Revisiting Stochastic Loss Networks: Structures and Algorithms

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1 Revisiting Stochastic Loss etwoks: Stuctues and Algoithms Kyomin Jung Depatment of Mathematics MIT Cambidge MA 39 USA Mayank Shama Mathematical Sciences Dept IBM Watson Reseach Cente Yoktown Y 598 USA mxshama@usibmcom Yingdong Lu Mathematical Sciences Dept IBM Watson Reseach Cente Yoktown Y 598 USA yingdong@usibmcom Mak S Squillante Mathematical Sciences Dept IBM Watson Reseach Cente Yoktown Y 598 USA mss@watsonibmcom Devavat Shah Depatment of EECS MIT Cambidge MA 39 USA devavat@mitedu ABSTRACT This pape consides stuctual and algoithmic poblems in stochastic loss netwoks The vey popula Elang appoximation can be shown to povide elatively poo pefomance estimates especially fo loss netwoks in the citically loaded egime This pape poposes a novel algoithm fo estimating the stationay loss pobabilities in stochastic loss netwoks based on stuctual popeties of the exact stationay distibution which is shown to always convege exponentially fast to the asymptotically exact esults Using a vaiational chaacteization of the stationay distibution an altenative poof is povided fo an impotant esult due to Kelly which is simple and may be of inteest in its own ight This pape also detemines stuctual popeties of the invese Elang function chaacteizing the egion of capacities that ensues offeed taffic is seved within a set of loss pobabilities umeical expeiments investigate vaious issues of both theoetical and pactical inteest Categoies and Subect Desciptos G3 [obability & Statistics]: Stochastic pocesses Makov pocesses Queueing theoy; F [onnumeical Algoithms & oblems]: Computations on discete stuctues Geometical poblems and computations; G6 [Optimization]: onlinea pogamming Geneal Tems Theoy Algoithms efomance This wok is patially caied out while the autho was visiting the IBM TJ Watson Reseach Cente Wok of Shah was suppoted in pats by SF CCF and SF CS emission to make digital o had copies of all o pat of this wok fo pesonal o classoom use is ganted without fee povided that copies ae not made o distibuted fo pofit o commecial advantage and that copies bea this notice and the full citation on the fist page To copy othewise to epublish to post on seves o to edistibute to lists equies pio specific pemission and/o a fee SIGMETRICS 8 June 6 8 Annapolis Mayland USA Copyight 8 ACM /8/6 $5 Keywods Loss netwoks multidimensional stochastic pocesses stochastic appoximations Elang loss fomula and fixed-point appoximation ITRODUCTIO As the complexities of compute and communication systems continue to gow at a apid pace pefomance modeling analysis and optimization ae playing an inceasingly impotant ole in the design and implementation of such complex systems Cental to these eseach studies ae the models of the vaious applications of inteest Fo almost a centuy stating with the seminal wok of Elang [5] stochastic loss netwoks have been widely studied as models of many divese compute and communication systems in which diffeent types of esouces ae used to seve vaious classes of customes involving simultaneous esouce possession and nonbacklogging wokloads Examples include telephone netwoks mobile cellula systems ATM netwoks boadband telecommunication netwoks optical wavelength-division multiplexing netwoks wieless netwoks distibuted computing database systems data centes and multi-item inventoy systems; see eg [ ] Loss netwoks have even been used ecently fo esouce planning within the context of wokfoce management in the infomation technology (IT) sevices industy whee a collection of IT sevice poducts ae offeed each equiing a set of esouces with cetain capabilities [7] In each case the stochastic loss netwok is used to captue the dynamics and uncetainty of the compute/communication application being modeled One of the most impotant obectives in analyzing such loss netwoks is to detemine pefomance measues of inteest most notably the stationay loss pobability fo each custome class The classical Elang fomula which has been thooughly studied and widely applied in many fields of eseach povides the pobabilistic chaacteization of these loss pobabilities Moe specifically given a stochastic netwok and a multiclass custome wokload the fomula endes the stationay pobability that a custome will be lost due to insufficient capacity fo at least one esouce type While the initial esults of Elang [5] wee fo the paticula case of oisson aivals and exponential sevice times Sevastyanov [3] demonstates that the Elang fomula holds unde geneal finitemean distibutions fo the custome sevice times The esults ae also known to hold in the pesence of dependencies among sevice 47

2 times fo a specific class [4] Recent esults [3] suggest that elaxations can be made to the custome aival pocess meely equiing that customes geneate sessions accoding to a oisson pocess and within each session blocked customes may ety with a fixed pobability afte an idle peiod of andom length A multi-peiod vesion of the Elang loss model has also been ecently studied [] Unfotunately the computational complexity of the exact Elang fomula and elated measues is known to be -complete in the size of the netwok [6] thus endeing the exact fomula of limited use fo many netwoks in pactice (Refe to [4] fo details on - complete complexity) The well-known Elang fixed-point appoximation has been developed to addess this poblem of complexity though a set of poduct-fom expessions fo the blocking pobabilities of the individual esouces that map the blocking pobability of each esouce to the blocking pobabilities of othe esouces In othe wods it is as if custome losses ae caused by independent blocking events on each of the esouces used by the custome class based on the one-dimensional Elang function The Elang fixedpoint appoximation has been fequently used and extensively studied as a tactable appoach fo calculating pefomance measues associated with the stochastic loss netwok including estimates fo stationay loss pobabilities Moeove the Elang fixed-point appoximation has been shown to be asymptotically exact in two limiting egimes one based on inceasing the taffic intensities and esouce capacities in a popotional manne [ ] and the othe based on inceasing the numbe of esouce types and numbe of custome classes [5 7] Despite being asymptotically exact in cetain limiting egimes it is equally well known that the Elang