Optimal resource capacity management for stochastic networks

Transcription

1 Submtted for publcaton. Optmal resource capacty management for stochastc networks A.B. Deker H. Mlton Stewart School of ISyE, Georga Insttute of Technology, Atlanta, GA 30332, S. Ghosh, M.S. Squllante Mathematcal Scences Department, IBM Thomas J. Watson Research Center, Yorktown Heghts, NY 10598, We develop a general framework for determnng the optmal resource capacty for each staton comprsng a stochastc network, motvated by applcatons arsng n computer capacty plannng and busness process management. The problem s mathematcally ntractable n general and therefore one typcally resorts to ether overly smplstc analytcal approxmatons or very tme-consumng smulatons n conjuncton wth metaheurstcs. In ths paper we propose an teratve methodology that reles only on the capablty of observng the queue lengths at all network statons for a gven resource capacty allocaton. We theoretcally nvestgate the proposed methodology for sngle-class Brownan tree networks, and further llustrate the use our methodology and the qualty of ts results through extensve numercal experments. Key words : capacty allocaton; capacty plannng; queueng networks; resource capacty management; stochastc networks; stochastc approxmaton Hstory : Ths paper was frst submtted on??? 1. Introducton Stochastc networks arse n many felds of scence, engneerng and busness, where they play a fundamental role as canoncal models for a broad spectrum of mult-resource applcatons. A wde varety of examples span numerous applcaton domans, ncludng communcaton and data networks, dstrbuted computng and data centers, nventory control and manufacturng systems, call and contact centers, and workforce management systems. Of partcular nterest are strategc plannng applcatons, the complexty of whch contnue to grow at a rapd pace. Ths n turn ncreases the techncal dffcultes of solvng for functonals of stochastc networks as part of the analyss, modelng and optmzaton wthn strategc plannng applcatons across dverse domans. A large number of such strategc plannng applcatons nvolve resource capacty management problems n whch resources of dfferent types provde servces to varous customer flows structured accordng to a partcular network topology. Stochastc networks are often used to capture the dynamcs and uncertanty of ths general class of resource management problems, where each type of servce requres processng by a set of resources wth certan capabltes n a specfc order and the customer processng demands are uncertan. The objectve of these resource management problems s to determne the capacty allocaton for each resource type comprsng the stochastc network that maxmze the expectaton of a gven fnancal or performance functonal (or both) over a longrun plannng horzon wth respect to the customer workload demand and subject to constrants on ether performance or fnancal metrcs (or both). It s typcally assumed that the plannng horzon s suffcently long for the underlyng multdmensonal stochastc process modelng the network to reach statonarty, where the multple tme scales nvolved n most applcatons of nterest provde both theoretcal and practcal support for such a steady-state approach. The objectve functon s often based on rewards ganed for servcng customers, costs ncurred for the resource capacty allocaton deployed, and penaltes ncurred for volatng any qualty-of-servce agreements. 1

2 2 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks Our present study of resource capacty management problems n stochastc networks s prmarly motvated by two partcular applcaton domans, although the same class of problems arse naturally n many other domans. The frst applcaton doman concerns capacty plannng across a wde range of computer envronments. Ths area has tradtonally receved a great deal of attenton wthn the context of hgh-performance computng and Internet-based computng envronments. However, wth the recent prolferaton of large-scale data centers and cloud computng platforms (see, e.g., Dkaakos et al. (2009), Armbrust et al. (2009), Gartner (2012), Iyoob et al. (2012)), ths applcaton doman has become an even more mportant source of resource capacty management problems n practce. For these problem nstances, the computer nfrastructure s modeled as a stochastc network reflectng the topology of the nfrastructure and the uncertanty of both the customer processng requrements and the overall demand. Varous companes, such as BMC Software and IBM, provde products that address these resource capacty management problems wthn the context of nformaton technology servce management, data center automaton, and computer performance management. The objectves of such solutons nclude mnmzng the costs of a computng nfrastructure whle satsfyng certan performance targets, as well as maxmzng performance metrcs wthn a gven computng nfrastructure budget. The second applcaton doman motvatng our present study s busness process management, whch s a key emergng technology that seeks to enable the optmzaton of busness process operatons wthn an organzaton. Here, a busness process generally conssts of any seres of actvtes performed wthn an organzaton to acheve a common busness goal, such that revenues can be generated and costs are ncurred at any step or along any flow comprsng the process. A smple busness process example s the processng of varous types of medcal clams by an nsurance company. Another recent representatve example s the flow of patents through the emergency department of a hosptal, where the goal s to address a chronc nablty of hosptals to delver emergency servces on demand n a hghly dynamc and hghly volatle envronment n whch the falure to match demand carres sgnfcant clncal rsks to patents and fnancal rsks to the hosptal; see Guarsco and Samuelson (2011). For these problem nstances, the busness process s modeled as a stochastc network reflectng the topology of the seres of actvtes to be performed and the uncertanty of both the processng requrements for each actvty and the overall demand. Varous companes, such as IBM and Oracle, provde products that address these resource capacty management problems wthn the context of busness process modelng and optmzaton. The prmary state-of-the-art approaches for solvng resource capacty management problems n stochastc networks from the two foregong applcaton domans fall nto two man categores. These two sets of soluton approaches also represent the state-of-the-art for a wde varety of applcaton domans beyond our motvatng applcatons. The frst category of soluton approaches s based on farly drect applcatons of product-form network results, despte the fact that the underlyng stochastc network does not have a product-form soluton. Indeed, t s only under strong restrctons that the statonary jont dstrbuton for the stochastc network s a product of the statonary dstrbuton for each queue n solaton (see, e.g., Baskett et al. (1975), Harrson and Wllams (1992)). Ths approach s often employed n computer capacty plannng applcatons, even though the requrements for product-form solutons almost always never hold n practcal capacty plannng nstances across a broad spectrum of computer envronments. Instead, the approach s used as a smple approxmaton typcally together wth a wde range of ad hoc heurstcs, whch nclude applyng product-form results for performance metrcs of nterest n a modfed verson of the orgnal stochastc network n an attempt to address characterstcs that yeld a network wth a nonproduct-form soluton; e.g., refer to Menasce and Almeda (1998, 2000), Menasce et al. (2004) and the references theren. One example of ths approach conssts of ncreasng the servce requrements of customers at a bottleneck resource n an attempt to have the results reflect the types of bursty arrval processes often found n computer envronments. Another example conssts of (artfcally)

