Validity of heavy-traffic steady-state approximations in multiclass queueing networks: The case of queue-ratio disciplines

Transcription

1 Validity of heavy-taffic steady-state appoximations in multiclass queueing netwos: The case of queue-atio disciplines Itai Guvich Kellogg School of Management, Nothwesten Univesity, Evanston, IL A class of stochastic pocesses nown as semi-matingale eflecting Bownian motions (SRBMs) is often used to appoximate the dynamics of heavily loaded queueing netwos. In two influential papes, Bamson (1998) and Williams (1998) laid out a geneal and stuctued appoach fo poving the validity of such heavy-taffic appoximations, in which an SRBM is obtained as a diffusion limit fom a sequence of suitably nomalized woload pocesses. Howeve, fo multiclass netwos it is still not nown in geneal whethe the steady-state distibution of the SRBM povides a valid appoximation fo the steady-state distibution of the oiginal netwo. In this pape we study the case of queue-atio disciplines and povide a set of sufficient conditions unde which the above question can be answeed in the affimative. In addition to standad assumptions made in the liteatue towads the stability of the pe- and post-limit pocesses and the existence of diffusion limits, we add a equiement that solutions to the fluid model ae attacted to the invaiant manifold at linea ate. Fo the special case of static-pioity netwos such linea attaction is nown to hold unde cetain conditions on the netwo pimitives. The analysis elucidates some inteesting connections between stability of the pe- and post-limit pocesses, thei espective fluid models and state-space collapse, and identifies the espective oles played by all of the above in establishing validity of heavy-taffic steady-state appoximations. Key wods: Steady-state ; Multi-class ; Heavy taffic ; Netwo ; Queue-atio MSC2 Subject Classification: Pimay: 6K25, 9B15 ; Seconday: 6F17, 6J2 OR/MS subject classification: Pimay: Queueing netwos, Limit theoems ; Seconday: Maov pocesses, Diffusion models 1. Intoduction and oveview of the main contibution 1.1 Motivation and the main question Queueing netwos ae commonly used to model communication netwos and complex sevice and manufactuing systems. In many cases moe than a single class of jobs can be pocessed at each station, and the model is then collectively efeed to as a multiclass queueing netwo. These models epesent a significant escalation in complexity elative to thei single class countepats and, fo all but the simplest cases, ae aely amenable to exact analysis. In an effot to establish tactable epesentations fo these types of complex systems, much of the eseach on stochastic pocessing netwos has focused on appoximate analysis. The most pevalent types of appoximations found in the liteatue fall into the following two categoies: (i) fluid appoximations that ae mostly used fo stability analysis; and (ii) diffusion appoximations that ae used fo pefomance analysis of heavily-loaded systems. Deiving diffusion appoximations fo queueing netwos has been the focus of eseach since the ealy 6 s; see, e.g. 28, 25, 23, 32]. The standad fomulation consides a sequence of systems in which time and space ae scaled in accodance with the functional cental limit theoem, and the taffic intensity (utilization) is made to appoach 1 at a suitable ate (fo this eason these ae often efeed to as heavytaffic appoximations). The seminal papes by Bamson 5] and Williams 39] povide a boad set of sufficient conditions fo the validity of such diffusion appoximations fo multiclass queueing netwos. In paticula, Williams 39] poves that as the taffic intensity appoaches one, the nomalized vecto of queue length pocesses conveges to a diffusion pocess nown as a semi-matingale eflecting Bownian motion (SRBM). This SRBM is often efeed to as the Bownian model o Bownian countepat of the oiginal queueing netwo. The main appeal of the Bownian system model is that it povides a elatively tactable and igoous appoximation fo the queue length dynamics. In addition, one can use the stationay distibution of the SRBM as a scaled poxy fo the steady-state behavio of the undelying queueing netwo. The advantages of this appoach ae evident: the steady-state behavio of the oiginal queueing netwo can typically only be chaacteized via exhaustive simulation, while the SRBM is a diffusion pocess whose stationay distibution can be obtained by solving patial diffeential equations. While these equations 1

2 2 Guvich: Validity of heavy-taffic steady-state appoximations will typically not give ise to closed-fom expessions fo the stationay distibution, they can nonetheless be solved elatively efficiently using a vaiety of numeical algoithms; see, e.g., Dai and Haison 16], Chen and Shen 9], and Saue, Glynn and Zeevi 34]. The use of a Bownian system model as a means to appoximate the netwo s steady-state distibution has been advocated by seveal papes in the liteatue. Haison and Nguyen 24] fomalized this pocedue, aticulating an appoximation scheme named QNET. The fist step in QNET constucts the Bownian system model fom the poblem pimitives chaacteizing the oiginal netwo. Then, the steady-state woload in the queueing netwo is appoximated by that of the Bownian model (suitably scaled). While this appoximation is clealy motivated by heavy-taffic theoy, thee is no igoous justification fo the tansition fom appoximations ove finite time intevals (the diffusion limits) to an appoximation ove an infinite time hoizon (steady-state vaiables). To bette explain the main issues undelying validity of heavy-taffic steady-state appoximations, conside a sequence of queueing netwos indexed by that satisfy the following heavy-taffic condition: (1 ρ j ) γ j as, (1) fo each station j, whee ρ j is the utilization in station j and γ j is a positive constant; a moe pecise definition will be given in 2.1. Let Ẑ (t) = Z (t)/ denote the popely scaled queue-length vecto in the th netwo at time t. To justify a Bownian appoximation of the steady-state distibution one must pove the following limit-intechange: lim lim E t f(ẑ (t)) ] ] = lim lim E t f(ẑ (t)), (2) fo all bounded and continuous functions f. This is expessed gaphically in Figue 1. Z ( t) II: t Z () IV: I: Z (t) III: t Z() Figue 1: The intechange-of-limits diagam The figue has fou components: (I) diffusion limits (pocess convegence); (II) stability and existence of a steady-state fo the pe-limit; (III) stability and existence of a steady-state fo the Bownian model; and (IV) convegence of the steady-state distibutions. To date, this limit-intechange poblem in open queueing netwos has only been woed out fo netwos with a single custome class othewise nown as genealized Jacson netwos. In paticula, the ecent papes by Gamani and Zeevi 21] and, subsequently, Budhiaja and Lee 6] deived such a esult, and consequently established the validity of the Bownian steady-state appoximation fo this class of netwos. It is woth pointing out that in the context of that poblem, edges (I), (II) and (III) wee nown, and the wo in 21] and 6] established (IV), hence poving that the limit intechange (2) is valid. The limit intechange has been established fo some instances of multiclass queueing systems in heavytaffic. Katsuda 27] (futhe discussed towads the end of this section) poved the limit intechange esult fo both the queue-length and woload pocesses in a multiclass single-seve queue with feedbac unde