fixed-point appoximation can povide elatively poo pefomance estimates in vaious cases The stochastic loss netwoks that model many compute and communication systems often opeate natually in a so-called citically loaded egime [6] Somewhat supisingly we find that even though the Elang fixed-point appoximation can pefom quite well in undeloaded and oveloaded conditions the fixedpoint appoximation can povide elatively poo loss pobability estimates when the netwok is citically loaded We establish such qualitative esults by means of estimating the convegence ate of the Elang fixed-point appoximation towad the exact solution unde lage netwok scalings This motivates the need to design bette algoithms fo estimating loss pobabilities In this pape we popose a novel algoithm fo computing the stationay loss pobabilities in stochastic loss netwoks which we call the slice method because the algoithm exploits stuctual popeties of the exact stationay distibution along slices of the polytope ove which it is defined Ou algoithm is shown to always convege and to do so exponentially fast Though a vaiational chaacteization of the stationay distibution we establish that the esults fom ou algoithm ae asymptotically exact We futhe estimate the convegence ate of ou algoithm whee compaisons between the convegence ates of the Elang fixed-point appoximation and the slice method favos ou appoach Using this vaiational chaacteization we also povide an altenative poof of the main theoem in [] which is much simple and may be of inteest in its own ight A collection of numeical expeiments futhe investigates the effectiveness of ou algoithm whee it convincingly outpefoms the Elang fixed-point appoximation fo loss netwoks in the citically loaded egime Anothe impotant obective in analyzing stochastic loss netwoks concens chaacteizing the fundamental elationships among the capacities fo evey esouce type and the loss pobabilities fo evey custome class In paticula the exact Elang fomula povides the loss pobabilities fo all custome classes given the capacity of each esouce and the wokload of each class Beezne et al [] conside the invese of this function in the one-dimensional single-esouce case and povide bounds fo the capacity equied to satisfy the given wokload and loss pobability constaint The coesponding bounds fo the multidimensional vesion of this invese function is a much moe difficult poblem The main eason is that depending upon the stuctue of the poblem and in paticula the esouce equiements of each custome class thee can be infinitely many possible capacities that satisfy the given vecto of loss pobabilities fo the custome classes Futhemoe the set that contains all of these possible capacities can be unbounded In this pape by exploiting lage netwok scalings ou esults fo vaious appoximation algoithms and pevious esults fo the one-dimensional poblem we establish stuctual popeties fo the multidimensional egion of capacities that ensues offeed taffic will be seved within a given set of loss pobabilities This egion of capacities is defined in tems of a system of polynomial equations and inequalities such that the capacities which coespond to the given loss pobability vecto lie with this egion These esults povide a pobabilistic chaacteization of the theoetical elationships between the link capacity and loss pobability vectos Ou esults also can be exploited to efficiently seach the feasible egion of vaious optimization poblems involving loss netwoks including many esouce allocation and capacity planning applications We make seveal impotant contibutions in this pape A new slice method fo estimating the stationay loss pobabilities in stochastic loss netwoks is poposed and shown to povide asymptotically exact esults The convegence ates of diffeent appoximation algoithms ae obtained unde lage netwok scalings A simple poof is povided fo a classical esult of Kelly which should be of independent inteest The stuctual popeties of the capacity vecto egion that achieves a given loss pobability vecto ae obtained unde lage netwok scalings While the poblems we conside ae of fundamental impotance fom the theoetical pespectives of stochastic loss netwoks in geneal and Elang loss model appoximations in paticula ou analysis and esults can suppot a wide ange of pactical applications involving loss netwoks This pape is oganized as follows The next section contains some technical peliminaies Section 3 descibes thee appoximation algoithms fo computing stationay loss pobabilities Ou main esults ae pesented in Section 4 with most of thei poofs consideed in Section 5 Some numeical expeiments ae povided in Section 6 and concluding emaks can be found in Section 7 Additional technical details can be found in [8] RELIMIARIES Model We investigate geneal stochastic loss netwoks with fixed outing using the standad teminology in the liteatue based on outes (custome classes) and links (esouce types); see eg [3] Conside a netwok with J links labeled J Each link has C units of capacity Thee is a set of K distinct (pe-detemined) outes denoted by R = {K} A call on oute equies A units of capacity on link A Calls on oute aive accoding to an independent oisson pocess of ate ν with ν =(ν ν K) denoting the vecto of these ates The dynamics of the netwok ae such that an aiving call on oute is admitted to the netwok if sufficient capacity is available on all links used by oute ; else the call is dopped To simplify the exposition we will assume that the call sevice times ae iid exponential andom vaiables with unit mean It is impotant to note howeve that ou esults ae not limited to these sevice time assumptions since 48

