Rare Events Prediction Using Importance Sampling in a Tandem Network

Transcription

1 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN Rare Evets Predcto Usg Importace Samplg a Tadem Network A. Behrouz Safaezadeh Departmet of Computer, Admeshk Brach, Islamc Azad Uversty, Admeshk, Ira Behrouz_safae@yahoo.com Abstract Importace samplg s a techque that s commoly used to speed up Mote Carlo smulato of rare evets. Estmatg probabltes assocated to rare evets has bee a topc of great mportace queueg theory, ad appled probablty at large. We aalyze the performace of a mportace samplg estmator for a rare evet probablty a Jackso etwork. The preset paper carres out strct deadles to a two-ode Jackso etwork whose arrval ad servce rates are modulated by a exogeous fte state Markov process. We derve a closed form soluto for the probablty of mssg deadle. The the results have employed a mportace samplg techque to estmate the probablty of total populato overflow whch s a rare evet. We have also show that the probablty of ths rare evet may be affected by varous deadle values. Keywords: Importace Samplg, Rare Evet, Jackso Network, Deadle, Performace. I. Itroducto Importace samplg s a varace reducto method for smulatg rare evets. The dea mportace samplg s to chage the samplg dstrbuto (ad modfy the Mote Carlo (Asmusse, 003) estmator accordgly) to reduce estmator varace. For more tha two decades, there has bee a growg of terest fast smulato wth mportace samplg for estmatg probabltes of rare evets(jueja et al,006) (Asmusse,003). Rare evet smulato volves estmatg extremely small but mportat probabltes. Such probabltes are of mportace varous applcatos: I moder packet-swtched telecommucatos etworks, -order to reduce delay varato carryg real-tme traffcs, the buffers wth the swtches are of lmted sze. Ths creates the possblty of packet loss f the buffers overflow. These swtches are modelled as queueg systems ad t s mportat to estmate the extremely small overflow probabltes such queug systems (Boer et al, 00) (Chag, 1994). The mportace samplg smulates the system uder a dfferet probablty dstrbuto.e., chage of measure (Boer et al, 006). I ths paper the evet of terest s total populato overflow a two-ode Jackso etwork (Dupus et al, 007) (Klerock, 1975) that allows feedback. Assumg the stablty codto (Asmusse, 003) esures that the overflow evet deed s a rare evet. The stablty assumpto says that the average arrval rate to the etwork s less tha the average servce rate at each ode. I (Dupus et al,007) a mportace samplg techque for estmatg probablty of the rare evet s used but the feedback probablty s cosdered costat ad o deadle s assumed the etwork. I (Movaghar, 006) (Movaghar, 000) a comprehesve study s gve o the probablty of mssg deadle of customers a M/M/1 queue ad we exted t for the two-ode Jackso etwork. I ths paper we assume that each customer the secod queue has a deadle utl the ed of ts servce at ode two. Uder such codto, we have exteded mportace samplg techque of (Dupus et al, 1