3 Guvich: Validity of heavy-taffic steady-state appoximations 3 vaious disciplines. Ye and Yao 41] studied a paallel-seve system with two custome classes and two seves. Gamani and Stolya 2] and Tezcan 36] poved the limit-intechange fo special instances of paallel-seve systems in the so-called Halfin-Whitt heavy-taffic egime. In this pape we add to the existing esults in studying multiclass open queueing netwos with queueatio disciplines. We will efe to these as queue-atio netwos. Queue-atio disciplines see to set the queues at each station equal to a linea combination of the queue in that station. These atios can be abitaily set, endeing this a faily geneal family of disciplines. The policies ae explicitly defined in Connections to antecedent liteatue and summay of the pape s main contibution Queue-atio netwos pesent new and significant challenges that did not appea in the case of genealized Jacson netwos. While we ae not able to develop a complete theoy that encompasses all disciplines fo which pocess-level convegence to an SRBM limit (edge (I) in Figue 1) has been igoously veified, we develop a systematic appoach and, fo the family of queue-atio disciplines, identify a simple set of sufficient conditions. To be a bit moe specific, yet speaing vey loosely at this stage, ou main esult states that given a sequence of stable queue-atio netwos, whose popely nomalized queue-length vecto conveges to an SRBM that is itself stable, the limit intechange is valid if solutions to the fluid model ae attacted to the so-called invaiant manifold at a linea ate. All of this will be caefully explained in what follows, but we will note that the theoetical constucts we use and develop build heavily on, and pesent inteesting connections to, the ey papes in the field that have established the existing thee edges (I-III) of the diagam in Figue 1. We next summaize some of the ey ideas elated to these edges and the manne in which the cuent pape builds on that theoy. Fluid models and stability analysis: Detemining whethe a queueing netwo is stable, i.e., whethe it admits a stationay distibution, is geatly simplified by educing the poblem to the study of stability of a deteministic countepat nown as a fluid model. An impotant esult due to Dai 15] (see also Stolya 35]) shows that if the fluid-model queues ae emptied in a finite time, then the oiginal queueing netwo is stable. Sufficient conditions that guaantee this fluid daining fo static pioity netwos wee povided in 12]. Fluid models play an analogous ole in studying stability of the Bownian countepat to the oiginal netwo. Dupuis and Williams 19] show that an SRBM is stable if its fluid model (a deteministic Soohod Poblem) dains to the oigin in finite time. Ou wo builds on the esults of 12]. Dai 15] and Dupuis and Williams 19]. The latte will play a ey ole in ou analysis, which hinges on identifying a suitable Lyapunov function fo the queue-length pocess. This illustates an impotant connection between stability of the SRBM, as viewed though the lens of a fluid model, and establishing tightness of the sequence of steady-state pe-limit queue-length pocesses. Diffusion limits and state-space collapse: Up until the wo of Williams 39], the standad appoach fo establishing diffusion limits, within the heavy-taffic famewo descibed ealie, elied on the Continuous Mapping Theoem. This, in tun, hinges on the continuity of an undelying Soohod mapping; one of the fist illustations of this appoach is Reiman s seminal pape on genealized Jacson netwos 32]. The continuous mapping appoach was used also in a handful of specific multiclass queueing netwo settings, such as static-pioity feed-fowad netwos 31], o e-entant static-pioity lines with deteministic outing 11]. Howeve, fo all but a vey small family of netwos, the continuous mapping appoach cannot be applied in multiclass settings due to the absence of a path-to-path mapping in the associated Soohod poblem; see 2, 3]. The two main assumptions in Williams 39] ae: (a) the egulato matix R is completely-s (see 3); and (b) the sequence of netwos in heavy-taffic admits a so-called state-space collapse (SSC) popety. The fist popety is necessay fo the existence of an SRBM pocess. The second popety guaantees that the queue-length vecto (whose dimension is equal to the numbe of job classes) is given, in the limit, as a linea mapping of the woload pocess (whose dimension is given by the numbe of stations). In othe wods, it is assumed that thee exists a matix, so that, unifomly on compact sets, as, Ẑ Ŵ, (3)

4 4 Guvich: Validity of heavy-taffic steady-state appoximations whee Ẑ and Ŵ ae the popely scaled queue-length and woload vectos. Consequently, in the limit the state-space collapses into one of lowe dimension and the matix is theefoe efeed to as a lifting matix. The limit pocess is then said to live on an invaiant manifold. The queue-atio disciplines that we study hee use only queue length infomation (athe than woload), yet a vesion of (3) emains cental to the analysis afte eplacing the scaled woload Ŵ with a cetain linea combination of the queues; see 2.2. SSC has been established fo specific cases (see, fo example, Whitt 37], Reiman 33]), but a unified famewo was fist povided by Bamson 5]. Thee, conditions fo SSC ae spelled out in tems of attaction of the fluid model to the so called invaiant manifold. The SSC assumption is cental to the poofs of Williams 39] and, togethe with cetain Oscillation inequalities, fills gaps ceated by the absence of a continuous mapping. Static-pioity netwos ae a special case of the family of disciplines that we conside hee. Diffusion limits fo static pioity netwos (building on state space collapse and the famewo in 39]) has been established in a sequence of papes by Chen and co-authos 1, 11, 13] whee explicit conditions ae also povided fo linea attaction of the fluid model to the invaiant manifold. We impose such a linea attaction as a condition towads limit intechange. SSC also plays an instumental ole in ou appoach to the limit-intechange poblem. While we adopt a moe stingent notion of state-space collapse, we build heavily on Bamson s famewo and in paticula on the connections between SSC and the netwo s fluid model. One of the ey steps in poving validity of heavy taffic steady-state appoximations is to show that SSC holds fo suitable sequences of steady-state quantities. Fo this step, we intoduce a tuncated analogue of the fluid model. The tuncated fluid model allows us to pove SSC in steady-state befoe (and independently of) poving the tightness of the scaled steady-state queues. Convegence of the steady-state distibutions: Povided that a diffusion limit is poved (I), and that the stability of the queueing netwo (II) and the Bownian model (III) have been established, it suffices to show that the sequence Ẑ ( ) is tight in ode to pove the limit-intechange esult. As indicated ealie, this has been established in the case of the single class genealized Jacson netwos in 21] and 6]. While the two papes diffe somewhat in tems of methodology, both ely on the continuous mapping appoach which can not be diectly extended to the multiclass case that we conside in the cuent pape. Ou analysis does, howeve, daw on 21], at least in tems of Lyapunov function aguments. It is also woth pointing out that ecent wo of Katsuda 27] has shown fo a lage family of multiclass queueing netwos that, povided that the sequence of scaled steady-state queues Ẑ ( ) o the sequence of steady-state woloads Ŵ ( ) ae tight, the esults of 5] and 39] can be extended to the case in which one initializes the system at time t = with its steady-state distibution. In tems of disciplines, ou scope is moe limited. The main focus of ou pape is on poving that fo netwos opeated unde queue-atio disciplines the sequence of steady-state queues Ẑ ( ) is indeed tight povided that a linea attaction condition holds fo elated fluid models. Summay of the pape s contibutions: The sufficient conditions in ou main esult, which is given in 3, educe the question of limit-intechange in multiclass queue-atio netwos to popeties of fluidmodels. Recall that if the fluid model coesponding to the pe-limit netwo is stable in the sense of Dai 15], and if the conditions in Williams 39] hold (namely, SSC in the sense of Bamson 5] and the egulaity of the eflection matix), and the fluid model of the coesponding SRBM is stable in the sense of Dupuis and Williams 19], then one has edges (I-III) of the intechange diagam. We add to this by identifying conditions that guaantee that the intechange (IV) holds. The main technical steps that ae used to establish this claim boil down to identifying a suitable Lyapunov function fo the Bownian model, and using this Lyapunov function as a constained Lyapunov function fo the sequence of queueing netwos in heavy-taffic. A steady-state vesion of state-space collapse (via tuncated fluid models) and cude peliminay bounds on the steady-state queue length ae then combined with the constained Lyapunov function to show the tightness of the sequence of diffusion-scale steady-state queue lengths. 2. Essential peliminaies