3 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 3 splttng the customer servce requrements at a network staton nto multple classes n an attempt to capture heavy tals n the servce tme dstrbutons (gnorng correlaton effects). Although the closed-form expressons render a drect soluton for the correspondng optmzaton problem n a very effcent manner, the serous accuracy problems nherent n ths smple approxmaton approach have been well establshed and thus are a great concern from both a theoretcal and practcal perspectve. The second category of state-of-the-art soluton approaches s based on smulaton-based optmzaton. Here, the lterature can be broadly dvded nto those that use a broad spectrum of metaheurstcs (e.g., tabu search, scatter search, neural networks) to control a sequence of smulaton runs n order to fnd an optmal soluton (see, e.g., Glover et al. (1999) and Chapter 20 n Nelson and Henderson (2007)), and those that apply several drect methods (e.g., stochastc approxmaton) whch have been wdely studed to address smulaton-based optmzaton problems wth a more rgorous theoretcal foundaton (refer to, e.g., Chapter 8 n Asmussen and Glynn (2007) and Chapter 19 n Nelson and Henderson (2007)). A great dsadvantage of all these smulatonbased optmzaton methods, however, s the often prohbtve costs n both tme and resources requred to obtan optmal solutons n practce for problems nvolvng multdmensonal stochastc networks. Ths s n large part due to the numerous parameters nvolved n each method that must be set va expermental tweakng for every problem nstance. In fact, a recent study llustrates how smulaton-based optmzaton can requre on the order of a few days to determne optmal resource capacty levels n a much smpler class of stochastc processes than those consdered n the present paper; see Hechng and Squllante (2013). The metaheurstc approach s often employed n busness process management applcatons, where there s essentally exclusve use of smulaton for the analyss and optmzaton of stochastc network representatons of busness processes; refer to, e.g., Laguna and Marklund (2004). More generally, the metaheurstc approach s employed n nearly all major smulaton software products that support optmzaton, such as those offered by Arena and AnyLogc through partnershp wth companes lke OptTek that provde a software control procedure whch ntegrates varous metaheurstc methods. At each step of the control procedure, there s a comparson between the smulaton results for the current set of decson varables (server capacty n our applcaton) and those for all prevous settngs, and then the procedure suggests another set of decson varable values for the next smulaton run. Once no further mprovement to the results s observed, the control procedure termnates and the best set of decson varable values found through ths sequence of smulaton runs are selected to be an (local) optmum. The metaheurstc approach gnores any structure of the underlyng stochastc network, and therefore can be prone to accuracy concerns wth respect to the true optmal soluton from both a theoretcal and practcal perspectve. The stochastc approxmaton algorthm for smulaton-based optmzaton has been extensvely studed n great generalty wth rgorous results avalable on the rates of convergence under reasonable condtons for the objectve functon. These methods are regrettably not as common n practce as the metaheurstc approach. One stumblng block has been that the method requres the settng of certan crtcal parameters to good values n order to realze an effcent mplementaton, where practtoner experence demonstrates that good values typcally depend on each nstance of the problem beng solved. As our results further demonstrate and quantfy, ths mportant requrement of stochastc approxmaton to fnd good values for certan crtcal parameters remans a key problem n practce for the general stochastc networks of nterest n ths study. In ths paper, our goal s to develop a general soluton framework that provdes the benefts of each of the above soluton approaches whle also addressng ther serous lmtatons. Namely, we seek to realze the effcency of analytcal methods together wth the accuracy of smulaton-based methods wthn a unfed framework for solvng resource capacty management problems. We devse a two-phase soluton framework n whch a new and general form of stochastc decomposton s

4 4 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks derved and leveraged as part of a fxed-pont teraton n the frst phase to obtan a nearly optmal soluton n a very effcent manner. The second phase, takng the frst-phase soluton as a startng pont, then explots advanced methods that deal drectly wth the orgnal network to obtan an optmal soluton. A good canddate for our second-phase soluton s the stochastc approxmaton algorthm, gven that t represents the only smulaton-based optmzaton approach wth a rgorous theoretcal foundaton. It s mportant to note, however, that any drect method can be used as the bass of the second phase of our framework. Ths second phase s much more accurate than the frst phase, at the expense of much hgher computatonal and temporal costs, but the nearly optmal startng pont from the frst phase allows us to leverage these detaled methods n a more surgcal manner. By establshng and explotng fundamental propertes of the underlyng multdmensonal process of the stochastc network throughout ths framework, we beleve that sgnfcant mprovements n computatonal costs and soluton accuracy are possble over varous state-of-the-art approaches for solvng resource capacty management problems n general. Our study ncludes a wde varety of numercal experments to nvestgate the performance of a partcular realzaton of our general soluton framework over a broad range of problem settngs. The results of these experments clearly and convncngly demonstrate the sgnfcant benefts of our general soluton framework over exstng state-of-the-art approaches, namely product-form and stochastc approxmaton solutons. In partcular, we show that the two-phase framework provdes vastly superor results than product-form solutons and converges much faster than stochastc approxmaton approaches wth respect to fndng (locally) optmal solutons. Moreover, the prncple new stochastc decomposton algorthm ntroduced as an accelerant n the frst phase performs very well n producng good approxmate solutons that are typcally wthn a tny percentage of optmalty for well-behaved problem settngs and wthn 5% of optmalty n more dverse settngs. Ths s of great advantage to the general user n practce because the algorthm has no parameters that must be tweaked to obtan such fast convergence to good approxmate solutons. These hgh qualty approxmatons support effcent exploraton of the entre resource capacty management space across varous assumptons, condtons, scenaros and workloads. Furthermore, these hgh qualty approxmatons as startng ponts obvously beneft the method used n the second phase. In contrast, our numercal experments clearly llustrate the dffculty encountered by users n tunng parameters to realze the best results from the sole use of the stochastc approxmaton algorthm. The remander of ths paper s organzed as follows. Secton 2 presents our general soluton framework. We then consder n Secton 3 a specfc nstance of our general problem, namely a sngle-class Brownan tree network, for whch we derve structural propertes that play a fundamental role n an analyss of our algorthm presented n Secton 4, ncludng results on unqueness, convergence and bounded optmalty gap n the tree-network settng. A representatve set of numercal experments are provded n Secton 5, whch ncludes a bref ntroducton to the stochastc approxmaton algorthm. Concludng remarks then follow, wth the appendces presentng addtonal techncal detals and some of our proofs. Notaton. All vectors n ths paper are L-dmensonal, where L represents the number of statons n the network. Throughout, we use boldface for vectors and dentfy ther elements through subscrpts: the -th element of the vector v s gven by v. We smlarly use boldface for vectorvalued functons regardless ther doman, and wrte for nstance f (x) for the -th element of f(x). We also wrte u, v for the nner product of vectors u and v. 2. A General Approach The complexty of solvng resource capacty management problems s due n great part to the techncal dffcultes of solvng for functonals of statonary stochastc networks. A major source of

5 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 5 dffculty n such analyss, modelng and optmzaton of stochastc networks concerns the multdmensonal aspects of the underlyng stochastc processes, whch nvolve varous dependences and dynamc nteractons among the dfferent dmensons of the multdmensonal process. In ths secton, we present our general approach to address these complextes and dffcultes n developng a novel resource capacty management soluton framework. We frst dscuss the basc dea to provde a proper context for our approach, and then we formally present our general soluton framework. Due to the hghly nonlnear and possbly nonconvex nature of resource capacty management problems n general, our focus les on fndng good local optma Basc dea Gven our goal of developng a soluton approach that provdes the effcences of analytcal methods together wth the accuraces of smulaton-based methods, we devse a general two-phase soluton framework for optmal resource capacty management problems n stochastc networks. The frst phase s based on a fxed-pont teraton approach where observed queue lengths at the current terate determne the resource capacty allocaton for the next terate. If one of the queue lengths s dsproportonally hgh n the current terate, the next terate wll allocate more resource capacty at the correspondng staton. Ths process repeats, formng the bass of an effcent fxedpont teraton that renders a nearly (locally) optmal soluton to the resource capacty management problem. Dependng on the stochastc network settng n whch our general soluton framework s appled, the requred queue length nformaton can be obtaned from (a combnaton of) advanced analytcal (e.g., Baskett et al. (1975), Deker and Gao (2011), Harrson and Wllams (1992), Pollett (2009)), numercal (e.g., Da and Harrson (1992), Saure et al. (2009)) or smulaton-based (e.g., Asmussen and Glynn (2007), Nelson and Henderson (2007)) methods. As a result, our frstphase approach can be appled to stochastc networks that are analytcally ntractable as long as they can be smulated. Our teratve algorthm updates resource allocatons based on the square root of the observed queue lengths, as motvated and formalzed mathematcally n the next subsecton. Roughly speakng, our updatng rule s derved from an approprate separable functonal form for performance metrcs of each staton n the network, such as expected steady-state queue length or expected steady-state sojourn tme at the queue. The functonal form s gven by τ/(β λ), where λ and β are the arrval and servce rates for the queue and τ s a functon of varous characterstcs of the arrval and servce processes at all statons n the network. Ths partcular functonal form naturally arses n all known queueng formulas; refer to, e.g., Kngman (1962) for an enlghtenng example of how t appears and see the next subsecton for a more detaled dscusson. Ths frst phase of our methodology s smlar n sprt to a tradtonal approach from the feld of appled probablty, whch approxmates the complex multdmensonal stochastc process through a set of smpler processes of reduced dmensonalty together wth fxed-pont equatons that capture the dependences and dynamc nteractons among the dmensons; see, e.g., Squllante (2011). A classcal example of ths basc approach s the well-known Erlang fxed-pont approxmaton n whch the multdmensonal Erlang formula s replaced by a system of nonlnear equatons n terms of the one-dmensonal Erlang formula; refer to, e.g., Kelly (1991). Our approach smlarly uses a functonal form for the one-dmensonal queueng processes as the bass of a multdmensonal approxmaton, but our fxed-pont teraton crtcally reles on the multdmensonal queueng dynamcs as captured through the means of, for nstance, numercal or smulaton methods. As a result, snce we do not rely on explct queueng formulas that hold under restrctve assumptons, our approach requres effectvely no underlyng assumptons and promses to be wdely applcable. The resource allocaton decsons from the frst phase subsequently serve as a startng pont for the second phase. Ths second phase s based on general search methods that deal drectly wth the orgnal stochastc network to further mprove upon the frst-phase startng pont and

