RESTART SIMULATION OF NETWORKS OF QUEUES WITH ERLANG SERVICE TIMES. José Villén-Altamirano

Proceedngs of the 009 Wnter Smulaton Conference M. D. Rossett, R. R. Hll, B. Johansson, A. Dunkn and R. G. Ingalls, eds. RESTART SIMULATION OF NETWORS OF QUEUES WITH ERLANG SERVICE TIMES José Vllén-Altamrano Dept. of Appled Mathematcs Polytechnc Unversty of Madrd Calle Arboleda s/n, 80 Madrd, Span ABSTRACT RESTART s an accelerated smulaton technque that allows the evaluaton of low probabltes. In ths method a number of smulaton retrals are performed when the process enters regons of the state space where the chance of occurrence of the rare event s hgher. These regons are defned by means of a functon of the system state called the mportance functon. Gudelnes for obtanng sutable mportance functons and formulas for the mportance functon of general Jackson networks were provded n prevous papers. In ths paper, we study networks wth Erlang servce tmes and wth the rare set defned as the number of customers n a target node exceedng a predefned threshold. The coeffcents of the mportance functons used here are the same as those obtaned wth the formula for Jackson networks but multpled by a constant obtaned heurstcally. Low probabltes are accurately estmated for dfferent network topologes wthn short computatonal tme. INTRODUCTION The study of crtcal events that occur very nfrequently s of nterest n many areas. The performance requrements of broadband communcaton networks and ultra relable systems are often expressed n terms of events wth very low probablty. Developng good estmates of qualty of servce provded by a network, requres studyng scenaros that may occur rarely durng the lfe of the network. Analytcal or numercal evaluaton of low probabltes s only possble for a very restrcted class of systems due to the model assumptons that are needed. Although smulaton s an effectve means of studyng such systems, acceleraton methods are necessary because crude smulatons requre prohbtvely long executon tmes for the accurate estmaton of very low probabltes. Importance samplng s a well known technque n rare event smulaton, see e.g., Rubno and Tuffn (009). The basc dea behnd ths approach s to alter the probablty measure governng events so that the formerly rare event occurs more often. One drawback of ths technque s the dffculty of selectng an approprate change of measure snce t depends on the system beng smulated. Most research has focused on fndng good heurstcs for partcular types of models. Dupus and Wang (008) deal wth the constructon of asymptotcally optmal mportance samplng schemes for queueng networks based on subsolutons to an assocated partal dfferental equaton. Importance samplng has dffcultes to deal wth large systems and/or systems wth strong feedback. Another known method s RESTART (REpettve Smulaton Trals After Reachng Thresholds), whch s based on a completely dfferent dea. In ths method a more frequent occurrence of a formerly rare event s acheved by performng a number of smulaton retrals when the process enters regons of the state space where the mportance s greater,.e., regons where the chance of occurrence of the rare event s hgher. These regons, called mportance regons, are defned by comparng the value taken by a functon of the system state, the mportance functon, wth certan thresholds. The retrals are klled f they go to a regon wth lower mportance than the regon where they were generated. Vllén-Altamrano and Vllén- Altamrano (99) coned the name RESTART and made a theoretcal analyss that yelds the varance of the estmator and the gan obtaned wth one threshold. A rgorous analyss of multple thresholds was made n Vllén-Altamrano and Vllén- Altamrano (00), where optmal values for thresholds and the number of retrals that maxmze the gan were derved. A precedent of much more lmted scope s the splttng technque descrbed n ahn and Harrs (95). See also chapter of Rubno and Tuffn (009) and the references theren. In ths method retrals are also performed but only the frst tme the process enters mportance regons and the retrals are not klled when they go to lower mportance regons. As a huge computatonal tme s wasted smulatng such unpromsng trals ths method s only vald for smulatons made by means of short replcas, e.g., regeneratve smulatons of very smple systems, or short transent smulatons. But even for smple cases Dean 978--444-577-7/09/$6.00 009 IEEE 46

