Bound Analyss of Closed Queueng Networks wth Workload Burstness Gulano Casale, Nngfang M, Evgena Smrn College of Wllam and Mary Department of Computer Scence Wllamsburg, VA, 2187-8795 {casale,nngfang,esmrn}@cs.wm.edu ABSTRACT Burstness and temporal dependence n servce processes are often found n mult-ter archtectures and storage devces and must be captured accurately n capacty plannng models as these features are responsble of sgnfcant performance degradatons. However, exstng models and approxmatons for networks of frst-come frstserved (FCFS) queues wth general ndependent (GI) servce are unable to predct performance of systems wth temporal dependence n workloads. To overcome ths dffculty, we defne and study a class of closed queueng networks where servce tmes are represented by Markovan Arrval Processes (MAPs), a class of pont processes that can model general dstrbutons, but also temporal dependent features such as burstness n servce tmes. We call these models MAP queueng networks. We ntroduce provable upper and lower bounds for arbtrary performance ndexes (e.g., throughput, response tme, utlzaton) that we call Lnear Reducton (LR) bounds. Numercal experments ndcate that LR bounds acheve a mean accuracy error of 2%. The result promotes LR bounds as a versatle and relable boundng methodology of the performance of modern computer systems. Categores and Subject Descrptors C.4 [Performance of Systems]: Modelng technques General Terms Algorthms, Performance, Theory Keywords Queueng networks, closed systems, bound analyss, burstness, nonrenewal servce, temporal dependence, Markovan arrval processes 1. INTRODUCTION Capacty plannng of modern computer systems requres to account for the presence of nonrenewal features n workloads, such as Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. SIGMETRICS 08, June 2 6, 2008, Annapols, Maryland, USA. Copyrght 2008 ACM 978-1-60558-005-0/08/06...$5.00. short-range or long-range temporal dependence whch sgnfcantly affect performance [2 25, 1, ]. A typcal example of temporal dependence s workload burstness, where the jobs processed by the system are not ndependent, e.g., the arrval of a short job s much more lkely to be followed by the arrval of another short job (and vce versa for long jobs). Tme-varyng workloads of ths type are naturally modeled as nonrenewal workloads wth temporal dependence among consecutve requests. Because of the complexty of ther analyss, only small nonrenewal models based on one or two queues have been consdered n the lterature, mostly n matrx analytc methods research [28]. We address the current lack of more general modelng technques for systems wth nonrenewal workloads by ntroducng and analyzng a new class of closed queueng networks whch can account for temporal dependence n ther servce processes. Our analyss enables for the frst tme the analytcal performance evaluaton of complex envronments wth nonrenewal workloads and mmedately fnds applcaton n the capacty plannng of mult-ter archtectures and storage systems. Capacty plannng based on product-form queueng networks [5] has been extensvely used n the past, snce these models enjoy smple soluton formulas and low computatonal cost of exact and approxmate algorthms [10, 22]. Queueng networks wth general ndependent (GI) servce [8, 16, 29, 2] have been proposed as a soluton, but although much more accurate than product-form networks, they reman nsuffcent for robust performance predctons f the servce process s nonrenewal. That s, because they completely gnore temporal dependence between servce tmes, they cannot be used to predct performance correctly n systems wth nonrenewal workloads. Ths paper overcomes the lmtatons of exstng modelng technques by provdng a bound analyss methodology for queueng networks wth nonrenewal workloads. We study a class of closed queueng networks where servce tmes are modeled by Markovan Arrval Processes (MAPs). We call these models MAP queueng networks. MAPs are a famly of pont processes whch can easly model general dstrbutons and nonrenewal features such as autocorrelaton n servce tmes [28]. Effcent fttng schemes for MAP parameterzaton from measurements are avalable, e.g., [1,11,20], and the resultng MAPs can approxmate effectvely long-range dependence [1]. Because of the well-known dffculty of extendng exact soluton formulas outsde the product-form case, we study bound analyss technques for MAP networks. Wth the excepton of the general ABA bounds [27] whch provde good estmates only for very low or very hgh populaton values, no boundng technques for nonrenewal networks exst and ths s due to the lack of exact results whch are usually needed to prove the boundng property. In ths
paper, we show that t s possble to obtan provable bounds on performance ndexes also n non-product-form models. Our nonrenewal bounds derve from the analyss of the Markov process underlyng the MAP queueng network. Because of the state space exploson, ts equlbrum behavor cannot be determned exactly, but we argue that t can stll be bounded accurately by descrbng the system wth reduced state spaces (whch we call margnal state spaces). Ths state space transformaton captures the behavor of the network condtoned on a gven queue beng busy or dle. The number of states n these margnal spaces grows lnearly wth the number of jobs n the network; thus, the proposed approach remans computatonally tractable also on models wth large populatons. We derve exact balance equatons for the equlbrum behavor of the reduced state spaces and llustrate how these formulas can be combned wth lnear programmng [6, 26] for the computaton of bounds on mean value ndexes. Because the number of reduced states grows lnearly wth the number of jobs n the network, we call these bounds Lnear Reducton (LR) bounds. The man contrbuton of ths paper s to present a new methodology for the effcent analytc soluton of queueng networks wth nonrenewal workloads. Ths methodology automatcally apples to queueng networks wth renewal workloads as well. The stated contrbutons and outlne of ths work are as follows. We provde evdence that exstng GI approxmatons and decomposton methods show unacceptably large errors on queueng network models wth temporal dependence n the servce process (Secton 2). We defne MAP queueng networks as a generalzaton of exstng queueng networks that can model nonrenewal workloads (Secton ). We develop the LR bounds on performance ndexes for nonrenewal MAP queueng networks that are based on a new margnal state space reducton (Sectons 4 and 5). We present an extensve set of representatve case studes showng that the LR bounds capture very well mean performance ndexes such as response tmes or utlzatons (Secton 6). We stress that MAP queueng networks are a superset of exstng non-product-form networks wth GI workloads. Therefore, the presented analytc methodology has a wde applcablty. The LR bounds are corroborated by extensve numercal valdaton, where we show that they acheve a mean accuracy error of approxmately 2% on a set of 10,000 random models, promotng MAP queueng networks as versatle models of modern computer systems. We conclude the paper by outlnng model generalzatons and extensons n Secton 7. The AMPL specfcaton [18] of the LR bounds s avalable at http://www.cs.wm.edu/mapqn/. 