3 the quantities of inteest emain unchanged in the stationay egime unde geneal sevice time distibutions due to the well-known insensitivity popety of this class of stationay loss netwoks Let n(t) =(n (t)n K(t)) K be the vecto of the numbe of active calls in the netwok at time t By definition we have that n(t) S(C) whee S(C) = n n Z K : n An C and C =(C C J) denotes the vecto of link capacities Within this famewok the netwok is Makov with espect to state n(t) It has been well established that the netwok is a evesible multidimensional Makov pocess with a poduct-fom stationay distibution [9] amely thee is a unique stationay distibution π on the state space S(C) such that fo n S(C) π(n) =G(C) Y ν n n R! whee G(C) is the nomalizing constant (o patition function) G(C) = Y n S(C) R ν n n! o oblems A pimay pefomance measue in loss netwoks is the pe-oute stationay loss pobability the faction of calls on oute in equilibium that ae dopped o lost denoted by L It can be easily veified that L is well-defined in the above model This model can be thought of as a stable system whee admitted calls expeience an aveage delay of (thei sevice equiement) and lost calls expeience a delay of (thei immediate depatue) Theefoe the aveage delay expeienced by calls on oute is given by D =( L ) +L = ( L ) Upon applying Little s law [5] to this stable system (with espect to oute ) we obtain which yields ν D = E[n ] L = E[n] () ν Thus computing L is equivalent to computing the expected value of the numbe of active calls on oute with espect to the stationay distibution of the netwok Even though we have an explicit fomula the computational complexity of the exact stationay distibution known to be -complete in geneal [6] endes its diect use of limited value in pactice We theefoe need simple efficient and (possibly) appoximate algoithms fo computing the stationay loss pobabilities One of ou goals in this pape is to design a family of such iteative algoithms that also have povably good accuacy popeties In the one-dimensional vesion of the foegoing stochastic loss netwok it is known that the capacity C fo the single esouce must satisfy the inequalities ν( L) < C < ν( L)+/L () in ode to ensue that the aivals at ate ν ae seved with a loss pobability of at most L [] The coesponding poblem in the multidimensional stochastic loss netwok of inteest is much moe difficult howeve and vey limited esults ae known On the othe hand undestanding the fundamental elationships among the link capacities and the loss pobabilities is citical to solving esouce allocation poblems in stochastic loss netwoks We theefoe need an effective chaacteization of these elationships which can be exploited to impove the efficiency and quality of solutions to a wide vaiety of optimization poblems [75] Anothe of ou goals in this pape is to detemine the capacity egion that ensues a given vecto of loss pobabilities will be satisfied 3 Scaling We conside a scaling of the stochastic loss netwok to model the type of lage netwoks that aise in vaious applications Although it has been well studied (see eg []) we will use this scaling both to evaluate analytically the pefomance of diffeent appoximation algoithms fo computing loss pobabilities and to obtain the capacity egion fo satisfying a set of loss pobabilities Given a stochastic loss netwok with paametes CA and ν a scaled vesion of the system is defined by the scaled capacities C = C =(C C K) and the scaled aival ates ν = ν =(ν ν K) whee is the system scaling paamete The coesponding feasible egion of calls is given by S(C) ow conside a nomalized vesion of this egion defined as ff S (C) = n : n S(C) Then the following continuous appoximation of S (C) emeges in the lage limit: S(C) ={x : Ax C x R K + } 3 ALGORITHMS We now descibe thee algoithms fo computing the stationay loss pobabilities L =(L ) [ ] K The well-known Elang fixed-point appoximation is pesented fist followed by a -point appoximation based on the concentation of the stationay distibution aound its mode in lage netwoks The thid algoithm is ou new family of slice methods that attempts to compute the aveage numbe of active calls on diffeent outes via an efficient exploation though slices of the admissible polytope S(C) 3 Elang fixed-point appoximation The well-known Elang fomula [5] fo a single-link singleoute netwok with capacity C and aival ate ν states that the loss pobability denoted by E(ν C) isgivenby " C # E(ν C) = νc ν i C! i! Based on this simple fomula the Elang fixed-point appoximation fo multi-link multi-oute netwoks aose fom the hypothesis that calls ae lost due to independent blocking events on each link in the oute Moe fomally this hypothesis implies that the loss pobabilities of outes L =(L L K) and blocking pobabilities of links E =(E E J) satisfy the set of fixed-point equations E = E(ρ C ) " # Y ρ = ν A ( E i) A i E i L = Y ( E ) A (3) i= 49

4 fo =J and R A natual iteative algoithm that attempts to obtain a solution to the above fixed-point equations is as follows: ERLAG FIED-OIT AROIMATIO Denote by t the iteation of the algoithm with t =initially Stat with E () =5 fo all J In iteation t + update E (t+) accoding to E (t+) = E(ρ (t) C) whee ρ (t) = ( E (t) ) Y ν A ( E (t) i ) A i : i:i 3 Upon convegence pe appopiate stopping conditions denote the esulting values by E E fo J Compute the loss pobabilities fom the Elang fixed-point appoximation L E Ras L E = Y ( E E ) A 3 -point appoximation Kelly [] established the asymptotic exactness of the Elang fixed-point appoximation in a lage netwok limiting egime by showing that the stationay distibution concentates aound its mode n given by n ag max π(n) n S(C) Such concentation suggests the following appoach which is the pemise of the -point appoximation: Compute the mode n = (n ) of the distibution and use n as a suogate fo E[n ] in the computation of L via equation () Befoe pesenting ou specific iteative algoithm we conside some elated optimization poblems upon which it is based Thedefinitionofthestationaydistibutionπ( ) suggests that the mode n coesponds to a solution of the optimization poblem maximize n log ν log n! ove n S(C) By Stiling s appoximation log n!=n log n n +O(log n ) Using this and ignoing the O(log n ) tem the above optimization poblem educes to maximize n log ν + n n log n ove n S(C) A natual continuous elaxation of n S(C) is n o S(C) = x R K + : Ax C which yields the following pimal poblem (): maximize x log ν + x x log x ove x S(C) The above elaxation becomes a good appoximation of the oiginal poblem when all components of C ae lage In ode to design a simple iteative algoithm we conside the Lagangian dual (D) to the pimal poblem whee standad calculations yield " minimize ν exp # y A + y C ove y Define the dual cost function g(y) as g(y) = " ν exp # y A + y C By Slate s condition the stong duality holds and hence the optimal cost of and D ae the same Standad Kaush-Kuhn-Tucke conditions imply the following: Letting (x y ) be a pai of optimal solutions to and D then (a) Fo each link g(y ) = o y =& g(y ) y y Equivalently " A ν exp # y A = C & y > o " A ν exp (b) Fo each oute " x = ν exp y A # C & y = y A # The above conditions suggest the following appoach: Obtain a dual optimal solution say y use it to obtain x and then compute the loss pobability as L = x /ν ext we descibe an iteative coodinate descent algoithm fo obtaining y Inwhat follows we will use the tansfomation z =exp( y ) given its similaity with the Elang fixed-point appoximation ote that z is minus the blocking pobability fo link E -OIT AROIMATIO Denote by t the iteation of the algoithm with t =initially Stat with z () =5 fo all J In iteation t + detemine z (t+) as follows: (a) Choose coodinates fom Jin a ound-obin manne (b) Update z (t+) whee g (t) by solving the equation n o g (t) (x) =min C g (t) () (x) = Q Aν i za i i with 8 >< z (t+) i fo i< z i = x fo i = >: fo i> z (t) i 4