Vllén-Altamrano and Dupus (009) showed that the computatonal tme of ther applcaton of RESTART (they called t RESTART/DPR) ranges from 4% to 4% of the correspondng tme requred wth splttng for the same smple systems. DPR (Drect Probablty Redstrbuton) was ntroduced by Haraszt and Towsend (999) and the authors clamed that t was a generalzaton of RESTART. The only true generalzaton s that DPR does not requre a nested sequence of mportance regons, but ths possblty has not been ever appled because t would lead to an neffcent algorthm. The other clamed generalzatons, as the possblty of umpng thresholds, or the possblty of reachng the rare set from several regons, were prevously contemplated wth RESTART. The applcaton of RESTART or Splttng to partcular models requres the choce of a sutable mportance functon. The problem of fndng the optmal mportance functon has been compared by Garvels et al. (00) wth the problem of fndng a good change of measure n mportance samplng, because n both cases knowledge about the behavour of the system leadng to the rare event s necessary. Glasserman et al. (999) stated that splttng ultmately reles on a detaled understandng of a process s rare event asymptotc, much as mportance samplng does. They suggested that t may be dffcult to use splttng or RESTART for systems wth multdmensonal state space. Vllén-Altamrano and Vllén-Altamrano (006) showed that t was possble to obtan heurstcally effectve mportance functons for multdmensonal systems wthout usng large devaton theory. Dean and Dupus (009) studed the constructon of asymptotcally optmal RESTART/ DPR algorthms based on subsolutons. Unlke wth mportance samplng, whch requres classcal sense subsolutons, RESTART requres subsolutons only n the vscosty sense. Vllén-Altamrano (009) obtaned a formula of the mportance functon for Jackson networks combnng heurstc arguments wth analytcal results. Dfferent types of networks wth dfferent loads of the nodes were smulated and very low overflow probabltes were accurately estmated wthn reasonable computatonal work. It s nterestng to study whether the mportance functon derved for Jackson networks would be ft for other networks. In ths paper we wll study networks wth Posson arrvals and Erlang servce tmes. We wll compare the results obtaned wth the mportance functon derved n Vllén-Altamrano (009) wth the results obtaned wth other mportance functons whose coeffcents are the same as those obtaned wth the formula for Jackson networks but multpled by a correcton factor obtaned heurstcally. The paper s organzed as follows: Secton presents a revew of the method, Secton descrbes the system under study and Secton 4 provdes the smulaton study. DESCRIPTION OF RESTART RESTART has been descrbed n several papers, e.g., Vllén-Altamrano and Vllén-Altamrano (00, 006). Nevertheless t s brefly descrbed here. Let Ω denote the state space of a process X(t) and A a rare subset of the state space whose probablty must be estmated. A nested sequence of sets of states C, C C,, CM s defned, whch determnes a partton of the state space Ω nto regons C C + ; the hgher the value of, the hgher the mportance of the regon C C +. These sets are defned by means of a functon Φ: Ω R, called the mportance functon. Thresholds T ( M) of Φ are defned so that each set C s assocated wth Φ T. RESTART works as follows: each tme the process enters a set C, the system state s saved and R trals of level are performed. Each tral of level s a smulaton path that starts wth the saved state and fnshes (except the last one) when