Large Deviations of Max-Weight Scheduling Policies...

MATHEMATICS OF OPERATIONS RESEARCHVol. 35, No. 4, November 2010, pp. 881–910issn 0364-765X !eissn 1526-5471 !10 !3504 !0881

informs ®

doi 10.1287/moor.1100.0462©2010 INFORMS

Large Deviations of Max-Weight Scheduling Policies onConvex Rate Regions

Vijay G. SubramanianHamilton Institute, National University of Ireland, Maynooth, Co. Kildare, Ireland,

[email protected], http://hamilton.ie/vsubramanian

We consider a single server discrete-time system with a fixed number of users where the server picks operating points froma compact, convex, and coordinate convex set. For this system, we analyse the performance of a stabilising policy that at anygiven time picks operating points from the allowed rate region that maximise a weighted sum of rates, where the weightsdepend on the work loads of the users. Assuming a large deviations principle (LDP) for the arrival processes in the Skorohodspace of functions that are right continuous with left-hand limits, we establish an LDP for the work load process using ageneralised version of the contraction principle to derive the corresponding rate function. With the LDP result available, wethen analyse the tail probabilities of the work loads under different buffering scenarios.

Key words : queueing theory; scheduling; max-weight policies; convex rate regions; large deviations; maximal monotonemaps; Skorohod problem

MSC2000 subject classification : Primary: 60F10, 60K25; secondary: 90B22, 90B36OR/MS subject classification : Primary: queues; secondary: limit theoremsHistory : Received November 18, 2009; revised July 19, 2010. Published online in Articles in Advance November 10, 2010.

1. Introduction. In this paper, we consider a multiclass discrete-time queueing system where the server isallowed to pick operating points from within a compact, convex, and coordinate convex set. Our motivation forconsidering compact, convex, and coordinate convex rate regions arises from information-theoretic analysis ofmultiuser channels as described in Cover and Thomas [11, Chapter 14] like the multiple-access channel (MAC)or the broadcast channel (BC). Such models are particularly useful for modelling wireless systems. To operatenear the capacity boundary of these channels, sufficiently long code words need to be used, which naturally leadsto a discrete-time operation: pick a time long enough such that at all operating points one can choose long enoughcode words, so that the probability of error of decoding the code words is small enough, and then schedule at thegranularity of the chosen time interval. For a simple class of such systems that includes the traditional single-server queue, where the rate regions are simplexes, a class of policies called the maximum weighted queue-lengthfirst policies were proposed in the context of wireless networks by Tassiulas and Ephremides [36] and switchesby McKeown et al. [24]. Under fairly general conditions, it was shown by Tassiulas and Ephremides [36],Andrews et al. [1], and McKeown et al. [24] that these policies are stabilising, i.e., if the average arrival ratevectors are strictly within the capacity region (in a manner to be defined later on), then the underlying Markovprocesses are positive recurrent (as described in Chen and Yao [10] and Foss and Konstantopoulos [19]). Fora network of nodes with fixed routes for each flow, a generalisation of the maximum weighted queue-lengthfirst policy was presented by Maglaras and Van Mieghem [23], where the flow with the largest weighted sumqueue length is given service over its entire route was again shown to be stabilising. A related class of policiesthat use the age of the head-of-the-line packet instead of queue length have been shown to exhibit optimalperformance in a large deviations sense over a general class of work-conserving stationary scheduling policiesfor a single node by Stolyar and Ramanan [35] and for a network of nodes by Stolyar [34] with fixed routesfor each flow, in all cases, the rate regions for the nodes were simplexes. In Stolyar and Ramanan [35] andStolyar [34], the queueing processes are embedded into the space of right-continuous functions with left-handlimits on the real line endowed with the topology of uniform convergence on compact sets. The authors thenanalyse the behaviour of the (stationary) maximum weighted end-to-end delay. They provide a large deviations(LDs) upper bound for the largest weighted delay first scheduling policy but only a LDs like lower bound usinginner measures over a general class of work-conserving stationary scheduling policies, because the (stationary)maximum weighted delay need not be measurable for all the policies considered. For the largest weighted delayfirst scheduling policy, the lower bound is exactly a LDs lower bound. The work in Stolyar and Ramanan [35]also extends the LDs, analysis to the maximum weighted queue length of a policy that serves the queue withthe largest weighted queue length, thus including serve the longest queue as a special case. In the context ofLDs analysis of scheduling policies, recent work by Venkataramanan and Lin [39] shows that for a class ofscheduling policies, a LDs principle (LDP) can be proved for a specific one-dimensional functional of the queuelengths; namely the Lyapunov function associated with each scheduling policy.

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Subramanian: Large Deviations of Max-Weight Scheduling Policies882 Mathematics of Operations Research 35(4), pp. 881–910, © 2010 INFORMS

The work in this paper is along the lines of the buffer overflow problem described in the work by Bertsimaset al. [3], we consider a specific (parametric) policy and analyse its performance. Instead of just consideringsimplex rate regions and two users as in Bertsimas et al. [3], we analyse a larger class of compact, convex, andcoordinate convex rate-regions for an arbitrary but finite number of users. For these rate regions, the maximumweighted queue-length first policies can be generalised to choosing an operating point that maximises (over therate region) the weighted sum of rates. In keeping with the original policy, we term these policies as max-weightpolicies. We prove an LDP result in the Skorohod space (as described in Billingsley [4], Ethier and Kurtz [17],and Jacod and Shiryaev [21]) of functions that are right continuous with left-hand limits. Our method of prooffollows the steps laid out in Puhalskii [27, 29] and Puhalskii and Vladimirov [30]. With the LDP result available,we can then analyse the tail behaviour of the work loads under different buffering scenarios by an applicationof the contraction principle (see Dembo and Zeitouni [13]). Other related work is by Shwartz and Weiss [33,§§15.6–15.9, pp. 443–454] where the authors prove a LDP (see Dembo and Zeitouni [13]) for a continuous-time problem where there are two queues and the server serves the longest queue (at a fixed rate). Anotherrelated work in the continuous-time context is in Dupuis et al. [16] where the authors extend the result fromShwartz and Weiss [33] to the case of many users for a (general) simplex rate region where the weighted longestqueue is served. They use the structural properties of max-weight policies to show that stability conditionsare automatically implied in the formulation of a general large deviations upper bound; one of our results,Theorem 3.1, can be viewed in the same vein.The paper is organised as follows. In §2, we briefly describe the theory in Puhalskii [27, 29], and Puhalskii

and Vladimirov [30] that is needed to prove our result. Our main results, the LDP result and applications of it,are presented in §3. The analysis proceeds by proving (in §4) certain properties of a deterministic problem thatemerges from the limiting procedure used to prove the LDP result. In §5, we apply the results to three two-userrate regions: (i) an elliptical rate region, (ii) a Gaussian BC, and (iii) a symmetrical MAC. We conclude in §6.

2. Background material. Here, we attempt to collect together, in brief, the mathematical background mate-rial necessary to understand and prove our result; because our paraphrasing of the material will necessarily berestrictive, for a cogent, detailed, and more general explanation, the reader is referred to Puhalskii [27] (andDembo and Zeitouni [13, Chapter 4]), to Puhalskii [29], and Puhalskii and Vladimirov [30] for other workedexamples of the method of proof, and to other references in this section. For the sake of consistency, as muchas possible, we will use notation similar to Puhalskii [27, 29] and Puhalskii and Vladimirov [30].

Let ! be a metric space with metric !E"·# ·$. A function % from ""!$ (the power set of !) to &0#1' is calledan idempotent probability (see Puhalskii [27, Definition 1.1.1, pp. 5–6]) if %""$ = 0, %"E$ = sup##E%"(#)$,E $ !, and %"!$= 1, and the pair "!#%$ is called an idempotent probability space. For ease of notation, wewill denote %"#$=%"(#)$ for # #!. A property $"#$# # #! about the elements of ! is defined to hold %-a.e.if %"(#* $"#$ does not hold)$ = 0. A function f from a set ! equipped with idempotent probability % to aset !% is called an idempotent variable. The idempotent distribution of an idempotent variable f is defined asthe set function %"f &1"E %$$=%"f # E %$#E % # !%. Let % be a collection of subsets of ! that contains the nullset ". Then, % is termed an % -idempotent probability measure (see Puhalskii [27, Definition 1.1.1, pp. 5–6])if %"infn Fn$ = infn%"Fn$ for every decreasing sequence of elements of % . From now onward, unless statedotherwise, we will take %C to be the set of all closed sets of !. Define an idempotent probability measure % to betight if for every +> 0, there exists a compact , $! such that %"!\,$' +. A tight %C-idempotent probabilitymeasure is called a deviability. By using Puhalskii [27, Lemma 1.7.4, pp. 51–52], an alternate characterisationof a deviability is a function % such that the sets (# #!* %"#$( -) are compact for all - # "0#1'. One definesanother function I from ! to &0#+)' that is deemed an action functional (or good rate function) if the sets(# #!* I"#$' x) are compact for x #*+ and inf ##! I"#$= 0 (termed lower compact). It is immediate that % isa deviability if and only if I"#$=& log%"#$ is an action functional. If f a mapping from ! to another metricspace !% is continuous on the sets (# #!* %"#$( -) for - # "0#1', then %"f &1" · $$ is a deviability on !%. Wedefine f to be a Luzin idempotent variable if %"f &1" · $$ is a deviability on !%.Let (&n#n # ') be a sequence of probability measures on ! endowed with the Borel .-algebra, and

let % be a deviability on !. Let (mn#n # ') be a sequence with mn + +) as n + +). The sequence(&n#n # ') LD converges at rate mn to % as n++) if the inequalities lim supn++) &n"F $

1/mn '%"F $ andlim infn++) &n"G$

1/mn ( %"G$ hold for all closed sets F and open sets G, respectively. Note that this is anequivalent means of describing a LDP for scale mn with good rate function I"e$ = & log%"e$, e # !, giventhe close association of deviabilities and action functionals. Equivalent definitions that make an association withconvergence of measures in the traditional sense can be found in Puhalskii [27, Theorem 3.1.3, pp. 254–255].As with the traditional weak convergence of measures, the LD convergence result is shown in two steps: (i) by

Subramanian: Large Deviations of Max-Weight Scheduling PoliciesMathematics of Operations Research 35(4), pp. 881–910, © 2010 INFORMS 883

claiming the existence of limit points by proving relative (sequential) compactness of the set of measures usingsome notion of tightness and (ii) by demonstrating uniqueness of the limit point. A deviability % is said tobe an LD limit point of the sequence (&n# n # ') for rate mn if each subsequence (&nt

# t # ') contains afurther subsequence (&ntu

#u #') that LD converges to % at rate mntuas u++). The notion of tightness from

weak convergence of measures theory translates to the notion of exponential tightness that holds as follows:the sequence (&n#n # ') is exponentially tight on order mn if for arbitrary + > 0 there exists a compact set, $! such that lim supn++) &n"!\,$1/mn < +. From Puhalskii [27, Theorem 3.1.19, pp. 262–263], exponentialtightness implies LD relative compactness of a sequence of measures, and therefore existence of limit points;furthermore, the limit points are all deviabilities. The LD convergence of probability measures can also bestated as the LD convergence in distribution of the associated random variables. A sequence of random variables((n#n # ') defined on probability spaces "/n#%n#&n$, respectively, and assuming values in ! LD convergesat rate mn as n++) to a Luzin idempotent variable ( defined on idempotent probability space "0 #%$ andassuming values in ! if the sequence of probability laws of (n LD converges to the idempotent distributionof ( at rate mn. If the convergence of probability measures is to a deviability, then we get LD convergencein the canonical setting. The role of the continuous-mapping principle in preserving convergence is played bythe contraction principle (see Dembo and Zeitouni [13, Theorem 4.2.1, pp. 126–127] and Puhalskii [27, Theo-rem 3.1.14 and Corollary 3.1.15, pp. 261–262]), whereby if a sequence of random variables (n LD converge indistribution to (, and f is a (%-a.e.) continuous function from ! to another metric space !%, then the sequencef "(n$ LD converges in distribution to f "($ in !%.In many cases of interest, the LD limit points belong to a subset !0 of !. Equip !0 with the relative topology.