5 Thus g (t) (x) is the evaluation of pat of the function g( ) coesponding to the th coodinate with values of components <being fom iteation t +values of components >fom iteation t and component being the vaiable itself 3 Upon convegence pe appopiate stopping conditions denote the esulting values by z fo J Compute the loss pobabilities fom the -point appoximation L Ras L = Y (z ) A 33 Slice method The Elang fixed-point appoximation and the -point appoximation essentially attempt to use the mode of the stationay distibution as a suogate fo the mean which woks quite well when the distibution is concentated (nea its mode) While this concentation holds fo asymptotically lage netwoks it othewise can be an impotant souce of eo and theefoe we seek to obtain a new family of methods that povide bette appoximations The main pemise of ou slice methods follows fom the fact that computing the loss pobability L is equivalent to computing the expected numbe of calls E[n ] via equation () By definition E[n ]= k[n = k] k= and thus E[n ] can be obtained though appoximations of [n = k] athe than by the mode value n ote that [n = k] coesponds to the pobability mass along the slice of the polytope defined by n = k An exact solution fo E[n ] can be obtained with ou slice method by using the exact values of [n = k] but obtaining the pobability mass along a slice can be as computationally had as the oiginal poblem Hence ou family of slice methods is based on appoximations fo [n = k] Todosowe will exploit simila insights fom pevious appoaches: Most of the mass along each slice is concentated aound the mode of the distibution esticted to the slice This appoximation is bette than the -point appoximation since it uses the -point appoximation many times (once fo each slice) in ode to obtain a moe accuate solution ext we fomally descibe the algoithm whee the cost function of the pimal poblem is denoted by SLICE METHOD q(x) = x log ν + x x log x Compute L fo oute Ras follows: Fo each value of k {n : n S(C)} use the -point appoximation to compute x (k ) as the solution of the optimization poblem maximize q(x) ove x S(C) &x = k Estimate E[n ] as E[n ]= k k exp(q(x (k ))) k exp(q(x (k ))) 3 Geneate L = E[n] ν 34 3-point slice method In the geneal slice method fo each oute we apply the - point appoximation to each slice defined by n = k k {n : n S(C)} In the scaled system this equies O() applications of the -point appoximation fo each oute Recall that in contast the Elang appoximation (o -point appoximation) equies only O() applications of the iteative algoithm To obtain a vaiation of the geneal slice method with simila computational complexity we intoduce anothe slice method appoximation whose basic pemise is as follows: Instead of computing x (k ) fo all k {n : n S(C)} we appoximate x (k ) by linea intepolation between pais of 3 points Fo a given oute fist apply the -point appoximation fo the entie polytope S(C) to obtain the mode of distibution x Define n max () max{n : n S(C)} ext obtain x (n max ()) the mode of distibution in the slice n = n max () using the -point appoximation as in the geneal slice method Finally obtain x () the mode of distibution in the slice n = using the -point appoximation ow fo k {n : n S(C)} unlike in the geneal slice method we will use an intepolation scheme to compute x (k ) as follows: (a) If k x then x (k ) =x k + x () x k x x That is x (k ) is the point of intesection (in the space R K ) of the slice x = k with the line passing though the two points x and x () (b) Fo x <k n max let x (k ) =x (n max k x ()) +x nmax () k n max () x n max () x ote that due to the convexity of the polytope S(C) the intepolated x (k ) ae inside the polytope ow as in the geneal slice method we use these x (k ) to compute the appoximation of E[n ] and subsequently L A pseudo-code fo the 3-point slice method can be found in [8] 4 OUR RESULTS In this section we pesent ou main esults most of the poofs of which ae postponed until the next section 4 Recoveing an old esult Conside a stochastic loss netwok with paametes A C and ν that is scaled by as defined in Section Kelly [] obtained a fundamental esult which shows that in the scaled system the stationay pobability distibution concentates aound its mode Theefoe the esults of the -point appoximation ae asymptotically exact We epove this esult using a vaiational chaacteization of the stationay distibution which yields a much simple (and possibly moe insightful) set of aguments THEOREM Conside a loss netwok scaled by paamete Let L be the exact loss pobability of oute RThen! ( L ) x = O (4) ν 4

6 Kelly established the asymptotic exactness of the Elang fixed-point appoximation by using the above esult togethe with the fact that the Elang fixed-point appoximation fo a scaled system essentially solves the dual D as inceases 4 Eo in Elang fixed-point appoximation The Elang fixed-point appoximation is quite popula due to its natual iteative solution algoithm and its asymptotic exactness in the limiting egime Howeve it is also well known that the Elang fixed-point appoximation can pefom pooly in vaious cases This is especially tue when the load vecto ν is such that it falls on the bounday of S(C) ie the stochastic loss netwok is in the citically loaded egime Moe pecisely this means ν is such that at least one of the constaints in Aν C is tight It can be eadily veified (at least fo simple examples) that when ν is stictly inside o stictly outside S(C) then the eo in the Elang fixed-point appoximation fo the scaled netwok is O(/ ) Howeve fo the bounday the qualitative eo behavio changes and in paticula we pove the following esult THEOREM When the vecto ν lies on the bounday of S(C)! L E L =Ω (5) whee L E =(L E ) is the vecto of loss pobabilities