t leaves set C. The last tral, whch contnues after leavng set C, potentally leads to new sets of trals of level f set C s reached agan. A set C + may be reached n a tral of level and an analogous process s set n moton: R + trals of level + are performed, and so on. In the case that the process crosses more than one threshold n a tme step, the retrals of all the crossed thresholds must be made. If, for nstance, a set C + s reached n a tral of level -, as set C s also reached, t s consdered that R trals of level are performed and that all of them reach threshold +. Thus, R R + trals of level + are performed, R of them fnsh when threshold s crossed down, and the other ones when threshold + s crossed down. Ths procedure for umpng thresholds was mplct n the descrpton of RESTART, see Vllén-Altamrano et al. (994). In fact, t was mplemented n the smulaton tool ASTRO (Advanced Smulaton Tool wth RESTART Optmzaton) descrbed n that paper. Some more notatons: P Pr{ A} = s the probablty of the system beng n a state of the set A at the nstant of occurrence of certan events denoted reference events (e.g., customer arrvals); C A; = M + 47

Vllén-Altamrano P h/ (0 h M + ) : probablty of the set C h at a reference event, knowng that the system s n a state of the set C at that reference event. For h M, as C h C, P h/ = Pr{ C } Pr{ C } ; r = R, M = Ω ( M) : set of possble system states x, when the process enters set C ; PA/ x ( M) : mportance of state x, defned as the expected number of events A n a tral of level startng wth that system state; P / ( M) : expected mportance when the process enters set C : A PA/ = E P A/ X P / ( ) = A xdf x Ω, where F(x ) s the dstrbuton functon of X ; V ( P ) ( M) / : varance of the mportance when the process enters set C : A X ( ) ( ) V P = P df( x ) ( P ). h A / X Ω A / x A / If the rare set A s ncluded n C M, the estmator of the probablty of A n a RESTART smulaton s: ˆ N A P =, where Nr M N A s the number of events A that occur n the smulaton and N the number of reference events smulated wthout countng the retrals. Otherwse, the estmator s: ˆ N M A P = = Nr, where N s the number of reference events A that occur n the regon C A C + (C M f = M). The formulas gven below n ths secton have been derved for the frst case. The gan (also called speedup) G obtaned wth RESTART can be defned as the rato of the computer cost tmes the varance of the crude smulaton estmator to the computer cost tmes the varance of the RESTART estmator, see Vllén- Altamrano and Vllén-Altamrano (00). In that paper, t s proved that G s gven by: G = ( ln + ) fv fo fr ft P P. () The factors f V, f O, f R and f T, all of them equal to or greater than (wth the excepton of f V whch may be smaller than n ( ) some cases), can be consdered neffcency factors that reduce the actual gan wth respect to the term P( ln P ) +. Ths term can be consdered the deal gan because t s the maxmum gan that can be obtaned (except n the cases where f V < ). Each factor reflects: f V : neffcency due to the non-optmal choce of the mportance functon. f O : neffcency due to the computer overhead of RESTART. f R : neffcency due to the non-optmal choce of the number of retrals. f T : neffcency due to the non-optmal choce of the thresholds. In Vllén-Altamrano and Vllén-Altamrano (00) crtera for mnmzng the factors f O, f R and f T were gven. A value of the factor f R equal to s acheved f the accumulated number of trals s chosen accordng to: r =, =,, M. () P P + / /0 As the number of trals R must be an nteger number, a value of f R very close to can be acheved wth the followng roundng algorthm: R s made equal to r rounded to an nteger number, R = r / R rounded to an nteger number,, 48