Then, we say that an idempotent probability is supported by !0 if %"!\!0$= 0. The sequence (&n#n # ') istermed !0-exponentially tight if it is exponentially tight and every LD accumulation point % is supported by !0.Important here from the point of view of the contraction principle is the definition of !0-closed and !0-opensets as well as !0-continuous functions. From Puhalskii [27, Definition 1.9.9, p. 68], we have the following1

(i) a set F , ! is !0-closed if it contains all its accumulation points that are in !0, i.e., cl"F $-!0 , F ; and(ii) a set G,! is !0-open if every point of G-!0 is an interior point of G, i.e., G-!0 , int"G$. Additionally,from Puhalskii [27, Definition 1.9.11, p. 69], we deem a function f * ! .+ * to be !0-continuous if h&1"G$ is!0-open for each open G, *, and alternatively, !0-continuity of f is equivalent to limn+) f ")n$= f ")$ forevery sequence ()n))n=1 ,! such that limn+) )n = ) #!0. From Puhalskii [27, Corollary 3.1.9, pp. 257–258] LDconvergence of sequence (&n#n #') to % that is supported by !0 only needs to be checked for all !0-open and!0-closed Borel measurable subsets of !. Then, the contraction principle (see Garcia [20] and Puhalskii [27,Corollary 3.1.22, p. 264]) applies to Borel-measurable but !0-continuous functions.

For the results of this paper, the space ! will be *"+$ *=*"&0#1'1+$ the space of +-valued, right contin-uous with left-hand limits functions x = "x"t$# t # &0#1'$, where + is a complete separable metric space. Ourresults carry-over if the setting was, instead, *"&0#T '1+$ for fixed T with 0< T <+). Equipping *"+$ withthe Skorohod J1-topology (see Billingsley [4], Ethier and Kurtz [17], and Jacod and Shiryaev [21]) and metris-ing it with the Skorohod-Prohorov-Lindvall metric (see Billingsley [4], Ethier and Kurtz [17], and Jacod andShiryaev [21]), we get a complete separable metric space. Following Billingsley [4, Chapter 3, §12] and Ethierand Kurtz [17, §§3.5–3.6], let , be the collection of (strictly) increasing functions f = (f "t$ # &0#1'# t # &0#1')mapping &0#1' onto &0#1'; in particular, f "0$ = 0, limt+1 f "t$ = 1, and f is continuous. Let /, be the set ofLipschitz-continuous functions f # , such that

-"f $ *= sup1(t>s(0

!!!!log

f "t$& f "s$

t& s

!!!!<+)2 (1)

For two functions x1#x2 #*"+$, the Skorohod-Prohorov-Lindvall metric is given by

!J1"x1#x2$ *= inff# /,

"

max#

-"f $# supt#&0#1'

!X"x1"t$#x2"f "t$$$$%

#

where !X"·# ·$ is the metric on +.The (closed) subset !0 of *"+$ that we will be dealing with in this paper is the set of all continuous functions

-"+$ *=-"&0#1'1+$ with the induced topology, which is the uniform topology with metric !C"x1#x2$ for twofunctions x1#x2 #-"+$ given by

!C"x1#x2$ *= supt#&0#1'

!X"x1"t$#x2"t$$2 (2)

1 From Puhalskii [27, Remark 1.9.10, p. 68], we note that both the interior and closure operation are taken in !.


For x #+ define -x"+$ *= (a #-"+$* a"0$= x), which is a closed subset of -"+$. We could prove our resultsfor *"+$ with the uniform topology. However, we prefer using the Skorohod J1-topology since *"+$ is then acomplete separable metric space where by using Puhalskii [27, Theorem 3.1.28, p. 268], exponential tightnessis equivalent to LD relative (sequential) compactness.Since -"+$ is a closed subset of *"+$ using Puhalskii [27, Corollaries 1.7.12 on p. 54 and 1.8.7 on p. 62],

we do not distinguish between deviabilities on -"+$ and deviabilities on *"+$ that are supported by -"+$.From Puhalskii [27, Remark 3.2.4 and Theorem 3.2.3, p. 278] (and Ethier and Kurtz [17, §§3.5–3.6]), we notethat a sequence of processes XN with trajectories in *"+$ is -"+$-exponentially tight on order N if and onlyif the two statements below hold; namely,

(i) (Exponential tightness of random variables) for every t # &0#1'#XN "t$ is exponentially tight, i.e.,

inf,#!

lim supN++)

&"XN "t$ #+\!$1/N = 0# (3)

where ! is the set of compact subsets of +; and(ii) (Continuous-limit points) for every T # "0#1'# +> 0 the following holds:

lim3+0

lim supN++)

&#

sups# t#&0#T '* !s&t!'3

!X"XN "t$#XN "s$$> +

$1/N

= 02 (4)

The above condition can also be stated as

lim3+0

lim supN++)

supt#&0#T '

&#

sups#&0#T '* !s&t!'3

!X"XN "t$#XN "s$$> +

$1/N

= 0# (5)

which is many times easier to verify.Instead, if the XN have trajectories in *"*K$, then by Puhalskii [27, Theorem 3.2.3, p. 278], -"*K$-

exponential tightness on order N holds if and only if

limL++)

lim supN++)

P "0XN "0$0>L$1/N = 0 and (6)

lim3+0

lim supN++)

P

#

sups# t#&0#T '* !s&t!'3

0XN "s$&XN "t$0> +

$1/N

= 0# T # "0#1'# +> 02 (7)

Similar to condition (4), the condition in (7) can also be stated as

lim3+0

lim supN++)

supt#&0#T '

P

#

sups#&0#T '*!s&t!'3

0XN "s$&XN "t$0> +

$1/N

= 0# T # "0#1'# +> 02 (8)

Alternate characterisations of -"+$-exponential tightness can be found in Feng and Kurtz [18, §4.4, pp. 65–67]and Puhalskii [28, Theorem 2.5, p. 15].By using the well-known characterisation that a sequence (xn))n=1 , *"+$ converging to x # -"+$ in the

Skorohod J1-topology also converges in the local uniform topology on *"+$ (c.f. Ethier and Kurtz [17,Lemma 10.1, Theorem 10.2, p. 148]), -"+$-open, and -"+$-closed sets as well as -"+$-continuous func-tions are also characterised using the local uniform topology on *"+$. Thus, we can then use Puhalskii [27,Corollary 3.1.9, pp. 257–258 and Corollary 3.1.22, p. 264] to characterise large deviations inequalities for theseobjects. We will use this property many times over.For some results, we will use + = *K

+, K # ' and for others, we will use + = *K+ 1."/"L$$, K#L # ',

where ."/"L$$ is the set of all finite nonnegative Borel measures on /"L$ a convex compact set in *L+. The set

of all finite nonnegative Borel measures ."!$ on a complete separable metric space !, is a complete separablemetric space with the Lévy-Prohorov metric and the topology of weak convergence. The Lévy-Prohorov metric(in the symmetric form, see Billingsley [4], Ethier and Kurtz [17], and Dembo and Zeitouni [13, Theorem D.8,pp. 355–356]) !P "41# 42$ for two measures 41# 42 #."!$ is given by the following:

!P "41# 42$ *= inf(+> 0* 41"C$' 42"C+$+ + and 42"C$' 41"C+$+ + 2C #0"!$)# (9)

where 0"!$ is the Borel .-algebra on !, and C+ *= (# #!* inf #1#C !E"## #1$< +). In practice, it is sufficient toconsider only %C instead of 0"!$ in the definition in (9). For t ( 0, define .t"!$ *= (4 #."!$* 4"!$' t) to be


the set of (nonnegative) finite measures assigning a measure at most t to !, and .t"!$ *= (4 #."!$* 4"!$= t)to be the set of (non-negative) finite measures assigning a measure exactly t to !. Then .t"!$ and .t"!$ arecompact if and only if ! is compact (see Doob [14, §VIII.5, p. 132] and Dembo and Zeitouni [13, Theorem D.8,pp. 355–356]). The topology of weak convergence also results by using the Kantorovich-Wasserstein metric(see Dembo and Zajic [12, Lemma A.1, p. 222], Dembo and Zeitouni [13, Theorem D.8, pp. 355–356], andDudley [15]). Let 1b"!$ denote the set of bounded continuous functions on ! that take values in *. Then, theKantorovich-Wasserstein metric !KL"41# 42$ for two measures 41# 42 #."!$ is constructed using bounded andLipschitz-continuous functions on !, and is given by

!KL"41# 42$ *= sup"!!!!

&

!f d41 &

&

!f d42

!!!!* f #1b"!$#0f 0) + 0f 0L ' 1

%

# (10)

where 0f 0) = sup##! f "#$ and 0f 0L *= sup(#1##2#!* #1 3=#2)"!f "#1$& f "#2$!$/"!E"#1# #2$$ is the Lipschitz constant

of f . We will use the Kantorovich-Wasserstein metric instead of the Lévy-Prohorov metric to prove (4) in ouranalysis. In our analysis, ! will be a compact convex subset of *L

+. In this setting, weak convergence of finiteBorel measures on ! is the same as weak4 convergence in a Banach space since the space of finite Borelmeasures on ! is the dual space of 1b"!$.Following Dembo and Zajic [12] for nondecreasing functions 5 in *"."!$$, i.e., functions such that

5"t$&5"u$ #."!$ for all t ( u with t#u # &0#1', we say that the right weak derivative exists at t # &0#1$ if"5"t+ +$&5"t$$/+ #."!$ weakly converges as ++ 0. Similarly, the left weak derivative exists at t # "0#1'if "5"t$&5"t& +$$/+ #."!$ weakly converges as ++ 0. For t # "0#1$ if both the right and left weak deriva-tive exist, then we deem 5 to be weakly differentiable at t with the limit denoted as 5"t$. For absolutelycontinuous 5, we have 5"t$&5"u$=

' t

u 5"6$d6 , where the integral is interpreted setwise, i.e., for C #0"!$,we have "5"t$&5"u$$"C$=

' t

u 5"6$"C$d6 .We make use of the terminology “strong” and “weak” in defining solutions (see Brézis [7, p. 64]) to differential

inclusions. For absolutely continuous function a # -"*L$ and set-valued function H" · $ such that H"x$ $ *L

for x #*L with domain D"H$ *= (x #*L* H"x$ 3="), we say that w #-"*L$ is a strong solution to differentialinclusion

w"t$ # a"t$+H"w"t$$ t # &0#1'# (11)

if w is absolutely continuous with w"t$ #Dom"H$ 2 t # &0#1' and satisfies

w"t$ # a"t$+H"w"t$$ for a2e2t # "0#1$2

Note that the absolute continuity of a automatically posits the (a.e.) existence of the integrable function a #21"&0#1'1*L$. One defines a function w # -"*L$ to be a weak solution to differential inclusion (11) forinput a #-"*L$ if there exist a sequence of absolutely continuous functions (aN #-"*L$)+)

N=1 and a sequence(wN #-"*L$)+)

N=1 such that each wN is a strong solution of the differential inclusion

wN "t$ # aN "t$+H"wN "t$$ t # &0#1'# (12)

and aN 5N++) a in 21"&0#1'1*L$ and wN 5N++) w uniformly in -"*L$.Finally, if a sequence of random variables (XN #N # ') defined on a complete probability space "/#% #&$

and assuming values in *K converges in probability to x # *K such that limN++) &"0XN & x0 > +$1/N = 0for all + > 0, then we deem the sequence as converging superexponentially in probability at rate N and write

XN

&1/N

&&+ x.