6 6 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks obtan a locally optmal soluton. When the orgnal stochastc network has a product-form soluton (see, e.g., Baskett et al. (1975), Harrson and Wllams (1992)), then the results of our frst-phase algorthm lead to the desred optmal resource capactes after exactly one teraton and the second phase of our general soluton framework s not needed Mathematcal formalzaton We now formalze our approach n a settng where the goal s to mnmze the sum of the weghted expected steady-state queue lengths n a stochastc network subject to a budgetary constrant. We gear the dscusson towards applcaton of our approach to generalzed Jackson networks (refer to, e.g., Chen and Yao (2001)) and ther Brownan counterparts (see, e.g., Harrson and Wllams (1987)); other settngs are dscussed n the next subsecton. Some addtonal notaton s needed. We wrte γ for the effectve arrval rate vector and β for the vector of servce rates. Further parameters of the network, such as the routng matrx and the exact external nterarrval and servce dstrbutons, need not be specfed to present our approach and thus we do not ntroduce them here. We wrte Z β for the steady-state queue length at the -th staton (alternatvely one can smlarly study sojourn tmes). The dependence on β s made explct snce we are nterested n comparng a functonal of the steady-state vector Z β as we change the servce-rate vector β. Assume that each unt of resource capacty at staton costs c, comprsng a cost vector c, and that we have a total budget of C for allocatng resources n the network. We am to mnmze the expected steady-state queue lengths weghted by a vector w, subject to the constrants that we cannot spend more than the budget C and that the queueng system s stable. Ths leads to the followng optmzaton problem: (OPT) mn β (0, ) L L w EZ β =1 s.t. c, β C, β > γ, = 1,..., L. Throughout, we shall assume c, γ < C so that the above mathematcal program s feasble. One can expect the soluton to satsfy c, β = C; see Secton 4.2 for a result n ths drecton. After defnng τ (β) = (β γ )EZ β, (1) the objectve functon takes the form t(β, w, τ (β)), where for β γ, w, τ > 0 we have t(β, w, τ ) = L k=1 w k τ k β k γ k. (2) For a queue n a sngle-class product-form network, τ s known to be equal to λ; see for nstance Baskett et al. (1975). Furthermore, τ equals λ(c 2 A + c 2 S)/2 n a sngle-class Brownan product-form network of GI/GI/1 queues, where c 2 A and c 2 S denote the second-order varaton terms for the arrval and servce process, respectvely; see Harrson and Wllams (1992). In general stochastc networks, however, τ (β) s mathematcally ntractable. Our approach reles on the dea that β t(β, w, τ (β)) can be reasonably approxmated locally by β t(β, w, τ ( β)) n the neghborhood of a gven pont β. Through ths functonal form, the -th summand n the approxmatng objectve functon only depends on β through the one-dmensonal quantty β, thus effectvely decomposng the objectve functon. The explct ncorporaton of β γ n the denomnator s motvated by the aforementoned natural occurrences of ths functonal form, whch ncludes product-form results that effectvely arse from one-dmensonal queueng

7 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 7 formulas. We note that the dea of locally approxmatng the objectve functon s a well-known prncple n trust-regon based optmzaton; refer to, e.g., Conn et al. (2000). Our approach, however, s very dfferent from tradtonal trust-regon methods n that we explot structural propertes of the stochastc network through a global functonal-form decomposton whose (few) unknown parameters are estmated locally, whereas trust-regon methods take a black-box approach n whch the parameters of an arbtrary polynomal approxmaton of the objectve functon are ftted locally. The followng lemma, whch s readly proved by applyng standard Lagrangan methods, then becomes an essental ngredent n our analyss. Klenrock (1964) and Wen (1989) used ths result n ther work on capacty allocaton for product-form networks. Lemma 1. The mnmum of t(β, w, τ ) over the feasble regon n (OPT) s β (w, τ ), where for l = 1,..., L β wl τ l /c l l (w, τ ) = γ l + (C c, γ ) L. k=1 wk τ k c k As an extenson of the dea that queue lengths may be approxmated locally by functons of the form (2), we propose to use the capacty allocaton β determned through the followng system of nonlnear equatons as the outcome of the frst phase of our approach: For l = 1,..., L, β wl τ l (β l = γ l + (C c, γ ) )/c l L. (3) =1 w τ (β )c Secton 4 shows that ths system of equatons s guaranteed to have a unque soluton for a certan class of stochastc networks, but we leave open the queston of exstence and unqueness for other settngs. In an attempt to numercally fnd a vector β satsfyng (3), assumng exstence, we propose the fxed-pont teraton scheme wth terates {β (k) : k 0} gven by β (k+1) l = γ l + (C c, γ ) wl τ l (β (k) )/c l L =1 w τ (β (k) )c. (4) Ths can be rewrtten n the followng nsghtful way. In vew of (1), we fnd that (4) mples β (k+1) γ = β(k) γ w β EZ (k) /c, (5) β (k+1) j γ j β (k) j γ j w j EZ β(k) j /c j whch les at the heart of the frst phase of our approach because ths equaton establshes an mportant connecton wth a resource capacty teraton scheme based on observed queue length nformaton. Snce we must allocate at least capacty γ to staton, (β γ )/(β j γ j ) s the rato of addtonal resource capactes allocated to staton and j, respectvely. Equaton (5) expresses ths rato n terms of the rato of mean queue lengths, so that more capacty s allocated n the next terate to statons wth dsproportonally long queue lengths n the current terate. The rght-hand sde of (5) can be nterpreted as the geometrc mean of two fractons, and thus we can thnk of (5) as a slowed down verson of the teratve scheme β (k+1) γ β (k+1) j β (k) = w EZ /c. (6) β (k) γ j w j EZj /c j From (5) t becomes evdent that we hope for our teratve scheme to converge to β satsfyng β γ = w EZ β /c. βj γ j w j EZ β j /c j