fom the Elang fixed-point appoximation and L =(L ) is the vecto of exact loss pobabilities both fo a loss netwok scaled by 43 Accuacy of the slice method The dastically pooe accuacy of the Elang fixed-point appoximation at the bounday (ie in the citical egime) fom Theoem stongly motivates the need fo new and bette loss pobability appoximations This led to ou development of the geneal slice method descibed in Section 33 fo which we establish its asymptotic exactness using the vaiational chaacteization of the stationay distibution THEOREM 3 Fo each oute RletL S be the loss pobability estimate obtained fom the geneal slice method fo the system scaled with paamete LetL be the coesponding exact loss pobability Then fo any system paamete values we have! L S L = O (6) This esult establishes the asymptotic exactness of the slice method ove all anges of paametes The poven eo bound which essentially scales as O(/ ) does not imply that it is stictly bette than the Elang fixed-point appoximation We ae unable to establish stict dominance of the slice method but numeical esults in Section 6 illustate that the slice method can convincingly outpefom the Elang fixed-point appoximation unde citical loading 44 Convegence of algoithms So fa cetain accuacy popeties have been established fo the iteative algoithms We now establish the exponential convegence of the iteative algoithm fo the geneal slice method It is sufficient to state the convegence of the -point appoximation since this is used as a suboutine in ou slice methods THEOREM 4 Given a loss netwok with paamete A C and ν letz (t) be the vecto poduced by the -point appoximation at the end of iteation t Then thee exists an optimal solution y of the dual poblem D such that z (t) z α exp ( βt) whee z =(z ) with z =exp( y ) and α β positive constants which depend on the poblem paametes The poof of Theoem 4 is povided in [8] 45 Capacity egion of invese function Suppose a desied vecto of loss pobabilities L =(L ) is given eithe diectly o though constaints We seek to identify the egion of capacities C that ensues the aival ate vecto ν will be seved with loss pobabilities of at most L This egion is of theoetical impotance because it chaacteizes fundamental popeties between the link capacity and loss pobability vectos It also can be exploited to efficiently seach the feasible egion in vaious optimization poblems involving stochastic loss netwoks Obviously the exact Elang loss fomula can not seve this pupose In fact even the Elang fixed-point equations which still have the basic stuctue of the oisson distibution function tun out to be too complicated fo this pupose We instead detemine the egion of inteest though the following esult THEOREM 5 Fo a loss system scaled by paamete thee exists a δ() such that fo any given feasible loss pobabilities L and any small positive numbe ɛ the capacity vectos C that achieve these loss pobabilities fall within the egion defined by the system of polynomial equations and inequalities log( L δ() /+ɛ ) A E ρ = Y ν A ( E i) A i i ρ ( E ) < C < ρ ( E )+/E The poblem of linea optimization ove a egion defined by polynomial equations and inequalities is known to be -had; note that this is in compaison with the complexity fo calculating loss pobabilities (efe to [4]) Moeove thee exist standad nonlinea optimization methods fo studying the geomety of such egions (see eg [4]) as well as polynomial appoximations fo solving vaious optimization poblems whose feasible egion is defined as above Theoem 5 theefoe can be instumental in impoving the efficiency fo solving a wide vaiety of optimization poblems involving stochastic loss netwoks Fo example the above polynomial equations and inequalities can be easily incopoated into the optimization poblems consideed in [] by adding the coesponding constaints on L and using the methodologies developed in [4] to obtain a nea-optimal solution 5 ROOFS We now conside the poofs of most of ou main esults above 5 oof: Theoem Vaiational chaacteization of π Recall that the stationay distibution π is epesented as π(n) := G(C) exp(q(n)) exp(q(n)) = Y R ν n n! 4

7 fo n S(C) DefineM(C) as the space of distibutions on S(C) Clealy π M(C) Foμ M(C)define F (μ) μ(n)q(n) μ(n)logμ(n) n S(C) n S(C) = E μ(q)+h(μ) ext we state a vaiational chaacteization of π which will be extemely useful thoughout This chaacteization essentially states that π is chaacteized uniquely as the maximize of F ( ) ove M(C) See [8] fo the poof of Lemma 6 LEMMA 6 Fo all μ M(C) F (π) F (μ) The equality holds iff μ = π FutheF (π) =logg(c) Scaled system: A useful appoximation ow conside the scaled system with paamete Foanyn S(C) this is equivalent to consideing n S(C) Then π fo a scaled system is equivalent to the distibution π on S (C) defined fo x S (C) as π (x) =π(x)= G(C) exp(q(x)) Upon consideing Q(x)wehave exp(q(x)) = Y = exp = exp = exp = exp (ν ) x (x )! x ν x + x + x x + x log ν log(x )!! x log ν x log ν x + x log x! log(x )! O(x ) + x + O(x ) whee the above calculations make use of Stiling s appoximation: log M! =M log M M + O(log M) It then follows fom these calculations that Q(x) = " # x log νe + log(x ) x whee = q(x)+o «q(x) = x log νe (7) x Concentation of π Given the above calculations we futhe obtain the following concentation fo the distibution π which will be cucial in poving Theoem Refe to [8] fo the poofs of Lemmas 7 and 8!! LEMMA 7 Given any ε> define the set A ε = y S (C) : y x >ε whee x =agmax x S(C) q(x) Then π (A ε)=o LEMMA 8 Fo any y S(C) ε q(y) q(x ) C y x «(8) q Completing poof of Theoem Using ε k = k fo the value of ε in the conclusion of Lemma 7 then fom (8) we obtain π ( x x >ε k ) = O which immediately implies! E [ x x ] =O O k Thus E[ x x ] =O and since L = E[x] ν wehave k! = O k! L L = E[ x x ] ν = O ν «(9)!! Additional esult: Value of log G(C) The above esults (specifically Lemma 6 and Lemma 7) lead to a shap chaacteization of log G(C) fo the scaled system as expessed in the following lemma See [8] fo the poof of Lemma 9 LEMMA 9 log G(C) max q(x) x S(C) = O 5 oof: Theoem Kelly poves in [] that fo any oute ( L ε ) x = O ν ««() Hence the following lemma togethe with () and () establishes Theoem LEMMA When the vecto ν lies on the bounday of S(C) ««! E[x ] x =Ω () ν ν whee ( E [x ] ν ) =( E [x ] ν E [x ] ν ) is the vecto consisting of the expectations fo the outes in the scaled (discete) system with paamete and ( x ν ) =( x ν x ν ) ROOF We shall biefly summaize a few of the technical details efeing to [8] fo the complete poof of Lemma Let us stat with the following claim the poof of which is povided in [8] 43