3. Model and a fluid limit. We consider a discrete-time queueing system with one server that can pickoperating points from a set /"K$ that is a compact, with nonempty interior int"/"K$$, and convex subset of*K

+ that includes the origin. We also assume that /"K$ is coordinate convex, i.e., if r #/"K$, then r #/"K$for all 0' r' r, where the inequalities hold coordinatewise.2 Since /"K$ is compact, there exists a rmax <+)such that for every r # /"K$, we have rk ' rmax for all k = 1#2# 7 7 7 #K. For user k, we assume an arrivalprocess of work brought into the system given by a sequence (Ak

m)+)m=0, where Ak

m #*+ is the work brought inat time m into the queue of user k. For &1'm1 'm2, integers define Ak"m1#m2' *=

(m2m=m1+1A

km, which is the

2 Henceforth unless specified otherwise, we assume that all vector inequalities hold coordinatewise.


total amount of work to arrive for user k after time slot m1 and until time slot m2. If m1 (m2, then we interpretAk"m1#m2' to be 0. Let the unfinished work in user k’s queue at time m( 0 be Wk

m. Then, work at time m+ 1in the kth user’s queue is given by Lindley’s recursion

Wkm+1 =max"Ak

m#Wkm +Ak

m & rkm$ = max"0#Wkm & rkm$+Ak

m

*= "Wkm & rkm$+ +Ak"m& 1#m'# (13)

where rm # /"K$ is the operating point chosen at time m. Note that we choose to wait for at least one slotbefore serving newly arrived work. Allowing the server to work on newly arrived work immediately does notchange the results.Our scheduling policy will be to choose a rate vector that maximises a (dynamic) weighted sum of rates over

this rate region, i.e.,2m( 1 rm # argmax

r#/"K$6"m# r7

with components rkm, where "m is given by

8km = 9kW k

m s.t.K)

k=1

9k = 1# 9k > 0 2k#

and where 6·# ·7 is the standard inner product in *K . Note that "m is the Hadamard/Schur product of # and Wm

and we will write this as "m = # 8Wm. Define the following (set-valued) functions for x #*K+:

H"x$ *= argmaxr#/"K$

6x# r7# (14)

/H"x$ *=H"# 8 x$2 (15)

We will fix on a specific solution if there is more than one maximiser. For a closed-convex set 3 $*K , definethe projection of element x #*K to be the unique element x4 #3 that solves

miny#3

0x& y02#

where 0·0 is the Euclidean norm given by*

6x#x7 for x #*K+. We define the function from x to x4 for a given

3 to be Proj3 "x$. For every x #*K+ it is clear that /H"x$ is a closed and convex set. Then, the specific operating

point that we choose at time m is given by

rm = Proj /H"Wm$"0$2

Based on the above definition, we call operating point rm the minimum norm solution.By using the operating point rm at time m, we get

Wkm+1 = "Wk

m & rkm$+ +Ak"m& 1#m'

= Wkm &min"Wk

m# rkm$+Ak"m& 1#m'

= Wk0 &

m)

l=0

Skl +Ak"&1#m'

= Wk0 & Sk"&1#m'+Ak"&1#m'# (16)

where for m ( 1, we define Skm *= min"Wk

m# rkm$ ' rmax, which is the amount of work from the queue of user

k served at time m. Coordinate convexity ensures that for every r # /"K$, every point min"x# r$ (with theminimum taken along every coordinate) belongs to /"K$ as x is allowed to vary in *K

+. Before proceeding, itis necessary to point out that our description of the max-weight policy is such that it is neither work conservingnor rate maximising. This can be best appreciated for a simplex rate region: if the chosen queue does not haveenough work, then the excess work is not applied to another set of queues that may have some work, i.e., theserver does not try to empty the system as much as possible.To aid in the proof of the result, we will define a few more quantities. Denote by Rk"m1#m2'=

(m2l=m1+1 r

kl ,

the total amount of service given to user k from m1 + 1 until m2. As discussed earlier, not all of this service


is used because the queues might not contain enough work to be served. Therefore we define Y k"m1#m2' *=(m2

l=m1+1"rkl &Wk

l $+ to be total amount of service not used by user k from m1 + 1 until m2. Note that this isanalogous to the server idle time (see Chen and Yao [10]) in a continuous-time queueing system. It is obviousfrom the definitions that Sk"m1#m2' = Rk"m1#m2'& Y k"m1#m2', where all three terms are nonnegative. Theform above is not very conducive to analysis. Therefore we modify it for m( 1 as follows:

Y k"m&1#m' = "rkm&Wkm$+=max"0#rkm&Wk

m$

= max"0#min"rkm+rkm&1&Akm&1&Wk

m&1#rkm&Ak

m&1$$

= max+

0#min+

Rk"&1#m'&Ak"&1#m&1'&Wk0 # min

1'i'm"Rk"i&1#m'&Ak"i&2#m&1'$

,,

= max+

0#Rk"&1#m$&Ak"&1#m&1'&Wk0 &max

+

0#max1'i'm

"Rk"&1#i&1'&Ak"&1#i&2'&Wk0 $,,

= max+

0# max1'i'm+1

"Rk"&1#i&1'&Ak"&1#i&2'&Wk0 $,

&max+

0#max1'i'm

"Rk"&1#i&1'&Ak"&1#i&2'&Wk0 $,

#

where we use the relationship in (13) many times over to unravel the recursive definition. Therefore we have

Y k"&1#m' = max+

0# max0'i'm

"Rk"&1# i'&Ak"&1# i& 1'&Wk0 $,

= max+

Wk0 # max

0'i'm"Rk"&1# i'&Ak"&1# i& 1'$

,

&Wk0 2 (17)

By using this, we can also write the following:

Wkm =

-

.

.

.

.

.

.

.

./

.

.

.

.

.

.

.

.0

Wk0 if m= 01

max1

Wk0 +Ak"&1#m& 1'&Rk"&1#m& 1'#

Ak"&1#m& 1'&Rk"&1#m& 1'

& min0'i'm&1

"Ak"&1# i& 1'&Rk"&1# i'$2

otherwise.

(18)

Assume that we are given a sequence (WN0 )N#' taking values in *K

+ that accounts for the vector of initialwork in the system. We then embed the sequences (Ak"&1#m'), (Rk"&1#m'), (Y k"&1#m'), (Sk"&1#m'), and(Wk

m) into functions in *"*K+$ by defining (scaling both space and time) for t # &0#1' the following: Ak#N "t$ *=

Ak"&1# 9Nt:'/N , Rk#N "t$ *=Rk"&1# 9Nt:'/N , Y k#N "t$ *= Y k"&1# 9Nt:'/N , Sk#N "t$ *= Sk#N "&1# 9Nt:'/N , andWk#N "t$ *=Wk#N

9Nt:/N for N # ', where 9t: is the largest integer less than or equal to t. The index N in Rk#N ,Y k#N , Sk#N , and Wk#N takes into account the different initial work load vectors given by WN

0 . Denote the vectorquantities by AN "t$, RN "t$, YN "t$, SN "t$, and WN "t$, respectively. Also, define the processes AN *= "AN "t$# t #&0#1'$, RN *= "RN "t$# t # &0#1'$, YN *= "YN "t$# t # &0#1'$, SN *= "SN "t$# t # &0#1'$, and WN *= "WN "t$# t #&0#1'$. The work load arrivals sequence (Am)m#' and the initial work load vector sequence (WN

0 )N#' areassumed to be defined on a common complete probability space "/#% #&$.Construct the random empirical measure :"&1#m'" · $ *=(m

l=0 $rl" · $, where $x" · $ is the Dirac measureconcentrated at x, i.e., for all C #0"*K

+$, we have

$x"C$=

-

/

0

1 if x #C1

0 otherwise.

Define the scaled empirical measure process : N "t$ *= : N "&1# 9Nt:'/N for t # &0#1'. Again, the index Naccounts for the different initial work load vector. Let ."/"K$$ be the set of finite (nonnegative) Borel measureson /"K$; when endowed with the topology of weak convergence of measures generated by the Kantorovich-Wasserstein metric, ."/"K$$ is a complete separable metric space. Then, the processes : N take values in*"."/"K$$$ again with the Skorohod J1 topology (see Ethier and Kurtz [17]). In fact, for every t # &0#1',we have : N "t$ #.t+1"/"K$$= (4 #."/"K$$* 4"/"K$$' t+ 1), where .t+1"/"K$$ is a compact subset of


."/"K$$. For a Borel measurable function f from /"K$ to *, denote the integral (if it exists) with respect toa measure 4 #."/"K$$ by

'

/"K$ f d4; this is a random variable taking values in * that we denote as 64# f 7.For our convergence proofs, we will be considering processes "AN #RN #YN #SN #: N #WN $ taking values in

the Skorohod space *"*K+ 1*K

+ 1*K+ 1*K

+ 1."/"K$$1*K+$; we denote the complete separable metric space

*K+ 1*K

+ 1*K+ 1*K

+ 1."/"K$$1*K+ by +. Denote the Euclidean metric on *K

+ by !E and the Kantorovich-Wasserstein metric on ."/"K$$ by !KL. Then, for two elements "a1#%1#&1# s1#51#w1$# "a2#%2#&2# s2#52#w2$#+, the distance between the two elements is given by the following metric:

!X""a1#%1#&1# s1#51#w1$# "a2#%2#&2# s2#52#w2$$

*=max"!E"a1#a2$#!E"%1#%2$#!E"&1#&2$#!E"s1# s2$#!KL"51#52$#!E"w1#w2$$2

The topology that results from this metric on + is the product topology.To prove the required LDs result, we will follow the programme outlined in Puhalskii [27, 29] and Puhalskii

and Vladimirov [30]. Loosely speaking, we first show in Theorem 3.1 that the sequence of measures on ametric space ! is LDs relatively compact using exponential tightness. In proving relatively compactness, wewill prove that all the limit points are supported on !0 a closed subset of !. The limit points are determinedby weak solutions to idempotent equations the properties of which are characterised by taking LD limits ofthe stochastic equations that determine the behaviour of the original system. Compactness of the rate regionsplays an important role in proving exponential tightness and characterising the LDs limit points. By using allthe characterised properties of the idempotent equations, we show the existence of unique weak solutions to theidempotent equations in Theorem 4.1 leading to an LDP result in Theorem 3.2. In effect, this step can also bereferred to as establishing uniqueness in idempotent distribution (in analogy to weak convergence of measures)and is accomplished by proving uniqueness of trajectories. The convexity of the rate regions and the natureof the scheduling policy (maximising a linear functional over a convex set) play an important part in not onlyproving the existence and uniqueness of solutions to the idempotent equations but also in providing a simpleexpression for the solution; coordinate convexity is then key in obtaining a further simplified expression forthe solution. The LDP result then follows directly from the uniqueness of trajectories proved in Theorem 4.1by using exactly the same argument as in the proof of Lemma 3.1 in Puhalskii and Vladimirov [30]; interestedreaders are also referred to the proof of Theorem 3.1 in Puhalskii [29].Assume we are given a function ;A* *K

+ + &0#+)' that attains zero at some ' #*K+ such that '<(4 with

(4 #/"K$, where the inequality holds coordinatewise. This is used to define function IA* *"*K+$+*+ by

IA"a$=& 1

0;A"a"t$$dt

if the function a = "a"t$# t # &0#1'$ # *"*K+$ is absolutely continuous such that a"0$ = 0, and where a #

21"&0#1'1*K$ is the (Lebesgue) a.e. derivative of a taking values in *K+; I

A"a$ is defined to be equal to +) ifthe function a #*"*K

+$ does not have the above properties. For x #*K+, we define by 4-x the set of absolutely

continuous functions in *"*K+$ such that for a #4-x, we have a"0$= x, and where a #21"&0#1'1*K$ takes

values in *K+. It is assumed that IA" · $ is an action functional on *"*K

+$, which, in turn, implies that ;A" · $ isan action functional on *K

+. The function ;A" · $ being convex and lower compact is sufficient (see Dembo andZajic [12, Lemma 8, p. 203]) for IA" · $ to be an action functional.We assume that the arrival process of work loads is such that (AN #N # ') satisfies an LDP at rate N as

N + +) in the space *"*K+$ with action functional IA" · $. If the arrival process of work loads is such that

(Am)m(0 is a sequence of i.i.d. *K+ valued random variables with 5"e6x#A07$<+) for all x # *K , where 6·# ·7

is the inner product on *K , then by the Borovkov-Mogul%skiı theorem (see Puhalskii [28, Theorem 2.15, p. 25],Borovkov [5], Mogul%skiı [25, 26]), we have (AN #N #') satisfying an LDP at rate N as N ++) in the space*"*K

+$ with action functional IA" · $ with

;A"x$= supy#*K

1

6y#x7& log5"e6y#A07$2

2 x #*K+2 (19)

See Dembo and Zajic [12, Theorem 5, p. 216] for general mixing conditions on a stationary sequence (Am), sothat (AN #N #') satisfies an LDP at rate N as N ++) in the space *"*K

+$ with action functional IA" · $.Under the above assumptions about the arrival processes, we now prove the -"+$-exponential tightness of

the sequence "AN #RN #YN #SN #: N #WN $.