5 Guvich: Validity of heavy-taffic steady-state appoximations The netwo model In this subsection we descibe the essential elements of the netwo model. Ou desciption follows mostly that of Williams 39]. The setting that we conside is moe esticted and we will point out wheeve ou constuction depats fom hes. We conside a queueing netwo with a set J = 1,..., J of single-seve stations and, a set K = 1,..., K of custome classes (with K J). The many-to-one mapping fom custome classes to stations is descibed by a J K constituency matix C whee fo j J and K, C j = 1 if class is seved at station j, and it equals othewise. Fo K, we let s() be the station at which class is seved, i.e., s() is the unique j J such that C j = 1. Fo each class K, E = (E (t), t ) counts the numbe of aivals to class fom outside the netwo that have occued by time t. Not all classes have exogenous aivals but we assume that the set K a = K : E is non-empty. Fo each K a, E is a (possibly delayed) enewal pocess constucted fom a sequence of nonnegative andom vaiables u (i), i = 1, 2..., whee u (i) denotes the time between the (i 1) st and the i th extenal aival of a class- custome so that u (1) is the time measued fom zeo until the fist extenal aival to class. It is assumed that u (i), i = 2, 3,... is a sequence of positive independent and identically distibuted (i.i.d) andom vaiables with distibution F a ( ), mean 1/α (, ) and coefficient of vaiation c a,, ). (The fist esidual inteaival time, u (1), is allowed to have a diffeent distibution.) To be able to apply the stability esults of 15] diectly, we futhe equie that the inte-aival times ae unbounded and speadout (see 1 of 15]). Letting U () = and U (n) = n i=1 u (i), fo n = 1, 2,..., the enewal pocess E satisfies, fo all t, E (t) = n : U (n) t. Fo convenience, we define E fo / K a and set E = E, K. In ou analysis, we will sometimes initialize the queueing netwo with its steady-state distibution, in which case u (1) will have the equilibium distibution of the coesponding enewal pocess. Fo each K we denote by v (i), i = 2, 3,... the sevice-time equiements of jobs in class in ode of thei entance to sevice, so that v (2) is the sevice time of the fist class custome to commence sevice afte time. The andom vaiable v (1) stands fo the esidual sevice time of the custome at the head of the class- queue at time if the sevice of that custome has aleady begun. We set v (1) = if thee is no such custome. Unde peemptive disciplines thee may be a custome whose sevice has begun but is not in sevice. It is assumed that v (i), i = 2, 3,... is a sequence of positive i.i.d. andom vaiables with distibution F s ( ), mean m (, ) and coefficient of vaiation c s,, ). We let M denote the K K diagonal matix with m as the th diagonal element. The paamete µ = 1/m then stands fo the long-un aveage ate at which class- customes would be seved if the seve in station s() wee neve idle and woed exclusively on class. The cumulative-sevice-time pocess fo class is defined by V () = and V (n) = n i=1 v (i), fo n = 1, 2,..., and we define the (possibly delayed) enewal pocess n : V (n) t, if v S (t) = (1) >, n : V (n) t 1, if v (1) =. The esidual sevice time of the class- custome in sevice at time, v (1), may have a diffeent distibution. Depating fom 39], we assume that the sevice time of a job is geneated when the seve commences pocessing that job (as opposed to assuming it is geneated upon aival to the pocessing station). Fo both the inteaival and sevice times it is assumed that, fo all p N, E (u (2) z) p u (2) > z] <, fo all K a, (4) z R + E (v (2) z) p v (2) > z] <, fo all K. z R + The outing in the netwo is assumed to be Maovian with a outing matix P so that P l is the pobability that a class- custome becomes a class-l custome upon its completion of sevice at station

