of followg dle perod Smlarly, for ay startg phase of a call C\S such that x+e A, we have p j(x) = 0 where j deotes ay phase of the precedg dle perod If the call s ot the last of the sesso, we also have p jk(x) = p j where k deotes the startg phase of the followg dle perod Other exteral arrval rates ad routg probabltes are the same as the absece of call blockg, as descrbed 3 Jump-over blockg s a stadard reroutg scheme queueg theory, see eg [5, 7] I partcular, t s kow that the codtos of Theorem 2 are satsfed We prove the result for the sake of completeess For all x such that x < C, C the soluto of the correspodg traffc equatos (3) s uque ad gve by λ (x) = λ Now let x be such that: x = C C We have λ (x) = 0 for all C It remas to prove that λ (x) = λ for all I Let, +,, k be the successve phases of the same dle perod We deote by s = s() the startg phase of the correspodg sesso The traffc equatos are λ (x) = ν s ad λ j+(x) = λ j(x)p j,j+, j =,, k It the follows from (6) that: λ j(x) = λ j, j =,, k The codtos of Theorem 2 are satsfed 42 Radom retrals We ow cosder radom retrals, as llustrated by Fgure 4 Each blocked call s reattempted wth a fxed probablty r after a dle perod of radom durato The sesso goes o as the jump-over blockg model wth probablty r Note that the retral behavor correspods to the jumpover blockg scheme whe r = 0 If r =, all calls of the sesso are reattempted utl they are accepted, except the frst call for whch there s o retral Thus ths case, ether the frst call s accepted ad all calls of the sesso are evetually accepted, or the frst call s rejected ad the etre sesso s rejected I the followg, we allow the retral probablty to deped o the sesso type For all S, we deote by r the retral probablty of the correspodg sesso Let x A be a admssble state For ay startg phase of a sesso S such that x + e A, we have ν (x) = 0 If the sesso cossts of more tha oe call, we also have ν j(x) = ν ( r ) where j s the startg phase of the followg dle perod Smlarly, for ay startg phase of a call C \ S such that x + e A, we have p j(x) = 0 where j deotes ay phase of the precedg dle perod, ad p jk(x) = p jr s where k deotes the startg phase of ths dle perod ad s = s() deotes the startg phase of the sesso If ths call s ot the last of the sesso, we also have p jl(x) = p j( r s) where l deotes the startg phase of the followg dle perod Other exteral arrval rates ad routg probabltes are the same as the absece of call blockg, as descrbed 3 For all states x such that x < C, C the soluto of the traffc equatos (3) s uque ad gve by λ (x) = λ Now let x be such that: x = C C Aga, we have λ (x) = 0 for all C It remas to prove that λ (x) = λ for all I Let, +,, k be the successve phases of the same dle perod We deote by s = s() the startg phase of the correspodg sesso The traffc equatos are: λ (x) = ν s( r s) + k λ j(x)p j,k+ r s, j=
ad λ j+(x) = λ j(x)p j,j+, j =,, k Usg the fact that p k,k+ = ad p j,j+ + p j,k+ = for all j =,, k, we deduce: λ (x) = ν s( r s) + λ (x)r s, If r s <, the soluto to ths equato s uque, gve by λ (x) = ν s If r s =, the subetwork correspodg to sesso s s a set of closed queueg etworks state x (oe per dle perod) The arrval rates are defed up to a multplcatve costat ad we choose λ (x) = ν s I both cases, we obta: λ j(x) = λ j, j =,, k The codtos of Theorem 2 are satsfed It s worth otg that the results exted to the case where a blocked call s reattempted wth a fxed probablty after a dle perod of costat durato equal to the precedg dle perod, stead of a radom durato wth the same dstrbuto By creasg the umber of queues N, the dstrbuto of call duratos ad dle duratos may deed be made as close to determstc as desred I the lmt, the radom ature of traffc reduces to the Posso sesso arrvals ad the probablty ν /ν a sesso s of type S 5 ETENSIONS We show ths secto that the sestvty property of the Erlag formula exteds to more geeral loss etworks ad to the Egset formula uder the same codtos o the retral behavor as descrbed the prevous secto 5 Mult-rate systems We frst cosder mult-rate systems that have bee extesvely studed the 70 s ad 80 s [7, 0,, 3, 5] ad are stll used to dmeso wred ad wreless etworks whe voce, data, audo ad vdeo flows are multplexed