8 8 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks We found numercal examples wth unque fxed ponts for whch the teraton process of (5) produces a convergng sequence whereas the teraton process of (6) produces an oscllatng sequence. Hence, slowng down the teratve scheme can mprove ts convergence propertes. Ths s n fact a well-studed phenomenon n the lterature on fxed-pont teraton processes, where takng an (arthmetc) average often produces better results; see, e.g., Mann (1953), Ishkawa (1974) and the vast body of subsequent work n ths area. The geometrc average n our case arses from the assumed functonal form gven n (2). As quantfed n Lemma 1, a further consequence of ths functonal form s that the terates n (5) avod the boundary of the feasble regon, unlke the terates of (6) Dscusson Our two-phase framework only reles on the capablty of evaluatng the queue lengths at all statons comprsng the stochastc network under any resource capacty allocaton, and thus t holds great promse to perform well n many stochastc network optmzaton problems beyond the settng of generalzed Jackson networks and ther Brownan analogs. The key approxmaton n our framework conssts of the separable functonal form τ/(β λ), whch consttutes a near unversal phenomenon n stochastc networks under a wde range of queueng dynamcs (e.g., processor sharng networks and mult-class networks under a varety of schedulng polces). As a result, we expect that resource capacty management optmzaton through an teratve algorthm based on ratos of observed queue lengths and slow-down through geometrc means s promsng for many dfferent settngs, such as several varants of the settng dscussed n the prevous subsecton. For nstance, one could have a dscrete decson space n whch to allocate a number of servers to each staton, and then use Lemma 1 n conjuncton wth a local search algorthm over the dscrete parameter space to generate an teratve scheme. In other examples, the feasble regon may be less explct and only descrbed through a stablty condton (snce t may not always be possble to determne ths regon, one could equp the queue-length evaluaton procedure wth an exploson check), n whch case the teratve scheme could requre new nsghts to perform an optmzaton of the approxmate objectve functon over ths space. Another nterestng varant s the dual formulaton of the problem, as dscussed n the ntroducton, where the am s to mnmze the total expendture subject to a bound on the sum of the weghted expected steady-state queue lengths. 3. Sngle-class Brownan Tree Networks We next ntroduce sngle-class Brownan tree networks, presentng several of ther structural propertes that play a key role n the analyss of our algorthm. The premse of Brownan network models s that they approxmate the queue-length (or watng-tme) dynamcs of stochastc networks, relyng on a central lmt theorem scalng. Such an approxmaton s often rgorously justfed n heavy traffc; refer to, e.g., Reman (1984), Harrson and Wllams (1987). The heavy-traffc assumpton s partcularly relevant n the context of resource capacty management problems, where t mples the often realstc assumpton that systems are operated close to the system capacty. As n the central lmt theorem, nterarrval and servce dstrbutons are approxmated n the Brownan model usng only ther frst two moments. Hence, a Brownan network model can be thought of as a two-moment approxmaton of the underlyng stochastc network. Ths dea les at the heart of the so-called QNET method proposed n Harrson and Nguyen (1990). Even though the queue-length process of a generalzed Jackson network s typcally non-markovan, the queue-length process of a Brownan network forms a Markov process known as reflected (or regulated) Brownan moton. Further background on Brownan tree networks s provded n Appendx A, together wth proofs for the two lemmas presented n ths secton. Our specfc focus n ths secton les on sngle-class Brownan tree networks, whch arse from an underlyng generalzed Jackson network (see, e.g., Chen and Yao (2001)) wth a tree network

9 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 9 topology. Wthn ths class of models, we are able to derve several qualtatve propertes of our framework. The network topology s represented by a rooted tree G = (V, E) comprsed of L = V vertces, at whch customers are served by a (Brownan) server. We dentfy the root as staton 1. For > 1, we wrte π() to denote the label of the unque staton adjacent to that s closer to the root. Customers are served at staton at rate β, meanng that the mean servce tme there equals 1/β ; we say β s the servce capacty at staton. After havng receved ther requred servce, customers are routed from staton to staton j wth probablty p j. The network topology mposes the restrcton that (, j) E f and only f p j > 0. We refer to Appendx A for second order (varance) parameters of ths model, whch are not used below. A path s defned to be a sequence of vertces such that from each of ts vertces there s an edge n E to the next vertex n the sequence. We wrte P = (1,..., π(), ) for the unque path from the root to staton. Gven a vector v R L, we wrte v for the vector obtaned by restrctng v to those elements n the path P. For nstance, β stands for (β 1,..., β π(), β ). Due to the specfc tree structure studed here, the queue length at staton, Z β, only depends on β through the upstream capacty vector β. We abuse notaton slghtly and wrte Z β j P. For notatonal convenence, we set p π(1),1 = 0. For any j P, defne q j = p π(k),k, k P,k P j nstead of Z β, smlarly usng Z β j where an empty product should be nterpreted as 1, so that q = 1. We wrte γ = k P q k λ k for the effectve arrval rate at staton, where λ denotes the external arrval rate at staton. Snce our nterest n ths paper s solely on stable networks, we mpose throughout that β > γ for = 1,..., L. The followng functons play a key role n the analyss of our algorthm for Brownan tree networks. We defne h : (0, ) P, for = 1,..., L, through h (x ) = j P q j EZ γ +x j, for wth the conventon h π(1) = 0. Ths defnton mples EZ β = h (β γ ) p π(), h π() (β π() γ π() ), (7) and thus the orgnal optmzaton problem (OPT) s readly reformulated n terms of these h functons for tree networks as follows. (OPT BTN) mn β (0, ) L L [ w h (β γ ) p π(), h π() (β π() γ π() ) ] =1 s.t. c, β C, β > γ, = 1,..., L. Ths formulaton s advantageous snce the functons h enjoy several useful structural propertes, whch are descrbed n the next two lemmas. Lemma 2. For any = 1,..., L, the functon h : (0, ) P R + s: () convex on ts doman; () nonncreasng n each coordnate; and () strctly decreasng n the last coordnate x unless the degeneracy condton of determnstc servce tmes at statons π() and, determnstc routng to staton, and no external arrvals at staton holds.

10 10 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks The monotoncty result n () mples that the mean queue length at the -th staton EZ β decreases n the servce capacty β. A more precse statement of the degeneracy condton n () s gven by Σ, defned n (18) of Appendx A, beng equal to 0. The lemma condton of Σ > 0 appearng n () prevents the capacty allocaton problem at the -th staton from beng degenerate,.e., addtonal capacty does not lead to lower (Brownan) queue lengths due to the determnstc settng stated n the lemma. In vew of Lemma 2, a wde varety of generc technques are avalable to study (OPT-BTN). The convexty property mples that (OPT-BTN) s a so-called dfference of convex functons (DC) programmng problem, as studed n for nstance An and Tao (2005). It also shows that (OPT-BTN) becomes a convex program under certan settngs of the weghts. Notce that the convex functon h has coeffcent (w L w j=1 jp j ) n the objectve functon of (OPT-BTN). Thus, f the weghts are constant (w 1 =... = w L ) or more generally the weghts are non-ncreasng (w 1... w L ), then the problem (OPT-BTN) s convex. Another mportant property s that h s homogeneous of degree 1, whch s a consequence of the Brownan scalng property and precsely stated n the followng lemma. Lemma 3. For any = 1,..., L, x > 0, and δ > 0, we have δh (δx ) = h (x ). 4. Analyss of our Approxmaton Algorthm for Brownan Tree Networks Ths secton analyzes the algorthm of Secton 2 n the context of the Brownan tree networks of Secton 3. In partcular, we prove that there s a unque fxed pont n ths case, and that (a mnor modfcaton of) our algorthm converges to ths fxed pont. We also establsh that the optmalty gap remans bounded n the budget C, further provng a desrable property of the step szes taken n our teratve capacty allocaton procedure Exstence and Unqueness of a Fxed Pont Our approxmaton to the optmal capacty allocaton s defned as a soluton of the fxed-pont equatons (3). In ths secton, we establsh the exstence and unqueness of such a fxed pont, whch s therefore a feasble soluton for (OPT). Standard technques for provng unqueness of a fxed pont typcally nvolve bounds on (frstorder) dervatves, or on a spectral radus n the present multvarate settng. Snce we were unable to derve such results for the τ functons, we take a dfferent approach. Our approach s to rewrte the fxed-pont equatons n an approprate form, and then use the structural propertes of the h functons from the precedng secton to show the exstence and unqueness of the fxed pont. Before proceedng, we need some addtonal notaton that s used throughout ths secton. Abusng notaton as before, defne x (β) = x (β ) = (β γ )/(β 1 γ 1 ) (8) and x(β) = (x 1 (β),..., x L (β)), so that, n partcular, x 1 (β) = 1 for all β. We wrte x (β ) for (x 1 (β 1 ),..., x π() (β π() ), x (β )). Note the dfference between the vector x (β ) and ts -th element x (β ). Lastly, we ntroduce the set S L 1, va the followng lemma, for our man result below. Lemma 4. Wrtng S L 1 = {β (γ 1, ) (γ L, ) : c, β = C}, the mappng β S L 1 x(β) {1} R L 1 + s one-to-one. Proof. It suffces to show that there s a unque β S L 1 correspondng to a gven vector x R L + wth x 1 = 1. By the defnton of x, we have β = γ + x (β 1 γ 1 ) and therefore c, β = c, γ +(β 1 γ 1 ) c, x. Settng the rght-hand sde equal to C yelds β 1, whch n turn fxes β 2,..., β L through x = (β γ )/(β 1 γ 1 ) for = 2,..., L.