6 6 Guvich: Validity of heavy-taffic steady-state appoximations s(). The matix P = P denotes the tanspose of P. To ensue that ou queueing netwo is open, the matix P (and, in tun, P ) is assumed to have spectal adius less than 1. Moe fomally, let e 1,..., e K be the unit basis vectos paallel to the K coodinate axes in R K, and let e be the K - dimensional vecto of all zeos. Fo each class K, ϕ (i), i = 1, 2,... is a sequence of i.i.d outing vectos whee ϕ (i) taes values in the set e, e 1,..., e K. The i th class- custome to depat fom station s() is outed to class l if ϕ (i) = e l, o it leaves the netwo if ϕ (i) = e. Accodingly, P l = Pϕ (i) = e l, fo K, l K. Then, fo K, Eϕ (i)] = P and Covϕ (i)] = Υ, whee P denoted the th column of P, and Υ is the K K matix defined by Υ Pl (1 P lm = l ) if l = m, P l P m if l m. Fo each K, we define the K-dimensional cumulative outing pocess fo class by n φ (n) = ϕ (i), n = 1, 2,..., i=1 (5) whee φ () =. Since P has spectal adius that is stictly smalle than 1, the matix Q = (I P ) 1 = I + P + ( P ) 2 + ( P ) 3 +, whee ( P ) n denotes the n th powe of P, is well defined. Finally, we assume that u l (i), i = 2, 3,..., v (i), i = 2, 3,..., and ϕ (i), i = 1, 2,..., fo l K a and K ae mutually independent sequences of andom vaiables (o vectos), and that collectively these ae independent of (Z (), v (1), u (1); K), whee Z () is the numbe of class- customes pesent in station s() at time t =. We shall efe to the stochastic pocesses E, V and φ as the pimitives fo the multiclass queueing netwo model. We assume that all the andom vaiables and stochastic pocesses intoduced thus fa ae built on a common pobability space (Ω, F, P). 2.2 Queue-atio disciplines and a Maovian state descipto A lifting matix is a K J matix such that δ if s() = j, j = othewise. (6) whee δ, K ae non-negative constants such that CM = I whee I is the identity matix. Recall that Z (t) is the queue length of class at time t. Given a lifting matix, we define ϵ(t) = Z(t) CMZ(t). The queue-atio discipline coesponding to then is defined as follows: at each time t, the custome in sevice in station j is the custome at the head of the highest index class that has a positive value of ϵ, namely, it is a custome fom class whee = maxi : C ji = 1, ϵ i (t) >. (7) If thee ae no such classes, the custome at the head of the highest index non-empty queue at that station is seved, i.e, = maxi : C ji = 1, Z i (t) >. (8) The tansition between jobs is made in a peemptive esume manne. Intuitively, the discipline sees to ende the queue of class popotional to a linea combination of the queues in station s(). This motivates the name queue-atio discipline. A queue-atio discipline can be defined fo any lifting matix. Once specified, this matix completely defines the discipline. Peemptive esume static pioities ae a special instance of queue-atio disciplines whee, fo each station j, δ = 1/m if = l(j) := mini : C ji = 1 and δ = othewise. In tun, ϵ (t) = Z (t) fo all l(j) and seve j seves the low pioity class only if all highe pioity queues ae empty. Fo K a. let R a (t) be the esidual time until the fist class- exogenous aival afte time t. Put R a = (R a, Ka ). If the sevice of the custome at the head of the class- queue at time t has aleady

7 Guvich: Validity of heavy-taffic steady-state appoximations 7 begun, we denote by R v (t) its esidual sevice time. If the pocessing of the head-of-the-line class- custome has not begun at time t we set R v (t) =. Putting Rv = (R v, K), let Ξ = (Z, R a, R v ), (9) and let X N K R Ka + R K + be the domain on which the pocess Ξ taes its values. We let T = (T (t), t ) be the allocation pocess so that the th component of T (t) is the cumulative sevice time allocated to class up to time t. Letting σ l l= be a stictly inceasing sequence of times at which successive aivals o depatues occu to o fom any class in the netwo, the pocess T (t) = ( T 1 (t),..., T K (t)), whee the dot stands hee fo the ight-deivative with espect to time, changes only on the event epochs σ l. Moeove, fo t σ l, σ l+1 ), T (t) = 1 if and only if = whee is as in (7) and (8). Thus, T (t) is a measuable function with espect to the σ-algeba on X and the Boel σ algeba of, 1] K. Since we geneate sevice times only upon commencement of sevice, it is evident that the pocess Ξ is, unde a queue-atio discipline, a Maov pocess. Queue-atio disciplines ae a special case of head-of-the-line (HL) disciplines. We efe the eade to of 39] fo a fomal constuction of HL disciplines as Maov pocesses. System dynamics Let A (t), K, count the numbe of aivals to class by time t (both exogenous and fom othe classes). Let D (t), K, count the numbe of sevice completions of class- customes by time t. The numbe of customes pesent in class K at time t is given by Z (t) and, fo j J, Y j (t) is the cumulative idleness at station j by time t. Thoughout, the matix is fixed and we define the nominal woload W = CMZ. Note that in 39] and 5] W is used fo the tue immediate woload of which we do not eep tac hee. This abuse of notation facilitates maing the needed connections to the antecedent liteatue. The pocess ϵ is then e-witten as ϵ = Z W. Fo each class, we denote by ϵ + (t) = i:c is() =1,i> ϵ i(t). This is the excess at station j = s() at time t coesponding to classes which ae seved in the same station as but have highe indices. Also, we let T + (t) be the aggegate time allocated to these classes by time t. With these definitions, the dynamics of the netwo must satisfy the following equations fo all t, A(t) = E(t) + φ (D (t)), (1) Z(t) = Z() + A(t) D(t), (11) W (t)dy (t) =, (12) Y (t) + CT (t) = et, (13) D(t) = S(T (t)), (14) t T + (t) can only incease when ϵ+ (t) =, = 1,..., K, (15) whee the integal in (12) should be ead componentwise and heeafte e denotes the J - dimensional vecto with all elements equal to 1. Equation (14) holds fo all HL disciplines. Equation (15) is equivalently witten as ϵ + (s)d(s T + (s)) =, = 1,..., K. Fo the special case of peemptive esume static pioity netwos this educes to the well nown condition whee Z + 15]. Z + (s)d(s T + (s)) =, = 1,..., K, coesponds to the total queue of classes with highe pioity than class ; see e.g. 5, page 2.3 Fluid model equations and thei tuncated countepats Thee types of fluid models ae used in the liteatue to specify sufficient conditions that guaantee edges (I-III) in the limit intechange diagam in Figue 1. In the context of poving SSC in 5] one consides fluid models that appoximate the evolution of queueing netwo ove shot time intevals unde a hydodynamic scaling. The appopiately