2. PREVIOUS WORK In Secton 2.1, we revew prevous work on non-product-form queueng network models wth FCFS queues and GI servce [7]. These models are the renewal specalzaton of the MAP queueng networks ntroduced n Secton. In Secton 2.2, we evaluate the applcablty of approxmaton algorthms for models wth GI servce to models wth nonrenewal servce. Due to lmted space, we pont the reader to [7,1,22] for general background on queueng network modelng and Markov processes. Throughout ths paper we assume that servce tme dstrbutons are modeled by the method of phases [7, 14]. 2.1 Analyss of Models wth Renewal Servce Closed networks of FCFS queues enjoy a product-form soluton f all servce tmes are exponentally dstrbuted [5]. If one or more servers have renewal (also called general ndependent (GI)) servce, such as hyperexponental or Coxan [14], the product-form theory does not apply and approxmate methods are used for evaluatng performance [7]. An approxmaton based on Markov renewal theory s developed by Reser n [29]. For each queue, the MVA arrval theorem [0] s generalzed to nclude the coeffcent of varaton (CV) of the GI servce process. Experments n [8, 16] show that ths approach, although smple, s prone to large approxmaton errors. In [2], Zahorjan et al. obtan an approxmate mean value analyss (AMVA) by decomposton-aggregaton [1]. The underlyng Markov process of the network s decomposed accordng to the actve phases at the GI servers. Each partton s evaluated n solaton by Mean Value Analyss [0] and the results are weghted to approxmate the GI network. Valdaton results of the AMVA decomposton-aggregaton show good accuracy. In [16], Eager et al. mprove the results n [29] and [2]. The response tme at the GI queue used n Reser s method s replaced by a more effectve nterpolaton whch also accounts for the response tme at the other queues. [16] also mproves the decomposton method n [2] and makes t compatble wth the teratve AMVA framework to acheve lower computatonal costs on networks wth several queues. Mare s method, the dffuson approxmaton (DA) method, and the maxmum entropy method (MEM) assume a product-form for the equlbrum state probabltes of the GI network and approxmate the model accordngly [7]. DA and MEM rely on formulas nvolvng only the mean and the coeffcent of varaton; Mare s method s more general and uses specalzed relatons for Coxan dstrbutons. Mare s method provdes good accuracy n models wth GI servers although ts convergence propertes have not been assessed [8]; DA and MEM are typcally less accurate. Fnally, the Chandy-Herzog-Woo (CHW) method [12, 22] replaces an arbtrary subsystem by a flow equvalent server whch preserves the mean throughput of the orgnal subsystem n each feasble state. If the subsystem ncludes GI servers, CHW s known to be less accurate than Mare s method [8]. 2.2 Applcablty to Nonrenewal Servce To the best of our knowledge, no results are avalable for analyzng closed networks wth nonrenewal servce, see [28] for related work n sngle queue systems. In ths secton, we establsh the applcablty of the methods descrbed n Secton 2.1 to closed networks wth nonrenewal servce. We consder dentcally dstrbuted servce processes whch are characterzed by temporal dependence. The temporal dependence of a nonrenewal processes can be approxmately modeled by the autocorrelaton functon ρ k whch captures the smlarty n magntude of samples spaced by k lags [14]. As an example, a servce process can have hyperexponentally dstrbuted samples wthout beng necessarly renewal; that s, the usual termnology hyperexponental servce mplctly refers to the renewal verson of the process, but n general nonrenewal processes wth hyperexponental dstrbuton can be defned. These are mmedately obtaned by changng the order of the samples wthout alterng ther dstrbuton, whch results n temporal dependence. A smple case of nonrenewal closed network s shown n Fgure 1. We use ths smple model to evaluate the applcablty of methods for GI queueng networks to models wth nonrenewal servce. Queue 1 s exponental wth rate µ 1 = 1; queue 2 has MAP(2)
M µ 1 queue 1 queue 2 MAP Fgure 1: Example model wth nonrenewal servce. The servce s an exponental process wth rate µ 1 at queue 1, at queue 2 s a possbly nonrenewal two-phase Markovan Arrval Process (MAP(2)) [28]. Queue 1 Utlzaton [0=0\%,1=100\%] 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 Evaluaton of the Nonrenewal Model n Fgure 1 Exact Decomp ABA Upper Bnd ABA Lower Bnd 0.55 0 50 100 150 200 250 00 50 400 450 500 N Job Populaton servce [17] whch can exhbt autocorrelaton n servce tme samples and thus be a nonrenewal process. In a MAP(2), the lag-k autocorrelaton ρ k geometrcally decays to zero wth rate 1 γ 2 accordng to the relaton ρ k = γ2 `1 k 1/CV 2 /2, [11]. In ths example we choose γ 2 = 0.75, the MAP(2) s % faster than the exponental queue (.e., mean rate µ 2 = 1.), has CV = 5, and skewness 15; the process s obtaned by the moment and autocorrelaton matchng algorthm n [11]. The results dscussed below are qualtatvely smlar for other models. Usng the model n Fgure 1, we have observed that several of the methods consdered n Secton 2.1 cannot be appled to models wth nonrenewal workloads; we dstngush two groups: Non-applcable methods. Ths group ncludes Reser s approxmaton, the AMVA methods n [16, 2], Mare s method, DA, and MEM. Intutvely, these methods cannot apply for the followng reasons. For the consdered example, all these methods depend only on the mean, the CV, and the probablty of startng servce n one of the two phases. For nstance, Mare s method apples correctons based on the two-phase Coxan renewal process model that s completely specfed by these three parameters [7]. The nformaton about the order of samplng, that s fundamental to nonrenewal servce, s gven only by the underlyng Markov process whch s not drectly evaluated by these methods. Snce these technques gnore the order of samplng of the servce tmes, they cannot account for the temporal dependence and therefore produce dentcal results f ρ k 0, k 1, or n the nonrenewal case ρ k 0, for some k 1. Yet nonrenewal models can have performance that s extremely dfferent from ther renewal counterpart [24, 25]; therefore these methods are unft for the analyss of nonrenewal models. Applcable methods. Decomposton-aggregaton [1] can nstead be used for the analyss of models wth nonrenewal servce. Snce ths method requres to evaluate all or part of the underlyng Markov process, t s not lmted to statstcal moments of the servce tme dstrbuton, but can account for changes n the phase transton rates of the MAP whch result n autocorrelated samples. Decomposton-aggregaton can be easly appled to the underlyng process by aggregatng states wth dentcal actve MAP phases; methods smlar to those n [16, 2] can be defned based on ths parttonng. However, we have found that decomposton-aggregaton can frequently exhbt severe errors f used n networks wth nonrenewal servce. Fgure 2 show the predcted utlzaton of decomposton (dashed lne) versus the actual utlzaton (sold lne) for the bottleneck queue 1 n Fgure 1. The actual utlzaton s obtaned by solvng the underlyng Markov process by global balance, therefore t s exact. The ABA