2 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN ). The paper s orgazed as follows. Secto gves a bref revew of the bascs of mportace samplg. I secto 3 ad aalyss of the probablty of mssg deadle the two-ode Jackso etwork, ts dyamcs ad the mportace samplg estmator are derved. Secto 4 exames some examples to llustrate the effcacy of our model. II. Bascs of mportace samplg We are terested effcet mportace samplg techque for estmatg the buffer overflow probablty p whe s large. Importace samplg smulates the system uder a dfferet probablty dstrbuto,.e., chage of measure. Deote by A the evet of buffer overflow, ad rewrte p = P( A). A mportace samplg scheme geerates samples from a ew probablty measure, sayq, such that P << Q. The estmator s the gve by the average of depedet replcatos of pˆ 1A dp dq =& (1) Where dp dq s the Rado-Nkodym dervatve (Asmusse, 003) or lkelhood rato ad 1A s the dcator of the evet A. Clearly p ˆ s ubased for ay suchq. The goal of mportace samplg s to choose Q to mmze the varace, or the secod momet of p ˆ. A obvous lower boud follows from Jese s equalty (Dupus et al,007) ad the large devatos propertes of p ˆ, 1 Q lm f log [ ˆ Q ] lm f log E p E [ pˆ] = lmf log p = γ () A mportace samplg scheme, or the chage of measureq, s sad to be asymptotcally optmal f ths lower boud s acheved,.e., f 1 Q ˆ lm sup log E [ p] γ (3) For future aalyss, t s worthwhle to ote that the secod momet equals Q P = (4) E [ pˆ ] E [ pˆ ] III. Aalyss of mssg the deadle Cosder a two-ode Jackso etwork whose servce ad arrval processes are modulated by a fte state Markov cha. I ths etwork the arrval process to the etwork s Posso wth rate λ ad customers are served the order of ther arrval,.e., servce dscple s Frst-Come-Frst-Served (FCFS) ad the arrval rate to the frst queue s λ 1. Servce tmes are expoetally dstrbuted wth rates µ 1 ad µ at ode oe ad two.

3 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN We assumed that wheever a customer ode oe has bee served completely, t receves a deadle oce eterg the secod queue, ad t should fsh ts servce at ode two ad leave the etwork before mssg the deadle. The dfferece betwee the deadle of a customer the secod queue ad ts arrval tme from ode oe, referred to as a relatve deadle, s a radom varableη kow as customer mpatece wth a probablty dstrbuto fucto D ( τ ). I ths paper, we cosder a model wth determstc customer mpatece that has already bee studed (Barrer, 1957). The probablty dstrbuto fucto of customer mpatece η s gve by D ( ητ, ) = 0, f τ < η, D ( ητ, ) = 1, f τ η, (5) where η s a costat deotg the mea customer mpatece ad τ s a varable wth values the set of o-egatve real umbers. I ths etwork customer servce tmes ad relatve deadles form sequeces of d radom varables that are mutually depedet. Let V the tme a arrvg customer from ode oe wth fte (o) deadle must wat before t completes (6) ts servce at ode two the log ru. V s called the offered sojour tme the etwork. The probablty dstrbuto fucto of V s FV ( τ) PV ( τ), = (7) Therefore the probablty of mssg deadle, defed as, ϕ P ( η V ) D ( τ) df ( τ) d 0 = < < = (8) V ϕd Represets the steady-state probablty that a customer msses ts deadle. A close-form soluto for ϕ d s derved as 1 1. (9) ϕ = qpv ( > η ) = 1 qpv ( η ) d = 0 = 0 V s the tme a arrvg customer from ode oe wth fte (o) deadle must wat the secod queue before t completes ts servce at ode two the log ru, gve t fds customers the secod queue. η s the relatve deadle of the th customer the secod queue, P s the probablty measure ad s the maxmum buffer sze of the secod queue. q s the probablty that there are customers the secod queue ad s gve by q µλ 1 0 µ = q Φ ( ), for 1!, (10) 3