8 8 Guvich: Validity of heavy-taffic steady-state appoximations defined limits (cluste points in the teminology of 5]) ae expected to satisfy the following fluid-model equations: Z(t) = Z() + αt (I P )M 1 T (t), (16) W (t) = CM Z(t), (17) ϵ(t) = ϵ() + (I CM)(αt (I P )M 1 T (t)), (18) T (t) is nondeceasing and stats fom zeo, (19) Ȳ (t) = et C T (t) is nondeceasing, (2) W (s)dȳ (s) =, (21) ϵ + (s)d(s T + (s)) =, = 1,..., K. (22) These ae natual deteministic countepats of (1)-(15). The equation (18) is hee edundant as it follows fom (16) and the definition of ϵ but it will be useful fo the discussions that follow. All solutions to (16)-(22) ae Lipschitz continuous and we let N be the Lipschitz constant (N is specified explicitly in 5.3). In the SSC famewo of 5] one equies that solutions X = ( W, Z, ϵ, T ) to the fluid-model equations (16)-(22) ae attacted to the invaiant manifold. In ou main esult 3.1 we stengthen this equiement to a linea dift equiement. In the following definition we say that t is egula fo X if X(t) exists at t. Definition 1 (piecewise-linea test functions fo SSC) A family of non-negative vectos h = (h 1,..., h n ) is said to induce a piecewise-linea Lyapunov function fo the fluid model if the following holds: Thee exists c > such that fo any solution X = ( W, Z, ϵ, T ) to the fluid-model equations (16)-(22) and any egula t, (a) max 1 l n h l, ϵ(t) = if and only if ϵ(t) =, and (b) max 1 l n h l, ϵ(t) c if ϵ(t) >. If such a family of vectos exists we say that the fluid model is attacted to the invaiant manifold at a linea ate. The linea attaction to the invaiant manifold guaantees, in paticula, that once ϵ is close to it stays thee egadless of, say, the specific values of Z o W. As queue-atio disciplines espond diectly to the distance, ϵ, fom the invaiant manifold such linea attaction seems plausible. In the special case of static pioities Chen and Ye 1, Poposition 3.5] and Chen and Zhang 13, Theoem 4] identifies explicit algebaic conditions on the netwo pimitives that guaantee that a piecewise linea test function exists fo the fluid model. To have meaningful appoximations unde the hydodynamic scaling one equies that the (sequence of) diffusion-scale queues at time t = fom a tight sequence (see e.g. Theoem 3 in 3]). To establish statespace collapse in steady-state, howeve, we will want to analyze the dift when the netwo is initialized with its steady-state distibution which we cannot assume apioi to be tight (as that is exactly what we see to pove). It will suffice fo ou puposes to captue the incements of ϵ. In essence, we will conside limits unde appopiate scaling of the (augmented and tuncated pocess) ( W Θ, Z Θ, T, ϵ Θ, H ), whee H (t) = ϵ (t) ϵ (), Θ is a tuncation constant and, fo abbeviation, we wite W Θ = (W 1 Θ,..., W J Θ ) and similaly fo the othe pocesses. Given a time inteval, L], Θ = 3NL whee N is the Lipschitz constant of the fluid model equations (16)-(22). The fluid model that emeges as an appopiate limit of these tuncated scaled pocesses (see 5.3) is the following modification of (16)-(22): (16 ) Z(t) Z() Θ + αt (I P )M 1 T (t), (17 ) W (t) = CM Z(t),

9 Guvich: Validity of heavy-taffic steady-state appoximations 9 (18 ) ϵ(t) ϵ() Θ + (I CM)(αt (I P )M 1 T (t)), (18a ) H(t) = (I CM)(αt (I P )M 1 T (t)), (18b ) H() =. (19 ) T (t) is nondeceasing and stats fom zeo, (2 ) Ȳ (t) = et C T (t) is nondeceasing, (21 ) W (s)dȳ (s) =, (22 ) ( ϵ+ (s) Θ)d(s T + (s)) =, = 1,..., K. We will efe to (16 )-(22 ) as the tuncated-fluid-model equations. Equations (16 )-(18 ) and (19 )- (22 ) ae natual (tuncated) countepats of equations (16)-(22). In (18a ) and (18b ), H is intepeted as captuing (in fluid scale) the incements of the pocess that tacs the distance fom the invaiant manifold ϵ(t) ϵ() (see below). We say that X = ( W, Z, T, ϵ, H) solves the tuncated fluid model if it satisfies equations (16 )-(22 ) fo all t L and if X(t 2 ) X(t 1 ) N t 2 t 1 fo all t 1 < t 2 L. Thee is an evident lin between the fluid-model equations and thei tuncated countepats. Given a solution ( W, Z, ϵ, T ) to the fluid model equations (16)-(22), X(t) = ( W (t), Z(t), T (t), ϵ(t), ϵ(t) ϵ()), whee ϵ(t) = Z(t) W (t), is a solution to the tuncated fluid model equations (16 )-(22 ). In paticula, fom evey solution to the fluid-model equations we can constuct a solution to the tuncated fluid-model equations, but the convese is, in geneal, false because the tuncated fluid model is unde specified compaed to the fluid model. We say that all solutions to the tuncated fluid model equations ae attacted to the invaiant manifold if thee exists c > such that, fo any solution X to the tuncated fluid model equations items (a) and (b) of Definition 1 hold with ϵ thee eplaced with H. The following lemma is poved in F of the appendix. Lemma 2.1 If all solutions to the fluid model equations (16)-(22) ae attacted to the invaiant manifold at linea ate then so do all solutions to the tuncated-fluid-model equations (16 )-(22 ). Thus, wheeas the attaction to the invaiant manifold of the tuncated fluid model plays a cucial ole in ou poofs, Lemma 2.1 allows us to state the sufficient conditions fo limit intechange in tems of the bette-undestood fluid-model equations (16)-(22). The discussion thus fa will suffice fo the statement of ou main esult in the next section which will be followed by futhe fomalization of some of the ey concepts. We end this section with some notational conventions that we use thoughout the pape. Additional notational conventions: Fo a Maov pocess Ξ = (Ξ(t), t ) on a locally compact sepaable metic space X we let P x be the pobability distibution unde which PΞ() = x = 1 fo x X and E x ] = E Ξ() = x] be the expectation opeato w..t. the pobability distibution P x. Let P π denote the pobability distibution unde which Ξ() is distibuted accoding to π and put E π ] to be the expectation opeato w..t. this distibution. A pobability distibution π defined on X is said to be a stationay distibution if fo evey bounded continuous function f E π f(ξ(t))] = E π f(ξ())], fo all t. It is said to be the steady-state distibution if fo evey such function and all x X, E x f(ξ(t))] E π f(ξ())] as t. We let C d, ) be the space of continuous functions fom, ) to R d. We let D d = D d, ) be the space of all RCLL (Right Continuous with Left Limits) R d -valued functions, equipped with the Soohod J 1 metic; see e.g. 38]. We use to denote wea convegence as with espect to this metic, and when discussing R d -valued andom vaiables will simply mean convegence in distibution as. Fo a vecto-valued pocess x D d, ), let x s,t = s t T x(t), whee x(t) = d =1 x (t) and we emove the subscipt s if s =. Finally, thoughout, we use the tem absolute constant to denote a finite and stictly positive constant that does not depend on the heavy-taffic index (but that may depend on othe paametes). We use c, c 1,... to denote such constants.