wth costat bt rate crcuts There are K call classes Class-k calls arrve as a Posso process ad requre a crcut of c k bt/s We deote by ϱ k the traffc testy of class-k calls Erlags Deotg by C the lk capacty bt/s, t s well kow that the statoary probablty that k class-k calls are progress, k =,, K, s depedet of the call durato dstrbutos ad gve by: wth π() = π(0) 0 π(0) = @ KY k= ϱ k k k, :c C k= c C KY ϱ k k A k where c deotes the scalar product: c def = K k c k k= The blockg probablty of class-k calls s the gve by: B k = π() :C c k <c C, I vew of Theorem 2, ths result stll holds uder the more geeral assumpto of Posso sesso arrvals provded the retral behavor s that descrbed Secto 4 Specfcally, cosder a partto of K subsets C,, C K of the set of queues C, correspodg to calls of class,, K, respectvely The set of admssble states s: 8 < A = : x NN : 9 K = x c k C ; C k k= The results of Sectos 3-4 readly apply provded sessos are homogeeous the sese that all calls belogg to the same sesso are of the same class We gve a example Secto 6 showg the sestvty of the blockg probablty whe sessos are composed of calls of dfferet classes 52 Loss etworks We ow cosder a geeral model wth several resources Specfcally, the etwork cossts of L lks of respectve capactes C,, C L bt/s There are K call classes Class-k calls arrve as a Posso process ad requre a crcut of c k bt/s through lks a k {,, L} We stll deote by ϱ k the traffc testy of class-k calls Erlags Aga, the statoary probablty that k class-k calls are progress, k =,, K, s depedet of the call durato dstrbutos ad gve by [, 4, 2]: wth π() = π(0) 0 π(0) = @ KY k= ϱ k k k, :A C k= A C KY ϱ k k A k where A s the L K-dmesoal matrx whose l, k etry s equal to c k f l a k ad to 0 otherwse, C s the L- dmesoal vector whose l-th compoet s equal to C l ad deotes the compoet-wse order The blockg probablty of class-k calls s the gve by: B k = π(), :C Ae k <A C where e k deotes the K-dmesoal ut vector wth compoet k ad 0 elsewhere ad for all vectors x, y, x < y meas x y compoet-wse ad x y Aga, the result remas vald for Posso sesso arrvals provded sessos are homogeeous ad the retral behavor s that descrbed Secto 4 53 The Egset formula Fally, we cosder the Egset model wth a fxed umber M of permaet sessos Though the results apply to geeral loss etworks as well, we focus o the smple example of a sgle lk of C telephoe les as orgally cosdered by Egset [6] We make the atural assumpto that M > C Let ϱ be the rato of the mea call durato to the mea dle durato Uder d assumptos, t s well kow that the statoary probablty that calls are progress s depedet of the dstrbutos of call duratos ad dle duratos beyod ϱ ad gve by: π() = π(0) M, ϱ, C (8)
wth π(0) = C M ϱ I addto, ew calls see a system of M permaet sessos steady state whe they arrve We deduce the call blockg probablty: B = C M C M ϱ C ϱ (9) Ths s the Egset formula We ow prove that the formula s vald for geeral traffc characterstcs cludg correlated call duratos ad dle duratos provded the retral behavor s that descrbed Secto 4 We cosder a closed etwork of N fte-server queues ad M customers Servce tmes at queue are depedet, expoetally dstrbuted of mea /µ, for some µ > 0 Oce served at queue, customers move to queue j wth probablty p j, wth: p j = j= Routg s rreducble the sese that each customer vsts all queues The arrval rate λ at queue s uquely defed, up to a multplcatve costat, by the traffc equatos: λ = λ jp j (0) j= We defe the traffc testy at queue by ρ = λ /µ The statoary dstrbuto of the Markov process (t) descrbg the etwork state s gve by: NY ρ x π(x) = G x, N where G s the ormalzg costat, gve by: G = M M ρ x = M, () We deote by C the set of queues correspodg to call phases ad by I the set of queues correspodg to dle phases, wth C I = {,, N} As explaed 3, the above queueg etwork may represet ay traffc characterstcs provded the umber of queues N s suffcetly large The rato of the mea call durato to the mea dle durato s gve by: P C ρ ϱ = P I ρ I vew of (), the statoary probablty that les are occuped s gve by: M π() = π(x) = π(0) ϱ (2) x: P C x = Aga, we assume that queues are umbered such a way that phases of successve calls ad dle perods correspod to creasg dces By coveto, we