Theorem 3.1. Assume that the arrival process is such that the sequence (AN #N # ') satisfies an LDP at

rate N as N + +) in the space *"*K+$ with action functional IA" · $. Also, assume3 that WN

0 /N&1/N

&&+ w"0$.Then, the sequence "AN #RN #YN #SN #: N #WN $ is -"+$-exponentially tight on order N in *"+$.If an idempotent process "a#%#&# s#5#w$ defined on an indempotent probability space "0 #%$ and having

trajectories in -"+$ is a limit point of "AN #RN #YN #SN #: N #WN $ for LD convergence in distribution at rateN , then the following properties hold for any limit point "a#%#&# s#5#w$:

(i) the function a is (%-a.e.) componentwise nonnegative and nondecreasing, and absolutely continuouswith a"0$= 0;

(ii) the function % is (%-a.e.) componentwise nonnegative and nondecreasing, and Lipschitz continuous with%"0$= 0;(iii) the function & is (%-a.e.) componentwise nonnegative and nondecreasing, and Lipschitz continuous with

&"0$= 0;(iv) the function s is (%-a.e.) componentwise nonnegative and nondecreasing, and Lipschitz continuous with

s"0$= 0;(v) the measure-valued function 5 is (%-a.e.) such that 5"t$ # .t"/"K$$ for all t # &0#1', absolutely

continuous with respect to the total variation norm (see Yosida [42, pp. 35–38, 118–119]) such that 5"t$&5"u$ # .t&u"/"K$$ for all 1 ( t ( u ( 0 with 5"0$"/"K$$ = 0, and possesses a weak derivative 5"t$ foralmost every t # &0#1'. The following inequality holds (%-a.e.) for all t#u # &0#1' with t ( u:

sk"t$& sk"u$' 65"t$&5"u$# ek7# (20)

where ek* /"K$+*+ is the kth-coordinate projection operator such that ek"r$= rk;(vi) the function w is (%-a.e.) componentwise nonnegative and absolutely continuous with w"0$ given such

that

w"t$= a"t$& s"t$ for (Lebesgue) a.e. t # &0#1'1 (21)

(vii) if wk"t$> 0 for t # &t1# t2' for t1# t2 # &0#1', then it follows that

sk"t$& sk"u$= 65"t$&5"u$# ek7 2 t ( u with t# u # &t1# t2'1 (22)

(viii) (%-a.e.) for (Lebesgue) almost every t # &0#1',

5"t$"/"K$\ /H"w"t$$$= 01 and (23)

(ix) (%-a.e.) for every k= (1#2# 7 7 7 #K), we have

<k"t$ = max#

0# sup0's't

"-k"s$& ak"s$$&wk"0$$

= max#

wk"0$# sup0's't

"-k"s$& ak"s$$

$

&wk"0$# (24)

and for (Lebesgue) almost every t # &0#1', it follows that

<k"t$=

-

/

0

"-k"t$& ak"t$$+ if wk"t$= 01

0 otherwise.(25)

Thus %-a.e. every limit point is an absolutely continuous solution (strong solution) of the following differentialinclusion for (Lebesgue) almost all t # &0#1':

w"t$& &"t$ # a"t$& /H"w"t$$ (26)

with w"0$ the initial condition such that &"t$( 0 and <k"t$wk"t$= 0 for all k= 1#2# 7 7 7 #K.

Proof. Refer to the appendix. !

3 This can be relaxed with a suitable LDP assumption—LDP with good rate function, for example.


We will show the existence of solutions of (26), and show uniqueness of the solution and other properties asa consequence of Theorem 4.1 in §4.Remarks. (i) The deterministic problem (26) is an instantiation of the Skorohod problem (see Chen and

Yao [10, Chapter 7]), and in that setting, one would term w the reflected process and & the regulator, respectively.In the language of Atar et al. [2], we are seeking solutions to a constrained discontinuous media problem wherethe domain 6 is *K

+, the constraint vector field 7 are the normal directions of constraint on the boundary of 6and the velocity vector field is determined by the maximal monotone map /H" · $ that has domain 6.

(ii) If we make the additional assumption that ;A"4$> 0 for all ) # *K+\(') and (Am)m(0 being stationary,

then we can construct a (regular) fluid limit (a functional strong law of large numbers result) that will obey arelationship similar to (26) given by absolutely continuous solutions to

w"t$& &"t$ #'& /H"w"t$$ (27)

with w"0$ the initial condition such that(K

k=1wk"0$ = 1. Now, one can easily argue for stability (existence

of stationary regime and stationary distribution, see Andrews et al. [1], Chen and Yao [10], and Foss andKonstantopoulos [19]) by using a quadratic Lyapunov function V "t$ *= 0;# 8w"t$02/2. The details are verysimilar to those in Andrews et al. [1] and are skipped for brevity. However, it is worth noting that one needs touse the property that for every k # (1#2# 7 7 7 #K), we have wk"t$<k"t$= 0. Uniform integrability (see Billingsley[4, Equation (3.15), p. 31]) of the sequence (AN "t$) (for some t > 0) would need to be proved for the resultto hold. Under the conditions for the Borovkov-Mogul%skiı theorem; namely, (Am)m(0 an i.i.d. sequence with5"e6x#A07$<+) for all x # *K , we can use Hölder’s inequality to prove the uniform integrability requirementusing a construction described in Botvich and Duffield [6]. The proof is as follows. We have for x > 0,

51

ex"Ak"&1# 9Nt:'/N $2 = 5

#9Nt:3

i=0

e"x/N $Aki

$

'9Nt:3

i=0

51

e"x"1+9Nt:$/N $Aki21/"1+9Nt:$

= 51

e"x"1+9Nt:$/N $Ak02

' 5"ex"1+t$Ak0$<+)2 (28)

Note that this bound only uses stationarity, and hence is applicable under the conditions of Dembo and Zajic[12, Theorem 5, p. 216]. For a real-valued nonnegative random variable X with distribution & using x ' yex&y

for x( y for all y ( 1, we have the following bound:& +)

yx d&"x$' ye&y

& +)

yex d&"x$' ye&y5"eX$2

By using this bound and the bound in (28) developed using Hölder’s inequality, uniform integrability follows.Now, we state our main result.

Theorem 3.2. Assume that the sequence of arrival processes (AN #N # ') satisfy an LDP at rate N as

N ++) in the space *"*K+$ with action functional IA" · $. Also, assume that WN "0$/N

&1/N

&&+w"0$. Then, thesequence WN obeys an LDP for scale N in the Skorohod space *"*K

+$ with action functional IWw"0$" · $ givenin (50).

We defer the proof and the identification of the action functional to §4. We can immediately write down acorollary to Theorem 3.2 that considers many applications of the result.

Corollary 3.1. Under the conditions of Theorem 3.2 with w"0$= 0 for x #*K+, x #*+, and t # &0#1', we

have

lim supN++)

log"&"W"9Nt:$(Nx$$N

'& infy#*K

+ *y(xJ"y# t$# (29)

lim infN++)

log"&"W"9Nt:$>Nx$$N

(& infy#*K

+ *y>xJ"y# t$# (30)

where

J"x# t$ *= infw#-0"*K

+$*w"t$=xIW0 "w$2 (31)


Furthermore, we also have

lim supN++)

log"&"maxk=1#2# 7 7 7 #K Wk"9Nt:$(Nx$$

N'& min

k=1#2# 7 7 7 #Kinf

y#*K+ *yk(x

J"y# t$# (32)

lim infN++)

log"&"maxk=1#2# 7 7 7 #K Wk"9Nt:$>Nx$$

N(& min

k=1#2# 7 7 7 #Kinf

y#*K+ *yk>x

J"y# t$2 (33)

lim supN++)

log"&"(

k=1#2# 7 7 7 #K Wk"9Nt:$(Nx$$

N'& inf

y#*K+ *(K

k=1 yk(x

J"y# t$# (34)

lim infN++)

log"&"(

k=1#2# 7 7 7 #K Wk"9Nt:$>Nx$$

N(& inf

y#*K+ *(K

k=1 yk>x

J"y# t$2 (35)

lim supN++)

log"&"maxk=1#2# 7 7 7 #K supt#&0#1'Wk"9Nt:$(Nx$$

N'& min

k=1#2# 7 7 7 #Kinf

y#*K+ *yk(x

inft#&0#1'

J"y# t$# (36)

lim infN++)

log"&"maxk=1#2# 7 7 7 #K supt#&0#1'Wk"9Nt:$>Nx$$

N(& min

k=1#2# 7 7 7 #Kinf

y#*K+ *yk>x

inft#&0#1'

J"y# t$2 (37)

lim supN++)

log"&"supt#&0#1'(

k=1#2# 7 7 7 #K Wk"9Nt:$(Nx$$

N'& inf

y#*K+ *(K

k=1 yk(x

inft#&0#1'

J"y# t$# (38)

lim infN++)

log"&"supt#&0#1'(

k=1#2# 7 7 7 #K Wk"9Nt:$>Nx$$

N(& inf

y#*K+ *(K

k=1 yk>x

inft#&0#1'

J"y# t$2 (39)

Proof. Since the coordinate projection map =t* *"+$+ *K+ for t # *+ given by =t""a#%#&# s#5#w$$ =

w"t$ for "a#%#&# s#5#w$ # *"+$ is -"+$-continuous and Borel measurable with the Skorohod J1 topology,the results in (29)–(30), and the expression for the rate function in (31) follow from a generalised version ofthe contraction principle in Puhalskii [27, Corollary 3.1.22, p. 264]. Now, the results in (32)–(33) follow by anapplication of the Principle of the Largest Term (see Dembo and Zeitouni [13, Lemma 1.2.15, p. 7] and Puhal-skii [27, Definition 1.1.1, Lemma 1.1.4, pp. 5–6] or equivalently the same generalised contraction principle fromPuhalskii [27, Corollary 3.1.22, p. 264]). Since the function that maps y #*K

+ to(K

k=1 yk (in *+) is continuous,

another application of the generalised contraction principle from Puhalskii [27, Corollary 3.1.22, p. 264] yields(34)–(35). Similarly, (36)–(39) follow from the -"+$-continuity (and Borel measurability with the Skorohod J1topology) of supt#&0#1'w

kt # 2k # (1# 7 7 7 #K) for "a#%#&# s#5#w$ #*"+$. !

The restriction w"0$ = 0 is only to simplify our further characterisation of J"x# t$. Note that (36)–(37) areuseful in calculating the tail probabilities of the work load when each user has a buffer to itself and (38)–(39)are useful when there is a shared buffer.

3.1. Polytope rate regions. If, in addition, the rate region /"K$ is a polytope, then one can use the methodof types (see Dembo and Zeitouni [13, §2.1.1]) to cast the result of Theorem 3.1 in a simpler setting.Let the extreme points of /"K$ be (rp)(p=1#2# 7 7 7 #P) with r1 equalling the all zero vector in *K ; in other words,

we include the origin in the set of extreme points. Since our policy either chooses the extreme points or choosesa minimum norm solution from the convex hull of a subset of extreme points, we can build up a bigger finite setof operating points (rq)(q=1#2# 7 7 7 #Q) with Q' 1+ 2P&1 such that the mapping from x to the operating points issingle valued. This we do by looking at the minimum norm solution corresponding to the largest subset /P "x$ of(1#2# 7 7 7 #P) such that rp # argmaxr#/"K$6x# r7 for every p # /P for given x #*K . Denote the label of operatingpoint corresponding to point x #*K by q"x$. Again, we assume that r1 equals the all zero vector in *K .

For q # (1#2# 7 7 7 #Q), define >qm as follows:

>qm =

-

/

0

1 if q = q"Wm$

0 otherwise##

then(m

l=0>ql counts the number of times until time m + 1 that operating point q is picked. By using thesequence (>qm), we can rewrite the queueing Equation (16) as follows:

Wkm+1 = Wk

0 & Sk"&1#m'+Ak"&1#m'

= Wk0 &

Q)

q=1

m)

l=0

>qlmin"Wkl # r

kq $+Ak"&1#m'2 (40)


Define for every q = 1#2# 7 7 7 #Q, the number of times in &0# 9t:' that operating point q is chosen to be>q"&1# 9t:' *=(9t:

l=0>ql. For scale N and starting work load vector WN0 , define >

Nq "t$ *= >N

q "&1# 9Nt:'/N (invector notation *N "t$= ">N

1 "t$#>N2 "t$# 7 7 7 #>

NQ "t$$); denote the process by *N *= "*N "t$# t # &0#1'$.