8 CLAIM Fo any R «E π [x ] E [x ] = O Then fom Claim to pove Lemma it suffices to show that «E π [x ] ν =Ω () Define S {v S K : S(C) (ν + tv) fo some t>} whee S K is the unit sphee in R K owfoagivenv S and t [t v) whee t v =sup{t R + :(ν + tv) S(C)}define g (v t) exp(q ((ν + tv))) Then fom spheical integation we obtain R R tv S E π [x] = (ν + tv)g(v t)tk dt dv R R vt g(v S t)tk dt dv RS = ν + v R t v g (v t)tk dt dv R R vt S g (v t)tk dt dv and thus RS E π [x] ν = v R t v g(v t)tk dt dv R R tv g(v (3) S t)tk dt dv ext we intoduce the following lemma which will be cucial in poving () See [8] fo the poof of Lemma LEMMA Let the polytope S(C) and an intege l be given If is lage enough then fo all v S Z tv g (v t)t l dt =Θ K Γ `! l+ exp(k) l+ whee Γ( ) is the Gamma function and the constant hidden in Θ( ) is unifomly bounded ove all v S Finally let T be a tangent plane of S(C) at the point ν and let w S K be a unit vecto that is pependicula to T and that satisfies v w foanyv S Then fom (3) we have RS E π [x] ν = v R t v g (v t)tk dt dv R R tv g S (v t)t K dt dv RS w v R t v g (v t)t K dt dv R R tv g(v S t)tk dt dv RS = w v R t v g(v t)tk dt dv R R tv g(v S t)tk dt dv Θ K Γ( K+ exp(k) ) = = Θ Θ K+ K exp(k) Γ( K ) K «R S v wdv «R S dv! (4) S whee we used Lemma and the facts that =Θ()and RS dv Γ( K+ ) =Θ() Fom (4) we obtain () which completes the Γ( K ) poof of Lemma R v w dv 53 oof: Theoem 3 Theoem implies that the actual loss pobability L Ris given by! L = x + O ν Theefoe the poof of Theoem 3 will be implied by showing that fo all R! L S = x + O (5) ν This esult is established next whee the poof cucially exploits ou concentation Lemma 7 Fom the definition of the slice method the estimated loss is defined as L S = k k exp(q(x (k ))) ν k exp(q(x (k ))) (6) Recall that x (k ) is the solution of the optimization poblem pobability L S maximize q(x) ove x S(C) &x = k futhe ecalling the definition of the function q( ) as q(x) = x log ν + x x log x ow conside a oute R In the est of the poof we will use C ε = whee C =max C Futhe define the following useful subsets S( ) {n : n S (C)} S ε( ) {k S( ) : x (k ) x ε} Sε( c ) {k S( ) : x (k ) x >ε} ext we note two facts that will be used to pove appopiate lowe and uppe bounds which yield the desied esult (5) Fist Lemma 8 and the above definitions imply that fo k Sε( c ) exp q(x k ) exp (q(x )) (7) Second it is easy to see thee exists k S ε( ) such that «x (k ) x = O Fo this k wehave exp (q(x (k ))) = Θ (exp(q(x ))) (8) Use of (7)-(8): Lowe bound Since S( ) = O() in the scaled system (7) and (8) imply that exp(q(x (k ))) k Sε c() «exp(q(x )) = O exp(q(x (k ))) A (9) k S ε() 44

9 Fom (9) the value of ε and the above subset definitions we obtain the following sequence of inequalities: k S() k exp(q(x (k ))) k S() exp(q(x (k ))) k S k ε() exp(q(x (k ))) k S() exp(q(x (k ))) k S k ε() exp(q(x (k ))) ( + O(/ )) k S() exp(q(x (k ))) +O(/ ) (x ε)! = x O () Use of (7)-(8): Uppe bound Fo all k Sε( c ) k is bounded by some constant and theefoe we have k exp(q(x (k ))) k Sε c() «exp(q(x )) = O exp(q(x (k )) A () k S ε() Fom () and the definition of ε we obtain k S() k exp(q(x (k ))) k S() exp(q(x (k ))) k S() k exp(q(x (k ))) k S ε() exp(q(x (k ))) k S ( + O(/ )) k ε() exp(q(x (k ))) k S ε() exp(q(x (k ))) ( + O(/ ))(x + ε)! = x + O () Finally equations () and () togethe with (6) imply (5) thus completing the poof of Theoem 3 54 oof: Theoem 5 Fom Theoem and () the eo fo the Elang fixed-point appoximation is O( p /) fo a scaled system with paamete amely the unique set of blocking pobabilities E satisfies L Y! ( E ) A = O Hence fo any small positive numbe ɛ > theeexists δ() > such that L Y ( E ) A δ() /+ɛ We theefoe have log( L δ() /+ɛ ) A log( E ) and then the inequality log( E ) E yields log( L δ() /+ɛ ) A E (3) Meanwhile we know that E is the solution to the Elang fixedpoint equations E = E(ρ C ) whee ρ = Y ν A ( E i) A i (4) i which is a polynomial of E Although the Elang fomula itself is in a complicated fom this connection enables us to apply the aguments fo the one-dimensional elationship between blocking pobability and capacity demonstated in () Hence we obtain ρ ( E ) <C <ρ ( E )+/E ; (5) see Theoem in [] ote that fo each = J (5) beaks into one linea inequality and one quadatic inequality of C and E This completes the poof of Theoem 5 6 EERIMETS The main contibutions of this pape ae the theoetical esults pesented in Sections 3 5 Howeve to illustate and quantify the pefomance of ou family of slice methods we conside two diffeent sets of numeical expeiments The fist is based on a small canonical loss netwok topology that is used to investigate the fundamental popeties of ou slice methods and pevious appoaches with espect to the scaling paamete Then we tun to conside a lage set of numeical expeiments based on wokfoce management applications in the IT sevices industy using eal-wold data 6 Small loss netwoks We conside a small canonical loss netwok topology compised of two outes and thee links as illustated in Figue Both outes shae link with links and 3 dedicated to outes and espectively Moe pecisely the netwok is defined by A = and C = Link Link Link 3 Route Route Figue : Illustation of the small canonical netwok model In ou fist collection of expeiments we set ρ = ρ = The loss pobabilities fo this netwok model instance ae then computed using ou geneal slice method the Elang fixed-point appoximation and the -point appoximation whee the loss pobabilities in each case ae consideed as a function of the scaling paamete ote that in this small model the esults fom the 3-point slice method ae identical to those fom the geneal slice method since the tace of the maximize point fo each slice in the geneal slice method indeed foms a linea intepolation of the thee 45