let C, N I ad p N = For ay call phase C, we have p,+ + p j = where j s the startg phase of the followg dle perod For ay dle phase I, N, we have p,+ + p j = where j s the startg phase of the followg call The startg phases of calls ad dle perods have the same arrval rate Sce arrval rates are defed up to a multplcatve costat, we let λ = for the correspodg queues For ay other queue j, we have: λ j = p,+p +,+2 p j,j (3) Now cosder the restrcted state space: ( ) A = x N N : x C x = M, C Deote by p j(x) the ew routg probablty from queue to queue j state x, wth p j(x) = 0 f x e + e j A Let λ (x) be the (relatve) arrval rate of customers to queue state x The correspodg traffc equatos are: λ (x) = λ j(x)p j(x + e j) (4) j= Let (t) be the Markov process descrbg the etwork state We have the aalog of Theorem 2: Theorem 3 If the arrval rates defed by: λ (x) = λ f x + e A, λ (x) = 0 otherwse are a soluto to the traffc equatos (4) for all queues ad all states x N N such that P N x = M, the the statoary dstrbuto π of the Markov process (t) s the restrcto of π to the set of admssble states A, that s: wth π (x) = G N Y ρ x x G = f x A, NY x A π (x) = 0 otherwse, ρ x x Proof It follows from the traffc equatos (4) that for all queues ad all states x such that P N x = M, π (x + e )µ (x + ) = π (x + e j)µ j(x j + )p j(x + e j) j= The equlbrum equatos follow by summato: π (x) µ x = π (x+e j e )µ j(x j +)p j(x+e j e ),,j= where we use the coveto that π (x) = 0 f x N N The correspodg Departures See Tme Average property s the followg Let: ( ) A = x N N : x C x = M, C
A queue s sad to be free f x+e A for all states x A Note that the set of free queues here correspods to the set of dle phases I Corollary 2 Uder the codtos of Theorem 3, customers leavg ay free queue see a etwork wth M customers steady state mmedately after ther departure Proof The probablty π (x) that customers leavg free queue see the etwork state x mmedately after ther departure s proportoal to π (x + e )µ (x + ), so that: π (x) = π (x + e )µ (x + ) π (y + e )µ (y + ) y A I vew of Theorem 3, we have: x A, π (x + e )µ (x + ) = G G π (x)λ, where π deote the statoary dstrbuto of the etwork wth M customers: wth π (x) = G We deduce: N Y ρ x x G = f x A, NY x A π (x) = 0 otherwse, ρ x x π (x) = π (x)λ = π (x) π (y)λ y A We verfy as Secto 4 that the codtos of Theorem 3 are satsfed for both jump-over blockg ad radom retrals It the follows from Corollary 2 that the blockg probablty s equal to the probablty that all les are occuped the presece of M customers, depedetly of the cosdered call Ths blockg probablty s gve by the Egset formula (9) 6 EAMPLES Fally, we llustrate the above sestvty results by some examples 6 Burst arrvals wth sessos The Erlag model wth sessos may clude call bursts, that s successos of calls separated by relatvely short dle perods, as llustrated by Fgure 5 It s the somewhat surprsg that the secod call of a sesso s blocked wth the same probablty as the frst call of the sesso Itutvely, gve the fact that the frst call was accepted, the secod call s also accepted wth a hgh probablty after a short dle perod Cosder for stace two-call sessos wth expoetal holdg tmes of mea /µ ad expoetal dle perods of mea τ, for some µ, τ > 0 Sessos arrve accordg to a Posso process of testy λ I vew of Theorem 2, the umber of dle sessos s depedet of the umber of ogog calls ad has a Posso dstrbuto of mea λτ I Fgure 5: Burst arrvals wth sessos partcular, the probablty that there s o other dle sesso at the ed of the frst call s equal to e λτ Moreover, the probablty that o other sesso s geerated durg a dle perod s equal to: + λτ We coclude that the probablty that the secod call of the sesso s accepted gve the fact that the frst call was accepted deed teds to whe the mea dle perod τ teds to 0 How to solve the paradox? I case of jump-over blockg, t s suffcet to observe that the probablty that the secod call of the sesso s blocked gve the fact that the frst call was blocked also teds to whe the mea dle perod τ teds to 0 Ths follows from the fact that the probablty that o ogog call eds durg a dle perod s equal to: + Cµτ, where C deotes the umber of les I the lmt τ 0, the blockg probablty of the secod call of