bounds, whch apply to general models [22, 27], 1 The value γ 2 s n a MAP(2) the second largest egenvalue of the Markov chan embedded at arrval nstants [11]. Fgure 2: Exact global balance soluton of the nonrenewal model n Fgure 1 compared wth the ABA bounds [22, 27] and the decomposton-aggregaton approxmaton [1]. Although accurate n renewal models, n nonrenewal models the decomposton s often unable to capture the trend of performance ndexes such as the utlzaton shown here. The saturaton of the approxmaton s due to the saturaton of the underlyng product-form models used n the decomposton. Also the ABA bounds, whch apply to general networks, are naccurate. are also depcted. We recall that other classes of bounds such as the popular balanced job bounds (BJB) apply only to product-form networks [22]. Although decomposton s very accurate n the renewal case, ts applcaton to the nonrenewal case results n ncreasngly large approxmaton errors for larger populatons. Cases smlar to the one plotted n Fgure 2 are easy to fnd for dfferent values of the model parameters. The low qualty of the results n the nonrenewal case s due to the quck saturaton of the productform models used n these approxmatons, whch reach maxmum utlzaton for lower loads than the nonrenewal model. The observatons of ths secton ndcate that the analyss of nonrenewal workloads cannot be performed accurately wth exstng technques for models wth GI servce tmes. Due to the large approxmaton uncertanty and the lack of an exact product-form soluton, boundng technques are desrable. In order to address ths lmtaton, n the followng sectons we ntroduce a bound analyss methodology for nonrenewal networks.. MAP QUEUEING NETWORKS We ntroduce the class of MAP queueng networks supportng nonrenewal servce whch s studed n the rest of the paper. A summary of the man notaton s gven n Table 1..1 Model Defnton We consder a closed network wth sngle-server queues, whch serve jobs accordng to a MAP servce tme process and under work-conservng FCFS schedulng. The servce process s ndependent of both the job allocaton across the queues and the state of other servce processes. The network s composed by M queues and populated by N statstcally ndstngushable jobs (sngle class model), whch proceed through the queues accordng to a statendependent routng scheme. That s, upon departure from a server, a job jons queue j wth fxed probablty p,j. Wthout loss of generalty, the average vst rato at j wth respect to the number of vsts at queue 1 s V j, thus V 1 = 1. The servce process at queue s modeled by a MAP wth K 1 phases. General servce can be approxmated accurately by a MAP [4]. If K = 1, then the MAP reduces to an exponental dstrbuton, otherwse t generates servce tme samples that
Bj k states ( n, k) where j s busy n phase k Cj k () mean queue-length of queue wthn Bj k e vector of zeros wth a one n the -th poston h, k, u, k phase ndexes, j, m queue ndexes Ij k states ( n, k) where j s dle n phase k Jj k (, h) utlzaton of queue n phase h wthn Bj k Ij k k actve phase at queue n k K number of phases n queue s MAP K max maxmum K, 1 M k phase vector,.e., actve phases M number of queues n the network µ mean servce rate of queue µ k,h completon rate of queue, phase k h N number of jobs n the network n number of jobs at queue n n n populaton vector,.e., job allocaton p,j routng prob. from queue to queue j π( n, k) prob. of state ( n, k) πj k (n, h) prob. of n jobs n queue n phase h wthn Bj k π j k (n, h) prob. of n jobs n n phase h wthn Ij k q k,h,j rate ( n, k) ( n e + e j, k ), k = k, k = h Q mean queue-length at queue Q k mean queue-length at queue n phase k U mean utlzaton of queue U k mean utlzaton of queue n phase k v k,h background trans. rate of queue, phase k h V mean vst rato at queue (V 1 = 1) X mean throughput (measured at queue = 1) Table 1: Summary of man notaton are phase-type (PH) dstrbuted [28]. That s, hyperexponental, hypoexponental, Erlang, and Coxan are all allowed servce tme dstrbutons; nonrenewal servce s also supported, e.g., Markov Modulated Posson Process (MMPP), Interrupted Posson Process (IPP) [17]. It should be nevertheless remarked that MAP fttng can be stll a challengng problem f the data has an rregular temporal dependence structure, see [20] for a revew. We pont to [11] for a new technque, called Kronecker Product Composton (KPC), that can provde MAP fttng of hgher-order moments and temporal dependence structure of arbtrary processes. The transton from phase k to phase h for the MAP servce process of queue has rate φ k,h and produces a servce completon wth probablty t k,h ; f h = k then t k,k = 1 accordng to the MAP defnton. We defne µ k,h = t k,h φ k,h to be the rate of job completons n phase k that leave the MAP n phase h; v k,h = (1 t k,h )φ k,h, k h s the complementary rate of transtons not assocated wth job completons that only change the MAP actve phase (background transtons). In ths representaton of queue s MAP, µ k,h s the element n row k and column h of the D 1 matrx; v k,h s n row k and column h of D 0. We pont the reader to [20] and references theren for background on MAPs and MAP fttng..2 Underlyng Markov Process General MAP servce requres to mantan nformaton at the process level on the current servce phase at each queue. A feasble network state n the queueng network underlyng Markov process s a tuple ( n, k), where n = (n 1, n 2,..., n M), 0 n N, M µ 1 Queue 1 p 1,1 p 1,2 p 1, M µ 2 Queue 2 Queue MAP p =1 2,1 p =1,1 Fgure : Example network composed by two exponental queues and a MAP queue. In the case where the MAP s a renewal two-phase hyperexponental process, ths reduces to Balbo s model consdered n the valdaton of approxmatons for GI servce [8]. =1 n = N, descrbes the number of jobs n each queue, and k = (k1, k 2,..., k M), 1 k K, specfes the actve phase for each servce process. Accordng to ths space, the Markov process transtons have rate q k,h,j from state ( n, k) to ( n e + e j, k ), k = k, k = h, where e t s a vector of zeros wth a one n the t-th poston; the rate s computed as ( p,jµ k,h q k,h,j = v k,h, j, + p,µ k,h, = j and k h. In (1), q k,h,j s for j the rate of departures from to j trggerng a phase transton n s servce process from phase k to h; otherwse t accounts for the background transtons v k,h and the rate of the self-loopng jobs p,µ k,h. Note that the case for = j and k = h s not explctly accounted snce t corresponds to the dagonal of the nfntesmal generator of the Markov process. Ths dagonal s computed to make each row sum to zero. The sze of the nfntesmal generator corresponds to the cardnalty of the related global balance equatons and s of the order of `N+M 1 `Kmax+M 1 N K max, where Kmax s the maxmum of K, 1 M; ths sze quckly becomes computatonally prohbtve. As a summarzng example, the MAP network n Fgure wth routng probabltes p 1,1, p 1,2, p 1, = 1 p 1,1 p 1,2 at the frst queue and p 2,1 = 1, p,1 = 1, at the remanng queues has underlyng Markov process as shown n Fgure 4. The state space descrpton s gven n the capton. For p 1,1 = 0.1 and p 1,2 = 0.7 the network reduces to Balbo s model used n the numercal experments n [8]; throughout the paper we llustrate some of the proposed technques usng ths model. 4. STATE SPACE REDUCTION General approxmaton technques for non-product-form models, such as decomposton, are revewed n Secton 2. These approaches often start from the dea of applyng a state space transformaton to reduce model complexty. For nstance, approxmate lumpng s used n decomposton to partton the state space nto macrostates that can be evaluated n solaton [7]. However, exstng state space reductons ntroduce approxmaton errors that cannot be bounded n sgn or n magntude. Ths leaves a hgh degree of uncertanty on the fnal approxmaton accuracy. In ths secton we develop a new famly of state space reductons that does not ntroduce any degree of approxmaton, whle stll smplfes model analyss. The proposed reducton s therefore exact, but because of several dfferences from exact lumpng, the transformaton cannot be reduced to lumpng or to any method presented n prevous work. (1)