(... ) Vllén-Altamrano R = r R R rounded to an nteger number. An alternatve for obtanng nteger number of retrals, proposed n Haraszty and Towsend (999), s to randomze the number of trals R for achevng an expected number of R equal to r /r -. The factor f T s mnmzed by choosng very close thresholds,.e., P + / close to one. An upper bound of f T s gven by, see Vllén-Altamrano and Vllén-Altamrano (00): f T P + P mn mn where P mn Mn ( P + / ) 0 M ( ln P ) mn =. () The factor f O affects to the computatonal tme but not to the number of events to be smulated. Ths factor usually takes moderate values. In Vllén-Altamrano and Vllén-Altamrano (006) the factor f V was analyzed and gudelnes for choosng the mportance functon were provded. One of the gudelnes for reducng f V s to reschedule the scheduled events at the begnnng of ( ) each tral. An upper bound of f V was also gven n the paper: fv Max V ( PA/ X) ( P / ) A M+ to mnmze V ( P A X). It can be acheved by a proper choce of the mportance functon. / +. Thus, the man concern s In Vllén-Altamrano (009) a formula of the mportance functon vald for estmatng overflow probabltes of a target node of any Jackson network was obtaned. The formula (that wll be gven at the end of the next secton) led to small values of factor f V n most cases, ncludng bg networks, networks wth strong feedback and networks wth hgh dependency. The most problematc cases were networks wth hgh dependency of the target node on the queue length of the other nodes and wth the load of the target queue much lower than the load of the other queues. SYSTEM UNDER STUDY A network wth any number of nodes s studed. Customer arrvals and departures are allowed n all the nodes. After beng served n node l, customers can go to any node m wth probabltes p lm or they can leave the network wth probablty p l0. The steady-state probablty of the number of customers exceedng a level at a target node, L, s estmated. Let us denote the number of nodes at dstance, that s, the nodes whch are drectly connected wth the target node ( pl 0 ), H the number of nodes whch are connected wth the target node through only one ntermedate node (nodes at dstance ), and N the number of nodes at dstance, for any value of, H and N. The nodes at dstance greater than, are not taken nto account n the formula of the mportance functon because the dependence of the target node on these nodes s usually very weak. Customers wth ndependent Posson arrvals enter each node from the outsde wth arrval rates γ, n,, N 0 n = to the nodes at dstance, γ,,, = H to the nodes at dstance, γ,,, = to the nodes at dstance and γ to the target node. The total arrval rates to each node (arrvals from the outsde + arrvals from the other nodes) are denoted by: λ, n =,, N, 0 n λ,,, = H, λ,,, = and λ, respectvely. The servce tmes at all the nodes are assumed to have an Erlang dstrbuton wth α phases ( or n the examples of Secton 4) and wth servce rates µ, n =,, N, 0 n µ,,, = H, µ,,, = and µ, respectvely. The buffer space n each queue s assumed to be nfnte. Let us ob- N H serve that λ = γ + λ0n pn + λ l pl + λ p + λ p, =,, H = n= l= = Q, λ = γ + λ p + λl pl + λ p, =,,, H = l= λ = γ + λ p + λ p. The loads of the nodes are ρ = λ µ, =,, H, ρ,,, = λ µ = and ρ = λ µ, respectvely. A formula of the mportance functon for estmatng the probablty that Q L for any Jackson network was obtaned n Vllén-Altamrano (009). Wth that formula, very low probabltes for dfferent network topologes were accurately estmated wthn short or moderate computatonal tmes. A smplfed verson of ths formula (also gven n that paper) that matches wth the general one n almost all cases s the followng: 49