the sesso s equal to the blockg probablty of the frst call of the sesso, whch s gve by the Erlag formula For radom retrals wth retral probablty r <, a fracto r of the frst calls blocked geerate a secod call Aga, the successve attempts of ths secod call wll be blocked wth hgh probablty for a short dle perod I the lmt τ 0, all these attempts are blocked Sce the average umber of attempts s equal to ( r), the average umber of secod calls blocked per frst call blocked s equal to The blockg probablty of the secod call s equal to the blockg probablty of the frst call of the sesso, whch s gve by the Erlag formula Fally, cosder the case where the retral probablty s r = The etre sesso s the blocked f the frst call of the sesso s blocked (cf Fgure 4) I ths case, we solve the paradox as follows: eve f the probablty that the secod call of the sesso s accepted codtoally to the fact that the frst call was accepted s close to for a short dle perod, the blockg probablty s ot equal to 0 Ad f the frst attempt of secod call s blocked, ext reattempts wll also be blocked wth a hgh probablty I the lmt τ 0, the blockg probablty of the secod call s equal to 0 but f blocked, the umber of reattempts s fte The blockg probablty s fact equal to the Erlag formula depedetly of the mea dle durato τ 62 Burst arrvals of calls It s worth observg that, eve f a sesso may geerate call bursts, each call (except the frst) starts after the prevous call of the same sesso eds Ths makes a sgfcat dfferece wth more artfcal traffc models where o-posso call arrvals are geerated based o statstcs o observed call terarrval tmes [4] I ths case, the blockg probablty may deed be strogly sestve to traffc
characterstcs, see eg [20] We llustrate ths sestvty o the smple example llustrated by Fgure 6 where calls arrve bursts of fxed sze b accordg to a Posso process of testy λ Fgure 6: Burst arrvals of calls Let C be the umber of telephoe les, wth C > b Assume the holdg tmes are depedet, expoetally dstrbuted of mea /µ, for some µ > 0 The statoary probablty π of the umber of ogog calls satsfes the equlbrum equatos: (λ + µ) π() = ( + )µ π( + ) for all = 0,, b, ad (λ + µ) π() = ( + )µ π( + ) + λ π( b) for all = b,, C The blockg probablty s the: B = b b π(c ) b Fgure 7 gves the blockg probablty wth respect to the traffc testy b λ/µ for C = 50 telephoe les We observe that the burst sze b has a strog mpact o the blockg probablty Note that a coservatve estmate of the blockg probablty s gve by the correspodg Erlag formula where each burst of b calls s cosdered as a sgle call holdg b les smultaeously durg a mea tme /µ (cf the fgure, the estmate s exact for b = ) The result s deed very sestve to the burst sze b Blockg probablty 0 00 000 Exact Erlag approxmato 0000 0 0 20 30 40 50 Traffc testy (Erlags) Fgure 7: Blockg probablty for varous values of the burst sze (b =, 2, 5, from bottom to top) 63 Impact of retral behavor We ow llustrate the mpact of retral behavor We focus o the frst call of a sesso, for whch there s o retral both schemes descrbed Secto 4 I the smple case of Posso call arrvals wth retral, the model reduces to a classcal retral queue, for whch a rch lterature exsts, see [9, 9] Such queues are kow to be sestve Cosder for stace the Erlag model wth C telephoe les, call arrval rate λ ad mea holdg tme /µ, for some µ > 0 We deote by ρ = λ/µ the correspodg traffc testy Erlags Assume that each call s reattempted utl t s accepted Successve attempts are separated by depedet, expoetally dstrbuted dle perods of mea τ The umber of dle sessos the creases cotuously f ρ > C If ρ < C, the umber of dle sessos remas stable but the blockg probablty of successve attempts s sestve to the mea dle perod τ I the lmtg case τ 0 for stace, a dle sesso becomes actve as soo as a telephoe le becomes dle The blockg probablty s the gve by the correspodg Erlag C formula [8]: B = C ρ C ρ C C C ρ + ρc C I the other lmtg case τ, the overall call arrval process (cludg reattempts) teds to a Posso process of testy λ/( B), where B deotes the correspodg blockg probablty The blockg probablty ca the be determed by the fxed pot equato: «ρ B = E B, B where E B(α) deotes the Erlag formula (also referred to as the Erlag B formula) for a traffc testy α: E B(α) = α C C C The results are llustrated by Fgure 8 for C = 0 telephoe les Blockg probablty 0 00 000 Ifte thk tme Null thk tme α 0000 0 2 4 6 8 0 Traffc testy (Erlags) Fgure 8: Sestvty of the blockg probablty to the mea dle perod betwee successve attempts
