Motivated by large-scale service systems with network structure, we introduced in a previous paper a

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1 Publshed onlne ahead of prnt Aprl 11, 213 INFORMS Journal on Computng Artcles n Advance, pp ISSN (prnt) ISSN (onlne) INFORMS Algorthms for Tme-Varyng Networks of Many-Server Flud Queues Yunan Lu Department of Industral and Systems Engneerng, North Carolna State Unversty, Ralegh, North Carolna, 27695, yunan_lu@ncsu.edu Ward Whtt Department of Industral Engneerng and Operatons Research, Columba Unversty, New York, New York 127, ww24@columba.edu Motvated by large-scale servce systems wth network structure, we ntroduced n a prevous paper a tme-varyng open network of many-server flud queues wth customer abandonment from each queue and tme-varyng proportonal routng among the queues, and showed how performance functons can be determned. The determnstc flud model serves as an approxmaton for the correspondng non-markovan stochastc network of many-server queues wth Markovan routng, experencng perods of overloadng at the queues. In ths paper we develop a new algorthm for the prevous model and generalze the model to nclude non-exponental servce-tme dstrbutons. In ths paper we report results of mplementng the algorthms and studyng ther computatonal complexty. We also conduct smulaton experments to confrm that the algorthms are effectve n computng the performance functons and that these performance functons provde useful approxmatons for the correspondng stochastc models. Key words: queues wth tme-varyng arrval rates; nonstatonary queues; queueng networks; many-server queues; determnstc flud models; flud approxmaton; nonstatonary networks of flud queues; customer abandonment; non-markovan queues Hstory: Accepted by Wnfred Grassmann, Area Edtor for Computatonal Probablty and Analyss; receved February 212; revsed May 212; accepted September 212. Publshed onlne n Artcles n Advance. 1. Introducton Servce systems typcally have tme-varyng arrval rates, wth sgnfcant varaton over the day, that nhbt applcaton of tradtonal stochastc modelng and analyss. Thus operatons researchers have developed a growng collecton of tools to cope wth the tme-varyng arrval rates to analyze and mprove the performance of these systems; see the recent survey by Green et al. (27). Servce systems are also becomng ncreasngly complex, exhbtng mportant network structure. Network structure s evdent n many applcatons, e.g., healthcare delvery systems, dstrbuted customer contact centers, and emergency response and relef organzatons. Because the customers typcally are people, these servce systems also commonly have customer abandonment, ncludng nonexponental patence dstrbutons. These factors motvated us n Lu and Whtt (211a) to develop a new model ncorporatng all these features. In partcular, we ntroduced a tme-varyng open network of many-server flud queues, whch we call a flud queue network (FQNet). The specfc model was the G t /M/s t +GI m /M t FQNet, whch has m flud queues, each wth a tme-varyng external arrval rate (the G t ), a tme-varyng staffng functon (the s t ) wth unlmted watng space, exponental servce (the M) and abandonment from queue accordng to a general dstrbuton (the +GI), plus tme-varyng proportonal routng from one queue to another (the fnal M t ). The general patence (tme-to-abandon) dstrbuton and servce dstrbuton (that appears n one algorthm) lead to consderng two-parameter performance functons at each queue, such as Q t y, the flud content n queue at tme t that has been so for at most tme y, as a functon of t and y. In ths paper we extend our prevous work n four mportant drectons. Frst, we solve the more general G t /GI/s t + GI m /M t FQNet wth nonexponental servce-tme dstrbuton, whch s mportant because servce tme dstrbutons are commonly found to be nonexponental (often lognormal); e.g., see Brown et al. (25). Second, we develop an entrely new algorthm based on solvng an m-dmensonal ordnary dfferental equaton (ODE) to fnd the vector of tme-varyng arrval rates at each queue, for the G t /M/s t + GI m /M t FQNet wth exponental servce tmes. Because the sngle-queue algorthm developed n Lu and Whtt (212a) requres solvng an ODE for the head-of-lne watng tme, ths new ODE method s valuable because t provdes a unfed 1

2 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 2 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS ODE framework for the entre analyss. Thrd, we show that the new ODE framework allows us to gve closed-form expresson for the arrval rates at each queue n the case of a two-queue network. Fnally, we mplement all the FQNet algorthms for the frst tme here and study ther computatonal complexty, thus verfyng that they can be effcently appled. In partcular, we compare the two dfferent algorthms for solvng the G t /M/s t + GI m /M t FQNet and reveal the advantages of each. We study how the algorthms perform for large networks by consderng a famly of networks wth m queues, wth m gong up to 16. The FQNets studed here are determnstc flud models so that the performance s necessarly descrbed by determnstc functons. Nevertheless, these flud models are ntended for applcatons to systems that evolve wth consderable uncertanty, as commonly captured by stochastc models wth stochastc arrval processes, servce tmes, abandonment, and routng. The flud models can provde useful nformaton when the predctable (determnstc) varaton n arrval rates and other model elements domnates or s comparable to the unpredctable (stochastc) varaton because of uncertanty. Ths tends to be the case when the system experences perods of overloadng. Accordngly, the flud models here are analyzed under the assumpton that the system alternates between successve overloaded (OL) and underloaded (UL) ntervals. Ths behavor commonly occurs when t s too dffcult or costly to dynamcally adjust staffng n response to tme-varyng arrval rates to precsely balance supply and demand at all tmes commonly occurng n healthcare. FQNets are legtmate models n ther own rght, but they also are ntended to serve as approxmatons for correspondng non-markovan stochastc queueng networks (SQNets), where the M t routng becomes tmevaryng Markovan routng; a departure from queue at tme t goes (nstantaneously) next to queue j wth probablty P j t, ndependent of the system hstory up to that tme. In the FQNet, a proporton P j t of the flud flow out of queue at tme t goes next to queue j. In the SQNet, servce tmes and patence tmes are random tmes for ndvdual customers. In the FQNet, they specfy flow proportons;.e., wth patence cumulatve dstrbuton functon (cdf) F at queue, F t represents the proporton of all flud that abandons by tme t after t jons the queue, f t has not already entered servce. For the assocated non-markovan SQNets, there are few useful analyss tools besdes dscreteevent stochastc smulaton. We envson the FQNets here beng used n performance analyss together wth smulaton of assocated SQNets. The FQNets can be analyzed much more rapdly, and so may be used effcently n prelmnary analyses, e.g., to effcently derve canddate staffng functons at all queues. Smulaton of SQNets can then be appled to verfy and refne the FQNet analyss. There s a body of mportant related lterature. Frst, there s a long hstory of flud queue models (Newell 1982). Second, among the lmted lterature on SQNets wth tme-varyng arrval rates, an mportant contrbuton was made by Mandelbaum et al. (1998), who establshed many-server heavy-traffc lmts for Markovan SQNets, showng that FQNets and assocated dffuson process refnements arse n the manyserver heavy-traffc lmt, n whch the arrval rate and staffng are both allowed to grow; see also Mandelbaum et al. (1999a, b). Detaled analyss can also be successfully performed for nfnte-server (IS) SQNets, havng nfntely many servers at each queue. Markovan IS SQNets were studed by Massey and Whtt (1993), and IS SQNets wth tme-varyng phasetype (PH t ) dstrbutons were studed by Nelson and Taaffe (24a, b). Nelson and Taaffe (24a, b) nvestgated PH t /PH t / m SQNets wth multple customer classes and tme-varyng phase-type arrval and servce processes. They showed that ths IS network wth k classes s mathematcally equvalent to k sngleclass IS networks, each of whch s furthermore equvalent to the PH t /PH t / IS model wth a modfed servce dstrbuton. They therefore drectly appled the numercal algorthm they frst developed for the PH t /PH t / model to the PH t /PH t / m SQNets. Parallelng that analyss technque, we demonstrate how the algorthm for the sngle G t /GI/s t + GI flud queue n Lu and Whtt (212a) can be appled to the G t /M t /s t + GI t m /M t FQNet. The motvaton and theory for non-markovan sngle many-server flud queues was gven by Whtt (26) and Lu and Whtt (211a, b; 212a; 213). Those works nclude extensve comparsons wth smulatons of stochastc models and supportng heavy-traffc lmt theorems. Kang and Pang (211) developed an alternatve algorthm for a flud queue based on a random-measure perspectve that does not requre alternatng OL and UL ntervals, but so far requres constant staffng (whch can be appled more generally n a pecewse-constant manner). We evaluate the performance of the algorthms by mplementng them and conductng smulaton experments for assocated SQNets for several examples. To relate the FQNets to assocated SQNets, we use many-server heavy-traffc scalng, as n Lu and Whtt (212b, 213) and references theren. Thus, for a stochastc queue ndexed by scale parameter n, we let the arrval rate be n t and the number of servers be ns t, where t and s t are the flud model counterparts, and x s the least nteger greater than or equal to x. We llustrate now wth an example of a twoqueue SQNet as depcted n Fgure 1 of Lu and

3 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 3 Fgure 1 w(t) Q(t) B(t) X(t) w 1 (t): Sm w 2 (t): Sm w 1 (t): Num w 2 (t): Num Q 1 (t): Sm Q 2 (t): Sm Q 1 (t): Num Q 2 (t): Num B 1 (t): Sm B 2 (t): Sm B 1 (t): Num B 2 (t): Num X 1 (t): Sm X 2 (t): Sm X 1 (t): Num X 2 (t): Num Tme t A Comparson of Performance Functons n the Two-Queue FQNet wth Sngle Sample Paths from a Smulaton of the Correspondng SQNet wth Scale Parameter n = 4 Whtt (211a); see 6.2 for detals about ths example. Fgure 1 compares the flud approxmaton (the dashed lnes) wth smulaton estmates of the performance n the stochastc model (the sold lnes) for n = 4. We plot sngle sample paths of the followng processes: () the elapsed watng tme of the customer at the head of the lne, W n t ; () the scaled number of customers watng n queue, Q n t Q n t /n; () the scaled number of customers n servce, B n t B n t /n; and (v) the scaled total number of customers n the system, X n t X n t /n. For ths extremely large value of n, there s lttle varablty n the smulaton sample paths. Fgure 1 shows that each smulated sample path falls rght on top of the FQNet approxmaton. The close agreement confrms that both the numercal algorthm and the smulaton must be done correctly, and t emprcally valdates the many-server heavy-traffc lmt. For more realstc stochastc models wth fewer servers, the flud performance functons serve as approxmatons for the mean values of the correspondng stochastc processes. A fgure nearly dentcal to Fgure 1 (Fgure 8 n the onlne supplement, avalable at shows that the flud model provdes excellent approxmatons for the mean values of the same example wth n = 5. Then the sold lnes become smulaton estmates of the mean of these scaled stochastc processes, obtaned by averagng multple ndependent sample paths. The rest of ths paper s organzed as follows. In 2 we revew the sngle G t /M t /s t + GI t flud queue studed n Lu and Whtt (211a, 212a). In 3 we revew the G t /M t /s t + GI t m /M t FQNet and ts results developed n Lu and Whtt (211a). We also specfy the frst fxed pont equaton (FPE)-based algorthm, Alg(FPE), n 3.2. In 4 we develop the alternatve algorthm, Alg(ODE), based on solvng an m-dmensonal ODE. In 5 we develop the new FPE-based algorthm, Alg(FPE, GI), for the G t /GI/s t + GI t m /M t model wth general servce-tme dstrbutons at each queue. In 6 we demonstrate the performance of the algorthms by consderng several examples. We also confrm conclusons drawn about the computatonal complexty. Addtonal materal appears n the onlne supplement, ncludng a dscusson about checkng for volaton of staffng feasblty. 2. The G t /M t /s t + GI t Sngle Flud Queue In ths secton we revew the G t /M t /s t + GI t flud queue model and ts performance; see Lu and Whtt (211a, 212a) for more detals.