Here, we would define + = *K+ 1 *K

+ 1 *K+ 1 *K

+ 1 *Q+ 1 *K

+. Then, we would show -"+$-exponentialtightness on order N in *"+$ of sequence "AN #RN #YN #SN #*N #WN $. Then, each limit point "a#%#&# s#+#w$in -"+$ would satisfy all properties from Theorem 3.1 %-a.e., with the following being modified to:

(v) the function + is componentwise nonnegative and nondecreasing, and Lipschitz continuous with+"0$= 0. The following inequality holds for all 1' t ( u( 0:

sk"t$& sk"u$'Q)

q=2

"?q"t$&?q"u$$rkq 1 (41)

(vii) if wk"t$> 0 for t # &t1# t2' for t1 > t2 # &0#1', then it follows that:

sk"t$& sk"u$=Q)

q=2

"?q"t$&?q"u$$rkq 2 t ( u with t#u # &t1# t2'1 and (42)

(viii) for any regular point t # &0#1' and for a chosen operating q # (1#2# 7 7 7 #Q) if6# 8w"t$# rq7< max

q=1#2# 7 7 7 #Q6# 8w"t$# rq7# (43)

then d?q"t$/dt = 0.

4. Analysis of fluid limit. Before addressing the main result of this section, we state and prove a fewpreliminary results that will be key for the analysis of the fluid limit. We are interested in the properties ofH"x$= argmaxr#/"K$6x# r7 and /H"x$= argmaxr#/"K$6# 8 x# r7 for x #*K .

Let ( be a Hilbert space. A set-valued map 8 from ( to ""($ (the power set of () with domain Dom"8$is monotone (see Brézis [7, Definition 2.1, p. 20] and Rockafellar and Wets [32, Chapter 12]) if and only if

2)1# )2 #Dom"F $# 29i #8")i$# i= 1#2# 691 & 92#)1 & )27 ( 0# (44)

where 6·# ·7 is the inner product on (. A monotone set-valued map 8 is maximal (see Brézis [7, Definition 2.2,p. 22] and Rockafellar and Wets [32, Chapter 12]) if there is no other monotone set-valued map /8 whosegraph strictly contains the graph of 8 . The reader is referred to Brézis [7], Browder [8] and Rockafellar [31],Rockafellar and Wets [32] for the properties of monotone maps, maximal monotone maps, and their connectionsto convex analysis, functional analysis, and semigroups of nonexpansive maps.

Lemma 4.1. H"x$= argmaxr#/"K$6x# r7 and /H"x$= argmaxr#/"K$6# 8 x# r7 for x #*K are maximal mono-tone maps from *K to /"K$,*K .

Proof. Since /"K$ is a proper, closed-convex set, from Rockafellar [31, Theorem 23.6, Corollary 23.5.1,Corollary 23.5.3, pp. 218–219], the definition of H"x$ characterises all the subgradients4 of a proper lower-semicontinuous and convex function maxr#/"K$6x# r7 from *K to * < (+)). Now, using Brézis [7, Exam-ple 2.3.4, p. 25], Rockafellar [31, Corollary 31.5.2, p. 340], and Rockafellar and Wets [32, Theorem 12.17,p. 542], we can assert that H"x$ is a maximal monotone map. The exact same proof applies to /H"x$ too. !

As described in §3, each policy in the class of max-weight scheduling policies can be associated with a uniquevector # # *K

+ such that # > 0 and(K

k=1 9k = 1. We now demonstrate that the performance of a max-weight

policy with a given # can be quantified by analysing the performance of a max-weight policy with weights# = "1/K#1/K# 7 7 7 #1/K

4 56 7

K times

,

, but with a new rate region that is a scaled version of the original rate region.

Consider the set-valued map /H"x$= argmaxr#/"K$6# 8 x# r7 for all x #*K+. Then, the differential inclusion (26)

that we need to analyse is w"t$& &"t$ # a"t$& /H"w"t$$, where a #4-0. Now, it is clear that

argmaxr#/"K$

6# 8 x# r7= argmaxr#/"K$

89

# 8 x#9

# 8 r:

#

4 Following Rockafellar [31, p. 214] for a convex function F "x$* *K + * with domain 6 $ *K (a convex set), a vector x # *K is asubgradient of F "x$ at x if

F "y$( F "x$+ 6x#y& x7# 2y #6#

where 6·# ·7 is the inner product in *K . We denote the set of subgradients of F "x$ at x by @F "x$.


where9

# is defined by taking square root coordinatewise. Define

/;#"K$ *=

"

r #*K *19

#8 r #/"K$

%

#

which is a scaled version of /"K$, and again convex, coordinate convex, and compact, with

=H;#"x$ *= argmax

r#/;#"K$

6x# r7#

which is equivalent to (r # *K * 1/9

# 8 r # /H"x$), i.e., a scaling of /H"x$. Therefore we can now claim thefollowing equivalence:

w"t$& &"t$ # a"t$& /H"w"t$$= a"t$& 19

#8 =H;

#

+9

# 8w"t$,

;<=

9

# 8 w"t$&9

# 8 &"t$ #9

# 8 a"t$& =H;#

+9

# 8w"t$,

2

Therefore setting /w"t$ *=9

# 8 w"t$, &"t$ =9

# 8 &"t$, and a"t$ *=9

# 8 a"t$ it suffices to analyse thefollowing differential inclusion with

/w"t$& ˙&"t$ # ˙a"t$& argmaxr#/;

#"K$

6 /w"t$# r7= ˙a"t$& =H;#" /w"t$$2 (45)

Note that Lemma 4.1 still applies and both the underlying domain 6=*K+ and the constraint vector field 7 do

not change. Thus, without loss of generality, from now onward we assume that #= "1/K#1/K# 7 7 7 #1/K4 56 7

K times

$ and

analyse the solutions of the following differential inclusion:

2 t # &0#1' w"t$& &"t$ # a"t$&H"w"t$$# (46)

where a #4-0, w"0$ # *K is given, H" · $* *K +""*K$ is a maximal monotone set-valued map with closeddomain Dom"H$ possessing a nonempty interior and &"t$ takes values in& @3"w"t$ !Dom"H$$, where

3"x !Dom"H$$=

-

/

0

0 if x #Dom"H$

+) otherwise

is the (convex) indicator function of Dom"H$ and @3"x ! Dom"H$$ is the set of all subgradients of 3"w"t$ !Dom"H$$. In other words, &"t$ takes values among the normal directions of constraint on the boundary ofDom"H$ depending on w"t$.Instead of just seeking a solution to (46) when a #4-0, in Cépa [9] (weak) solutions for a #-w"0$"*K

+$ wereanalysed. By using the results of Cépa [9], we then have the following theorem where we define cl"3 $ to be theclosure of set 3 and the (a.e.) right derivative at time t # &0#1' of an absolutely continuous function w #4- tobe "d+w/dt$"t$.

Theorem 4.1. Let 8 be a maximal monotone map such that its domain Dom"8$ has a nonempty interiorint"Dom"8$$. If w"0$ # cl"Dom"8$$, then for a #-0"*K

+$, there exists a unique (weak) solution of

w"t$ # a"t$&8"w"t$$ 2 t # &0#1'# (47)

with w #-w"0$"*K+$ taking values in cl"Dom"8$$. The map from w"0$+ a to w is continuous with the uniform

topology on -"*K+$. If, in addition, a #4-0, then x #4-w"0$ (and is a strong solution) with right derivative

d+w/dt given by

d+wdt

"t$= a"t$&Proj8"w"t$$"a"t$$2 (48)


Proof. The first part is sufficient to prove Theorem 3.2 and follows from Cépa [9, Theorem 3.2]. However,the second part aids explicit calculations and follows from a combination of Brézis [7, Theorems 3.4–3.5,Proposition 3.8]. Also, see Browder [8, Theorems 9.25–9.26]. !

Remarks. (i) For a comprehensive survey of the Skorohod problem and the state of the art, the reader isreferred to Atar et al. [2].(ii) From the above discussion, it is also clear that the results of Cépa [9] answer in the affirmative questions

about the existence and uniqueness of fluid limits (e.g., Wischik [41]) such as (27) in the context of max-weightscheduling for the single-server setup that we have assumed. Additionally, following (48), the solution is suchthat the right derivative is given by

d+wdt

"t$='&Proj8"w"t$$"'$2 (49)

Now, we spell out the details of the proof of Theorem 3.2.Proof of Theorem 3.2. First note that H" · $ is maximal monotone with Dom"H$=*K

+. Additionally, fromRockafellar [31, pp. 215–216, 226] and Brézis [7, Example 2.3.4, p. 25], Rockafellar [31, Corollary 31.5.2,p. 340] and Rockafellar and Wets [32, Theorem 12.17, p. 542], we have @3"· !Dom"H$$ also being a maximalmonotone map, again with domain *K

+. Therefore using Brézis [7, Corollary 2.7], we have H" · $ + @3"· !Dom"H$$ also being a maximal monotone map.5 If we take the map 8" · $ to be H" · $+@3"· !Dom"H$$, then wecan apply Theorem 4.1. Now, if one considers the function in -w"0$"*K

+$ given by a=w"0$+a for a #-0"*K+$,

then using the results of Cépa [9] summarised in Theorem 4.1, one can show the existence of a unique continuous(weak) solution to (46) for continuous input a such that the map from a to w is continuous with the uniformtopology on -"*K

+$. With the uniqueness of trajectories for every a #-0"*K+$, we can now follow the argument

of the proof from Puhalskii and Vladimirov [30, Lemma 3.1] to prove the LD convergence result; since theargument is exactly the same, we do not reproduce it here. Owing to relative compactness (established usingexponential tightness), one gets LD convergence along subsequences. Uniqueness of trajectories then allows oneto show that the limit along every subsequence has to be the same, which automatically yields convergence.The action functional is also immediate from the argument of the proof from Puhalskii and Vladimirov

[30, Lemma 3.1]. Define the composite map from a #-0"*K+$ to w through w"0$+ a to be : . Let :A be the

image under : of absolutely continuous functions a #4-0. Note that :A is a subset of the set of absolutelycontinuous functions from &0#1' with initial value w"0$. Then the action functional for the LDP result IWw"0$" · $is given as follows: if w #:A, then

IWw"0$"w$= IA": &1"w$$= infa#4-0*: "a$=w

IA"a$# (50)

and for every other w #-"*K+$, we set IWw"0$"w$ to +). !

Remark. Following the proof from Puhalskii [29, Theorem 3.1], it is sufficient to prove the existence ofunique solutions to (46) using results from Brézis [7] for a #4-0 such that IA"a$<+), for the LDP resultto hold or to an even smaller set of functions that determine the rate function. The characterisation in (48)would also still apply. However, we feel that using the results of Cépa [9] provides us with a more completecharacterisation.For x # *K

+, let ;%"x$ *= (k* xk = 0) be the set of coordinates that are zero. Then, using the coordinate

convexity of /"K$ it is immediate that H"x$ is coordinate convex in each of the coordinates in ;%"x$. From thisobservation and noting that @3"x ! *K

+$= (y # *K * yk ' 0 2k #;%"x$ and yk = 0 2k # (1#2# 7 7 7 #K)\;%"x$),we obtain the following lemma.

Lemma 4.2. For every x #*K+ and a #*K

+, we have

ProjH"x$+@3"x!*K+$"a$= ProjH"x$"a$2 (51)

Proof. Let y4 *= ProjH"x$"a$ #H"x$. We know that y4 is the unique element in H"x$ such that

6y4 & a#y& y47 ( 0 2y #H"x$2 (52)

Fix k #;%"x$. Since H"x$ is coordinate convex along k, we have y #H"x$, where given + # &0#1$, we set

2k% # (1#2# 7 7 7 #K) yk% =

-

/

0

y4k% if k% 3= k1

+y4k% otherwise.

5 H" · $+ @3"· !Dom"H$$ at x is the set (y #*k* y= y1 + y2 s.t. y1 #H"x$ and y2 # @3"x !Dom"H$$).


Therefore, substituting y in (52), we get

"y4k & ak$y

4k ' 02 (53)

This holds for every k #;%"x$.Since /"K$ is coordinate convex with nonempty interior, it is easy to argue using (53) the following for any

given x #*K+:

(i) if ak% = 0 for some k% #;%"x$, then y4k% = 0;

(ii) if we have y4k = 0 for some k #;%"x$ for which ak > 0, then for all y #H"x$, we must have yk = 0; and

(iii) if we have a y #H"x$ such that yk > 0 for some k #;%"x$ for which ak > 0, then y4k > 0 and y4

k ' ak.Therefore, in all cases, we have y4

k ' ak for all k #;%"x$.Now, any y% #H"x$+ @3"x ! *K

+$ is such that we can write y% = y+ =y, where y #H"x$ and =y # @3"x ! *K+$.