64 Loss etworks wth o-homogeeous sessos We ow show that successve calls of the same sesso must requre the same etwork resources to preserve sestvty Cosder for stace a mult-rate system as descrbed 5 wth K = 2 call classes ad c < c 2 Sessos arrve as a Posso process ad cosst of two calls, a class- call followed by a class-2 call after a radom dle perod Blocked calls are reattempted wth probablty r as descrbed Secto 4 Such a system s sestve I the lmtg case c = 0, frst calls are always accepted ad the system behaves for secod calls as a retral queue, cf 63 I the partcular case where r =, c 2 = ad C = 0, the sestvty of the blockg probablty to the dle durato s gve by Fgure 8 65 Mult-class Egset formula Fally, we cosder the Egset model wth K sesso classes All sessos of the same class have the same traffc characterstcs Let ϱ k be the rato of the mea holdg tme to the mea dle perod for class-k sessos There are M k such sessos The statoary probablty that k classk calls are progress, k =,, K, s depedet of the holdg tme ad dle perod dstrbutos ad gve by: KY M k K π() = π(0) ϱ k k, k M k, k C k wth 0 π(0) = @ k= KY : k M k, P K k= k C k= B = M k k k= ϱ k k I addto, ew class-k calls see a system wth M k class-k sessos ad M l class-l sessos, l k, steady state whe they arrve We deduce that the blockg probablty depeds the sesso class As above, the correspodg multclass Erlag formula s vald for geeral traffc characterstcs cludg correlated call duratos ad dle duratos provded the retral behavor s that descrbed Secto 4 The sestvty to the traffc mx s llustrated o Fgure 9 for C = 0 telephoe les ad K = 2 classes, wth M = 20, M 2 = ad ϱ 2 The class-2 sesso s ether actve or cotuous retral f blocked The class- blockg probablty s gve by the Egset formula for C telephoe les ad M class- sessos: M ϱ C C C M The class-2 blockg probablty s gve by the Egset formula for C telephoe les ad M + class- sessos: M ϱ C C B 2 = C M ϱ It s worth otg that the average blockg probablty depeds o the traffc mx through the umber of sessos ϱ A Blockg probablty 0 00 000 Class Class 2 0000 0 2 4 6 8 0 Traffc testy (Erlags) Fgure 9: Sestvty of the blockg probablty to the sesso class of each class, M,, M K, ad the correspodg ratos of the mea call durato to the mea dle durato, ϱ,, ϱ K Egset cojectured hs 95 report that the worst case s obtaed the homogeeous case where ϱ = = ϱ K [6] Ths result was proved by Dartos much later [3] 7 CONCLUSION The multplexg of varous traffc streams lke voce, data, audo ad vdeo flows may geerate o-posso flow arrvals Ths rases the ssue of the applcablty of stadard teletraffc models lke the Erlag model whe these flows are multplexed physcal or vrtual costat bt rate crcuts We have demostrated that these models rema applcable provded calls are geerated wth depedet sessos, whch s typcally the case practce These sessos may arrve as a Posso process as the Erlag model or be permaet as the Egset model, coverg the whole rage of traffc codtos Thus these models satsfy eve stroger sestvty propertes tha geerally beleved A crtcal role s played by the retral behavor We have descrbed two schemes, amely jump-over blockg ad radom retrals, for whch the sestvty property s deed satsfed We have also gve examples of other schemes for whch the sestvty property s volated A atural questo of terest s to fd the whole set of schemes for whch the sestvty property holds Ths wll be the subject of future research 8 REFERENCES [] DY Burma, JP Lehoczky ad Y Lm, Isestvty of blockg probabltes a crcut-swtchg etwork, J Appl Probab 2 (984) 850 859 [2] JW Cohe, The Geeralzed Egset Formula, Phllps Telecommucatos Revew 8 (957) 58 70 [3] JP Dartos, Lost call cleared systems wth ubalaced traffc sources, : Proc of 6th Iteratoal Teletraffc Cogress, 970 [4] Z Dzog, JW Roberts, Cogesto probabltes a crcut-swtched tegrated servces etwork, Performace Evaluato 7-4 (987) 267 284
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