4 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 4 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 2.1. Specfyng the Model A sngle flud queue s a servce faclty wth fnte capacty and an assocated watng room or queue wth unlmted capacty. Flud s a determnstc, dvsble, and ncompressble quantty that arrves over tme. Flud nput flows drectly nto the servce faclty f there s free capacty avalable; otherwse t flows nto the queue. Flud leaves the queue and enters servce n a frst-come, frst-served (FCFS) manner whenever servce capacty becomes avalable. There cannot be smultaneously free servce capacty and postve queue content. The staffng functon (servce capacty) s an absolutely contnuous postve functon s t wth dervatve s t. The servce capacty s exogenously specfed, provdng a hard constrant. In general, there s no guarantee that some flud that has entered servce wll not be later forced to leave wthout completng servce, because we allow s to decrease. We drectly assume that phenomenon does not occur;.e., we drectly assume that the gven staffng functon s feasble. However, Lu and Whtt (211a, Theorem 6) show how to construct a mnmum feasble staffng functon greater than or equal to an ntal nfeasble staffng functon. The total flud nput over an nterval t s t, the ntegral of a postve arrval rate functon t. Servce and abandonment occur determnstcally n proportons. Because the servce s M t, the proporton of flud n servce at tme t that wll stll be n servce at tme t + x s Ḡ t x = e M t t+x where M t t + x +x t y dy (1) for t and x. The cdf of the servce tme of a quantum of flud that enters servce at tme t s G t 1 Ḡ t x ; Ḡ t x s the complementary cdf (ccdf). The cdf G t has densty g t x = t + x Ḡ t x and hazard rate h Gt x = t + x, x. The model allows for abandonment of flud watng n the queue. In partcular, a proporton F t x of any flud to enter the queue at tme t wll abandon by tme t + x f t has not yet entered servce, where F t s a cdf wth densty f t y for each t. Let h Ft y f t y / F t y be the hazard rate assocated wth the patence (abandonment) cdf F t. System performance s descrbed by a par of two-parameter determnstc functons ˆB ˆQ, where ˆB t y ( ˆQ t y ) s the total quantty of flud n servce (n queue) at tme t that has been so for a duraton at most y, for t and y. (Alternatvely, ˆB ˆQ can be regarded as a par of tme-varyng measures.) These functons were shown to be absolutely contnuous n the second parameter, so that ˆB t y y b t x dx and ˆQ t y y q t x dx (2) for t and y. Performance s prmarly characterzed through the par of two-parameter flud content denstes b q. Let B t ˆB t and Q t ˆQ t be the total flud content n servce and n queue, respectvely. Let X t B t + Q t be the total flud content n the system at tme t. Because servce s assumed to be M t, the performance wll prmarly depend on b va B. (We wll not drectly dscuss ˆB.) The total servce completon rate and abandonment rate at tme t are t b t x h Gt x dx = B t t t (3) t b t x h Ft x dx (4) respectvely. The total amount of flud to complete servce n the nterval t s S t y dy = B y y dy t (5) Because flud n servce (queue) that s not served (does not abandon or enter servce) remans n servce (queue), the flud content denstes b and q must satsfy the equatons b t + u x + u = b t x Ḡt x x + u Ḡ t x x = b t x e M t t+u (6) F q t + u x + u = q t x t x x + u F t x x x + u < w t (7) for t, x, and u, where M s defned n (1), and w t s the boundary watng tme (BWT) at tme t, w t nf { x > q t y = for all y > x } (8) (By Assumptons 7 9 of Lu and Whtt 211a, we never dvde by zero n (6) and (7). Because the servce dscplne s FCFS, flud leaves the queue to enter servce from the rght boundary of q t x.) Let A t be the total amount of flud to abandon, and let E t be the total amount of flud to enter servce n t. For each t, we have the flow conservaton equatons Q t = Q + t A t E t The abandonment satsfes A t B t = B + E t S t y dy t and (9) q t y h Ft y y dy (1) for t, where t s the abandonment rate at tme t and h Ft y s the hazard rate assocated wth the

5 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 5 patence cdf F t. (Recall that F t s defned for t extendng nto the past.) The flow nto servce satsfes E t b u du t (11) where b t s the rate flud enters servce at tme t. If the system s OL, then the flud to enter servce s determned by the rate that servce capacty becomes avalable at tme t: t s t + t = s t + B t t t (12) Then t concdes wth the maxmum possble rate that flud can enter servce at tme t: t s t + s t t (13) To descrbe watng tmes, let the BWT w t be the delay experenced by the quantum of flud at the head of the queue at tme t, already gven n (8), and let the potental watng tme (PWT) v t be the vrtual delay of a quantum of flud arrvng at tme t under the assumpton that the quantum has nfnte patence. Proper defntons of q, w, and v are somewhat complcated, because w depends on q, and q depends on w, but that has been done n 7 n Lu and Whtt (212a). The ntal condtons are specfed va the ntal flud denstes b x and q x, x. Then ˆB y and ˆQ y are defned va (2), and B ˆB and Q ˆQ as before. Let w be defned n terms of q as n (8). We assume that B Q and w are fnte. In summary, the sextuple t, s t t F t x b x q x of functons of the varables t and x specfes the model data that we assume s sutably smooth; see Assumpton 5 of Lu and Whtt (211a). The system performance s characterzed by b t x q t x w t v t t t. We analyze the flud queue by consderng alternatng ntervals over whch the system s ether UL or OL, where these ntervals nclude what s usually regarded as crtcally loaded. In partcular, an nterval startng at tme t wth () Q t > or () Q t =, B t = s t and t > s t + t, s OL. Let R denote the current system regme; e.g., we wrte R t OL. The OL nterval ends at the OL termnaton tme: T OL t nf { u t Q u = and u s u + u } (14) Case (), where Q t = and B t = s t, s often regarded as crtcally loaded, but because the arrval rate exceeds the rate that new servce capacty becomes avalable, s t + t, we must have the rght lmt Q t + >, so that there exsts > such that Q u > for all u +. Hence, we necessarly have T OL t >. An nterval startng at tme t wth () Q t < or () Q t =, B t = s t, and t s t + t s UL, desgnated by R t = UL. The UL nterval ends at UL termnaton tme: T UL t nf { u t B u = s u and u > s u + u } (15) As before, case (), n whch Q t =, and B t = s, s often regarded as crtcally loaded, but because the arrval rate t does not exceed the rate that new servce capacty becomes avalable, t s t + t, we must have the rght lmt Q t + =. The UL nterval may contan subntervals that are conventonally regarded as crtcally loaded;.e., we may have Q t =, B t = s t, and t = s t + t. For the flud models, such crtcally loaded subntervals can be treated the same as UL subntervals. However, unlke an overloaded nterval, we cannot conclude that we necessarly have T UL t > for a UL nterval. Moreover, even f T UL t > for each UL nterval, we could have nfntely many swtches between OL ntervals and UL ntervals n a fnte nterval. Thus we make assumptons to ensure that those pathologcal stuatons do not occur; see 3 of Lu and Whtt (211a). In general, the termnaton tme of the current nterval s defned by T R t T OL t 1 R t =OL + T UL t 1 R t =UL (16) 2.2. The Performance Formulas From the basc performance vector ˆ t b t q t and the defntons n 2.1, we can easly compute the performance vector t ( ˆ t w t v t B t Q t X t t S t t A t E t ) (17) We now revew the way the basc functons b q w v can be computed from the model data s F ˆ. For the flud model wth unlmted servce capacty startng at tme, B t = b t x = e M t x t t x 1 x t + e M t b x t 1 x>t (18) e M t x t t x dx + B e M t t for M n (1). The same formulas apply to a UL fnte-capacty system over T, where T nf t : B t > s t, wth T = f the nfmum s never obtaned. In an OL nterval, B t = s t and b t x = ( s t x + s t x t x ) e M t x t 1 x t + b x t e M t 1 x>t (19)

6 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 6 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS Let q t x be q t x durng an OL nterval T under the assumpton that no flud enters servce from queue. Durng an OL nterval, q t x = t x F t x x 1 x t F + q x t t x x F t x x t 1 t<x q t x = q t x F t x x 1 x w t t F + q x t t x x F t x x t 1 t<x w t (2) = t x F t x x 1 x w t t F + q x t t x x F t x x t 1 t<x w t We characterze the BWT w appearng n the formula for q above by equatng the quantty of new flud admtted nto servce n the nterval t t + to the amount of flud removed from the rght boundary of q t x that does not abandon n t t +. By careful analyss (Lu and Whtt 212a, Theorem 3), that leads to the nonlnear frst-order ODE w t = t w t 1 t q t w t (21) for n (13). (By Assumptons 6 9 of Lu and Whtt 211a, there s no dvson by n (2) and (21). Overall, w s contnuously dfferentable everywhere except for fntely many t.) The end of an OL nterval s the frst tme t that w t = and t s t + s t t. Durng an OL nterval, the PWT v s fnte and s characterzed as the unque soluton of the equaton v t w t = w t for all t (22) 2.3. The Flud Algorthm for Sngle Queues The prevous results yeld an effcent algorthm to compute the basc performance four-tuple b q w v over a fnte nterval T that we call the flud algorthm for sngle queues (FASQ). Frst, for each UL nterval, we compute b drectly va (18), termnatng the frst tme we obtan B t > s t. Second, for each OL nterval, we compute b va (19), q va (2), and then the BWT w by solvng the ODE (21). We consder termnatng the OL nterval when w t =. We actually do termnate the OL nterval f t s t + s t t. The proof of Theorem 5 n Lu and Whtt (212a) provdes an elementary algorthm to compute v durng an OL nterval from (22) once w has been computed. Theorem 6 of Lu and Whtt (212a) shows that v satsfes ts own ODE under addtonal regularty condtons. The key step beyond drect computaton s to control the swtchng between UL and OL ntervals. Ths can be done by selectng a fxed swtchng step sze T over whch to perform all calculatons before checkng to see f there s a regme change. Startng at tme t n regme R t, the calculatons are performed over the nterval t t + T. Then the algorthm fnds the frst tme s n t t + T at whch there s a regme change, f any, and that becomes the new ntal tme t. If the swtchng step sze T s too large, then there can be much wasted computaton. Otherwse, the algorthm tends to be nsenstve to the choce of T, as we show n C of the onlne supplement. A formal statement of the sngle-queue algorthm appears n C of the onlne supplement. For a tme nterval T wth regme swtches, examples show that the runnng tme of the FASQ tends to be lnear n both T, for fxed, and, for fxed T, and ndependent of T, provded that T s sutably small, e.g., f T T /, assumng that the swtchng ponts are approxmately unformly dstrbuted throughout the nterval T. Thus, for a fxed densty of swtches per tme, the run tme should be O T 2, because would be proportonal to T. These observatons are llustrated by a numercal example n C of the onlne supplement. 3. The G t /M t /s t + GI t m /M t Flud Network We now revew the G t /M t /s t +GI t m /M t FQNet ntroduced by Lu and Whtt (211a) and the FPE-based algorthm to compute all transent performance functons proposed there Model Propertes There are m queues, where each queue has model parameters as gven n 2.1. In addton, a proporton P j t of the flud output from queue at tme t s routed mmedately to queue j, and a proporton P t 1 m j=1 P j t 1 s routed out of the network. Consstent wth the termnology, we assume that P t s substochastc for each t. If two nput streams are combned to form a sngle nput (superposton), then the arrval rate functons are added. If one stream wth arrval rate functon s splt, such that a proporton p t of that stream goes nto a new splt stream at tme t, then the arrval rate functon of the splt stream s p t t p t. Smlarly, f the departure flow from one queue becomes nput to another, then the resultng arrval rate functon s. (We do not let the abandonment flow from one queue become nput to another.) We next dscuss convertng departure rate nto new nput rate. As n open queueng networks, there s an external exogenous arrval rate functon to each queue (from outsde the network, whch could be null at some queues), denoted by j, and there s a total arrval rate (TAR) functon to each queue (whch we smply