v 2,1 v1,2 µ 1,1 µ 1,1 µ µ 2,2 2,2 200,1 101,1 002,1 002,2 101,2 200,2 p 1, µ 1 p 1, µ 1 p p 1, µ 1, µ 1 1 v2,1 p 1,2 µ 1 µ 2 p 1,2 µ 1 µ 2 v1,2 p 1,2 µ 1 µ 2 p 1,2 µ 1 µ 2 110,1 µ 1,1 µ 2,2 011,1 011,2 p 1, µ 1 p 1, µ 1 110,2 p 1,2 µ 1 µ 2 Busy condton subspace of v2,1 Busy condton subspace of p 1,2 µ 1 µ 2 queue n phase 1 queue n phase 2 020,1 Fgure 4: Underlyng Markov process of the network n Fgure n the smple case when the MAP s a MMPP(2) process; the job populaton s N = 2. Two queues are exponental wth rates µ 1 µ 1,1 1 and µ 2 µ 1,1 2, respectvely; the thrd queue s a MAP wth K = 2 phases havng µ k,h = 0 for k h, that s a MMPP(2) process. (002, 1) ndcates that the exponental queues are dle and the MAP queue has two jobs and s n phase 1; n (110, 2), the phase 2 s the phase left actve by the last served job. v1,2 020,2 p 1,2 µ1 p 1,2 µ 1 µ 1,1 101,1 002,1 p 1, µ 1 queue 2 µ 2 length 0 011,1 queue 2 length 1 Busy condton subspace of queue n phase 1 101,1 011,1 µ 2 (a) µ 1,1 p 1, µ 1 queue length 1 002,1 queue length 2 Busy condton subspace of queue n phase 1 (c) unknown rate 0 1 Margnal State Space µ 2 queue 2 Busy condton subspace of queue n phase 1 1 (b) Margnal State Space µ 1,1 unknown rate 2 queue phase 1 Busy condton subspace of queue n phase 1 (d) 4.1 Busy Condton Reducton We ntroduce a state space reducton that scales lnearly wth the populaton sze. We use the term busy condton" to dentfy the set of states where a gven queue s busy n a certan phase, whch s ntutvely smlar to a condtonal state space. For each model we generate the followng O(K 2 maxm 2 ) reduced state spaces wth dmenson O(N) as follows. DEFINITION 1 (MARGINAL STATE SPACES). Let the busy condton subspace Bj k = {( n, k ) : n j 1, k j = k} be the set of states of the MAP network where queue j s busy and n phase k. The margnal state space of queue n phase h wthn Bj k s the state space descrbng the observaton wthn Bj k of queue s queuelength whle ts phase s h, 1 h K, (the cases = j and h = k are both consdered). Snce n a non-product-form network the state of a queue mplctly depends also on the actvty of the rest of the network, the margnal state spaces allow to explore n a compact way the mutual relatons between any two queues and j. A probablstc defnton of margnal state space s gven later n Secton 4.1.1. Two example margnal spaces for the model n Fgure 4 obtaned for the busy condton subspace B 1 are shown n Fgure 5. Fgures 5(a)- (b) are obtaned by observng the exponental queue = 2 n ts only phase h = 1 wthn B 1. Snce queue s always busy n B 1, t has queue-length n 1 and the queue-length of queue 2 can only be n 2 = 0 or n 2 = 1. Note that the rate of transtons from n 2 = 1 to n 2 = 0 depends only on queue 2 s servce rate µ 2; the rate from n 2 = 0 to n 2 = 1 depends nstead on job completons at the other queues and n the orgnal state space s equal to π(101, 1)p 1,2µ 1 whch s unknown 2 wthout the equlbrum probablty π(101, 1). Fgure 5(b) smlarly descrbes the queue-lengths of queue n phase 1 wthn B 1, whch can be only n = 1 or n = 2 snce queue s busy. The unknown transton rate s n ths case π(101, 1)p 1,µ 1. Fgure 5 clearly shows that our approach dffers from an exact lumpng or a decomposton-aggregaton for at least three reasons: the latter technques are appled to the entre state space and not to busy subspaces only, the aggregates are always non-overlappng 2 We henceforth assume that global balance solutons for MAP network s prohbtvely expensve, therefore the equlbrum probabltes are all unknown. Fgure 5: Example of margnal state spaces for the model n Fgure 4. Fgures (a) and (c) show the observaton of ether queue 2 or queue n phase 1 durng the busy subspace B 1 of queue n phase 1. The dashed ovals ndcate states n the orgnal state space n Fgure 4 that are mplctly accounted for n the margnal state spaces for queue 2 and for queue n phase 1 depcted n fgure (b) and (d), respectvely. Note that the reducton s not a lumpng or a decomposton-aggregaton snce we are completely gnorng the transton from/to the busy condton subspace that are present n Fgure 4; also some rates would requre the equlbrum probablty of the state (101, 1) n Fgure 4 whch s unknown, and therefore the margnal state space cannot be solved n solaton. (two busy subspaces nstead can overlap, e.g., B,1 and B 2,1), and the aggregates result n a reduced state space where all rates are known so that t s later analyzed by other technques (e.g., decomposton solves each macrostate n solaton by global balance or mean value analyss). The man dea motvatng the busy condton reducton s as follows. Even f some rates are unknown, we can obtan balance equatons both for the equlbrum nsde each margnal space or between the probabltes of multple margnal spaces. We show n Secton 5 how the busy condton reducton can be used to defne the LR performance bounds. 4.1.1 Margnal Probabltes The margnal probablty π k j (n, h) of havng n jobs n queue durng phase h, 1 h K, whle queue j s busy n phase k, 1 k K j, completely characterzes the margnal state spaces. Each margnal probablty can be computed as π k j (n, h) = P {( n, k ) B k j :n =n,k =h} π( n, k ), where Bj k s the busy condton subspace of queue j n phase k. By defnton, t s πj k (n = N, h) 0 for j, πj k (n j = 0, k) 0, πj k (n j, h) 0 for h k, and πj k (n j, k) P K h=1 πh (n j, k) for j, n j 1. The last nequalty follows mmedately by observng that πj k (n j, k) accounts for all states n P K h=1 πh (n j, k) plus the states wthn Bj k where s dle. Because any event n the underlyng Markov process nvolves at most two-phases and two queues, that s, source and destnaton queues for job departures wth a possble phase transton at the