11 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 11 The frst step s to observe that the fxed-pont system (3) n terms of β can be wrtten as a system of equatons n terms of x (β ). For = 1,..., L, we obtan from (1), (7) and the homogenety of h (Lemma 3) that τ (β) = (β γ )EZ β = x (β )(β 1 γ 1 )[h (β γ ) p π(), h π() (β π() γ π() )] = x (β )[h (x (β )) p π(), h π() (x π() (β π() ))]. (9) In partcular, τ (β) s a functon solely of x (β ) (assumng all network parameters are fxed except for the servce capactes). Namely, the tree structure yelds that τ (β) only depends on β, and the Brownan nature of the network further reduces the dependency to only x (β ). Hence, abusng notaton, we can wrte τ (x ) = x [h (x ) p π(), h π() (x π() )]. (10) We next rewrte the fxed-pont system n terms of x. If β solves the system of fxed-pont equatons (3), we readly obtan C c, γ w τ c (β (x (β ))c = γ ) L wk τ k (x k (βk ))c k, mplyng that w τ (x (β ))c /(c (β γ )) does not depend on. Thus, we have w τ (x (β ))c w1 τ 1 c 1 = c (β γ ) c 1 (β1 γ 1 ), whch s equvalent to τ (x (β )) c /w = τ x (β 1 x (β ). ) c 1 /w 1 The next step s to ntroduce the h functons by notng that (9) mples k=1 h (x (β )) p π(), h π() (x π() (βπ())) = τ (x (β )). x (β ) We then conclude that β satsfes (3) f and only f, for = 1,..., L, h (x (β )) p π(), h π() (x π() (β π())) = τ 1 c /w c 1 /w 1 x (β ). (11) We now prove that there s exactly one soluton x to (11), whch mmedately shows that there s exactly one soluton β of (3) on the boundary of the feasble set. Theorem 1. The system of fxed-pont equatons (3) has exactly one soluton on the set S L 1. Proof. We frst show that there s exactly one soluton to the fxed-pont equatons n (x 2,..., x L ) (0, ) L 1,.e., to the equatons h (x ) p π(), h π() (x π() ) = τ 1 c /w c 1 /w 1 x (12) for = 1,..., L. To see ths, frst consder all equatons correspondng to statons at a dstance of c 1 from the root. These equatons are of the form h (1, x ) = p 1 τ 1 + τ /w 1 c 1 /w 1 x. Snce the left-hand sde s nonncreasng from + n x > 0 and the rght-hand sde s strctly ncreasng n x, there s exactly one x -value for whch equalty holds. Next assume that the x l correspondng to statons at a dstance of n 1 from the root have been determned from (12), and consder a staton at a dstance of n from the root. As before, the left-hand sde of (12) s nonncreasng n x whle the rght-hand sde s strctly ncreasng n x. Thus, there s a unque x for whch equalty holds.

12 12 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 4.2. Optmalty Gap We next study the relatonshp between the soluton to (OPT-BTN) and the proposed fxed-pont approxmaton. Usng the homogenety of h from Lemma 3 and the defnton of x n (8), one readly fnds that the objectve functon of (OPT-BTN) becomes mn β 1 >γ 1,x 2 >0,...,x L >0 1 β 1 γ 1 L w [h (x ) p π(), h π() (x π() )] =1 subject to (β 1 γ 1 )[c 1 + L l=2 c lx l ] C c, γ. Snce the objectve functon s decreasng n β 1 γ 1, the constrant wll be bndng n any optmal soluton. After castng the resultng problem back nto the form of (OPT-BTN), we mmedately obtan the followng lemma. Lemma 5. Any soluton β to (OPT-BTN) satsfes c, β = C. Upon substtutng the constrant (β 1 γ 1 )[c 1 + L l=2 c lx l ] = C c, γ nto the objectve functon, we see that the soluton of (OPT-BTN) s equvalent to c, x mn x 1 =1,x 2,...,x L C c, γ L w [h (x ) p π(), h π() (x π() )]. (13) =1 Defne the optmalty gap as the rato of the objectve functon at the allocaton gven by the fxedpont approxmaton and by the soluton to (OPT-BTN). Now we can present the man result of ths subsecton. Proposton 1. Our fxed-pont approxmaton has the followng propertes when appled to Brownan tree networks. () As C, the optmalty gap remans bounded. In fact, t has a lmt n [1, ). () The dfference between the fxed-pont approxmaton and the optmzng argument n (OPT- BTN) s O(C) n each coordnate as C. Proof. For (), we use the fact that the soluton of (OPT-BTN) s of order 1/C as C, as shown n (13). The fxed-pont soluton approxmates (13) by evaluatng the objectve functon at the fxed pont x n x-space. Thus, t s also of order 1/C by constructon. For (), we note that the one-to-one correspondence from Lemma 4 between ponts x wth x 1 = 1 and ponts β S L 1 satsfes x β = γ + (C c, γ ) c, x. Snce nether a true optmal soluton for (13) nor the fxed-pont approxmaton depends on C n x-space, the correspondng quanttes n β-space are of order C. The argument n the proof shows that part () of ths proposton can be further refned by sayng that, unless the fxed-pont approxmaton s exact (as n the product-form case), t s exactly order C away from the optmal capacty allocaton Analyss of the Iterates Ths subsecton analyzes the terates of our fxed-pont algorthm wthn the context of Brownan tree networks. In partcular, we show that the relatve step sze of our algorthm s larger when the current terate s further away from the fxed pont, whch can be a key advantage of our fxed-pont approxmaton snce generc optmzaton algorthms do not possess ths property (as t would requre knowledge of the target pont). Based on ths result, we also dscuss how our algorthm can be modfed slghtly to guarantee that t converges to the unque fxed pont.