4 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN q 0 = 1 1 = 1 1 µ λ Φ ( µ )! (11) Where Φ ( µ ) s the Laplace trasform of τ 0 (1 Dx ( )),.e., τ µτ ( µ ) (1 ( )) τ. 0 0 (1) Φ = D x e d The proof of equatos ths secto s straghtforward ad ca be foud ( Movaghar,006) ( Movaghar,000) that had bee used for a sgle queue ad we apply t for our etwork. A. The system dyamcs Let N = { N ( k): k = 0,1,,... } be the embedded dscrete tme Markov cha that represets the queue legths at the trasto epochs of the etwork ad suppose that N ( k) = ( N1( k), N ( k)) where N ( k ) s the legth of the queue at ode after the kth trasto. Obvously, N ca oly take values at ad p equals the probablty that N1 N reaches before returg to 0, gve that the system s tally empty. The the dyamcs of N ca be modelled by N ( k 1) = N ( k) π[ N ( k), Y ( k 1)] where { Y ( k )} are d radom varables takg values Ω= { ω = (1,0), ω = ( 1,1), ω = (0, 1), ω = (1, 1) }, ad the mappg π s defed for every N N N = ( 1, ) as 0, f N1 = 0 ad y= ω1 π ω ω y, otherwse [ N, y] = & 0, f N = 0 ad y = or. (13) 3 The dstrbuto of N s completely determed by that of the sequece Y { Y ( k) } =. Defe P ( Ω ) = & { θ = ( θ0, θ1, θ, θ3) }, θ s a probablty o Ω ad θ = θω [ ] for every = 0,1,,3 uder the orgal probablty measure P, the dstrbuto of Y ( k ) s just Θ= ( λµ,,(1 ϕ ) µ, ϕµ ) Ρ ( Ω) & (14) 1 d d To be more precse, for a gve threshold, defe the scaled state process X = N /, where N s defed above. Sce the defto of π mples π[ x, y] = π[ x, y] for every x, t s ot dffcult to see that X satsfes the equato 1 X ( k 1) = X ( k) [ X ( k), Y ( k 1)] π (15) wth tal codto X (0) N (0)/ 0 = =. 4

5 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN B. The mportace samplg estmator ad ts asymptotcs The mportace samplg geerates { Y ( k )} as follows. The codtoal probablty of Y ( k 1) = ω, that s { Y ( j ): j 1,,..., k } descrbed the prevous secto, gve =, s just Θ [ ω X ( k)] for each = 0, 1,. The mportace samplg scheme we cosder use state-depedet chages of measure that ca be characterzed by stochastc kerels Θ [..] o Ω gve,.e, [. x] P Θ ( Ω ) for every x. Defe the httg tmes T = & f{ k 0: X ( k) X ( k) = 1} 1 T = f{ k 0: X ( k) X ( k) = 0} & (16) 0 1 Let A be the evet of terest, that s, A = X X Reaches 1 before returg to 0} = { T < T0} (17) { 1 The mportace samplg estmator s just pˆ = 1. Θ [ Y ( k 1)] (18) [ Y ( k 1) X ( k )] T 1 A k = 0 Θ whose proof ca be foud (Dupus et al,007). The secod momet of p ˆ, thaks to (4), equals E P [ p ˆ ]. The goal s to choose a stochastc kerel Θ so that ths secod momet (whece the varace of p ˆ ) s as small as possble. Aother mportat cosderato s that oe would lke Θ to be smple ad easy to mplemet. Before we proceed to costruct mportace samplg techque, we collect some otato ad termology. We Defe fgure 1, D = & {( x, x ): x 0, x x < 1} 1 1 δ = & {(0, x ):0< x < 1} 1 δ = {( x,0):0< x < 1} & (19) 1 1 δ = & {( x, x ): x 0, x x = 1} e 1 1 Sometmes we refer to δ e as the ext boudary. Fgure. 1. Domas ad boudares of the etwork 5