source queue, the margnal probabltes πj k (n, h) stll capture all departures and phase changes n the model. Therefore, the knowledge of all πj k (n, h) s s suffcent to compute all mean performance ndexes of nterest n the orgnal state space, ncludng: the utlzaton of queue,.e., U = P K k=1 Uk, where we denote by U k the utlzaton of n phase k, that s U k = P N P Kt n t=0 h=1 πk (n t, h) (2) where t, 1 t M, s an arbtrary queue snce the summaton s always equal to the probablty of the busy subspace B k ; the throughput whch by the Utlzaton Law [22] s K 1 K X X 1 MX X = q k,h 1,j Uk 1 = U 1µ 1/V 1, k=1 h=1 j=1 that s the mean rate of jobs flowng out of queue 1 assumed as reference for network completons and where µ 1 denotes the mean rate of the MAP servce process at queue 1; the mean queue-length of queue s Q = P K k=1 Qk, wth Q k = P N n =1 n πk (n, k) () beng the mean queue-length of n phase k. Note that these ndexes are also suffcent to compute response and resdence tmes by Lttle s Law, see [22]. In partcular, the response tme s R = N/X. 4.1.2 Sngle Busy Subspace of a Sngle Queue We characterze the equlbrum reached at steady state by margnal spaces. We focus on the margnal state spaces whch descrbe a sngle busy subspace Bj k and use the populaton constrant =1 n = N. Although an obvous condton, t s mpossble to mpose t f the state space s transformed n such a way to hde some of the n s, as n the margnal state spaces. We therefore defne a new populaton constrant for the busy condton subspace. THEOREM 1. Defne Cj k () = P N P K n =1 h=1 nπk j (n, h), (4) as the mean queue-length of n the busy condton subspace B k j, thus C k j (j) = Q k j. Then wthn B k j the C k j () sum to NU k j,.e., =1 Ck j () = NU k j, (5) 1 k K j. PROOF. Usng (2) and the populaton constrant we have NUj k = P N P Kt =1 n n t=0 h=1 πk j (n t, h) and choosng the arbtrary queue t equal to NUj k = P N P K =1 n =1 h=1 nπk j (n, h) = =1 Ck j (). 4.1. Multple Busy Subspaces of a Sngle Queue We obtan a constrant for multple busy subspaces whch resembles the global balance equatons of the MAP servce process consdered n solaton. THEOREM 2. The utlzatons of queue n ts K phases are n equlbrum,.e., j=1 PK h=1 h k f j= q k,h,j Uk = j=1 for all 1 M, 1 k K. PK h=1 h k f j= q h,k,j Uh, (6) PROOF. (Sketch of the proof, see [9] for a complete dervaton.) Consder the cut separatng the group of states G k where queue s n phase k from the complementary set of states C k where queue s n phase h k. The outgong probablty flux from G k s the left hand sde of (6) and must be balanced at steady state by an equal ncomng flow generated by the phase change transtons n C k. Ths probablty flux s exactly the rght hand sde of (6), whch completes the proof. The derved equaton mposes that the MAP n solaton and the MAP observed n the busy subspaces of queue have the same stochastc propertes, whch s expected f the servce process of queue s ndependent of the job allocaton across the network and of the servce processes of the other queues. 4.1.4 Margnal Balance Condtons Compared to the prevous balances whch only nvolve means such as queue-lengths or utlzatons, the balances descrbed n ths secton, called margnal balances, are more nformatve as they relate ndvdual margnal probabltes. We have found that there exsts a form of partal balance between margnal state spaces, although the class of models consdered n ths paper s non-product-form. Ths new class of balances, called margnal balances, shows that MAP servce mposes an equlbrum between the departure and the arrval process of queue n groups of states belongng to dfferent busy subspaces. Margnal balance derves from global balance, but characterzes only the set of margnal queue-length probabltes whch makes t always computatonally tractable. The balance s expressed as follows. THEOREM (MARGINAL BALANCE). The arrval rate at queue when ts queue-length s n jobs, 1 n N 1, s balanced by the rate of departures when the queue-length s n + 1, that s, j=1 j P Kj P Kj P K k=1 h=1 u=1 qk,h j, πk j (n, u) = j=1 j P K k=1 P K h=1 qk,h,j πk (n + 1, k), (7) for all 1 M. In the case n = 0 the margnal balance specalzes to the more nformatve relaton j=1 j P Kj P Kj k=1 h=1 qk,h j, πk j (n = 0, u) = j=1 j P K k=1 qk,u,j πk (n = 1, k), (8) whch holds for each phase u, 1 u K, wth 1 M. PROOF. (Sketch of the proof, see [9] for a complete dervaton.) The statement s a consequence of the state parttonng that separates the states where has no more than n enqueued jobs from the states where the queue-length s at least n + 1 jobs. Ther exchanged probablty flux must be balanced at steady state. The flux from the partton for states n to the partton for state n + 1 s equal to the rate of a job completed anywhere n the network beng routed to queue. Ths s the left hand sde of (7), whch also accounts for all possble phases of the job s departng queue j and the destnaton queue. The opposte flux from n + 1 to n has rate equal to the rght hand sde of (7), whch s the set of all possble departures from that are not routed to tself. Followng the proof of the margnal balance condtons, we obtan an addtonal balance between margnal probabltes.
COROLLARY 1. Let k, 1 k K, be a phase of queue ; the followng balance holds for each queue-length n, 1 n N 2, j=1 j P Kj P Kj k=1 h=1 (qk,h j, πk j (n + 1, k ) + P K u=1 q k,h u k,j πk j (n, u)) + P K = j=1 j (q k,k,j + q k,k,j π k k=1 q k,k k k, π k (n + 1, k ) (n + 2, k ) + P K k=1 (q k,k k k,j πk (n + 1, k) π k (n + 2, k) + P K + P K k=1 q k,h,j πk (n + 1, k))) h=1 h k q k,k k k, π k (n + 1, k), (9) for all 1 M. For n = N 1 the balance reduces to P K k=1 q k k k,,k + j=1 π k (n + 1, k ) j = P K j=1 k=1 (q k,k j k k P Kj P Kj P K k=1 h=1 u=1 q k,h u k,j πk j (n, u),j πk (n + 1, k)+ P K + P K k=1 h=1 h k q k,k k k q k,h,j πk (n +1, k)), π k (n + 1, k), (10) for all 1 M. PROOF. The proof follows smlarly to the proof of Theorem by now consderng the set of states where has no more than n enqueued jobs except for phase k, 1 k K, where ts populaton can be no more than n + 1. The theorem follows mposng the equlbrum at the nterface wth the set of states where the margnal queue-length s at least n + 1 and n phase k k and at least n + 2 and n phase k. 4.2 Idle Condton Reducton Ths state space reducton can be regarded as the complementary of the busy condton reducton descrbed n the prevous secton. We consder the dle condton subspace I k j where queue j s empty and the last served job has left the MAP process at j n phase k, 1 k K j. We obtan a set of O(K maxm 2 ) reduced state spaces wth dmenson O(N) by descrbng the evoluton wthn I k j of the queue-length of durng phase h, 1 h K. The related margnal probablty functon s π k j (n, h) = P ( n, k ) S k j (n,h) π( n, k ), (11) where the margnal space s S k j (n, h) = {( n, k ) I k j : n = n, k = h}, the dle subspace s I k j = {( n, k) : n j = 0, k j = k}. Further, by the gven defntons, π k j (n j, h) 0 f n j 1 or h k and smlarly to the busy condton reducton π k j (n j, k) P K h=1 πh (n j, k) for j, n j 1. Note that from the complementarty of π k j (n, h) and π k j (n, h), the total state space probablty s mmedately obtaned as P K h=1 P N n =0 (πk j (n, h) + π k j (n, h)) = 1, (12) for all 1 M. Moreover, let the utlzaton of queue n phase h wthn B k j I k j be J k j (, h) = P N n =1 (πk j (n, h) + π k j (n, h)). (1) where by defnton the second term n the summaton may be rewrtten as P N n =1 πk j (n, h) = π h (n j = 0, k), (14) whch smlarly to (12) relates the busy and dle reductons. Balances smlar to those gven for the busy condton reducton can be derved for the dle tme reducton. For nstance, followng the proof of (5) one mmedately obtans the populaton constrant =1 C k j () = N π k j (n j = 0, k), (15) where π j k (n j = 0, k) s the probablty of Ij k and C j k () = P N P K n =1 h=1 n πk j (n, h) (16) s the mean queue-length of n phase h wthn I k j. The balance equatons obtaned for the dle reducton are often