7 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 7 call the arrval rate functon), takng nto account the flow from other queues, denoted by j. The external arrval rate functons are part of the model data. The arrval rate functons satsfy the system of traffc rate equatons where j t = j t + m t P j t (23) =1 t = B t t t (24) Equatons (23) and (24) produce a system of equatons, wth j dependng upon for 1 m, whereas n turn depends on for each, because B depends on. The formulas for B as a functon of have been gven n 2.2, provded that we know whether the queue s OL or UL. That requrement s the major source of complexty. Because (23) s a lnear equaton, t can be wrtten n matrx notaton as = + P by omttng the argument t as below, provded that the product P s nterpreted as n (23). Moreover, we can combne (23) and (24) to express as the soluton of a fxed pont equaton. Hence the vector B t B 1 t B m t s a functon of over t and the model data. Hence, we can express (23) and (24) abstractly as =, where x t depends on ts argument x only over t for each t. Here the functon depends on all the model data s F b q P, 1 m. We assume that there are only fntely many swtches between OL and UL ntervals n each fnte nterval T. Under that assumpton, the operator mentoned above s a monotone contracton operator, by Lu and Whtt (211a, Theorem 1). Therefore, a recursve algorthm can be developed. If the recurson starts wth ntal vector =, the vector of external arrval rate functons, then the kth terate k j s the arrval rate of flud that has prevously experenced k transtons n the flud network. Wth ths notaton, we can wrte the recursve formulas n j t = n j t m = j t + =1 n 1 t P j t n 1 (25) where n t = B n t t, n. Because necessarly 1 for each, ths recurson converges monotoncally to the fxed pont The FPE-Based Algorthm Alg(FPE) The algorthm Alg(FPE) conssts of two successve steps: () solvng the traffc-rate Equatons (23) and (24) and () solvng for the performance vector b q w v at each queue usng the algorthm n 2.3. For step (), we start wth an ntal vector of arrval rate functons, whch can be an ntal rough estmate of the fnal arrval rate functons or the gven external arrval rate functons. We then apply the performance formulas n 2.2 to determne the performance functons B and at each queue to determne a new vector of arrval rate functons. We then teratvely calculate successve vectors of arrval rate functons untl the dfference (measured n the supremum norm over a bounded nterval) s sutably small. Then we apply step (). Gven a desred duraton T of an nterval T, we specfy the followng nput data: () the model parameter vector ( s G F ) ( t s t G F 1 m t T ) (26) where the ntal performance vector (at tme ) of queue, 1 m, s ( b q B Q w v ) and () the algorthm parameters: the teraton error tolerance parameter (ETP) and the swtchng step sze T, both assumed to be strctly postve. (We assume that the swtchng step sze s the same for all queues, whch usually provdes lttle loss of generalty.) We gve a formal statement of the algorthm n the onlne supplement. From the structure of algorthm Alg(FPE), we can drectly determne the computatonal complexty (computer-dependent requred run tme) FPE FPE T m as a functon of the ETP, number of queues m, length of the tme nterval T, and the number of regme swtches per queue, but we wll also confrm t n numercal examples. Proposton 1 (Computatonal Complexty of Alg(FPE)). The computatonal complexty of Alg(FPE) s FPE FPE m T = O mt log 1/ (27) If we may regard = O T, as s the case wth perodc models, then FPE m T = O mt 2 log. Proof. Let I I be the number of teratons of the FPE as a functon of the ETP. Roughly, we need to apply the FASQ for each of the m queues I tmes, although the full FASQ s not needed n the steps before the fnal one needed to compute the actual performance functons at each queue. Let be the number of regme swtches at queue over

8 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 8 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS T. Thus the overall complexty should be FPE = O IT m =1 S. Assumng that for all, wth the swtches at dfferent queues occurrng at dfferent tmes, that yelds FPE I m T = O ITm. Moreover, I = O log 1/ where s the ETP, because the convergence to the fxed pont n successve teratons s geometrcally fast. Unfortunately, unlke the other parameters, the number of regme swtches per queue cannot be drectly observed from the model data. However, f the model parameters, such as and s, are perodc functons wth perods and s, then the total number of swtchngs s usually bounded by 2T / + 2T / s so that we may regard = O T makng FPE m T = O mt 2 log. Proposton 1 s supported by the examples n The Alternatve ODE-Based Algorthm Alg(ODE) Now we develop the new algorthm Alg(ODE) for the G t /M t /s t + GI t m /M t FQNet. Agan, the key s to compute total arrval rates for all queues and then treat each queue ndependently. In some specal cases, analytc formulas are avalable Fndng the Total Arrval Rate Vector Instead of solvng the fxed-pont equaton, as n 3, to fnd the TARs, we now solve an m-dmensonal ODE. To do that, we need to work over subntervals where all queues are n specfed regmes. So now we consder successve swtchng tmes for any queue n the network. We recursvely solve the ODE n each of these ntervals. The key s to characterze and update the system regme n dfferent ntervals and recursvely advance n t. We descrbe the system regme at t wth two sets: t s the set of ndces of queues that are UL, and t s the set of ndces of queues that are OL. In other words, t { 1 m B t s t Q t = } (28) t { 1 m B t = s t Q t > } (29) Of course, t s smply the complement of t wthn the set 1 m. Gven t and t, consder 1 m. () If queue s UL,.e., t, flow conservaton mples that B t = t + j t P j t B j t j t + k t P k t s k t t B t k t If t, B t = s t. We partton and regroup the ndces of queues so that B t B t B t T, t t t T, t t t T, t t t T, s t s t s t T, t dag t, t dag t, t dag t t, and [ ] P t P t P t P t P t where P t (P t, P t, and P t ) denotes the transton probablty from a state n,, and ) to a state n,, and ) at tme t. Let P t = P t = P t = when P t = P t (.e., all queues are UL), and let P t = P t = P t = when P t = P t (.e., all queues are OL), where denotes an empty matrx (wth rank ). Therefore, n matrx notaton we have B t = C t B t + D t and B t = s t (3) where D t t + PT t t s t C t P T t I t If the servce rates and the routng probablty matrx are ndependent of tme, t = and P j t = P j,.e., the model becomes the G t /M/s t + GI t m /M network, then t = dag, C C t = P T I, and (3) has the unque soluton B t = e Ct ( ) e Cu D u du + B In all cases, the TAR vector can be represented as t = t + P T t t B t (31) 4.2. Explct Formulas for m = 2 The ODE-based approach yelds analytc solutons when m = 2. Consder the followng four system regmes: () When queue 1 s OL and queue 2 s UL (.e., B 1 t = s 1 t, Q 1 t, B 2 t < s 2 t ), B 1 t = s 1 t B 2 t = 2 t +P 1 2 t 1 t s 1 t + P 2 2 t 1 2 t B 2 t whch has a unque soluton [ B 2 t = e P 2 2 u 1 2 u du e u P 2 2 v 1 2 v dv 2 u ] + P 1 2 u 1 u s 1 u du + B 2