6 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN The mportace samplg chage of measure s determed by 3 εδ, * Θ. x = & ρk Θ ( rk), f x D k= 1 (0) ad 3 εδ, * Θ. x = ρk Θδ ( r ), k & f x δ (1) k= 1 * * εδ, where r k s defed as (3), the formulae for Θ ad Θδ ca be foud () ad ρk s probablty vector that ca be foud (6). Accordg to theorem 1 we wll allow ε ad δ to be -depedet, deoted by ε, δ. Theorem 1. The mportace samplg estmator p ˆ s asymptotcally optmal f δ 0, ε δ 0, ad ε. Oe ca also use a fxed par of parameters ε ad δ for all. A good choce s to set δ = ε logε. p1 p1 p p p p1 * p 1 p p1 p λ e µ 1e ϕ d µ e ϕ dµ e Θ ((, )) =ϒ((, )).(,,(1 ), ) p1 p p p1 * δ p 1 1 p 1 p1 p λ e µ 1 ϕ d µ e ϕ dµ e Θ ((, )) =ϒ ((, )).(,,(1 ), ) () p1 p1 p * δ p 1 p p1 p λ e µ 1e ϕ d µ ϕ d µ Θ ((, )) =ϒ ((, )).(,,(1 ), ) where ϒ (( p1, p)), ϒ (( p1, p)) are ormalzg costats so that all these vectors are probablty vectors (.e., elemets P ( Ω )). The value of r k for k = 1,,3 s gve by r1 = & γ( 1, 1) r = & γ( 1,0) ( γ α)(0, 1) 3 (0,0) where α s gve by r =& (3) log µ 1 ( µ 1 λ (1 ϕ d ) µ ), f µ 1 µ α = log µ 1 ( λ ϕ dµ 1), f µ 1 < µ & (4) C. Pecewse affe subsolutos ad mollfcato Smooth subsoluto for the rare evet ths paper s smoothed verso of pecewse affe fucto ad lead to effcet ad easly mplemetable mportace samplg techque. Oe of the key characterstcs of the subsoluto based mportace samplg schemes s that, as opposed to the mportace samplg schemes wth d cremets, the dstrbuto of the cremet uder them depeds o the curret state of 6

7 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN the radom walk represetg the queue legths ad deed chage as the queues trasto betwee beg empty ad oempty. As (Dupus et al, 005) (Dupus et al, 005), the costructo of classcal subsoluto s dvded to two steps. We frst detfy a subsoluto as the mmum of affe fuctos ad the mollfy t to obta a classcal subsoluto. There are may dfferet choces the costructo of pecewse affe subsolutos. We cosder δ δ δ δ pecewse affe subsolutos that take the form W = & W1 W W3 where δ W1 ( x) = & r1, x γ δ δ W ( x) = r, x γ δ & (5) δ W3 ( x) = & r3, x γ (1 γ / αδ ). Note that, r x, = 1,,3, s the er product betwee r ad x. There are dfferet ways to mollfy the pecewse affe subsoluto W δ. We wll adopt a mollfcato called expoetal weghtg that s specalzed here to the mmum of a fte set of smooth fuctos. The expoetal weghtg produces a smooth mollfcato of W δ by ρ εδ, ( x) = 3 k = 1 δ { W x ε} δ { Wk x ε} exp ( ) & (6) exp ( ) where ε s obtaed accordg to the theorem 1. D. Computato of exact probabltes The trasto probabltes the etwork are related to each other learly through the Markov property (Dupus et al,007). For small values of, oe ca accurately compute p smply by teratg ths relato ad usg the tal codto of p.the same method ca be used prcple for ay Jackso etwork, however, because the sze of the state space grows lke d where d s the umber of queues, t s oly feasble for small values of d ad. The exact values of excessve backlog probabltes ths paper were computed usg ths method. The precso these computatos were up to seve dgts after the floatg decmal pot. IV. Numercal results ad dscusso Two expermets have bee vestgated to show the relato betwee the relatve deadle,η, ad the probablty of mssg deadle, ϕ d the we use the results the mportace samplg techque to estmate the probablty of total populato overflow the etwork. I the frst expermet µ = 0.5, ad for varous values of λ, t s show that how a chage η affects the probablty of mssg deadle. As llustrated Fg., the crease relatve deadle reduces ϕ d lke a decay fucto. Further crease the relatve deadle moves the probablty of mssg deadle towards zero, such case, the possblty 7