Vllén-Altamrano H ln ( ρ ) ρ ln ( ρ ρ ) α Q α Q Q, (4) Φ= + + ln ρ ln ρ = = where: ρ = ρ { ( ), } γ + Mn λ + µ λ p µ p + λ p = λ γ + µ p + λl pl + λ p l ρ = = ρ + µ λ ( µ λ ) p α = + γ p p + γ p + γ l l l = = µ p p = ; α = + H γ p p + γ p + γ l l l l = l l µ p. ρ and ρ are, approxmately, the loads of the target queue when a node at dstance from the target node or a node at dstance, respectvely, are not empty. It s more dffcult to get nsght of the meanng of α and α wthout followng the dervaton of Equaton (4). Nevertheless, the formulas are easy to apply because all ther terms are parameters of the system. Ths mportance functon was derved equatng the mportance of one extreme state (a system state when only one queue s not empty) wth the mportance of each of the other extreme states. We wll study whether the mportance of the extreme states would be affected n a smlar manner when the servces tmes are not exponentally dstrbuted and, as a consequence, the mportance functon derved for Jackson networks would be ft for networks wth Erlang servce tmes. As a partcular case of the prevous model, we wll study a three-queue tandem network wth the thrd node as the target node. For ths network, Equaton (4) matches wth the followng equatons: If ρ < ρ < ρ then ln ρ ln ρ Q Q Q ln ρ ln ρ (5) ln ρ If ρ ρ < ρ or f ρ ρ < ρ then Φ= ( Q+ Q) + Q. ln ρ (6) If ρ ρ ρ or f ρ ρ ρ then Φ= Q+ Q + Q. (7) ln ρ If ρ < ρ < ρ then Φ= Q+ Q + Q. ln ρ Remark: Ф s a functon of the current state of the system that could change at each nstant t. Ths dependence on t could Φ ln ln ( t ρ ) = ( ) ( ) ( ) ln Q t ρ be wrtten n the formulas. For example, Equaton (5) could be wrtten as: ρ + ln ρ Q t + Q t. 4 TEST CASES We conducted several smulaton experments on networks wth dfferent topologes and loads. The rare set A was defned as Q L, where Q s the number of customer at the target node. The steady state probablty of A was of the order of 0-5 n all the examples. Thresholds T were set for every nteger value of Φ between and M, where M vares between L and L+ dependng on the case beng smulated. Therefore, the rare set can be reached n retrals from the last, or 4 regons C C + (C M f = M). Plot runs (one or two for each case) were made to estmate the probabltes P,,..., /0 = M +, and 50

Vllén-Altamrano then to set the number of retrals accordng to Eq. (), followng the gudelnes gven n Secton for roundng to nteger values. The nterval wdth of each confdence nterval was evaluated usng the ndependent replcaton method. Each replcaton (sample) fnshed after a fx number of arrvals (usually between 00.000 and 500.000). After each sample the half wdth of the 95% confdence nterval dvded by the estmate (relatve error) was calculated and the smulaton fnshed when the relatve error was smaller than 0.. For each case we made smulatons, and we wrote n the tables the results correspondng to the medan of the computatonal tmes. All the experments were run on a Pentum(R) D CPU.0 GHz. 4. Three-Queue Tandem Network Customers arrve to the frst queue of ths network accordng to a Posson dstrbuton wth arrval rate equal to, then go to the second queue, then to the thrd and then they leave the network. The servce tme at each node follows an Erlang dstrbuton wth shape parameter equal to α ( or ). Intally, the mportance functon Φ was chosen accordng to Eqs. (5), (6), and (7), that s, Φ = aq+ bq + Q, wth values of a and b dependng on the loads ρ, ρ and ρ. Then, the coeffcents a and b of Φ were multpled by a correcton factor k, for k = 0.6, 0.7, and 0.9. The comparson among the mportance functons was made for less rare set probabltes (of the order of 0-9 ) because the results obtaned are also vald for much lower probabltes, and so we could do the comparson wth shorter computatonal effort. We have checked that the best value of k for estmatng probabltes of the order of 0-9 was also the best for estmatng probabltes of the order of 0-5. The results, summarzed n Table, correspond to the values of a and b for whch the computatonal tme was lowest. Table : Results for the three-queue tandem network wth Posson arrvals and Erlang (α, β) servce tmes. Rare set probablty: P( Q L). Relatve error = 0.. ρ = /. Φ = aq+ bq + Q. ρ ρ α L ˆP k a b Events Tme Gan f V Gan f 0 (mllons) (mnutes) (events) (tme) / / / / /5 /5 / / / / /4 /4 8 0 6 4.8 0-5 5. 0-5.9 0-5 9.7 0-6.7 0-5.99 0-5 0.7 0.6 0.7 0.9 0.6 0. 0.50 0.44 0.9 0.44 0.50 0.44 0.9.5.0.9 4. 0.6 9.6.8.8.9 0.4 0.6. 0 0.6 0 9. 0 8. 0 0 4.8 0. 0 4.9 0.9.6 7.6 0.7.6 7.6 0 8.4 0 8 9.4 0 9 5.6 0 9.6 0 0 9. 