9 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 9 () When queue 1 s UL and queue 2 s OL (.e., B 1 t < s 1 t, B 2 t = s 2 t, Q 2 t ), B 1 t = 1 t + P 1 1 t 1 1 t B 1 t +P 2 1 t 2 t s 2 t B 2 t = s 2 t whch has a unque soluton [ B 1 t = e P 1 1 u 1 1 u du () When both queues are OL, B 1 t = s 1 t e u P 2 1 v 1 1 v dv 1 u ] +P 2 1 u 2 u s 2 u du + B 1 B 2 t = s 2 t (v) When both queues are UL, B 1 t = 1 t + P 1 1 t 1 1 t B 1 t +P 2 1 t 2 t B 2 t B 2 t = 2 t +P 1 2 t 1 t B 1 t + P 2 2 t 1 2 t B 2 t or where B t = t + C t B t (32) C t P T t I t and t After B t s obtaned, the TARs are [ ] 1 t 2 t 1 t = 1 t + P 1 1 t 1 t B 1 t + P 2 1 t 2 t B 2 t 2 t = 2 t + P 1 2 t 1 t B 1 t + P 2 2 t 2 t B 2 t 4.3. The Overall Algorthm and Its Complexty Just as for FASQ n 2.3, the key step beyond drect computaton s to control the swtchng between regmes. Because each queue can be ether UL or OL, there are overall 2 m dfferent network regmes. We say that the system changes ts regme at some tme f one or more of the queues changes ts regme, ether from UL to OL or from OL to UL. We provde the followng regme termnaton tme: T R t T 1 t T 2 t where T 1 t nf { t > t some t s.t. Q t = t t } T 2 t nf { t > t some j t s.t. B j t = s j t j t > j t } (33) wth t beng the startng tme of the desred nterval and the nfmum of an empty set understood to be nfnty. Wthn each regme, we use an ODE to compute the TARs t and the servce content functons B t, based on (3) and (31). Gven the TARs at all queues, we use the FASQ to calculate the performance functons. We gve a formal algorthm statement n E of the onlne supplement. The computatonal complexty clearly depends largely on the computatonal complexty of the ODE solver. Fortunately the ODEs arsng n the present context tend not to be computatonally dffcult; e.g., they are rarely stff. Let ODE m t be the computatonal complexty for solvng an m-dmensonal ODE over an nterval of length t. For the conventonal solvers we use (see 6.1), we should have approxmately ODE m t = O mt. From the structure of algorthm Alg(ODE), we can determne the computatonal complexty ODE ODE T m as a functon of the number of queues m, length of the tme nterval T, and number of regme swtches per queue, but we wll also confrm t n numercal examples. Proposton 2 (Computatonal Complexty of Alg(ODE)). If the computatonal complexty of the ODE solver s ODE m t = O mt, then the computatonal complexty of Alg(ODE) s ODE ODE T m = O m 2 T (34) Proof. As n 3.2, the parameter par m T s drectly observable, but s not. Let be the number of regme swtches at queue over T. Hence the total number of regme swtches for any queue n the network s m =1. Assumng that for all as before, we see that the ODE must be solved m tmes over subntervals, whose combned length s T. In addton, there s some computatonal cost of carryng out the swtchng n each regme swtch. For the ODE porton of the algorthm, the computatonal complexty s m T = m j=1 m T where m j=1 T = T (35) Hence, the overall computatonal complexty for the ODE solver s O mt. But we must factor n the regme swtchng, whch has computatonal effort proportonal to the number of network regme swtches, O m. Assumng that these components each contrbute sgnfcantly, we get the overall computatonal complexty n (35). We fnd that Proposton 2 s consstent wth numercal examples; e.g., see Fgure 2.

10 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 1 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 15 1,2 1,2 Computaton tme (secs): Algorthm Number of queues m Computaton tme (secs): Algorthm 3 1, Number of queues m Fgure 2 Computng Tmes of Algorthms Alg(FPE) and Alg(ODE) for the m-queue FQNet as a Functon of m, 2 m Allowng GI Servce Dstrbutons: Alg(FPE, GI) We now generalze the model, allowng the servce dstrbuton at each queue to be GI nstead of M. We need a new algorthm because nether the FPE-based algorthm Alg(FPE) n 3 nor the ODEbased algorthm Alg(ODE) n 4 s drectly applcable. For smplcty, we focus on the G t /GI/s t + GI m /M t FQNet, where the servce and patence dstrbutons are not tme varyng; the analyss can be easly generalzed to G t /GI t /s t + GI t m /M t. As part of the model data, we let G 1 m be the general servce cdfs of the G t /GI/s t + GI m /M t FQNet, and let Ḡ 1 G be the assocated ccdf; e.g., Ḡ x = e x for M servce A New FPE for the TAR Vector The key s to obtan the TAR t for 1 m and t T. Once t s obtaned, the sngle-queue algorthm for GI servce developed n Lu and Whtt (212a) can be appled to compute all other performance measures; see 8 and Appendx G n Lu and Whtt (212a). Ths sngle-queue algorthm for GI servce s a generalzaton of FASQ, whch requres solvng another FPE to fnd the rate at whch flud enters servce b t (whch we call the rate nto servce (RIS)) durng each OL nterval. For M servce, ths FPE for RIS smplfes to (19) wth x =. We next analyze the transent dynamcs of the G t /GI/s t + GI m /M t model at arbtrary tme t assumng the knowledge of the current system status. We refer to the explct formulas for b t x developed n Lu and Whtt (212a) durng our analyss. The formulas for q t x and w t are dentcal to those n 2. Consder a queue j that s UL,.e., j t. From Proposton 2 of Lu and Whtt (212a) we have that Computaton tme (secs): Algorthm 3 1, (as a generalzaton of (18)), m Ḡ j x b j t x =Ḡ j x j t x 1 x t + Ḡ j x t b j x t 1 x>t j t = = + b j t x h G j x dx g j x j t x dx g j x + t b j x dx (36) Ḡ j x Note that formula (36) for queue j s n terms of the TAR, whch s unknown. Consder a queue k that s OL,.e., k t. From Equatons (17) (2) of Lu and Whtt (212a) we obtan k t = b k t s k t (37) where the RIS b k t satsfes the FPE (as a generalzaton of (19)) wth b k = b k (38) y t â k t + y t x g k x dx â k t s k t + b k y g k t + y dy Ḡ k y Moreover, we have shown n Lu and Whtt (212a, Theorem 2) that s a contracton operator under mld condtons, and thus mples that the FPE (38) has a unque soluton. We note that the RIS for an OL queue depends on the rate at whch the servce capacty becomes avalable (defned n (12)) and s ndependent of the TAR,