Such a decomposition need not be unique and we do not need uniqueness for the proof. Note that =yk = 0 for allk # (1#2# 7 7 7 #K)\;%"x$ and =yk ' 0 for k #;%"x$. Therefore we have

6y4 & a#y% & y47= 6y4 & a# y& y47+)

k#;%"x$

"y4k & ak$=yk ( 6y4 & a# y& y47 ( 0#

where the penultimate inequality follows because y4k & ak ' 0 and =yk ' 0 for all k #;%"x$. This completes the

proof. !

An elementary but extremely useful conclusion from Lemma 4.2 is the following corollary.

Corollary 4.1. If a #4-0, then w #4-w"0$ such that w=: "a$ has right derivative d+w/dt given by

d+wdt

"t$ = a"t$&ProjH"w"t$$+@3"w"t$ !*K+$"a"t$$

= a"t$&ProjH"w"t$$"a"t$$2 (54)

Applying this to the fluid limit says that the fluid limit (27) and (49) has an even simpler characterisationwhere the right derivative is given by

d+wdt

"t$='&ProjH"w"t$$"'$2

Now that the underlying rate function IWw"0$" · $ has been specified, we set out to derive an alternate expressionfor J"x# t$ (see (31)) that converts the calculus of variations problem to a finite-dimensional optimisation; notethat w"0$= 0 in the definition of J"x# t$. In the process, we will show that for determining the rate function, itsuffices to consider piecewise linear functions (illustrated in Figure 1) a #4-0 determined by two parametersu # &0# t' for t # "0#1' and ( #*K

+\/"K$ such that for v # &0#1', we have

a"v$=

-

.

.

.

./

.

.

.

.0

'v if v # &0#u'1

("v& u$+'u if v # &u# t'1

'"v& t+ u$+("t& u$ if v # &t#1'2

(55)

This is proved in the following lemma.

t –u t0

!1

!2

!1

!2

Arr

ival

rat

e "2

"1

Figure 1. Typical element of class of a considered for optimisation.


Lemma 4.3. If ;A"x$* *K+ +*+ is convex with ;A"'$= 0 for some ' #*K

+ with ' #/"K$ such that thereexists (4 #/"K$ with '<(4, then for x #*K

+ and t # "0#1', we have

J"x# t$=

-

.

./

.

.0

infu#"0# t'

u infA#argmaxr#/6x#r7

;A

#

xu+(

$

if x 3= 01

0 otherwise2

(56)

Proof. We can rewrite J"x# t$ for x #*+ and t # "0#1' as

J"x# t$= infa#4-0* : "a$"t$=x

& 1

0;A"a"u$$du2 (57)

Since we are only interested in the behaviour of the work load in &0# t', it suffices to consider a #4-0 suchthat : "a$"t$= x and a"u$= ' for all u # "t#1'; note that

' 1t ;

A"a"u$$du will be 0 with this restriction, andsince ' is strictly inside /"K$ (as given by the assumptions above), after a finite time (depending on x), w willreduce to 0 and remain there.6

For a given a # 4-0, define u4 = sup(u # &0# t'* w"u$ = 0), i.e., the last time w is zero in all coordinatesbefore time t. Since

' u4

0 ;A"a"u$$du ( 0 and w"u4$ = 0 (by continuity), we can reduce the cost by settinga"u$=' for all u # &0#u4' all the while ensuring w"u4$= 0. Thus, for x 3= 0, it suffices to solve the followingcalculus of variations problem for u # "0# t':

infa#4-0*: "a$"u$=x#: "a$"v$3=0 20<v'u

& u

0;A"a"v$$dv2 (58)

From Corollary 4.1, any a #4-0 that achieves : "a$"u$= x and, in particular, any a #4-0 such that : "a$"v$ 3=0 20< v' u, satisfies the following equation:

x= a"u$&& u

0ProjH"w"v$$"a"v$$dv2 (59)

Now, using Jensen’s inequality (see Rockafellar [31]) and (59), we have& u

0;A"a"v$$dv ( u;A

#

1u

& u

0a"v$dv

$

= u;A

#

a"u$u

$

= u;A

#

xu+' u

0 ProjH"w"v$$"a"v$$dvu

$

2

If one can now find a constant ( # *K+ such that x/u+ ProjH"xv/u$"($= ( for all v # &0#u', then it suffices to

optimise over the possible values (B"x#u$) of the constant ( to solve the calculus of variations problem in (58).The answer to (58) would then be u infA#B"x#u$ ;A"($ with w"v$= xv/u for v # &0#u', and

J"x# t$= infu#"0# t'

u infA#B"x#u$

;A"($2 (60)

The work load trajectories (up to time t) that result from the considered class of arrivals are illustrated inFigure 2, where for the sake of illustration, for v # &0# t', we take a"v$= 'min"v# t & u$+("v& t + u$+ forsome ( #*K

+\/"K$.Since we are maximising a linear functional, it is clear that H"xv/u$=H"x$ for v # "0#u' and H"xv/u$=

/"K$ for v= 0. Thus we need to solve for ( that solves both

xu+ProjH"x$"($=(# and (61)

xu+Proj/"K$"($=(2 (62)

Pick r4# r #/"K$, then>>>

xu+ r4 & r

>>>

2= 0r4 & r02 +

>>>

xu

>>>

2+ 2?

xu# r4@

& 2?

xu# r@

2

6 As mentioned in §3, this can be shown using the Lyapunov function V "t$= 12 0w"t$02.


t –u t0

x1

x2

0

Wor

k lo

ad

Figure 2. Typical workload trajectory until time t for the analysed class of input functions a.

Therefore

minr#/"K$

>>>

xu+ r4 & r

>>>

2(>>>

xu

>>>

2+ 2?

xu# r4@

+ minr#/"K$

0r4 & r02 & 2 maxr#/"K$

?

xu# r@

=>>>

xu

>>>

2+ 2?

xu# r4@

& 2 maxr#/"K$

?

xu# r@

#

since r4 #/"K$. Now minr#/"K$ 0x/u+ r4 & r02 ' 0x/u02 since we can set r= r4. From this, it follows that ifr4 #H"x$, then r4 = Proj/"K$"x/u+ r4$.If r4 = Proj/"K$"x/u+ r4$, then 26r4 &x/u& r4# r4 & r7 ' 0 for all r #/"K$, which is another way of saying

that r4 #H"x$.For r4 #H"x$, we also have

minr#H"x$

>>>

xu+ r4 & r

>>>

2=>>>

xu

>>>

2+ min

r#H"x$0r4 & r02

#

Since?

xu# r4@

=?

xu# r@$

=>>>

xu

>>>

2#

and r4 = ProjH"x$"x/u+ r4$Thus, any ( # x/u+H"x$ solves (61)–(62), and there are no other solutions, i.e., B"x#u$= x/u+H"x$ and

the result holds. !

Remarks. (i) Lemma 4.3 informs us that for a given /"K$ to quantify, the performance one needs tocharacterise H"x$ for all x #*K

+. Note that this is equivalent to a complete characterisation of the boundary of/"K$.(ii) It is also worth noting that J"x# t$ need not be convex in its arguments. We will explicitly show this lateron using some examples.(iii) The starting point being 0 was critically exploited in the characterisation of the critical sample paths.However, it is possible to argue with polytope rate regions that one only needs to consider piecewise linearsample paths. The reason for this is that by using Corollary 4.1, it is clear that under max-weight policies, thestate space can be partitioned into a finite set of cones where the scheduling policy is fixed. Now, for any samplepath, Jensen’s inequality can be used to show that in every interval where the work load remains within eachof these cones, the cost of the path can be bettered by choosing an arrival rate vector that is constant in thisinterval. This then parallels Dupuis et al. [16, Lemma 4.1].By using the expression for J"x# t$, we can now write down simpler expressions for (36)–(39). First consider

(36) and without loss of generality let k= 1. Then, we have the following:

infy#*K

+ *y1(xinf

t#"0#1'J"y# t$ = inf

y#*K+ *y1(x

inft#"0#1'

infu#"0# t'

u infA#H"y$

;A

#

yu+(

$

= infy#*K

+ *y1(1inf

t#"0#1'inf

u#"0# t'u infA#H"yx$

;A

#

yxu

+(

$


= infy#*K

+ *y1(1inf

t#"0#1'x inf

v#"0# t/x'v infA#H"y$

;A

#

yv+(

$

= x infy#*K

+ *y1(1inf

t#"0#1'infz(x/t

infA#H"y$ ;A"yz+($

z

= x infy#*K

+ *y1(1infz(x

infA#H"y$ ;A"yz+($

z

= x infz(x

infy#*K+ *y1(1 infA#H"y$ ;A"yz+($

z2 (63)

The rest of the terms in (36) have a similar form. By using a similar logic, we have one of the terms of (37)given by

infy#*K

+ *y1>xinf

t#"0#1'J"y# t$= x inf

z(x

infy#*K+ *y1>1 inf(#H"y$ ;A"yz+($

z2 (64)

Similar logic applied to (38)–(39) yields

infy#*K

+ *(K

k=1 yk(x

inft#"0#1'

J"y# t$ = x infz(x

infy#*K+ *(K

k=1 yk(1 infA#H"y$ ;A"yz+($

z1 and (65)

infy#*K

+ *(K

k=1 yk>x

inft#"0#1'

J"y# t$ = x infz(x

infy#*K+ *(K

k=1 yk>1 infA#H"y$ ;A"yz+($

z2 (66)

These expressions can be simplified further with additional assumptions on the arrival processes (such as inde-pendence) and the rate region /"K$.

5. Examples. We will conclude by presenting three example rate regions to show how the analysis developedabove applies. The first is a two-user elliptical rate region. The last two examples are obtained from Coverand Thomas [11]. The first of these considers a two-user Gaussian BC channel and the second a symmetricaltwo-user MAC. For the remainder of this section, we will set the scheduling weight vector # to "1/2#1/2$. Notefrom the analysis in §4 that other values of # can be analysed by modifying the parameters of /"2$.

5.1. Example I: A two-user queue with an elliptical rate region. Consider a specific enunciation of ourmodel with two users such that the rate region /"2$ is a quadrant of an ellipse with parameters rM # rm > 0, i.e.,

/"2$="

"r1# r2$ #*2+*

#

r1

rM

$2

+#

r2

rm

$2

' 1%

2 (67)

Now, let us solve the scheduling policy generation problem; namely, H"x$= argmaxr#/"2$6x# r7 for x # *2+.

We will only consider the case of at least one coordinate being positive because H"0$ =/"2$. The optimalsolution is the unique point "r1# r2$ # //"2$ that satisfies

r1

rM= rMx1

*

"rMx1$2 + "rmx2$2and

r2

rm= rmx2

*

"rMx1$2 + "rmx2$22 (68)

This can be derived easily using Lagrange multipliers. The rate region and the solution of the scheduling problemare illustrated in Figure 3.The expression for J"x# t$ with x 3= 0 for this example is given by


u;A##

x1

u+ rM

rMx1

*

"rMx1$2 + "rmx2$2#x2

u+ rm

rmx2

*

"rMx1$2 + "rmx2$2

$$

2 (69)


rM

rm x1r1 + x 2r 2=cx1r 2 – x 2r1=c

Operatingpoint

Figure 3. Illustration of an elliptical rate region with the solution of the scheduling problem shown.

5.2. Example II: Two-user Gaussian BC. The BC (see Cover and Thomas [11, §14.6]) models a commu-nication system where there is one transmitter and multiple receivers who can all listen to the transmitter. Thecapacity region (in natural units, i.e., nats) of a two-user Gaussian BC (see Cover and Thomas [11, §14.6]) isdetermined by two parameters (signal-to-noise ratios) P1 >P2 > 0 and is given as follows:

/"2$=A

-#&0#1'

"

"r1# r2$ #*+2* r1 ' 1

2log"1+-P1$# r

2 ' 12log#

1+P2

1+-P2

$%

2 (70)

If P1 = P2 > 0, then one gets a simplex.The scheduling rule H"x1#x2$ with at least one coordinate positive is then given by " 12 log"1 + -4P1$#

12 log""1+P2$/"1+-4P2$$$ with

-4 =

-

.