0 9 4.5 5.0.8 4.5.4 4.0 We can observe that for α =, the value of k s closer to than for α =. As the coeffcent of varaton of Pearson of the Erlang dstrbuton s α, t seems that the more smlar s ths coeffcent to that of the exponental dstrbuton, the closer s the mportance functon to that gven by the formula. It s also observed that lower values of the loads of the two frst nodes lead to mportance functons closer to those derved for the exponental dstrbuton. Future studes wll gve us more nsght for estmatng n advance a good value of k. To evaluate approxmately the gan n events or the gan n tme wth respect to a crude smulaton, the data of the crude smulatons were estmated by extrapolatng the measured values for lower probabltes, see e.g., Vllén-Altamrano and Vllén-Altamrano (006). The extrapolaton was made takng nto account that wth crude smulaton the relatve error for estmatng a probablty P wth n samples s proportonal to P ( P) n P np. Thus, the number of samples for achevng a gven relatve error s nversely proportonal to the probablty that s to be estmated. Factors f V s estmated by comparng the measured gan n events wth the theoretcal one derved from Eq. (), followng the procedure explaned n that paper and f O s estmated as the rato between gan n events and gan n tme. The values of factor f O are much greater than those obtaned n Vllén-Altamrano (009) for the same network but wth exponental servce tmes. The reason s that reschedulng s straghtforward only for the exponental dstrbuton due to the memoryless property of ths model. Reschedulng servce tmes of any other dstrbuton s more tme consumng. We can proceed as follows: a random value of the whole servce tme of a costumer s obtaned. If that value s greater than the servce tme at the current tme, the remanng servce tme of the customer s obtaned as the dfference between the two amounts. Otherwse a new random value s obtaned and so on. As we must reschedule the servce tmes when any threshold s reached, the computatonal tme wth Erlang servce tmes s around four tmes greater than wth exponental tmes for estmatng a probablty of the order of 0-5. The values of factor f V are smlar to those obtaned n Vllén-Altamrano (009) for the same network but wth exponental servce tmes. The low or moderate values of f V show that the choce of the mportance functon s approprate and that the applcaton s not far from the optmal, at least for the tested cases. We observe that the worst results (greatest 5

Vllén-Altamrano computatonal tmes and greatest values of factor f V ) were obtaned when ρ < ρ < ρ, but even n these cases the computatonal tmes are moderate (lower than mnutes). For the other four cases, the computatonal tmes needed for estmatng probabltes of the order of 0-5 were lower than 4 mnutes. We have studed the senstvty of the results to changes n the mportance functon, n partcular for varous correcton factors. We have observed that values of f V for values of a and b up to 0% (0% n the last two cases) lower or greater than those gven n Table, became less than double those of the best result. The mportance functons gven by Eqs. 5-7 lead to greater computatonal tmes. In the last four cases these tmes are moderates (lower than 0 mnutes), but n the second case the computatonal tme was greater than one day. 4. A Network wth Seven Nodes Let us now consder a network wth 7 nodes. Customers from the outsde arrve at any node of the network wth a rate γ =, =,, 7. After beng served n each node, a customer leaves the network wth probablty 0.. Otherwse, the customer goes to another node n accordance to the followng transton matrx: 4 5 6 0. 0. 0. 0. 0. 0. 0 0. 0. 0. 0. 0. 0. 0 0. 0. 0. 0. 0. 0. 0 4 0. 0. 0. 0. 0. 0. 0 5 0. 0. 0. 0. 0 0. 0. 6 0. 0. 0. 0. 0. 0 0. 0. 0. 0. 0. 0. 0. 