redundant wth the balances of the busy ones. Therefore, we are not nterested n developng a comprehensve characterzaton of ths reducton. We pont out two relatons dervng from manpulatons of the global balance equatons whch characterze B k j I k j where j s n phase k; these formulas cannot be expressed wthn the probablty space of the busy subspace only. THEOREM 4. The sum of mean queue-lengths durng the subspace B k I k satsfes t=1 (Cj k (t) + C j k (t)) N P K h=1 Jk j (, h), (17) for all 1 M, 1 j M, 1 k K j. PROOF. Lettng P B k j Ik j N P B k j Ik j P ( n, k) B j k, we have Ik j π( n, k) = P t=1 B j k n Ik j tπ( n, k) = t=1 (P B j k n tπ( n, k) + P I j k n tπ( n, k)) = t=1 (Cj k (t) + C j k (t)), where the last passage follows by defnton of C j k (t) and C j k (t) as mean queue-lengths n Bj k and Ij k. Startng from the same term we also have N P B k j Ik j π( n, k) N P K h=1 Jk j (, h) snce the utlzaton of any queue, 1 M, durng B k j I k j cannot be greater than the sum of the probabltes of all states of B k j I k j. THEOREM 5. The performance ndexes n busy and dle subspaces are related by the followng equaton j=1 qk,h,j Qk + P K j=1, h=1 qh,k,j Uh j P K h=1, h k = j=1, j P Kj P Kj h=1 u=1 qh,u for all 1 M, 1 k K. j, Jk (j, h) + P K h=1, h k j=1 qh,k,j Qh, (18) PROOF. (Sketch of the proof, see [9] for a complete dervaton.) The proof follows smlarly to that of Theorem 2 by weghtng the contrbuton of each group of states by n. We pont to the techncal report [9] for an extensve dervaton. 5. LINEAR REDUCTION BOUNDS We obtan the LR bounds usng the results for the busy and the dle condton reductons. We determne the values of the margnal probabltes π = {π k j (n, h),, j, k, h, n } { π k j (n, h),, j, k, h, n }
f mn = mn f(π) subject to: /* prelmnary defntons */ eq. (2),(),(4),(1),(16); Cj k (j) = Q k j ; πj k (n j, k) P K h=1 πh (n j, k), f n j 1, j; πj k (n j, k) P K h=1 πh (n j, k), f n j 1, j; πj k (n j, h) = 0, f n j = 0; πj k (n j, h) = 0, f h k; πj k (n, h) = 0, f n = N, j; π j k (n j, h) = 0, f n j 1; /* exact characterzaton */ eq. (5), (6), (7), (8), (9), (10), (15), (17), (18); /* reducton constrants */ eq. (12), (14); /* feasblty of results */ πj k (n, h) 0, for all πj k (n, h) π. π j k (n, h) 0, for all π j k (n, h) π. Fgure 6: Lnear program determnng a lower bound on an arbtrary lnear performance ndex f exact = f(π exact). For nstance, f exact can be ether a mean ndex such as a throughput or a more detaled descrptor such as a margnal probablty π k j (n, h). so that the lnear functon f(π) s a bound on a performance metrc f exact f(π exact), where π exact s the set of exact equlbrum probabltes of the MAP network. In the case of lower bounds f mn f exact, the values of the margnal probabltes n π can be determned usng lnear programmng [6] as follows. PROPOSITION 1 (LR LOWER BOUND). The program n Fgure 6 returns a lower bound f mn f(π exact). PROOF. All the relatons n the lnear program are exact as we have proved n the prevous sectons; therefore π = π exact s a feasble soluton. Snce lnear programmng always returns an optmum mn f(π) = mn{f(π) feasble π}, we conclude that mn f(π) f(π exact ) because π exact s a feasble value of π. The last proposton generalzes mmedately f the lnear program s reformulated to compute an upper bound (LR Upper Bound) f max = max f(π) f(π exact); therefore the same constrants n Fgure 6 can be used both for upper and lower bounds and only the objectve functon has to be modfed. The accuracy of the LR bounds s valdated n the Secton 6. The computatonal costs of the LR technque are ndeed feasble for practcal applcatons, e.g., we have solved the lnear program for a model wth 10 MAP(2) queues and N = 50 jobs usng an nteror pont solver n approxmately four mnutes; for N = 100 the soluton of the same model s found n approxmately ten mnutes suggestng good scalablty. In general, the complexty of computng bounds wth the lnear program n Fgure 6 grows as O(lp(M 2 K max + MN, KmaxM 2 2 N)), where lp(r,c) s the computatonal cost of solvng a lnear program wth r rows and c columns. The number of rows s ether domnated by the number of possble margnal balances for the case n 1 that s O(MN) or by the number of nequaltes (17) whch grows as O(M 2 K max); total states total states M N K max margnal spaces orgnal space 50 2 1.84 10 5.0 10 100 2.64 10 2.06 10 4 200 2 7.24 10 8.12 10 4 5 50 2 5.10 10 1.27 10 6 5 100 2 1.01 10 4 1.84 10 7 5 200 2 2.01 10 4 2.80 10 8 10 50 2 2.04 10 4 5.0 10 10 10 100 2 4.04 10 4 1.71 10 1 10 200 2 8.04 10 4 7.04 10 15 Table 2: State Space Reducton Effectveness. Comparson of the total number of states n the margnal state spaces wth the orgnal state space of the queueng network. All queues have MAP servce tmes wth K max phases. The number of states n the margnal state spaces grows lnearly n the populaton sze, whereas the growth for the orgnal state space s combnatoral. the number of columns s O(K 2 maxm 2 N) because the cardnalty of π s upper bounded by 2K 2 maxm 2 N. To apprecate the reducton of the state space, Table 2 compares the number of states n the margnal state spaces wth the orgnal state space sze n models wth larger populaton and number of queues. The reduced spaces have cardnalty that can be several orders of magntude smaller than the orgnal state space. 5.1 Dscusson The balances obtaned n the Secton 4 provde a rch characterzaton of the underlyng Markov process of the MAP network. However, the number of exact relatons remans much smaller than the number of the margnal probabltes π k j (n, h) and π k j (n, h). We stress that our exact characterzaton s n general underdetermned and descrbes a famly of possble equlbra for the underlyng Markov process, among whch we cannot dstngush the real one. The lnear programmng approach allows to select the equlbrum that provdes a worst-case or best-case bound on a gven performance metrc. Because of the complexty of the feasblty regon descrbed by (2)-(18), t s also very hard to establsh the relatve mportance of each equaton wth respect to the others, as well as determnng analytcal lnear ndependence condtons among the balance equatons. In our experments, we have frequently observed that removng ether equaton (7) or (18) reduces sgnfcantly the qualty of the bounds. Conversely, we have found that (9) and (10) mprove accuracy only on certan models. Standard senstvty analyss of lnear programs [6] may be used as a tool for nvestgatng the relatve mportance of a certan balance for the model under study n order to mnmze the sze of the lnear program. 6. ACCURACY VALIDATION We assess the accuracy of the LR bounds usng the followng methodology. We use both randomly-generated models and representatve case studes, see Table for a descrpton of the employed nput parameters. In order to assess the accuracy of the LR bounds, we evaluate ther maxmal relatve error wth respect to the exact soluton of the MAP network computed by global balance. Due to the state space exploson, the expermentaton usng exact global balance solutons s often prohbtve for MAP networks wth more than three queues and populaton N 100. Gven ts mean, CV, skewness, and autocorrelaton decay rate γ 2, a MAP(2) s generated usng the exact moment and autocorrelaton matchng formulas n [11].