10 points We also diectly compute the exact loss pobability by bute foce and then obtain the aveage eo (ove both outes) fo each method These esults ae pesented in Figue Eo point slice Elang Section 6 fo a much simple canonical model which captues fundamental popeties of stochastic loss netwoks In the emainde of this section we shall focus on two epesentative model instances and pesent the details of ou compaative findings among the slice methods and pevious appoaches The fist model instance consists of 37 outes and 84 links wheeas the second model instance consists outes and 3 links In both data sets the aival ate vecto ν happens to lie on the bounday of S(C) Figue 3 depicts the spasity plot fo the A matix of the fist model togethe with the coesponding distibutions fo the numbe of outes pe link and the numbe of links pe oute Figues 4 and 5 pesent the same infomation fo the second model instance 4 5 Distibution of outes pe link 4 Distibution of links pe oute Figue : Aveage eo of loss pobabilities computed fo each method as a function of the scaling paamete Fist we obseve that the slice method pefoms bette than the -point appoximation method fo evey scaling value Thise- sult is as expected since the slice method utilizes moe infomation about the pobability distibution of the undelying polytope than the -point method Second it is quite inteesting to obseve that the Elang fixed-point method initially pefoms bette than the slice method in the small scaling egion wheeas the slice method eventually povides the best pefomance among the appoximation methods in the lage scaling egion and the pefomance odeing among the methods shown fo =7continues to hold fo >7 To undestand this phenomena note that as a function of the scaling with espect to the output of the Elang fixed-point method conveges to that of the -point appoximation method on the ode of O ` and the eos of the -point ap- q poximation method ae given by Ω as established in Theoem Moeove when becomes lage the eo of the slice method becomes smalle than that of the Elang fixed-point method because the eo of the -point appoximation method is oughly a constant times that of the slice method fo evey sufficiently lage (as seen in Figue ) Finally while the asymptotic exactness of the Elang fixed-point appoximation is associated with the -point appoximation method Figue also illustates some of the complex chaacteistics of the Elang fixed-point appoximation in the non-limiting egime We also conside a second collection of expeiments epesenting the symmetic case of ρ = ρ =5 These esults exhibit the same tends as in the asymmetic case and hence ae omitted 6 Lage eal-wold netwoks umeical expeiments wee also conducted fo a lage numbe of eal-wold loss netwok instances taken fom vaious esouce planning applications within the context of wokfoce management in the IT sevices industy In each of these applications the netwok outes epesent vaious IT sevice poducts and the netwok links epesent diffeent IT esouce capabilities The vaious data sets compising these model instances wee obtained fom actual IT sevice companies Fist we geneally note that ou esults fom such eal-wold model instances exhibit tends with espect to the scaling paamete that ae simila to those pesented in 84 Links Routes Figue 3: Spasity plot and distibutions of outes/link and links/oute fo the fist model instance 66 Links Routes Figue 4: Spasity plot fo the second model instance The loss pobabilities ae computed fo each loss netwok model instance using ou geneal slice method ou 3-point intepolation slice method and the Elang fixed-point appoximation Since all of the eal-wold model instances ae too lage to numeically compute the exact solution we use simulation of the coesponding loss netwok to estimate the exact loss pobabilities within tight confidence intevals The aveage eo (ove all outes) and the individual pe-oute eos ae then computed fo each method in compaison with the exact loss pobabilities whee the fome esults ae summaized in Table 46

11 4 3 Distibution of outes pe link 8 Eo impovement fo model instance aveage impovement eo impovement pe oute Distibution of links pe oute Relative eo impovement Figue 5: Distibutions of outes/link and links/oute fo the second model instance Routes Figue 6: Relative impovement of the appoximation eos in loss pobabilities fo model instance Elang slice method 3-point slice Model instance Model instance Table : Aveage eo of loss pobabilities fo each method The impovements in the appoximation eos povided by the geneal slice method and the 3-point slice method ove the Elang fixed-point method ae pesented in Figues 6 and 7 Specifically we plot the elative impovement I in the appoximation eo fo each oute that is obtained with both slice methods whee I := LE L L S L (6) L with L E and L S denoting the oute- loss pobability fom the Elang fixed-point appoximation and fom one of the slice methods espectively and L denoting the exact loss pobability fo oute Hence a positive elative impovement I quantifies the benefits of the slice method a negative elative impovement I quantifies the benefits of the Elang fixed-point appoximation and I = indicates equivalent esults fom both methods The aveage elative impovement (ove all outes) of I = 49 fo the 3-point slice method in model instance is shown by the hoizontal line in Figue 6 and the aveage elative impovements (ove all outes) of I =76 and I =7 fo the geneal and 3-point slice methods in model instance espectively ae shown by the points on the y-axis in Figue 7 We note that in the fist model instance the geneal slice method and the 3-point slice method povide identical loss pobabilities fo all outes with one exception whee the diffeence between the loss pobabilities fom the slice methods fo this one oute is quite small Theefoe only the esults fo the 3-point slice method ae pesented in Figue 6 It can be clealy obseved fom the esults in Figues 6 and 7 that the aveage elative impovements of ou slice methods ove the Elang fixed-point appoximation ae quite significant Even moe impotantly we obseve that the elative impovements fo the individual outes ae consistently and significantly bette unde both slice methods In paticula the geneal (espectively 3-point) slice method povides the exact loss pobabilities fo 98 (espectively 93) of the outes while the Elang fixed-point appox- imation neve povides exact esults in model instance and the 3-point slice method povides the exact loss pobabilities fo of the 37 outes while the Elang fixed-point appoximation povides the exact esults fo 5 of these outes in model instance ote that when L S = L the elative eo fo the Elang fixed-point appoximation L E /L is equal to +I (espectively I ) when L E >L (espectively L E <L ) In all of the cases whee I = which epesents a consideable numbe of outes in model instance and the ovewhelming maoity of outes in model instance both slice methods povide the