10 1 Guvich: Validity of heavy-taffic steady-state appoximations 3. Statement and discussion of the main esult To state ou main esult, we let Ẑ (t) = Z (t)/ be the diffusion scaled queue-length in the th netwo at time t and let Ŵ = CMẐ. The (scaled) distance fom the invaiant manifold is then ϵ = Ẑ Ŵ. Also, we let T = T (t)/ be the diffusion scaled allocation pocess at time t, Ŷ (t) = Y (t)/ be the scaled cumulative idleness at time a, t and R () = Ra, ()/ v, and R () = Rv, ()/ be, espectively, the scaled esidual inte-aival and sevice times at time. Finally, α is the exogenous-aival-ate vecto in the th system. We assume that (α α) = β fo some β (, ) so that, in paticula, α α as ; additional details egading the scaling and the heavy-taffic conditions ae povided in 4. Some of the assumptions made in ou main esult below ae boowed diectly fom the liteatue and wee shown to be sufficient fo edges (I-III) in Figue 1. Specifically, (1) fo the existence of the limit SRBM we impose cetain stuctue on the data matices, most impotantly, that the eflection matix R = (CMQ ) 1 satisfies a completely-s condition. This is Assumption 7.1 in 39] which we will flesh out as Assumption 1 in 4. (2) fo the positive ecuence of the SRBM we equie that all solutions to a Soohod poblem (the fluid model of the SRBM) ae attacted to the oigin in finite time. This is the ey assumption in Theoem 2.6 of 19] that we epeat hee as Assumption 2 in 5.2. (3) fo the positive ecuence of the queueing netwo we equie that, fo each index along the sequence of netwos, the coesponding fluid model is stable. This is the ey assumption in Theoem 4.2 of 15] that we flesh out as Assumption 3 in 5.4; When added to the above, the linea attaction to the invaiant manifold, guaantees the validity of (IV) in the limit intechange diagam. Theoem 3.1 (The main theoem) Conside a sequence of queue-atio netwos in heavy-taffic and pose that Assumptions 1, 2 and 3 hold and that any solution to the fluid model equations (16)-(22) is attacted to the invaiant manifold at linea ate. We then have the following: (Ẑ I. If (), R a, (), R ()) v, (Z(),, ), and ϵ then Ẑ Ẑ, whee Ŵ = CMẐ is an SRBM. II. Fo all N, the pocess Ξ has a unique stationay distibution which is also its steady-state distibution. III. The SRBM Ŵ has a unique stationay distibution which is also its steady-state distibution. IV. Steady-state convegence: The sequence of steady-state queue-length vectos conveges wealy Ẑ ( ) Ẑ( ), whee CMẐ( ) has the steady-state distibution of the SRBM Ŵ. Futhe, fo any m N, E Ẑ ( ) m ] E Ẑ( ) m ]. Discussion of the main esult: On top of Assumptions 1-3 that follow pevious liteatue, we impose in Theoem 3.1 two futhe equiements on queue-atio netwos. Fist, wheeas the typical condition fo state-space collapse is mee attaction to the invaiant manifold, we equie linea attaction. As mentioned above, fo the special case of static pioity netwos, conditions that guaantee these have been explicitly elated to algebaic popeties of the undelying matices but this emains to be veified fo geneal queue-atio disciplines. Second, the equiement that inteaival and sevice times have finite moments of all odes is an atifact of ou poof techniques and it is plausible that this condition can be tightened. In fact, ou poofs do not necessitate the existence of all such moments but we do equie moments of significantly geate ode than the mee second moment equied in 39] and 5]. In ou poofs we mae explicit the dependence on p so as to undescoe the souces of this equiement. The numbe of moments, m, fo which the convegence in item (IV) of the theoem holds does depend on the value of p. Fo the same easons explained above, howeve, the elation between the value of p in (4) and the numbe of moments, m, fo which the convegence holds is not as clean as in the Genealized Jacson case (see 6]) whee it was shown that such convegence holds fo all m < p 1.

11 Guvich: Validity of heavy-taffic steady-state appoximations Outline of the poof Hee we povide an infomal outline of the poof that highlights the ey steps and ingedients fo the poof of item (IV) in Theoem 3.1. Each step in this outline will be expanded upon and spelled out in detail in 5. Step 1: Inclusion sets and Lyapunov functions Let Ξ ( Ξ(t), t ) be a continuous-time Maov pocess defined on a locally compact sepaable metic state space X. Fo the special case of the queue-atio netwos, Ξ would be the scaled vesion of (9). The following notion will be useful: Definition 2 A function Φ : X R+ is said to be a constained Lyapunov function of ode q 1 fo Ξ with dift-size paamete δ <, dift-time paamete t >, exception paamete κ, and inclusion set A X, if E x Φ q ( Ξ(t ))] Φ q (x) x A:Φ(x)>κ Φ q 1 δ. (23) (x) The equiement that the initial state x belongs to the inclusion set A is the distinguishing featue of constained Lyapunov functions. In Poposition 5.1 we establish that, if Φ( ) is a constained Lyapunov function fo a Maov pocess Ξ that has a unique stationay distibution π, then unde suitable conditions fo constants ε 1, ε 2 >. ] ( E π Φ q 1 ( Ξ()) 1 + ε 1 δ ) E π Φ q 1 ( Ξ())1 Ξ() / A ] + ε 2 δ, (24) The intoduction of constained Lyapunov functions is motivated by paticula chaacteistics of multiclass queueing netwos in heavy taffic. Roughly speaing, as inceases, the queueing netwo exhibits state-space collapse and as a esult lives in a small neighbohood of the invaiant manifold. This neighbohood is expected to seve as an inclusion set fo an appopiately chosen Lyapunov function. Let X be the domain on which the pocess Ξ taes its values; see 2.2. Given N and ϵ >, define Bϵ = x = (z, ϱ a, ϱ v ) X : ϱ a + ϱ v ϵ, K, (25) and A ϵ = x B ϵ : z CMz ϵ. (26) In othe wods, A ϵ is the intesection of an ϵ-neighbohood of the invaiant manifold with the family of states in which the (scaled) initial esiduals ae well behaved. A fist step in the poof of Theoem 3.1 will be to identify a suitable constained Lyapunov function Φ( ) and show that the bound (24) holds fo the diffusion-scaled queueing-netwo pocess Ξ (see 4), with ε 1 thee eplaced by c q 2 and with ε 2, δ not depending on. We will then deduce tightness fom (24) using popeties of Φ( ) and showing that ] lim q 2 Eπ Φ q 1 ( Ξ ())1 Ξ () / A ϵ <. (27) Step 2: Identifying the constained Lyapunov function Ou point of depatue hee is the stability analysis of SRBM caied out by Dupuis and Williams 19] and summaized in Theoem 5.2 hee. In that wo, a (non-constained) Lyapunov function Ψ( ) is used fo the SRBM. Ou constained Lyapunov function is constucted fom that function. Specifically, we will establish (see Poposition 5.2) that, fo any constant b > 1, the function Φ( ) = b + Ψ( ) is a constained Lyapunov function fo the scaled queueing-netwo pocess Ξ, with inclusion set A ϵ as defined in (26). Intuitively, the distance between the queueing-netwo and its appoximating SRBM is mostly captued by the distance of the queueing netwo fom the invaiant manifold. If the queueing netwo emains close to the invaiant manifold, one expect that a negative dift fo the SRBM (with the coesponding Lyapunov function) will tanslate into a simila dift fo the queueing netwo. This logic assumes that, stating in the inclusion set, the netwo indeed emains close to the invaiant manifold. This is the subject of step 3.