13 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 13 In order to explot the structural propertes of h, t s convenent to study the terates {β (k) } through {x (k) = x(β (k) )} as defned n (8); the two sequences are n one-to-one correspondence n vew of Lemma 4. The system of fxed-pont equatons (4) then becomes x (k+1) = w τ (x (k) )/c. (14) w 1 τ 1 /c 1 Namely, the fxed-pont terates n (4) can be equvalently descrbed through (14), where the defnton of τ (x ) s as gven n (10). Note that x (k) 1 = 1 for any k 1. We also wrte x for x(β ) Relatve Step Szes. We now show that the terates move n the drecton of the fxed pont, and that the magntude of the step s larger when the terate s further away from the fxed pont. Frst, we consder the terates correspondng to chldren of the root. Lemma 6. Let be a chld of the root. If x (k) Smlarly, f x (k) > x, we have x (k+1) x (k) x (k) < x, then we have x 1 > 0. (15) x (k) x (k) x (k+1) x 1 > 0. (16) x (k) x (k) Proof. Wrte α = c 1 w /(c w 1 τ 1 ). If x (k) < x, we have h (1, x (k) ) p 1 τ 1 h (1, x ) p 1 τ 1 = x by Lemma 2. In vew of (14) and (10), ths leads to α 1 x (k+1) = Smlarly, x (k) > x mples x (k+1) α x (k) [ ] h (1, x (k) ) p 1 τ 1 x x (k) > x (k). x x (k) < x (k). An analogous result holds for an arbtrary staton n the network, but the formulaton s slghtly more ntrcate. The formal statement of the result s gven n the followng proposton, whch s llustrated n Fgure 1, and whose proof s provded n Appendx B. Proposton 2. Let > 1 be fxed. Suppose that lm k x (k) = π() x π(). Then for any η > 0, there exsts a δ = δ(η) > 0 such that (15) holds for x (k) L η and (16) holds for x (k) U η, where L η U η = [x 1 ± δ(η)] [x π() ± δ(η)] (0, x η), = [x 1 ± δ(η)] [x π() ± δ(η)] (x + η, ). The convergence assumpton on x (k) π() prevents us from applyng ths lemma nductvely, snce the statement of the proposton does not exclude the possblty that the terates overshoot the fxed pont ndefntely Convergence of the Iterates. Snce Lemma 6 and Proposton 2 do not exclude the possblty of oscllatons occurrng under our fxed-pont teraton algorthm, the terates may not converge. Even though we have not encountered any nstance n whch the terates do not converge throughout our many numercal experments, t s worthwhle to explore how the algorthm can be slghtly modfed to ensure that the terates converge to the fxed pont.

14 14 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks Fgure 1 Illustraton of Proposton 2 for three queues n seres. The arrows ndcate that terates move n the specfed drecton. In our modfed algorthm, we addtonally mantan an nterval I (k) wthn whch x as well as all modfed terates x (k+1), x (k+2),... wll le. The dea s easest to explan when staton s a chld of the root, n whch case the argument reles on Lemma 6. If the sequence {x (k) } s a monotone sequence, then t must converge to x n vew of Lemma 6 and no modfcatons are needed; let us therefore suppose that the sequence s not monotone. The left endpont of I (k) s defned as the largest terate among x (0),..., x (k) that s smaller than x, and the rght endpont s smlarly defned as the smallest terate exceedng x. Then, f x (k+1) calculated from (14) les outsde of I (k), we overwrte x (k+1) wth the center of the nterval and contnue the algorthm on the subnterval contanng x. By addng ths bsecton step, the length of the nterval I (k) shrnks as k. To ensure that the nterval length shrnks to zero, we add the addtonal requrement (agan enforced by bsecton) that ether the left endpont grows by a factor 1 + ξ or the rght endpont shrnks by a factor 1 ξ n each teraton, where ξ > 0 s a parameter that s of smaller order than the desred level of accuracy for the fxed-pont approxmaton. Smlarly, f s not a chld of the root, then Proposton 2 shows that the modfed terates {x (k) } must converge. 5. Numercal Experments In ths secton we descrbe an extensve collecton of numercal experments performed to evaluate our general two-phase soluton framework, ntroduced n Secton 2, under a varety of stochastc network settngs. A prmary goal s to gan an understandng of how well the frst phase of the framework, employng our fxed-pont teraton algorthm, approxmates a locally optmal soluton to varous nstances of the stochastc optmzaton problem (OPT). Ths evaluaton s based on comparng and contrastng aganst both the second phase of our framework and an approach based solely on the stochastc approxmaton (SA) algorthm, each of whch dentfes local optma. In the settng of convex optmzaton problems, we obvously have unque globally optmal solutons and, from our results n Secton 4.1, the fxed pont dentfed n the frst phase of our approach s unque for convex nstances of (OPT) n the Brownan tree-network settng. Hence, n convex settngs, we are nterested n the qualty of the fxed-pont approxmaton vs-à-vs the globally optmal soluton to the problem (OPT) dentfed by the second phase of our approach. For settngs when

15 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 15 the optmzaton problem s possbly non-convex and may therefore have multple local optma, we also nvestgate whether the lmt pont(s) of our frst-phase terates changes under dfferent startng ponts for the algorthm. The key step n our fxed-pont approxmaton method s the estmaton of the expected queue lengths under the capacty values for the current terate. A smulaton-based mplementaton of queue-length estmaton n the fxed-pont teraton under the orgnal stochastc network settng yelds a consstent estmaton. Our expermental results are therefore generated from such a smulaton-based mplementaton. We note, however, that a numercal soluton for the fxed-pont queue-length estmaton under a Brownan approxmaton of the orgnal stochastc network can be used as the bass of the frst-phase soluton wthn our framework, provdng results comparable to smulaton n a much more effcent manner when the Brownan network s a reasonable approxmaton. The second phase of our framework then searches for a locally optmal soluton close to the lmt pont, by startng the smulaton-based optmzaton from the lmt pont dentfed n the frst phase. We chose to use the SA algorthm because of ts rgorous theoretcal foundaton, but note that any drect method can be used as the bass of the second-phase soluton wthn our framework. The remander of ths secton s organzed as follows. After provdng a bref overvew of the SA algorthm, Secton 5.2 detals the settngs over whch the numercal experments were performed. A summary of the observatons from these experments s then dscussed n Secton Stochastc Approxmaton (SA) Denote the objectve functon of (OPT) by z(β) = L mn β (0, ) L w =1 EZ β. We then consder the followng teratve smulaton algorthm to solve the optmzaton problem (OPT): ( ) β (n+1) = β (n) ɛ n K W (n+1) W(n+1), c c, (17) c, c where the varable W (n+1) s an estmator of the gradent of z(β) wth respect to β, the scalng matrx K s taken by common practce to be the dentty matrx I, and the term n parentheses s the projecton of the gradent estmate W (n+1) onto the hyperplane { c, β = C}. (Recall from Lemma 5 that the optmal soluton satsfes the budget nequalty constrant strctly n the Brownan tree-network settng.) Ths teratve scheme, known as the SA algorthm, has been well studed n the lterature; refer to Asmussen and Glynn (2007), Kushner and Yn (2003). The SA algorthm s effectvely the stochastczaton of a Newton-type teratve optmzaton (or rootfndng) algorthm. Assumng that the estmates W (n+1) are all generated wth the same sample sze (.e., ndependent of the teraton number n) and that the ɛ n -sequence satsfes ɛ n 0 and n ɛ n, t follows from known results (see Kushner and Yn (2003)) that the terates β (n) of (17) converge to the set of local mnmzers of (OPT). Furthermore, the rate of convergence s O(n 1/2 ) when the gradent estmator W (n+1) s consstent, wth a slower convergence rate otherwse. Unbased methods of gradent estmators can be constructed for specal stochastc network settngs, e.g., tandem (Ho et al. (1983)) and Jackson-lke (Glasserman (1991)) networks, whch mprove the asymptotc rate of convergence over based methods such as those based on fnte dfference. However, fnte-dfference methods were chosen for our comparsons because of the wder applcablty of fnte-dfference methods to general stochastc network settngs. In addton, the dffculty n choosng the parameters s unaffected by the choce of the estmator. We modfy the generc fnte-dfference gradent estmaton method for our numercal experments. In partcular, we employ a central-dfference gradent estmator: W (n+1) = z(β + h n+1 e ) z(β h n+1 e ) 2h n+1, = 1,..., L,