.

.

.

.

.

./

.

.

.

.

.

.

.0

1 if x1

#

1+ 1P2

$

( x2

#

1+ 1P1

$

1

0 if x1P1 ' x2P21

x1/P2 & x2/P1

x2 & x1otherwise2

(71)

Again H"0#0$=/"2$.From this exercise, we can now write down expressions for J"x# t$ for x 3= 0 as follows:

J"x# t$=

-

.

.

.

.

.

.

.

.

.

.

.

.

.

./

.

.

.

.

.

.

.

.

.

.

.

.

.

.0

infu#"0# t'

u;A#

x1

u+ 1

2log"1+P1$#

x2

u

$

if x1

#

1+ 1P2

$

( x2

#

1+ 1P1

$

1

infu#"0# t'

u;A#

x1

u#x2

u+ 1

2log"1+P2$

$

if x1P1 ' x2P21

infu#"0# t'

u;A

B

CCC

D

x1

u+ 1

2log#

x1"P1 &P2$

P2"x2 & x1$

$

#

x2

u+ 1

2log#

"1+P2$P1"x2 & x1$

x2"P1 &P2$

$

E

FFF

G

otherwise2

(72)

5.3. Example III: Centralised MAC. Consider the rate region /"K$ to be the capacity region of a K-userMAC (see Cover and Thomas [11, Section 14.3]) and Tse and Hanly [37]. At the beginning of every transmissioninterval, each of the users communicate their queue lengths to a centralised scheduler that then determines theoperating point to be used. It was shown in Tse and Hanly [37, Lemma 3.4] that /"K$ is a polymatroid (seeChen and Yao [10, §11.1]) where the rank function is given by conditional mutual information terms. As shownin Tse et al. [38], this rate region is also applicable in the asymptotic regime of (very) high signal-to-noise ratioof the multiple-input, multiple-output, MAC with fading such that the receiver has perfect channel informationand the transmitters have no channel information. Such a model (Tse et al. [38]) exhibits a nice trade-off between


diversity and multiplexing that was used to provide performance bounds based on the tails of the queue lengthsfor max-weight-type scheduling algorithms in Kittipiyakul and Javidi [22]. In Kittipiyakul and Javidi [22], thetwo-user case was analysed completely when the rate-region is a simplex, and bounds were presented when therate region is a symmetric polymatroid. The analysis presented here can be used to improve on the bounds ofKittipiyakul and Javidi [22] in the general case.Define ; = (1#2# 7 7 7 #K), and suppose that we are given a function f * "";$+*+ from the power set of

; to the (nonnegative) real line. Then, the polytope

/f "K$ *="

r #*K+*)

i#<r i ' f "<$# < $;

%

(73)

is a polymatroid (see Chen and Yao [10, §11.1]) if the function f satisfies the following properties:(i) (normalised) f ""$= 0;(ii) (increasing) if <1 $ <2 $;, then f "<1$' f "<2$; and(iii) (submodular) if <1#<2 $;, then f "<1$+ f "<2$( f "<1 <<2$+ f "<1 -<2$.A function f with these properties is called a rank function. Let = be a permutation of ;, then the vector r=

defined by

r="1$= = f "(="1$)$

r="2$= = f "(="1$#="2$)$& f "(="1$)$

222

r="K$= = f "(="1$#="2$# 7 7 7 #="K$)$& f "(="1$#="2$# 7 7 7 #="K& 1$)$

belongs to /f "K$ for all permutations =. Along with 0, the points r= are the extreme points of /f "K$. Also,for any pair of sets <1 , <2 $;, there exists a point r #/f "K$ such that

)

i#<1r i = f "<1$ and

)

i#<2r i = f "<2$2

Maximising a linear functional (6x# r7) over a polymatroid is very easy (see Chen and Yao [10, §11.1.2]) and isgiven by the following:

(i) without loss of generality, assume that xk ( 0 for all k #;. Otherwise, we simply set the correspondingrk = 0 for the optimal solution;

(ii) let = be a permutation of ; such that the weights are in decreasing order, i.e.,

x="1$ ( x="2$ ( · · ·( x="K$ ( 0# (74)

then r= is an optimal solution; and(iii) the set of optimal solutions is the convex hull of r= for all permutations of ; that yield the ordering

in (74).With /f "K$ as the rate region for our system, this completely specifies H"x$ for all x # *K

+. To understandthis better, we will look at a class of two-user channels.

5.3.1. Symmetric two-user case. Let K = 2 and given two parameters rM > rm > 0 define the rank functionf as follows:

f "<$ *=

-

.

.

.

.

.

.

./

.

.

.

.

.

.

.0

0 if < ="1

rM if < = (1)1

rM if < = (2)1

rM + rm if < = (1#2)2

(75)

Then, our rate region is /f "2$. The edge cases for the parameters rM # rm do not give any new insights: ifrM = rm > 0, then /f "2$ is a square; and if rM > rm = 0, then /f "2$ is a simplex. An example of this rateregion is shown in Figure 4.


rm

rM

rm rM

Figure 4. An example of a symmetrical two-user polymatroidal rate region.

We now solve for H"x$= argmaxr#/f "2$6x# r7 for w # *2

+. We need to partition *2+ into six regions; these

and the corresponding sets H"x$ are as follows:(i) Region A= (0). Here, it is clear that H"0$=/"2$;(ii) Region B= (x1 > 0#x2 = 0). Then H"x$= (r1 = rM # r2 # &0# rm');(iii) Region C = (x1 > x2 > 0). Then H"x$= (r1 = rM # r2 = rm);(iv) Region D= (x1 = x2 > 0). Then H"x$= ("r1# r2$ # &rm# rM '2* r1 + r2 = rM + rm);(v) Region E = (x2 > x1 > 0). Then H"x$= (r1 = rm# r2 = rM); and(vi) Region F = (x2 > 0#x1 = 0). Then H"x$= (r1 # &0# rm'# r2 = rM).

The different scheduling regions are shown in Figure 5.By using the above, we can write down expressions for J"x# t$ as follows:(i) if x # B, then


u infr2#&0# rm'

;A#

x1

u+ rM # r2

$

1 (76)

DF

BA

C

E

Figure 5. Partitioning of *2+ into regions where the same scheduling action results.


(ii) if x #C, then


u;A#

x1

u+ rM #

x2

u+ rm

$

1 (77)

(iii) if x #D, then


u inf"r1# r2$#&rm# rM '2*r1+r2=rM+rm

;A#

x1

u+ r1#

x2

u+ r2$

1 (78)

(iv) if x #E, then


u;A#

x1

u+ rm#

x2

u+ rM

$

1 (79)

(v) if x # F , then


u infr1#&0# rm'

;A#

r1#x2

u+ rM

$

2 (80)

5.4. Numerical and simulation results. Here, we demonstrate how our results can be used to predict certaintail exponents. To validate our results, we compare the numerical results to ones obtained using simulations. Forsimplicity, we restrict our attention to two-user scenarios. We assume that the arrival processes are independentacross users and across time slots. In fact, the work brought in per slot for user 1 is assumed to be Bernoulliwith mean 0.4955 and that for user 2 is assumed to be Bernoulli with mean 024. We consider three rate regions:circular, symmetric simplex and symmetric MAC. The rate regions are assumed to be such that "024955#024$ isstrictly in the interior so as to ensure stability. The simplex rate region is such that x-asymptote and y-asymptoteare "029#0$ and "0#029$, respectively. The MAC rate region is obtained using the rank function from (75) whererM = 025 and rm = 024. Note that this corresponds to the simplex rate region being restricted to a maximum of025 in either coordinate. For the circular rate region, we set rM = rm =

;0252 + 0242 =

;0241> 026403 in (67).

These are illustrated in Figure 6 where we also show the location of the average arrival rate vector.The numerical calculations involve evaluation of (56) for each of the rate regions and for every chosen value

of x, where the local action functional ;A" · $ is constructed using the rate function of a Bernoulli randomvariable and is given by

;A"x$=

-

.

./

.

.0

x1 log#

x1024955

$

+ "1& x1$ log#

1& x1025045

$

+ x2 log#

x2024

$

+ "1& x2$ log#

1& x2026

$

# if x # &0#1'21

+)# otherwise,

where 0 log"0$ is assumed to be 0 by continuity. The rate functions for the simplex rate region, the MAC rateregion, and the circular rate region are shown in Figures 7–9, respectively. Note that the polytope nature of the

0.35 0.40 0.45 0.50 0.550.35

0.40

0.45

0.50

0.55

r1

r 2

Rate regions

Avg arrival rateCircularSimplexMAC

Figure 6. Rate regions used for the numerical analysis and simulations.


Symmetric simplex rate region

00.1

0.20.3

0.4

0

0.1

0.2

0.3

0.40

0.5

1.0

1.5

2.0

w1

w2

Rat

e fu

nctio

n

Figure 7. Rate function of the simplex rate region.

Symmetric MAC rate region

00.1

0.20.3

0.4

0

0.1

0.2

0.3

0.40

0.1

0.2

0.3

0.4

0.5

w1

w2

Rat

e fu

nctio

n

Figure 8. Rate function of the MAC rate region.

Circular rate region

00.1

0.20.3

0.4

0

0.1

0.2

0.3

0.40

0.5

1.0

1.5

w1

w2

Rat

e fu

nctio

n

Figure 9. Rate function of the circular rate region.

simplex and MAC rate regions results in the discontinuity of the rate functions. In the case of the simplex rateregion, it is clear that the work loads being equal is much more likely than any other configuration.We also compare the numerical results with empirical calculations of the negative of the tail asymptote

obtained from simulations; all for the case of # = 1. For this purpose, we set N = 100 for the computationof the fluid limit. Thereafter, starting from 0, we run 109 trials to estimate the probabilities in (29) for manyvalues of x. Note that the smallest value of the tail asymptote that we can estimate is log"10&9$/100>&022072.The comparison of the numbers obtained from numerical optimisation and simulations for the quantities in (29)for the simplex rate region, the MAC rate region, and the circular rate region are shown in Figures 10–12,


0 0.05 0.10 0.15 0.20 0.250

0.05

0.10

0.15

0.20

0.25

Tai

l asy

mpt

ote

Symmetric simplex rate region

w1empw1calcw2empw2calcdiagempdiagcalc

Work load

Figure 10. Comparison of numerical results with simulations for the symmetric simplex rate region.

0 0.05 0.10 0.15 0.20 0.250

0.05

0.10

0.15

0.20

0.25

Tai

l asy

mpt

ote

Symmetric MAC rate region


Work load

Figure 11. Comparison of numerical results with simulations for the symmetric MAC rate region.

0 0.05 0.10 0.15 0.20 0.250

0.05

0.10

0.15

0.20

0.25

Tai

l asy

mpt

ote

Circular rate region


Work load

Figure 12. Comparison of numerical results with simulations for the circular rate region.

respectively. For ease of comparison, we only present three cases on the graphs: w2 = 0, which is tagged asw1emp/calc; w1 = 0, which is tagged as w2emp/calc; and w1 = w2, which is tagged as diagemp/calc. In all cases,“emp” stands for the numbers obtained using simulations, while “calc” stands for the numbers using numericaltechniques. In the case of the simplex rate region, the numerical results seem to indicate that the preferred wayto achieve a large work load is to increase both work loads such that they are equal. However, in the case of


the MAC rate region or the circular rate region, the preferred way to achieve a large work load only for thefirst user is different from the preferred way to achieve a large work load only for the second user; the latterseems to follow a trajectory above where both work loads are increased simultaneously in a way such that theyremain equal.

6. Conclusions. In this paper, we proved an LDP for max-weight scheduling where the server can chooserate points from within a compact, convex, and coordinate-convex set. Because the rate points chosen by thescheduling policy are closely related to the set of subgradients of a convex function that is constructed usingthe rate region, compactness, and convexity of the rate region is sufficient to prove the LDP and identify therate function. Coordinate convexity of the rate region, which is a natural assumption for queueing systems, wasthen used to simplify the rate function.