12 12 Guvich: Validity of heavy-taffic steady-state appoximations Step 3: Tuncated fluid models and state-space collapse in steady-state To establish the concentation of π in A ϵ we will show in Theoem 5.3 that unde the conditions of Theoem 3.1, the sequence of steady-state queues Ẑ ( ), N satisfies Ẑ ( ) CMẐ ( ). The tuncated fluid model equations play a ey ole hee. These allow us to pove SSC befoe (and independently of) poving the tightness of the scaled steady-state queues. We also show that, initializing the netwo in the inclusion set A ϵ, the netwo pocess emains close to the inclusion set; see Theoem 5.4. This is instumental in establishing that the Lyapunov function, identified in step 2 above, is indeed a constained Lyapunov function fo the queueing netwo pocess. Step 4: Cude steady-state bounds Establishing (27) equies moment bounds fo Z ( ) and the stating point of this pape is that such bounds ae not available a pioi. Fotunately, if the pobability P π Ξ () / A ϵ decays sufficiently fast, cude moment bounds ae sufficient. To that end, we will show in Theoem 5.6 that fo suitably lage constants c, l and all N E π Ẑ () q] c l, and, as a coollay, obtain (27) which will conclude the poof. The emainde of the pape In 4 we define in detail the heavy-taffic scaling and eview elevant diffusion-limits esult fom 39]. The main contibution of this pape is embedded in pat IV of Theoem 3.1. This pat is e-stated and poved in 5. We conclude in 6 with some diections fo futue eseach. Thoughout, poofs of auxiliay esults ae elegated to the appendix. 4. The queueing netwo in heavy-taffic and diffusion limits 4.1 Heavy-taffic conditions and scaling We add the heavy-taffic index N to all elevant pocesses and quantities defining the netwo to mae the dependence on this index explicit, and omit it in the absence of such dependence. Fo example, the aival ate of class is given by α (so that Eu (2)] = 1/α ). The taffic equations fo the th queueing netwo ae given by o equivalently by λ = α t + P λ, λ = Qα, whee λ, the th components of λ, denotes the total aival ate fo class in the th system. We define the total taffic intensity ρ j fo the jth station as ρ j = m λ, C(j) o in matix fom: ρ = CMQα whee ρ = (ρ 1,..., ρ J ). Thoughout we will assume that M = diag(m 1,..., m K ), the coefficients of vaiation c a,, K and c s,, K, as well as the outing matix P, emain fixed and do not scale with. This is assumed fo simplicity of pesentation and the analysis can be extended to the case in which these paametes ae obtained as limits of coesponding sequences, M, P, c a, and c s, (see e.g. the analysis in 39] and 5]). The sequence of systems defined above is said to be in heavy-taffic if α = α + β/ fo some vecto β, and a stictly positive vecto α so that, fo all N, (1 ρ j ) = γ j, (28) fo some vecto γ = (γ 1,..., γ J ). Since we estict attention to cases in which the diffusion limit is stable, we will assume that γ has stictly positive enties. Let D (t) = D (t), T (t) = T (t),

13 Guvich: Validity of heavy-taffic steady-state appoximations 13 denote the fluid-scale depatue and time-allocation pocesses, espectively, and define the following diffusion-scale pocesses: Ê (t) = E (t) α t, φ, (t) = φ, ( t ) P t, D (t) = D (t) µt, and We also wite Ẑ (t) = Z (t), Ŷ (t) = Y (t), and Ŵ (t) = CMẐ. R a, (t) = Ra, (t), Rv, (t) = Rv, (t). Finally, we define the scaled vesion of (9) Ξ = (Ẑ, R a,, R v, ), (29) and let X denote its domain. We wite R = (CMQ ) 1, (3) ϵ (t) = Ẑ (t) Ŵ (t), (31) η (t) = CMQ P ( ϵ () ϵ (t)), (32) ( ) ξ (t) = CMŜ ( T K (t)) + CMQ Ê (t) + φ, ( D (t)) γt, (33) X (t) = Ŵ () + R( ξ (t) + η (t)), (34) =1 so that by (1)-(14), Ŵ (t) = X (t) + RŶ (t). (35) The lifting matix in (3) and (31) is as in (6). Put ( ( ) ) K H = C ΛΣ + MQ Π + λ Υ Q M C, (36) whee Λ = diag(λ 1,..., λ K ), Π = diag(α 1 c 2 a,1,..., α K c 2 a,k), Σ = diag(m 2 1c 2 s,1,..., m 2 Kc 2 s,k), and Υ is defined in (5). The above constuction implicitly pesumes, then, that the matix CM Q is invetible. This is fomally stated in Assumption 7.1 of 39] that we epeat below and fo which it is said that a J J matix R is completely-s, if and only if fo each pincipal submatix Ř of R, thee is a vecto ν > such that Řν >. =1 Assumption 1 (Data Matices) (i) The matix CMQ is invetible and R = (CMQ ) 1 completely-s. (ii) The matix H given in (36) is stictly positive definite. is Assumption 1 completes the desciption of the system paametes, dynamics and scaled pocesses. 4.2 The Bownian system model The natual diffusion analogue of the queueing netwo is captued mathematically by means of a semi-matingale eflecting Bownian motion (SRBM). The definition of SRBM with data (S, θ, Γ, R, ν) is given below (we efe to 6 of 39] fo futhe discussion of the SRBM and elevant efeences). Thoughout this section we fix S = R J + and a filteed pobability space (Ω, F, F t, P). Let B be the σ-algeba of Boel subsets of S. Let θ be a constant vecto in R J, Γ a J J non-degeneate covaiance matix (symmetic and stictly positive definite), and R a J J matix.