16 16 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks where h n+1 s the dfference ncrement sequence and e the vector of all zeros except for a one n the -th element. Such an estmator s known to converge at the best possble rate of O(n 1/3 ) under a certan optmal choce for the ɛ n and h n. Observe the much slower optmal order of convergence for ths scheme compared to that of the unbased gradent-estmator. When such a fnte-dfference gradent estmator s used, the resultng SA algorthm s called a Kefer-Wolfowtz scheme. Ths scheme suffers from several sgnfcant sources of potental errors and neffcences when employed to solve (OPT). Frst, the quanttes z(β) that need to be evaluated are steady-state measures, and standard batch-means technques for estmatng such measures suffer from an addtonal source of bas due to ntal transence. Second, although the SA theory specfes an optmal order-of-magntude result for ɛ n and h n, the specfc choce for these constants can be problematc. Ths s especally true when the gradent estmates are created from a fxed sample sze, whch s assumed n the theoretcal results on convergence. (Note that, n ths context, sample sze s counted n terms of batches needed by the standard steady-state batch-means technques.) In our experments, we use an equvalent smulaton termnaton crteron that tracks the standard confdence nterval (CI) for the estmator of the objectve functon z(β) and checks f the nterval satsfes a desred relatve sze. Ths makes the sample sze senstve to the current terate β (n) va the varance of the gradent estmator W (n). However, snce the varance can be expected to be well-behaved n the nteror of the budget-defned smplex, the sample szes at each step reman ndependent of the teraton count n of the SA algorthm, and thus the asymptotc analyss of Kushner and Yn (2003) contnues to hold. Our experments demonstrate that f the sample sze s chosen to be too small, then the nose n the terate sequence s hgh (n the sense that terates show lttle systematc mprovement n soluton qualty wth ncreasng n) and the SA algorthm needs to execute a large number of teratons n before the decreasng step-sze sequence ɛ n can ensure convergence. On the other hand, f the sample sze s chosen to be too large, the algorthm expends a great deal of computer tme n each step and s very slow to converge. Our experence here suggests that the SA teraton scheme works best n practce when started wth a small sample sze whch s then slowly ncreased wth the teraton count n. Table 1 descrbes the tradeoff observed for one specfc expermental settng. Ths graduated-ncrease approach however does not ft any framework of analyss for SA algorthms, and a convergence analyss of such a scheme s beyond the scope of ths paper Stochastc Network Confguratons We present results for two stochastc network confguratons, namely a fve-staton tree network depcted n Fgure 2(a) and a sx-staton feedforward network wth a non-tree structure depcted n Fgure 2(b). Both network structures arse naturally n a varety of computer archtectures that serve Internet traffc, as well as varous canoncal busness processes. In each confguraton, there are three ters of servce for the processng of ncomng requests, whch we characterze n the context of a generc data center to clarfy the presentaton. The servers comprsng the frst ter (e.g., web servers) provde ntal processng (e.g., access to web pages that may requre updatng of tmely nformaton such as stock prces). If a request requres addtonal processng (and the request s deemed to be legtmate), then the frst-ter server ether routes the request to a specfc server comprsng the second ter (e.g., applcaton servers) or performs a form of load balancng of the arrvng requests among these second-ter servers. The second ter of servers n turn ether completes the processng of the request n ts entrety usng locally avalable nformaton or provdes another level of processng before forwardng the request for addtonal servce by the next ter of servers (e.g., database servers). Probablstc routng s often used to model the flow of traffc through these feedforward stochastc networks, and the specfc probabltes used n our representatve experments presented below are gven n Fgure 2. All nterarrval tmes and servce tmes follow

17 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 17 Fgure 2 (a) The Tree Network (b) The Non-Tree Feedforward Network Network structures explored by the experments. The parameters of the probablstc routng are gven. 2-stage Coxan dstrbutons. Each server s assumed to have unt cost. The total capacty budget s set to 5 unts for the tree-network confguraton and 10 unts for the feedforward confguraton. Three settngs are defned for each network confguraton by varyng the weghts assgned to each server. Recall from the dscusson followng Lemma 2 that the Brownan tree network verson (OPT-BTN) of the optmzaton problem (OPT) s convex f the server weghts satsfy w π() w. We therefore consder three settngs that consst of () unt weghts, () weghts satsfyng the nonncreasng condton, and () weghts arbtrarly assgned to yeld a (possbly) non-convex nstance of (OPT-BTN). For each of the resultng sx model settngs, numercal experments are conducted over multple combnatons of coeffcents of varaton (CoVs) for the arrval and servce processes, none of whch satsfy the product-form requrements. The tree network s analyzed n Secton 3, where we establsh that the optmzaton problem (OPT-BTN) s convex when the server-weghts w satsfy the non-ncreasng condton, and thus ths problem has a unque globally optmal soluton. Expermental results suggest that the terates (4) of our fxed-pont approxmaton method have a unque lmt pont for more general stochastc networks than those satsfyng the condtons of Theorem 1. Under these crcumstances, the test then becomes a straghtforward comparson of the qualty of the lmt pont obtaned by teratons (4) aganst the global optmal soluton dentfed by the SA algorthm n the second phase when started from the dentfed lmt pont. For the same network confguraton when the weghts w yeld a dfference-of-convex functons for the objectve functon of (OPT-BTN), the problem may have multple local optma and our fxedpont teraton algorthm tself may have multple lmt ponts. Ths s a possblty for the fnal four model settngs. To test the hypothess that multple lmt ponts could exst, we execute the frst-phase algorthm from a collecton of startng server capacty values sampled unformly from the feasble set. In addton, we execute the SA algorthm from each of these same startng ponts to nvestgate f the algorthm dentfes multple local optma for the orgnal problem (OPT) Observatons Fgure 3 plots the relatve optmalty gap for the lmt ponts dentfed by the frst phase of our framework based on the fxed-pont teraton (4) n comparson wth the locally optmal soluton dentfed by the second phase of our framework phase based on the SA algorthm startng from the frst-phase lmt pont. Ths two-phase framework was appled to each of the sx model settngs under multple CoV combnatons for the nterarrval and servce tme dstrbutons. It s evdent from the results for each settng, plotted n Fgure 3, that the second-phase SA algorthm s able to mprove upon the soluton qualty by at most 5% for convex problem nstances, wth the majorty of such mprovements lmted to %. The relatve performance of our fxed-pont teraton s reduced a bt for the non-convex case, where the worst-case mprovement provded by the second phase rses to about 10% wth the average-case mprovement n the 3 5% range. In contrast, the