Network Param. Case Studes Random Models M 2 p,j varable random [0,1] N 10, 25, 100 all n [10, 1000] # of nonren. MAPs 1, 2, 1 MAP(2) Param. Case Studes Random Models mean varable random [0, 1] CV 0.5, 2, 4, 8 random [0.5, 10] skewness 0 random [2, 250] γ 2 [.00,.99] random [.00,.99] Table : Input parameters used n the valdaton study. For each model, we use the lnear program n Fgure 6 to compute upper and lower lmts X max and X mn on the mean throughput f(π) = X. Then, usng Lttle s Law we get the response tme bounds R mn = N/X max and R max = N/X mn whch are used to compute absolute relatve errors from the exact response tme R. We do not report errors on other measures due to lack of space, but we remark that they are typcally n the same range as those of response tme. We used the GNU Lnear Programmng Kt [19] to solve the lnear program on an Intel Xeon.7Ghz usng an AMPL specfcaton [18] of the lnear program n Fgure 6. The AMPL specfcaton s avalable for download at [15]. 6.1 Random Models In order to evaluate the general qualty of the LR bounds, we evaluate 10, 000 random models. The models are generated accordng to the specfcatons n Table. Each random model s solved for all feasble populatons and the followng absolute value of the maxmal relatve error s computed bnd = max R exact(n) N Rbnd(N), R exact(n) where R exact(n) s the response tme of the exact soluton computed for the network consdered wth populaton N and R bnd (N) s the LR bound evaluated wth the same populaton, ether R max(n) or R mn(n). We stress that the error functon s a conservatve estmator snce t returns the maxmum error of R bnd over all evaluated populatons. The converge of the bounds to the exact asymptotc value s not accounted by ths metrc and only the worst case error s measured. We see that the varablty n the routng matrx makes t possble to evaluate dfferent levels of balancng n the mean servce demand at the queues. Table 4 ndcates that the proposed bounds perform extremely well also for ths class of models. The mean error s 1 2% for both bounds wth a standard devaton of 0.02; the medan s less than the mean, ndcatng that the asymmetry of the error dstrbuton s more concentrated on small errors. The maxmum error s found to be 14.2% for the response tme upper Maxmal Relatve Error M mean std dev medan max R max 0.01 0.021 0.004 0.141 R mn 0.022 0.020 0.019 0.126 Table 4: Results of Random Experments Absolute maxmal relatve error (0 0%, 1 100%) over 10, 000 random queueng networks for the response tme R = N/X (R mn =lower LR bound, R max=upper LR bound). E.g., the mean of the error s 1.% for R max and 2.2% for R mn. bound and 12.6% for the lower bound. We have nspected carefully these cases and found that models wth more than 10% error n at least one of the two bounds account for only the 1% of the total number of experments. The burstness n these cases ncreases the response tmes at the autocorrelated staton n a way that cannot be easly captured. Furthermore, the lower bound seems to be more senstve to ncreased varablty and autocorrelaton than the upper bound, where the worst case error s for a MAP wth moderate burstness. The dfference n senstvty to MAP parameters s a postve property of the LR bounds, because large naccuraces n one bound can be compensated by the relatve accuracy of the other. Detaled numercal senstvty of the two bounds wth respect to the model parameters supportng these ntutons s dscussed n the next subsecton. 6.2 Representatve Case Studes We consder sx representatve case studes llustratng the accuracy propertes of the LR bounds; the results presented n ths sectons are typcal of the actual bound accuracy as we have shown n the random model valdaton secton. Dependng on the experment, the MAP can be ether an Erlang- 2 (E 2), a renewal two-phase hyperexponental (H 2), a Posson process (M), or a nonrenewal MAP(2) (MAP ). As we show n Case, the boundng s more dffcult for ncreasng values of CV; therefore we consder the E 2 process wth CV < 1 only n Case 1. To focus on the effects on accuracy of the most mportant moments (.e., mean and CV), we also fx n the case studes the skewness to 0. In the random experments, we have spanned all feasble skewness values for the consdered MAPs (range [2, 250]). 6.2.1 Case 1: Senstvty to Renewal and Nonrenewal Servce Processes The network s composed by two queues n seres. The servce process can be E 2, M, H 2 or MAP. For all processes, the mean rate s µ 1 = 1 at the bottleneck queue 1, µ 2 = 2 at the nonbottleneck queue 2. The MAP has CV = 5 and autocorrelaton decay rate (γ 2 = 0.5); the H 2 has the same moments of the MAP, but beng renewal ρ k 0, for all lags k 1. Results. Table 5 reports the maxmal relatve error on response tmes for all possble combnatons of servce processes. It s found that: (1) nonrenewal models are sgnfcantly more dffcult to evaluate than renewal models, e.g., on the most dffcult renewal case, the H 2/H 2 model, the LR bounds progressvely converge to the exact as N ncreases, whle on the MAP/MAP case the error remans at 10% also for N = 100; (2) hyper-exponental CVs are typcally more dffcult to approxmate than hypo-exponental CVs; () the error of R mn s no greater than 4% and s qute nsenstve to the servce process type; (4) R max s more senstve to nonrenewal servce where t acheves a worst-case error of 11%. 6.2.2 Case 2: Senstvty to Network Routng We evaluate the mpact on accuracy of network routng, showng the counter-ntutve fact that nonrenewal balanced networks are dffcult to approxmate. We consder the three queue model n Fgure wth the mean servce rates consdered n [8],.e., µ 1 = 1/0.028, µ 2 = 1/0.04, and µ = 1/0.28. We evaluate the accuracy of the LR response tme bounds n the case where the network s perfectly balanced (p 1,1 = 0.2, p 1,2 = 0.7, p 1,1 = 0.1), partally unbalanced (.e., queue s bottleneck, queue 1 and queue 2 are balanced, p 1,1 = 0.1, p 1,2 = 0.7, p 1,1 = 0.2, ths case corresponds to Balbo s model n [8]), or unbalanced (queue s bottleneck, queue 2 s slower than queue 1, p 1,1 = 0., p 1,2 = 0., p 1, = 0.4). The MAP queue has CV = 4 and γ 2 = 0.5.