14 14 Guvich: Validity of heavy-taffic steady-state appoximations Definition 3 (SRBM) Given a pobability measue ν on (S, B), an SRBM associated with the data (S, θ, Γ, R, ν) is an F t -adapted, J-dimensional pocess W such that (i) W = X + RY, P-a.s., (ii) P-a.s., W has continuous paths and W (t) S fo all t, (ii) unde P, (a) X is a J-dimensional Bownian motion with dift vecto θ, covaiance matix Γ and X() has distibution ν, (b) (X(t) X() θt, F t, t ) is a matingale, (iv) Y is an F t -adapted, J-dimensional pocess such that P-a.s. fo each j J, (a) Y j () =, (b) Y j is continuous and non-deceasing, (c) Y j can incease only when W is on the face F j x S : x j =, i.e., W j (s)dy j (s) =. When discussing steady state, the initial distibution ν is immateial and we will efe to the SRBM with data (S, θ, Γ, R). The following is an adaptation of the main esult in Williams 39]. Theoem 4.1 (II: Diffusion limits) Suppose that Assumption 1 holds and that (Ẑ (), R a, (), R ()) v, (Ẑ(),, ), whee Ŵ () = CMẐ() has distibution ν. Suppose futhe that state-space collapse holds, i.e, that Then, ϵ. Ẑ Ŵ, whee Ŵ is an SRBM associated with the data (S, θ, Γ, R, ν) fo Γ = RHR and θ = Rγ. Recall that ou constuction of the queueing netwo is diffeent than that of 39] in that we geneate the custome sevice times only upon entance to sevice, athe than upon aival of a custome to a station. Fo queue-atio disciplines ou constuction is, howeve, equivalent to that of 39] in that, stating empty, the pocess Z has the same pobability law unde both constuctions and, in tun, both constuctions will shae the same diffusion limits. This equivalence holds also if both constuctions ae initialized at time with the same distibution of esiduals and povided that, in 39], the sevice times of the customes in queue at time ae i.i.d. and distibuted accoding to F s ( ). Thus, Theoem 4.1 is a diect coollay of Theoem 7.1 of 39]. Theoem 4.1 hints to the applicability of queue-atio disciplines. In the moe geneal esult of 39], given state-space collapse, the law of the diffusion limit is detemined by the initial distibution ν, the data matices R, Γ, and the vecto γ. (R itself is also defined though ). In tun, since a queue-atio discipline can be defined fo abitay lifting matices as in (6), it stands to eason that, asymptotically, any law fo the queue length vecto that is coveed by the geneal esults of 39] can be achieved via the coesponding queue-atio discipline povided. The fomalization of this statement is beyond the scope of this pape; see futhe discussion in Re-statement of the main esult and completion of the poof Ou main contibution is concened with the steady-state appoximation as embedded in statement IV of Theoem 3.1 which we now tun to estate and pove. Theoem 5.1 (IV: Steady-state convegence) Unde the conditions of Theoem 3.1, it holds that Ẑ ( ) Ŵ ( ), whee Ŵ ( ) has the steady-state distibution of the SRBM with data (S, Rγ, Γ, R).

15 Guvich: Validity of heavy-taffic steady-state appoximations 15 We pove Theoem 5.1 by elaboating on the outline povided in 3. Sections 5.1, 5.2, 5.3 and 5.4 ae dedicated, espectively, to steps 1-4 in that outline. Section 5.5 combines all the steps to conclude the poof of this theoem. 5.1 Inclusion sets and Lyapunov functions Given a Maov pocess Ξ = ( Ξ(t), t ) on a locally compact sepaable metic state space X, a subset A X and a function Φ( ) : X R+ we define fo all q N, ϕ Ξ q (t, A) = Φ (q 1) (x)e x (Φ q ( Ξ(t)) Φ q (x)) +], (37) x A whee the expectation may be infinite. Below, the notion of constained Lyapunov function is as in Definition 2. Poposition 5.1 Suppose that the Maov pocess Ξ possesses a stationay distibution π. Assume that Φ is a constained Lyapunov function of ode q 1 with dift-size paamete δ <, dift-time paamete t >, exception paamete κ and inclusion set A X, such that: (a) ϕ Ξ q (t, A) is finite; (b) E π Φ q ( Ξ())] < ; ] (c) E π (Φ q ( Ξ(t )) Φ q ( Ξ()))1 Ξ() / A ε >. ] ε E π Φ q 1 ( Ξ())1 Ξ() / A fo some constant Then, ] ( E π Φ q 1 ( Ξ()) 1 + ε ) ] E π Φ q 1 ( Ξ())1 Ξ() / A + κq 1 ϕ Ξ q (t, A). (38) δ δ ] If, in addition, thee exists ε 1 such that E π Φ q 1 ( Ξ())1 Ξ() / A ε 1, then P π Φ q 1 ( Ξ()) > y ε 1 y ( 1 + ε δ ) + κq 1 ϕ Ξ q (t, A). (39) δy 5.2 Identifying the queueing-netwo constained Lyapunov function The fluid model of the SRBM is infomally obtained by emoving the Bownian tem to get a Soohod Poblem (SP) as defined below and whee, as befoe, S = R J +. Definition 4 (Soohod poblem (SP)) A pai (ϕ, η) C J, ) C J, ) solves the SP with espect to (S, θ, R, x) if the following holds: (i) ϕ(t) = x + θt + Rη(t) S, fo all t ; (ii) η is such that, fo i = 1,..., J, (a) η i () =, (b) η i is nondeceasing and (c) t 1ϕ i(s) dη i (s) = fo all t. A solution (ϕ, η) is said to be attacted to the oigin in finite time if fo any ϵ > thee exists t ϵ < such that ϕ(t) ϵ fo all t t ϵ. The following is the main assumption made in 19] fo puposes of stability of the SRBM. Assumption 2 Assumption 1 holds and fo any initial state x, the ϕ component of all solutions to the SP with data (S, θ, R, x) is attacted to the oigin in finite time. Resolving the question of attaction to the oigin is not a tivial tas (see e.g. 8]), but is not a focal point fo the pesent pape. Fo ou puposes, the following esult is petinent.