18 18 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks Spread n Qualty of Fxed Pont Iteraton Approxmaton for Sx Network Confguratons Fgure 3 Improvement n Objectve Functon Value (%) Tree Unt Weghts Tree Decreasng Weghts Tree Non-Convex Weghts FeedFwd Unt Weghts FeedFwd Decreasng Weghts FeedFwd Non-Convex Weghts Qualty of fxed-pont teraton. The relatve optmalty gap s wthn 5%, typcally 1% for settngs where the Brownan network s convex. For settngs where the Brownan network s possbly non-convex (smply referred to as non-convex n ths dagram), the relatve gap can be as much as 10%, whle the average s around 4 5%. objectve functon value for the optmal capacty allocaton obtaned usng a correspondng productform approxmaton (.e., settng all CoVs to 1) was observed to have a relatve optmalty gap n the range of 75% to 350%, clearly ndcatng the very poor qualty of such smplstc assumptons. Table 1 Sm Termnaton Num Converged Average Num Total Sm Crteron Iteratons Tme 1 Fxed at Geometrc Decrease: to n 20 ters. 3 Fxed at Settng stochastc approxmaton parameters. Not all 20 trals ran to convergence because the maxmum teraton count, set to 25, was reached. The smulaton experments for both the fxed-pont teraton and the SA algorthm were termnated wth the crteron that the relatve CI of the estmator of the objectve functon z( ) falls below a desred value. (The smulatons for the fxed-pont teraton were also executed wth an alternatve stoppng rule for relatve CI gaps on the estmaton of the average ndvdual queue lengths, and the algorthm was found to be ndfferent to the stoppng rule chosen.) An mplementaton of the SA algorthm requres the user to select good parameter values, and a crtcal parameter s the stoppng crteron for the smulaton runs. Table 1 further underlnes the computatonal savngs from the fxed-pont approxmaton method n the frst phase of our framework, before employng the SA algorthm n the second phase, by llustratng the dffculty faced n makng ths stoppng crteron choce. Multple runs of the fxed-pont approxmaton and the SA algorthm were ntalzed wth 20 unformly sampled startng ponts for a partcular network settng from the Feedforward non-convex weghts set n Fgure 3. When the fxed-pont approxmaton of the frst phase s executed wth a smulaton stoppng target of as the relatve

19 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks 19 CI wdth, the method termnated after an average of 4.75 teratons over the 20 trals, wth an average total smulaton-tme of tme-unts. The SA algorthm was executed from the same 20 startng ponts wth three dfferent termnaton crtera: the frst executes all smulatons wth relatve CI-wdth to match , the second gradually strengthens ths requrement n a geometrc sequence to over the frst 20 teratons, and the last executes all smulatons to match Table 1 shows that the weaker CI bound helps the method converge faster, but the teratons of the SA algorthm tend to wander and fewer trals complete wthn the maxmum count crteron. Strengthenng the CI bound ncreases the number of successfully completed trals and decreases the average number of teratons requred, but each tral takes much longer to execute n the aggregate. A practcal parameter choce seem to be the approach that gradually strengthens the CI requrement and strkes a good balance between the run length and the accuracy. Algorthm Soluton Returned Num. Trals Objectve Value Percentage n β-space Converge Here Estmate Improvement Frst phase (Fxed-pont) Second phase % (SA) % Table 2 Performance of two-phase procedure over a non-convex settng wth multple locally optmal solutons. Note that the SA algorthm n the second-phase dd not converge wthn reasonable computatonal budget for two of the trals. Fnally, n settngs that do not satsfy the suffcency condtons for a convex objectve functon n (OPT-BTN), we observe that the teratve approxmaton (3) always produces a unque lmt pont ndependent of the startng pont. These results suggest that the unqueness of our frstphase lmt pont holds more generally than the network condtons of Secton 4. Ths s n stark contrast to the SA algorthm executed over the same randomly chosen startng ponts, whch n some nstances produces multple local optma. In these cases, one expects to fnd locally optmal solutons spannng a range of soluton qualty, as llustrated by Table 2 whch provdes results from such a problem settng. Here, the fxed-pont approxmaton and the SA algorthm were executed from 20 unformly sampled startng ponts. Whle the fxed-pont approxmaton always produces the same lmt pont, the overall two-phase framework fnds two locally optmal solutons due to the fact that the SA algorthm can fnd more than one local optma even f t s started close to the same lmt pont. The qualty mprovement between the two local solutons does not exceed 10%, and thus one expects the locally optmal soluton from our two-phase framework to be close to the seemngly unque fxed pont dentfed n the frst phase. In comparson, when the SA algorthm was executed from the same 20 startng ponts, many addtonal local optma were obtaned that were of both better and worse qualty than the local optma dentfed by our two-phase framework. 6. Conclusons In ths paper we developed a general framework for determnng the optmal resource capacty allocaton at each staton comprsng a stochastc network, motvated by computer capacty plannng and busness process management applcatons. These problems are well known to be very dffcult from both a mathematcal and practcal perspectve. Our soluton framework s based on an teratve methodology that reles only on the capablty to observe the queue lengths at all network statons under any gven resource capacty allocaton. We theoretcally nvestgated ths methodology for sngle-class Brownan tree networks, and further demonstrated the benefts of our methodology through extensve numercal experments. The latter show that the frst phase of our

20 20 Deker, Ghosh, and Squllante: Optmal resource capacty management for stochastc networks methodology renders approxmatons to locally optmal solutons that are wthn 5% of optmalty on average. In addton to these soluton-qualty benefts, our framework does not requre the fnetunng of parameters and appears to be nsenstve to the chosen smulaton stoppng crteron. Our methodology further provdes reductons n computaton of multple orders of magntude over a purely smulaton-optmzaton approach based on stochastc approxmaton. In fact, regardless of the parameter settngs consdered, all of the stochastc approxmaton algorthms requred orders of magntude more teratons than our methodology to converge to an optmal soluton. Appendx A: Propertes of Brownan Tree Networks Ths appendx brefly revews elements of the constructon of a sngle-class Brownan tree network as t arses from a generalzed Jackson network wth a tree network topology. Further detals on constructng sngle-class Brownan networks n general can be found, for nstance, n Harrson and Wllams (1987). We also establsh n ths appendx several mportant propertes of the networks of nterest, ncludng proofs of Lemmas 2 and 3 from Secton 3. Both queue-length dynamcs and steady-state behavor are consdered, and thus we use slghtly dfferent notaton from the body of the paper. In partcular, we wrte Z β (t) for the queue-length vector at tme t, and use Z β ( ) nstead of Z β for the correspondng steady-state vector. A Brownan tree network reles on an L-dmensonal Brownan moton {X(t)} wth zero mean and covarance structure determned by where Σ j s gven by, for = 1,..., L, and, for j, Cov(X (t), X j (t)) = Σ j t, Σ = λ c 2 A, + γ c 2 B, + γ π() p π(), (1 p π(), ) + γ π() p 2 π(),c 2 B,π(), (18) Σ j = [ γ c 2 B,p j + γ j c 2 j,bp j + γ π() p π(), p π(j),j (1 c 2 B,π())1 {π()=π(j)} ]. Here the notaton ntroduced n Secton 3 s used, and the parameters c 2 A, and c 2 B, correspond to the squared coeffcent of varaton of the external arrval process at staton and the squared coeffcent of varaton of the servce tmes at staton, respectvely, n the underlyng (pre-lmt) generalzed Jackson network. We wrte Z β for the vector-valued process of queue lengths n the Brownan network under the servce-rate vector β. By constructon, as n Harrson and Wllams (1987), the process Z β arses from the soluton of a hgh-dmensonal Skorokhod reflecton problem wth X plus a drft term (dependent on β) as nput. More precsely, followng the conventon that I π(1) (t) = γ π(1) = β π(1) = 0, (Z β, I) s the (unque) process satsfyng: Z β (t) = X (t) + [(β γ ) p π(), (β π() γ π() )]t + I (t) p π(), I π() (t) 0 for any = 1,..., L, t 0; I s contnuous and nondecreasng, wth I (0) = 0, for = 1,..., L; Z 0 (t)di (t) = 0 for = 1,..., L. Usng the explct soluton to ths Skorokhod problem, we can establsh the followng lemma. Smlar results appear n varous places, but we provde a proof here for completeness. Lemma 7. For = 1,..., L and x > 0, we have [ ] ( ) h (x ) = E sup q j Xj (t j ) [x j p π(j),j x π(j) ]t j. 0 t t π() t 1 < j P