11 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 11 unlke durng a UL regme. Hence, havng k t and b k t avalable (by solvng the FPE (38) and (37)) for all OL queues (.e., for all k t ), the TAR of queue satsfes the followng traffc-rate equaton: t = t + P k t k t + P j t j t k t j t = ˆ t + ( ) P j t g j x j t x dx (39) where j t ˆ t t + P k t k t k t + j t P j t g j x + t b j x dx Ḡ j x wth ˆ not dependng on the TAR and determned by the FPE (38) and the second equalty holdng by (36). Equaton (39) expresses the TAR vector as the soluton of an FPE,.e., where m m wth u t ˆ t + ( P j t j t = (4) ) g j x u j t x dx 1 m (41) where u u 1 u m m. Under regularty condtons, we can show that there exsts a unque soluton to Equaton (39) by applyng the Banach contracton theorem. We wll use the complete (nonseparable) normed space m wth the unform norm over the nterval T,.e., u T m sup u t (42) =1 t T Theorem 1 (TAR for GI Servce). Assume the system regme does not change n a small nterval T, then the operator n (41) s a monotone contracton operator on n wth norm defned n (42). Proof. Assume that T > s small enough so that the system regme does not change,.e., t = and t = for t T. Then u 1 u 2 T [ m = sup P j t g j x u 1 j t x u 2 j t x dx] =1 t T j m sup u 1 j u 2 j T P j t G j t =1 t T j m max 1 j m G j T sup t T j u 1 j u 2 j T C T u 1 u 2 T where C T m max 1 j m G j T. Ths provdes what we need, because we can make C T < 1 for suffcently small T >, because G t as t for all 1 m by our assumpton on the exstence of the servce denstes The Overall FPE-Based Algorthm wth GI Servce Algorthm Alg(FPE, GI) has two parts: () regme swtchng and () the new FPE wthn each fxed network regme. The regme swtchng can be managed just as for the FASQ and Alg(ODE). As before, we work wth a regme swtchng step sze T. Gven a tme t, we apply the new FPE n 5.1 to fnd a new TAR vector over the nterval t t + T. However, after dong that calculaton, we must check to see f there s a regme swtch at any queue n the network. If such a regme swtch occurs at tme s t t + T, then we replace t wth s and repeat. In ths way, we move forward n tme untl we compute the TAR vector for all of T. Wthn each nterval wth fxed network regme, we calculate the TAR usng FPE (4). Gven that TAR wthn each nterval wth fxed network regme, we apply the sngle-queue algorthm from Lu and Whtt (212a) to calculate the queue performance at each queue. Ths s more complcated than the FASQ n 2, because t s necessary to solve the FPE (38) at each queue that s OL n that partcular network regme. For ths last algorthm, the computatonal complexty s more dffcult to determne from the algorthm structure, because the algorthm s more complcated. Just as for Alg(ODE), there are O m network regmes, so that regme swtchng should have complexty of order O m. The new FPE s more complcated, requrng an FPE wthn the overall FPE at each queue. Because the frst-step FPE (38) s done at each queue throughout T, we can estmate ts complexty as O mt. The second-step FPE (4) may also have complexty of order O mt. In addton, these FPEs depend on the ETPs. Because both operators are contracton, the rate of convergence s geometrc. Hence the computatonal complexty of both teratons as functons of are O log 1/. Thus, we estmate that the computatonal complexty should be ( ( )) 1 FPE GI m T = O mt + mt m log = O m 2 T log 1/ (43) 6. Examples In ths secton we report the results of mplementng the algorthms n 3 5 and applyng them to three examples: () a Markovan M t /M/s + M 2 /M twoqueue FQNet, () a Markovan M t /M/s + M m /M

12 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues 12 INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS FQNet wth m queues, 2 m 16, and () a non- Markovan G t /LN /s + E 2 2 /M model. For smplcty, n these examples we make only the arrval rate tme varyng. The extenson to tme-varyng staffng s of course very mportant and s not dffcult to do as well, as we llustrate wth an example n the onlne supplement. Addng tme-varyng functons to the servce, abandonment and routng are less mportant, so we do not drectly llustrate those extensons. The thrd algorthm apples to all three examples, but the frst two algorthms only apply to the frst two examples. In 6.1 we frst provde detals about our mplementaton Implementaton Detals Before dscussng the examples, we brefly explan how we mplemented the numercal algorthms and conducted the smulaton experments. For both, we used MATLAB on a personal computer. To numercally solve ODEs both one-dmensonal for w t at each queue as n (21), and multdmensonal for the TAR as n (3), we used the MATLAB solvers ode23 and ode45, whch employ automatc stepsze Runge Kutta Fehlberg ntegraton methods. The frst one, ode23, uses a par of smple second-order and thrd-order formulas. The second, ode45, uses a par of fourth-order and ffth-order formulas. See Thomas (1995) for detals on fnte-dfference methods for numercally solvng dfferental ODEs. As a base case for the examples, we consdered a system startng empty over the tme nterval T wth T = 2. In that framework, we dvded the contnuous tme nterval T nto dscrete ntervals wth length 2. Care s needed n estmatng the varous tmedependent performance functons n the smulaton experments. For the mean head-of-lne watng tme E W t, the mean queue length E Q t, and the mean number of busy servers E B t, we dvde the nterval T nto subntervals or bns. For E W t, we keep track of all customer arrvals n each sample path. For a customer n, we keep track of the arrval tme A n and