0. We can observe n the matrx that nodes 5 and 6 are at dstance from the target node, and that nodes to 4 are at dstance from the target node. We can also observe that the network s a full mesh network but only nodes 5 and 6 are drectly connected to the target. The overall arrval rate to each node s λ = 4.5 for nodes at dstance, λ = 5.7 for nodes at dstance and λ = 5.55. The rare set s Q L, where Q s the target queue. The load of the target node s 0.6. Servces rates were chosen such that nodes 5 and 6 have the same load, gven by ρ n Table, and also nodes to 4 (ρ n Table ). The mportance functon Φ was chosen accordng to Eq. (4) wth the coeffcents gven n Secton, that s, wthout applyng any correcton factor. The gan and the factors f V and f O were estmated as n Secton 4.. The results, whch are summarzed n Table, show that very accurate results were obtaned n short computatonal tme, less than mnutes n all the cases. Table : Results for a Jackson network wth 7 nodes, Posson arrvals and Erlang (α, β) servce tmes. Rare set probablty: ( L) P Q. Relatve error = 0.. ρ = 0.6. Φ = a Q + b Q + Q 4 6. = = 5 ρ ρ α L ˆP a b Events (mllons) 0.50 0.4 6.69 0-5 0. 0.45. 0.50 0.4 5.85 0-5 0. 0.45.7 0. 0.4 6.80 0-5 0.6 0.45. 0. 0.4 5.76 0-5 0.6 0.45. 0.0 7.95 0-5 0.40 0.6. 0.0 6.7 0-5 0.40 0.6.8 Tme (seconds) 77 9 40 66 40 59 Gan (events).9 0. 0. 0.7 0 4.4 0.0 0 f V.4.7 0.9.5. Gan (tme).9 0 0. 0 0.6 0 0.9 0 0 7. 0 0 4.8 0 0 We can also observe that the worst results (greater computatonal tmes) were obtaned when ρ < ρ < ρ, though the computatonal tmes needed for estmatng probabltes of the same order of magntude s much lower than n the prevous f 0 6.6 6.8 5.8 5.8 6. 6. 5

Vllén-Altamrano network. The results are better than n ths network because many customers of the other queues never go to the target queue. Thus, the dependence of the target queue on the queue length of the other queues s weaker and the effcency of RESTART s greater. Unlke wth mportance samplng, the effcency of RESTART may mprove wth the complexty of the systems and t does not seem affected by the feedback. The values of factor f O are also greater than those obtaned wth exponental servce tmes, but the dfferences are lower than n the prevous network because n complex networks the proporton of computatonal tme that the smulaton program s makng reschedulng s lower than n smple networks. In the three cases of α = the best results were obtaned wth the mportance functon gven by Eq. (4), whle for α = the best results were obtaned wth coeffcents of Q and Q around 0% lower than those gven by Eq. (4), that s, wth a correcton factor of k = 0.9. Nevertheless, very good results were also obtaned wthout any correcton factor, as can be seen n the table. The very low values of f V acheved show that the applcaton s very close to the optmal one, at least for the tested cases. The senstvty of the choce of these coeffcents s smaller than n the prevous network, because acceptable results (values of f V less than double those of the best results) are obtaned for values of a and b up to 0% lower or greater than the optmal ones. The three cases were also smulated for α = 0 wth the mportance functon gven by Eq. (4). The parameter β of the Erlang dstrbuton was also changed for achevng the loads of Table. The computatonal tmes for estmatng probabltes of the order of 0-5 were very smlar to those obtaned wth α =. It seems to ndcate that ths mportance functon leads to very good results n ths network for Erlang servce tmes wth any number of phases α. 5 CONCLUSIONS The choce of the mportance functon s the most crtcal feature when the method RESTART s appled to multdmensonal systems. Ths paper has focused on fndng formulas of effectve mportance functons for two types of networks: a three-queue tandem network and a more complex network wth seven nodes, n both cases wth dfferent loads and wth Posson arrvals and Erlang servce tmes. In the second network the formulas of the mportance functons were the same as those derved n Vllén-Altamrano (009) for exponental servce tmes, whle for the tandem network the coeffcents of the mportance functons were multpled by a correcton factor equal to 0.6, 0.7, or 0.9. Ths correcton