exact loss pobability fo oute while the Elang fixed-point appoximation yields L E = even though the exact loss pobabilities fo these outes span the full ange of values in ( ) The loss pobability estimates fo a few outes ae bette unde the Elang fixed-point appoximation than unde the slice methods but such outes ae clealy in the minoity epesenting a single oute in model instance and less than 65% of the outes in model instance The above esults fo two epesentative examples of a lage numbe of loss netwoks taken fom eal-wold wokfoce management applications clealy illustate and quantify the benefits of ou family of slice methods ove the classical Elang fixed-point appoximation at least fo the class of loss netwoks consideed 7 COCLUSIO Stochastic loss netwoks have emeged in ecent yeas as canonical models fo a wide vaiety of multi-esouce applications including telephone and communication netwoks compute systems and inventoy management and wokfoce management systems One of the main pefomance measues of inteest in such applications is the stationay loss pobability fo each custome class The Elang fixed-point appoximation is the most popula appoach fo computing these loss pobabilities Howeve it is well known that this appoximation can povide elatively poo esults fo vaious model instances In paticula we found that the Elang fixed-point appoximation can povide elatively poo loss pobability estimates when the netwok is citically loaded which is often the natual egime fo stochastic loss models of many applications Given this motivation we poposed a geneal algoithm fo estimating the stationay loss pobabilities in loss netwoks based on 47

12 Relative eo impovement Eo Impovement fo model instance 3 point pe oute 3 point aveage Geneal slice pe oute Geneal slice aveage Routes Figue 7: Relative impovement of the appoximation eos in loss pobabilities fo model instance the popeties of slices of the exact stationay distibution along the polytope ove which it is defined We established that ou algoithm always conveges with an exponentially fast ate whee convegence compaisons favo ou slice method appoach ove the Elang fixed-point appoximation Though a vaiational chaacteization of the stationay distibution we futhe established that the loss pobabilities fom ou slice method ae asymptotically exact Using this chaacteization we also povided an altenative poof of an impotant esult due to Kelly [] which is simple and of inteest in its own ight umeical expeiments investigate vaious issues of both theoetical and pactical inteest Ou geneal slice method povides an effective appoach fo computing accuate estimates of the stationay loss pobabilities in loss netwoks Stochastic loss netwoks ae often the undelying model in a wide vaiety of esouce allocation and capacity planning poblems One of the difficulties of such optimization poblems concen thei lage feasible egions and the lack of known stuctual popeties on the elationships among esouce capacities and loss pobabilities fo seaching though the feasible egion Although bounds on the esouce capacity equied to achieve a loss pobability constaint have been established in the single-esouce single-class case the coesponding bounds fo the multidimensional vesion epesent a significantly moe difficult poblem To addess this poblem we detemined stuctual popeties fo the egion of esouce capacities that ensues offeed taffic will be seved within a given set of loss pobabilities In addition to the theoetical chaacteization of elationships between the link capacity and loss pobability vectos ou esults can be exploited to efficiently seach the feasible egion of many optimization poblems involving stochastic loss netwoks Acknowledgments The authos thank Sem Bost fo helpful comments on Section 6 8 REFERECES [] S Beezne A Kzesinski Taylo On the invese of Elang s fomula J Appl ob 35: [] S Bhada Y Lu M Squillante Optimal capacity planning in stochastic loss netwoks with time-vaying wokloads In oc ACM SIGMETRICS [3] T Bonald The Elang model with non-oisson call aivals In oc SIGMETRICS-efomance [4] D Buman J Lehoczky Y Lim Insensitivity of blocking pobabilities in a cicuit-switching netwok J Appl ob : [5] A Elang Solution of some poblems in the theoy of pobabilities of significance in automatic telephone exchanges In E Bockmeye H Halstom A Jensen eds The Life and Woks of AK Elang Denmak 948 [6] Hunt F Kelly On citically loaded loss netwoks Adv Appl ob : [7] Jelenkovic Momcilovic M Squillante Scalability of wieless netwoks IEEE/ACM Tans etwoking 5: [8] K Jung Y Lu D Shah M Shama M Squillante Revisiting stochastic loss netwoks: Stuctues and algoithms Reseach Repot IBM Reseach ov 7 [9] F Kelly Revesibility and Stochastic etwoks John Wiley & Sons ew Yok 979 [] F Kelly Stochastic models of compute communication systems J Royal Stat Soc B 47: [] F Kelly Blocking pobabilities in lage cicuit-switched netwoks Adv Appl ob 8: [] F Kelly Routing in cicuit-switched netwoks: Optimization shadow pices and decentalization Adv Appl ob : [3] F Kelly Loss netwoks Ann Appl ob : [4] J Lassee Global optimization with polynomials and the poblems of moments SIAM J Opt : [5] J Little A poof of the queuing fomula L = λw Ope Res 9: [6] G Louth M Mitzenmache F Kelly Computational complexity of loss netwoks Theo Comp Sci 5: [7] YLuARadovanović M Squillante Optimal capacity planning in stochastic loss netwoks ef Eval Rev 35 7 [8] Z Luo Tseng On the linea convegence of descent methods fo convex essentially smooth minimization SIAM J Cont Opt 3: [9] D Mita J Moison K Ramakishnan ATM netwok design and optimization: A multiate loss netwok famewok IEEE/ACM Tans etwoking 4: [] D Mita Weinbege obabilistic models of database locking: Solutions computational algoithms and asymptotics J ACM 3: [] K Ross Multisevice Loss Models fo Boadband Telecommunication etwoks Spinge-Velag 995 [] A Saleh J Simmons Evolution towad next-geneation coe optical netwok J Lightwave Tech 4: [3] B Sevastyanov An egodic theoem fo Makov pocesses and its application to telephone systems with efusals Theo ob Appl :4 957 [4] L Valiant The complexity of computing the pemanent Theo Comp Sci 8: [5] W Whitt Blocking when sevice is equied fom seveal facilities simultaneously AT&T Bell Lab Tech J 64: [6] S u J Song B Liu Ode fulfillment pefomance measues in an assemble-to-ode system with stochastic leadtime Ope Res 47: [7] I Ziedins F Kelly Limit theoems fo loss netwoks with divese outing Adv Appl ob :