Servce N = 10 N = 25 N = 100 renewal servce processes only bnk nonbnk mn max mn max mn max E 2 E 2 0.01 0.00 0.01 0.00 0.00 0.00 E 2 H 2 0.01 0.08 0.02 0.05 0.02 0.00 H 2 E 2 0.01 0.00 0.00 0.00 0.00 0.00 H 2 H 2 0.01 0.09 0.01 0.06 0.01 0.01 at least one nonrenewal servce process bnk nonbnk mn max mn max mn max MAP E 2 0.01 0.00 0.01 0.00 0.00 0.00 MAP H 2 0.01 0.09 0.01 0.06 0.01 0.01 E 2 MAP 0.00 0.11 0.00 0.11 0.00 0.10 H 2 MAP 0.00 0.05 0.00 0.05 0.00 0.06 MAAP 0.00 0.11 0.00 0.11 0.00 0.10 Table 5: Case 1 - Senstvty to Renewal/Nonrenewal Servce Processes. Absolute value of the maxmal relatve error (0 0%, 1 100%) on the response tme R = N/X of two queues networks ( mn =lower bound error, max=upper bound error). Results. Fgure 7 shows the LR bounds on response tmes and queue utlzaton n the balanced and partally unbalanced cases; the utlzaton bounds follows mmedately from the response tme bounds by Lttle s Law and the Utlzaton Law [22] and are useful to evaluate the LR bound accuracy as a functon of the bottleneck queue congeston level. We have found that the unbalanced case s extremely smlar to the partally unbalanced and therefore s not plotted. It s found that: (1) the LR bounds of both utlzaton and response tmes are very close to the exact value on most populatons; (2) both bounds progressvely converge to the asymptotc exact, a feature that s not always found n standard bounds for queueng networks (e.g., the ABA lower utlzaton bound never converges asymptotcally f M 2); () the slower asymptotc convergence n the balanced case makes the approxmaton more challengng, but the maxmal error remans less than 11% of response tme and 12% of bottleneck utlzaton. 6.2. Case : Senstvty to Servce Varablty and Autocorrelatons We consder the nonrenewal MAP/MAP model n the last row of Table 5 and we vary the CV and the autocorrelaton decay rate γ 2 for the two dentcal MAPs. Results. Table 6 reports the maxmal relatve error on response tmes. It s found that: (1) the error of R mn s loosely senstve to CV and γ 2; (2) the error of R max s proportonal to both CV and γ 2, but decreases wth the populaton sze; () the maxmum error of R max (1%) s compensated by the mnmum error of R mn (0%). Note that the errors are hgher than n the random models snce we are now consderng two MAPs nstead of one. 6.2.4 Case 4: Applcablty to Real Workloads We llustrate the applcablty to real workloads evaluatng the model n Fgure usng Balbo s partally unbalanced confguraton (p 1,1 = 0.1, p 1,2 = 0.7, p 1,1 = 0.2) and consderng at queue the nonrenewal MMPP(16) ftted n [1] from the classc long-range-dependent (LRD) Bellcore-pAug89 trace of [2]. Ths trace s often consdered n the lterature as representatve of many long-range dependent processes found n modern computer, communcaton, and multmeda systems. We scale the mean of the MMPP(16) so that µ = 1/0.28; CV, skewness and autocorrelatons are unchanged (see [1] for detals on ths MMPP(16) and ts autocorrelatons ρ k ). Results. Table 7 llustrates results for dfferent populatons. We consder N = 50 nstead of N = 100 because global balance N = 10 N = 25 N = 100 CV γ 2 mn max mn max mn max 2 0.000 0.00 0.01 0.00 0.01 0.00 0.00 2 0.250 0.00 0.02 0.00 0.01 0.00 0.00 2 0.500 0.00 0.02 0.00 0.01 0.00 0.00 2 0.750 0.00 0.02 0.00 0.02 0.00 0.01 2 0.990 0.00 0.02 0.00 0.02 0.00 0.02 4 0.000 0.00 0.05 0.01 0.0 0.01 0.00 4 0.250 0.00 0.05 0.01 0.04 0.01 0.01 4 0.500 0.00 0.06 0.00 0.04 0.01 0.02 4 0.750 0.00 0.06 0.00 0.05 0.01 0.0 4 0.990 0.00 0.06 0.00 0.06 0.00 0.06 8 0.000 0.00 0.12 0.01 0.10 0.02 0.04 8 0.250 0.00 0.12 0.01 0.10 0.02 0.05 8 0.500 0.00 0.12 0.01 0.11 0.02 0.07 8 0.750 0.00 0.12 0.00 0.12 0.01 0.09 8 0.990 0.00 0.1 0.00 0.1 0.00 0.1 Table 6: Case - Senstvty to Burstness and Autocorrelatons. Absolute value of the maxmal relatve error (0 0%, 1 100%) on the response tme R = N/X for a MAP/MAP network ( mn =lower bound error, max=upper bound error). N mn max 1 0.00 0.00 2 0.05 0.0 10 0.0 0.02 25 0.01 0.01 50 0.00 0.01 Table 7: Case 4 - Applcablty to Real LRD Workloads. We use the Bellcore-Aug89 trace [2] ftted n [1] by a MMPP(16). Absolute value of the maxmal relatve error (0 0%, 1 100%) on the response tme R = N/X for the network n Fgure wth the MMPP(16) at queue ( mn =lower bound error, max=upper bound error). s prohbtvely expensve n the second case. The results ndcate that: (1) the maxmal error s of 5% and the accuracy mproves wth the populaton sze. Ths result s consstent wth the accuracy levels n the other case studes and ndcates the applcablty of the LR bounds also wth models of real workloads; (2) because of the large order of the MMPP, the experment also llustrates the low senstvty of accuracy to changes n the number of phases. 6.2.5 Case 5: Senstvty to Multple MAP Queues We consder a closed network wth three queues n seres. The mean servce rate at queue s µ =. Servce s ether exponental or MAP(2) wth CV = 4 and γ 2 = 0.5. Durng the -th experment, 0, the frst queues are exponental, whle the remanng M are MAP(2). Results. Table 8 reports results of the four experments. It s found that: (1) the worst case error of 11% s acheved when all three queues are MAP(2); (2) both LR bounds are senstve to the ncrease n the number of MAP queues; () wth zero or one MAP(2) the bounds yet converge asymptotcally to the exact value; (4) the same concluson s not mmedate for the other cases, but computng R mn and R max for N = 250, 500 reveals (not shown n the table) that at N = 250 the gap between the bounds s R max/r mn 1 11%, whle for N = 500 t drops to 6% suggestng convergence. 6.2.6 Case 6: Senstvty to Network Sze In general, we have observed that for large models wth dense routng matrces the lnear constranng of the nevtably very small margnal probabltes may lead to numercally dffcult problems.