factor s closer to for α = than for α =. It could be due to the coeffcent of varaton of Pearson that s more smlar to that of the exponental dstrbuton for α =. Although t has been easy to obtan heurstcally the correcton factor, future studes wll gve us more nsght for estmatng n advance a good value of ths factor. Overflow probabltes lower than those needed n practcal problems (around 0-5 ) have been accurately estmated wthn short computatonal work. The worst results are obtaned when the dependence of the target queue on the length of the other queues s very hgh (as t occurs n a tandem network) and the load of the target queue s much lower than the other ones. As a consequence, the effcency of RESTART often mproves wth the complexty of the system because the dependence of the target queue on the other queues s weaker. The effcency does not seem affected by the feedback. It would be nterestng to see whether the mportance functon derved for Jackson networks would be ft for other networks or at least f a sutable mportance functon can be obtaned by multplyng the coeffcents of the mportance functon (except the target coeffcent) by the same correcton factors. The methodology followed n ths paper could be used n future researches to determne the types of non-markovan mult-dmensonal networks for whch those mportance functons are vald. ACNOWLEDGEMENTS I should lke to thank Mr. Iván García-Jménez for hs assstance n wrtng the smulaton program and Mr. Manuel Vllén-Altamrano for hs helpful comments. REFERENCES Dean, T., and P. Dupus. The desgn and analyss of a generalzed DPR/RESTART algorthm for rare event smulaton. Submtted to Annals of Operaton Research. Dupus, P., and H. Wang. Importance samplng for Jackson networks. Submtted to QUESTA. Garvels, M.J.J., J..C.W. Ommeren, and D.P. roese. 00. On the mportance functon n splttng smulaton. European Transacton on Telecommuncatons (4): 6-7. 5

Vllén-Altamrano Glasserman, P., P. Hedelberger, P. Shahabuddn, and T. Zac. 999. Multlevel splttng for estmatng rare event probabltes. Operaton Research 47(4): 585-600. Haraszt, Z., and J.. Townsend. 999. The theory of drect probablty redstrbuton and ts applcaton to rare event smulaton. ACM Transacton on Modellng and Computer Smulaton 9(): 05-40. ahn H, and T.E. Harrs. 95. Estmaton of partcle transmsson by random samplng. Natonal Bureau of Standards Appled Mathematcs Seres : 7-0. Rubno, G., and B. Tuffn (edtors). 009. Rare even smulaton usng Monte Carlo methods. Chchester: Wley. Vllén-Altamrano J. 009. Importance functons for RESTART smulaton of general Jackson networks. To appear n European Journal of Operaton Research. Vllén-Altamrano M., A. Martínez-Marrón,, J. Gamo, and F. Fernández-Cuesta 994. Enhancement of the accelerated smulaton method RESTART by consderng multple thresholds. Proceedng of the 4th Internatonal Teletraffc Congress. In The Fundamental Role of Teletraffc n the Evoluton of Telecommuncatons Networks, 787-80. Elsever Scence Publsher, Amsterdam. Vllén-Altamrano M., and J. Vllén-Altamrano. 99. RESTART: a method for acceleratng rare event smulatons. Proceedng of the th Internatonal Teletraffc Congress. In Queueng, Performance and Control n ATM, 7-76. Elsever Scence Publsher, Amsterdam. Vllén-Altamrano M., and J. Vllén-Altamrano. 00. Analyss of RESTART smulaton: theoretcal bass and senstvty study. European Transacton on Telecommuncatons (4): 7-86. Vllén-Altamrano M., and J. Vllén-Altamrano. 006. On the effcency of RESTART for multdmensonal state systems. ACM Transacton on Modellng and Computer Smulaton 6(): 5-79. AUTHOR BIOGRAPHY JOSÉ VILLÉN-ALTAMIRANO s Professor of the Department of Appled Mathematcs at the Polytechnc Unversty of Madrd. He receved M.S. degree n Mathematcs from Complutense Unversty of Madrd n 977 and he receved Ph. D. degree n Computer Scence from Polytechnc Unversty of Madrd n 988. Hs research nterests are focused on relablty, queueng theory and rare event smulaton. Hs emal s vllen@eu.upm.es. 54