the tme that the customer enters servce E n. Therefore, one value for ths sample path s t Ŵ t = E n E n A n. Of course, ths customer may have already abandoned by tme E n. Because we are nterested n the potental watng tme, assumng nfnte patence, we keep track of the tme that the customer would enter servce even after they abandon; Table 1.e., our procedure ncludes the behavor of vrtual customers. The bn sze for E W t s 1, whereas the bn sze for E Q t and E B t s 5. Thus, we sampled the queue length once every 5 unts of tme A Two-Queue FQNet Example We frst consder the two-queue M t /M/s + M 2 /M FQNet dscussed n 1. It has snusodal external arrval rates t = a + b sn c t + = 1 2 (44) exponental servce and patence dstrbutons Ḡ x = e x and F x = e x, = 1 2, respectvely; constant staffng functons s, = 1 2; and a constant 2 2 Markov transton probablty matrx P wth elements P 1 2 = P 2 1 = 2 and P = 3, so that P = 5, = 1, 2. Let a 1 = a 2 = 5, b 1 = 25, b 2 = 35, c 1 = c 2 = 1, 1 =, 2 = 3, 1 = 1, 2 = 5, 1 = 5, 2 = 3, s 1 = 1, and s 2 = 2. We let the network be ntally empty. We frst show how the FPE-based algorthm Alg(FPE) from 3 works. It s based on an FPE for the TARs 1 t and 2 t for t T. Fgure 6 n Secton G.1 of the onlne supplement dsplays the arrval rates n successve teratons, dramatcally showng both the monotone convergence and the geometrc rate of convergence of the operator n 3.1. Alg(FPE) termnates after teraton I, where > s the prespecfed ETP, and { } I nf n E T n max j n 1 j T j=1 2 n yeldng fnal TARs I, = 1 2. For ths example, we show how the number of teratons I, the total run tme, and the termnatng error E T I depend on the EPT n Table 1. Fgure 7 n Secton G.1 of the onlne supplement shows plots of all the standard performance functons n the flud network usng Alg(FPE), ncludng, Q, w, B, X, and b, = 1 2. Fgure 1 compares the flud approxmatons wth results from a smulaton experment for a very large-scale queueng system. The queueng model has nonhomogeneous Posson external arrval processes wth snusodal rate functons n t = n t, = 1 2, wth n = 4. We compare the flud model predctons to a sngle sample path of the queueng system (one smulaton The Number of Iteratons I, Computaton Tme, and Termnatng Error E T I for Algorthm Alg(FPE) as a Functon of the ETP 1 n, n 1, for the Two-Queue FQNet Example Usng T = 2 and T = 2 log I E T I E 4 4.8E 5 4.9E 6 2.8E 7 5.2E 8 8.3E 9 1.4E 1

13 Lu and Whtt: Algorthms for Tme-Varyng Networks of Many-Server Flud Queues INFORMS Journal on Computng, Artcles n Advance, pp. 1 15, 213 INFORMS 13 Table 2 The Number of Iteratons I m and Computaton Tme m (Seconds) as a Functon of m, the Number of Queues, Usng Alg(FPE) wth Fxed EPT = 1 5 m I m m m I m m run). In Fgure 1 the sold lnes are the smulaton estmatons of sngle sample paths appled wth flud scalng, and the dashed lnes are the flud approxmatons. When the scale of the queueng model s not large,.e., when n s smaller, sngle sample paths of the queueng functons typcally do not agree closely wth the flud functons because of stochastc fluctuatons. However, the mean functons of these processes can be well approxmated, as shown n Secton G.1 of the onlne supplement, Fgure 8, for the case n = 5. In ths example, the two queues do not become OL (UL) at the same tme because of the phase dfference of the external arrval rates (.e., 1 =, 2 = 3). We also consder dfferent phases n another example n Secton G.1 of the onlne supplement. All three algorthms were run on ths example; the resultng dentcal performance functons confrm all of the algorthms. For ths small FQNet example, the most mportant characterstc s ease of mplementaton, for whch Alg(ODE) from 4 tends to be easest, whereas Alg(FPE, GI) from 5 s hardest. For all examples, Alg(FPE, GI) tends to have the longest run tme, as expected because t nvolves an FPE for each queue as well as an FPE for the TARs. For two-queue examples lke the one just consdered, the runnng tme of Alg(FPE, GI) tends to be twce as long as that of Alg(ODE) A Network wth Many Queues We next evaluate the performance of algorthms Alg(FPE) and Alg(ODE) as a functon of the number of queues m. To do so, we consder a smple dealzed network wth m queues. Each queue has a tmevaryng arrval rate as n (44), exponental servce and patence tmes wth rates and, constant staffng level s, and constant routng probabltes P j, where a = 5 b = a /m = 1 5 /m = 5 c = s = = 1 P j = 1/2m 1 m 1 j m Table 3 The Computaton Tme m (Seconds) as a Functon of the Number of Queues m Usng Alg(ODE) Fgure 13 n Secton G.2 of the onlne supplement shows plots of the performance functons for m = 1. Table 2 shows the number of teratons I m and computaton tme m n seconds as a functon of the number of queues m, 2 m 16, usng algorthm Alg(FPE) wth fxed EPT = 1 5. In ths example we observe that () the number of teratons I m does not grow wth the number of queues m, and () the computaton tme m grows lnearly n m. We also analyzed the performance of ths same model usng Alg(ODE). Table 3 shows the computaton tmes m as a functon of m. Because we used the ODE solvers ode23 and ode45, whch are O m algorthms, the runnng tme for Alg(ODE) becomes O m 2. Fgure 2 dramatcally shows the dfference n the algorthm performance. We conclude ths secton wth some general observatons comparng the performance of the two algorthms Alg(FPE) and Alg(ODE). For small m (e.g., 2 m 8) and small (e.g., < 1 5 ), Alg(ODE) runs faster than Alg(FPE); for bg m and medum, Alg(FPE) runs faster than Alg(ODE). Of course, the complexty of Alg(ODE) depends on the choce of the multdmensonal ODE solver. The polynomal growth n m as shown n Table 3 s attrbuted to the specfc numercal scheme (such as Runge Kutta Fehlberg) of the ODE solver A G t /LN /s + E 2 2 /M Non-Markovan Example We now consder an example wth a nonexponental servce-tme dstrbuton for whch only the fnal algorthm Alg(FPE, GI) ntroduced n 5 apples. To llustrate ths example, we consder the G t /LN /s +E 2 2 /M model wth lognormal servce dstrbutons at each queue (the LN ) and Erlang-2 patence dstrbutons at each queue (the E 2. Specfcally, we let the servce tme at staton be S e Z, where Z s a normal random varable wth mean ˆ and varance ˆ 2,.e., m m m m