8 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN of mssg deadle of customers s very low. O the cotrary, assgg low values for η creases the probablty of mssg customers' deadle leadg to rasg the feedback rate the etwork. The customers' arrval rate at the secod queue s λ 1, thaks to the stablty assumpto the etwork, thus the value of µ 1 wll ot affect the probablty of mssg deadle. I the secod expermet we evaluate the performace of the etwork for arbtrary values of the offered µ λ load ( ). Fgure 3 represets the probablty of mssg deadle for a set of offered loads, = 1 ad varous values ofη. I ths fgure the etwork has the best performace for η = 6 for 1 η =. ad t has the worst oe Fgure.. Probablty of mssg deadle for varous arrval rates Fgure. 3. Probablty of mssg deadle for varous servce rates The above results have bee used the mportace samplg techque to estmate the probablty of total populato overflow the etwork for arbtrary values ofϕ d. For the buffer sze of the etwork we set = 1, ad each estmate cossts of replcatos. We ru smulatos for ϕ d = 0., 0.4 ad 0.6. The theoretcal value s obtaed usg a umercal teratve algorthm descrbed secto

9 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN TABLE I IMPORTANCE SAMPLING ESTIMATION, λ = 0.1, µ 1 = 0.5, µ = 0.4 TABLE II IMPORTANCE SAMPLING ESTIMATION, λ = 0.1, µ 1 = 0.4, µ = 0.5 I Table 1 we set λ = 0.1, µ 1 = 0.5 ad µ = 0.4, I Table we set λ = 0.1, µ 1 = 0.4 ad µ = 0.5. I all tables, "Std. Err" stads for "Stadard Error" ad "C.I." for "Cofdece Iterval". The results of smulatos dcate that the performace of the mportace samplg techque s stable across dfferet smulatos, wth estmates that are close to the theoretcal value wth small stadard errors. V. Cocluso I ths paper, we deal wth the cocept of deadle o a two-ode Jackso etwork wth feedback whch arrval ad servce rates are modulated by a exogeous fte state Markov process. Based o the defto of deadle for customers the secod queue, we have calculated the probablty of mssg deadle, ad have show how the feedback rate of the etwork s affected by the deadle value, the results have used a mportace samplg estmator. It was foud that a crease the probablty of mssg deadle rases the probablty of total populato overflow. Lkewse, t was kow that how the probablty of total populato overflow affected by servce rates the etwork. 9

10 Iteratoal joural of Computer Scece & Network Solutos September.013-Volume 1.No ISSN Refereces. A. Movaghar, O queueg wth customer mpatece utl the ed of servce, Stochastc Models., vol., o. 1, May. 006, pp v. A. Movaghar, O queueg wth customer mpatece utl the ed of servce, Proc. 4th IEEE Iteratoal Computer Performace ad Depedablty Symposum, Chcago, 000, pp C.S. Chag, P. H, S. Jueja ad P. Shahabudd, Effectve badwdth ad fast smulato of ATM tree etworks, Performace Evaluato., vol. 0, o. 1, May. 1994, pp De Boer, P. J, Aalyss of state-depedet mportace samplg measures for the two-ode tadem queue, ACM Tras. Modelg Comp. Smulato, vol.16, Jul. 006, pp v. D.Y. Barrer, Queug wth Impatet Customers ad Ordered Servce, Operatos Research., vol. 5, o. 5, Oct. 1957, pp v. L. Klerock, Queueg Systems, Volume 1: Theory. New York, USA: Joh Wley & Sos, 1975, pp v. v. x. P. Dupus, A. Sezer, ad H. Wag, Dyamc mportace samplg for queueg etworks, The Aals of Appled Probablty., vol. 17, o. 3, Ja. 007, pp P. Dupus ad H. Wag. Subsolutos of a Isaacs equato ad effcet schemes for mportace samplg: Covergece aalyss. Preprt, 005. P. Dupus ad H. Wag. Subsolutos of a Isaacs equato ad effcet schemes for mportace samplg: Examples ad umercs. Preprt, 005 x. S. Asmusse, Appled Probablty ad Queues. New York: Sprg, 003, ch.4. x. x. S. Jueja, P. Shahabudd, Hadbook o Smulato, Amsterdam: Elsever, 006, ch.11. Peter-Tjerk De Boer, Vctor F. Ncola, Adaptve state-depedet mportace samplg smulato of markova queueg etworks, Europea Trasactos o Telecommucatos., vol.13, Apr. 00, pp