Appendix. ProofsProof of Theorem 3.1. From §2, we need to prove (3)–(4) to show -"+$-exponential tightness of the

net process XN *= "AN #RN #YN #SN #: N #WN $. Now, it is clear using metric !X"·# ·$ that it suffices to demon-strate both statements (3)–(4) for AN , RN , YN , SN , : N , and WN separately so as to prove the result for XN .Equivalently, the same result follows from the contraction principle (see Puhalskii [27, Corollary 3.2.7, p. 283]).Since both these properties for AN are a consequence of the assumptions on the arrival process, there is

nothing to prove there.Proof of exponential tightness (3). Fix t # &0#1'. From the compactness of /"K$, it is clear that

0SN "t$0 'Krmax9Nt:+ 1

N'Krmax"t+ 1$' 2Krmax#

which immediately yields exponential tightness. The same bound can be used to prove exponential tightness ofRN and YN . Similarly, for every t # &0#1', the exponential tightness of the sequence of measures : N "t$ is adirect consequence of .t+1"/"K$$ being a compact set. Finally, we have the following bound:

WN "t$'WN "0$+AN "t$#

and the exponential tightness of WN follows (using the superexponential convergence of WN0 /N to w"0$).

Proof of continuous-limits points (4). Fix T # "0#1'# +> 0, u # &0#T $, and t # "u#min"T #u+ 3$'. We have

0SN "t$&SN "u$0= 0SN "9Nu:# 9Nt:'0N

' Krmax9Nt:& 9Nu:

N

' Krmax

#

t& u+ 1N

$

'Krmax

#

3+ 1N

$

2 (81)

Therefore (4) holds for SN . The very same logic can be used to prove that (4) holds for RN and YN .For any Borel set C #/"K$, we have

: N "u$"C$ = 1N

9Nu:)

i=0

$rNi"C$

' 1N

9Nt:)

i=0

$rNi"C$=: N "t$"C$

= 1N

9Nu:)

i=0

$rNi"C$+ 1

N

9Nt:)

i=9Nu:+1

$rNi"C$

' : N "u$"C$+ 9Nt:& 9Nu:N

' : N "u$"C$+ 3+ 1N2 (82)

By using the relationships in (82) and upper bounding the Kantorovich-Wasserstein metric using only boundedcontinuous functions, we can assert (4) for : N . Since (4) also holds for AN and SN , from (16) and the super-exponential convergence of WN

0 /N to w"0$, we can assert (4) for WN too.


Since we have LD relative compactness, there exist limit points "a#%#&# s#5#w$ taking values in -"+$,which are obtained as limits along subsequence (Nn)

+)n=1 with limn++)Nn =+). We will now prove the required

properties of the limit points. Note once again that (i) follows from the assumptions, and there is nothing toprove.Proof of properties (ii)–(vi). For given 1 ( t > u ( 0, and + > 0, consider the set C+ *=

("a1#%1#&1# s1#51#w1$ # *"+$* 0s1"t$ & s1"u$0 ' Krmax"t & u$ + +). Note that the set C+ is -"+$-closed.Defined O+ *= *"+$\C+, which is -"+$-open. Note that we are interested in %"C$ and %"O$, where C *=H

+>0C+ (C+s decrease as + + 0) and O *= I+>0O+ (O+s increase as + + 0). Since O+ is -"+$-open, byPuhalskii [27, Corollary 3.1.9, pp. 257–258], we have lim infn++) &"XNn # O+$

1/Nn ( %"O+$ and by (81) forevery + > 0 and for all n such that Nn > 9Krmax/+: we have &"XNn #O+$= 0. Thus %"O+$= 0, and therefore%"O$= sup+>0%"O+$= 0. Therefore %-a.e., we have s also being Lipschitz.The relationships in (82) are also sufficient to show for 1( t > u( 0 that 5"t$&5"u$ #.t&u"/"K$$ and

that the total variation norm (see Yosida [42, p. 35–38, 118–119]) of 5"t$&5"u$ is t & u. Since .1"/"K$$is compact, we can follow the second part of the proof of Dembo and Zajic [12, Lemma 4, pp. 198–200] toconstruct 5"t$ #.1"/"K$$ for almost every t ( 0. By using ideas similar to those mentioned above, we canalso show that for every limit point %, &, and s are componentwise nondecreasing, and %"0$=&"0$= s"0$= 0and 5"0$"/"K$$= 0.By using the assumptions for WNn

0 /Nn and results for SNn and ANn and (16), we now prove results for WNn . Itis clear from the convergence of each term in (16) (and using the -"+$-continuity of addition and subtraction)that every limit point satisfies the following for k= 1#2# 7 7 7 #K and for all t # &0#1'

wk"t$=wk"0$+ ak"t$& sk"t$# (83)

for absolutely continuous ak"t$, and where sk"t$ is a limit point of Sk#N "t$. Note that this is the same as (21).Since a is absolutely continuous and s is Lipschitz continuous, we have w being absolutely continuous for everylimit point. The (componentwise) nonnegativity of w is proved using the nonnegativity of the original sequenceWNn , just as illustrated above.

Proof of (20). First, we show that 6: Nn"t$# ek7 LD converges to 65"t$# ek7 for every t # &0#1'. Since %-a.e. the limit points are in -"+$ and since the projection operator "a1#%1#&1# s1#51#w1$+ 51"t$ is -"+$-continuous (and Borel measurable under the Skorohod J1 topology), we get LD convergence of : Nn"t$ to 5"t$in ."/"K$$ by an application of the contraction principle (see Puhalskii [27, Corollary 3.1.22, p. 264]). Sinceek is a (bounded) continuous function from a compact set /"K$ to *+, the LD convergence of 6: Nn"t$# ek7 to65"t$# ek7 follows from Puhalskii [27, Corollary 3.1.9, pp. 257–258].Since no more than rmax can be served from any user no matter which operating point is chosen, we claim

that for all 1( t ( u( 0,

sk"t$& sk"u$' 65"t$&5"u$# ek7 "%-a2e2$2 (84)

The proof of this follows from the observation that for any subsequence "ANn#RNn#YNn#SNn#: Nn#WNn$ thatLD converges to "a#%#&# s#5#w$, we have

0= lim infn++)

&""Sk#Nn"t$& Sk#Nn"u$$&6: Nn"t$&: Nn"u$# ek7> 0$1/Nn

(%""sk"t$& sk"u$$&65"t$&5"u$# ek7> 0$#

which implies the result in (84) since

%

#A

1(t>u(0

(sk"t$& sk"u$&65"t$&5"u$# ek7> 0)$

= sup1(t>u(0

%"sk"t$& sk"u$&65"t$&5"u$# ek7> 0$= 02

Above we used the fact that ("a1#%1#&1# s1#51#w1$ # *"+$* "sk1"t$ & sk1"u$$ & 651"t$ & 51"u$# ek7 > 0) is-"+$-open.


Proof of property (vii). Without loss of generality, it suffices to prove this for k = 1. We firstrephrase the statement that needs to be proved. Define -"+$-open set O *= ("a1#%1#&1# s1#51#w1$ #*"+$* mint#&t1# t2'w

1"t$> 0), =Et#u *= ("a1#%1#&1# s1#51#w1$ #*"+$* t ( u# s1"t$& s1"u$= 65"t$&5"u$# e17),and /E *= ("a1#%1#&1# s1#51#w1$ # *"+$* s1"t$ & s1"u$ = 65"t$ & 5"u$# e17# 2 t ( u# t#u # &t1# t2') =H

t(u# t#u#&t1# t2'=Et#u. Now =Et2# t1

$ =Et#u for all t ( u with t#u # &t1# t2' from which it follows that /E = =Et2# t1. Also,

define /Ec and =Ect#u to be the complement of /E and =Et#u in *"+$, respectively. Then, we need to prove that

%"O- /Ec$= 0, in other words, %"O- =Ect2# t1

$= 0 holds. For ease of notation from now onward, we set E *= =Et2# t1

with Ec being the complement. For future use, we note that E is a -"+$-closed set.For + > 0, define (increasing in + as + + 0) -"+$-open sets O+ *= ("a1#%1#&1# s1#51#w1$ #

*"+$* mint#&t1# t2'w1"t$ > +); it is clear that O *= I+>0O+. Therefore we have %"O$ = sup+>0%"O+$ and

%"O -Ec$= sup+>0%"O+ -Ec$. Therefore we will prove that %"O+ -Ec$= 0 for all + > 0. Also, define setsO"n$# + *= (mint#&t1# t2' /W 1#Nn"t$ > +) $ *"+$ and En *= ( /S1#Nn"t2$& /S1#Nn"t1$ = 6: Nn"t2$& : Nn"t1$# e17) (withEn $*"+$). Now En ? (mint#&t1# t2' /Wk#Nn"t$> rmax/Nn) since any service will be at the allocated rate if there issufficient work to be done. If Nn ( 9rmax/+:+1, then En ?O"n$# +. Let Ec

n be the complement of En with respectto *"+$. Therefore, for n large enough (such that Nn ( 9rmax/+:+ 1), we get Ec

n -O"n$# + = ", and therefore&"Ec

n -O"n$# +$= 0. The result follows since we can derive the following:

%"O+ -Ec$' lim infn++)

&"O"n$#+ -Ecn$

1/Nn = 0 "O+ -Ec is -"+$-open$2

Proof of property (viii). If w"t$= 0, then the result trivially holds since /H"0$=/"K$.Following Billingsley [4, p. 8] for +> 0, define functions f +#g+ from /"K$1*K

+ to &0#1' as follows:

f +"r#x$=#

1& maxr#/"K$6r## 8 x7& 6r## 8 x7+

$

+# (85)

and g+"r#x$ = 1 & f +"r#x$. From the definition, it is clear that f +"r#x$ = 1 if and only if r # /H"x$, andf +"r#x$ = 0 if and only if r # F + *= (r # /"K$* 6r## 8 x7 ' maxr#/"K$6r## 8 x7 & +). As + decreases to 0,F + increases to (open) /"K$\ /H"x$, and g+"r#x$ converges to 1(/"K$\ /H"x$)"r$ yielding the indicator function of/"K$\ /H"x$. Both f +"·# ·$ and g+"·# ·$ are continuous functions. In fact, it is easy to prove that

!f +"r1#x1$& f +"r2#x2$! '2Krmax

+0# 8 "x1 & x2$0+

0# 8 ""x1 + x2$/2$0+

0r1 & r202

Given 1( t > u( 0 for element "a#%#&# s#5#w$ #*"+$, define the following functional:

h+u# t"w$ *=& t

u

&

/"K$g+"r#wv$d5"v$"r$2 (86)

For every + > 0, we have'

/"K$ g+"r#x$d4"r$ being a bounded and continuous function on "x# 4$ # *K

+ 1."/"K$$ (and also on +), and hence by Ethier and Kurtz [17, Problem 13, p. 151] and continuity of integrals(see Whitt [40, Theorem 11.5.1, p. 383]), h+u# t"w$ is a bounded and continuous function on *"+$. Thus we candirectly apply the definition of LD convergence (see Puhalskii [27, Definition 3.1.1, pp. 253–254]) by goingback to the LD converging subsequence. By using the property that our scheduling policy enforces rm # /H"Wm$for all m( 0, it is easy to argue that %-a.e. we have h+u# t"w$= 0 for all +> 0.If we further assume the absolute continuity and Lipschitz continuity assumptions from the already established

properties (i)–(vi), then using Fubini’s theorem, we can write the following alternate expression for h+u# t"w$;namely,

h+u# t"w$=& t

u

#&

/"K$g+"r#wv$d5"v$"r$

$

dv2 (87)

Under exactly these assumptions, we can further prove (by the bounded convergence theorem) that h+u# t"w$converges to hu# t"w$=

' t

u 5"v$"/"K$\ /H"wv$$dv as + @ 0. Now h+u# t"w$= 0 for all limit points, implying thathu# t"w$ = 0. Therefore we have for (Lebesgue) almost all t # &0#1' the result that 5"t$"/"K$\ /H"wt$$ = 0.


Note that this is another way of saying that for every limit point %, it holds that %"t$ # /H"w"t$$ for (Lebesgue)almost all t # &0#1'.Proof of property (ix). First, we fix k # (1#2# 7 7 7 #K). From (17), we can write /Y k#N "t$ as follows:

/Y k#N "t$=max#

0# sup0's't

" /Rk#N "s$& Ak#N "s& 1/N $$& /Wk#N "0$$

2

Since (Ak#N "s&1/N $))N=1 is exponentially equivalent (see Dembo and Zeitouni [13, Definition 4.2.10 and The-orem 4.2.13, p. 130]) to (Ak#N "s$))N=1, which satisfies an LDP with a good rate function, and since function<"t$ *=max"0# sup0's't"-"s$& a"s$$& =w"0$$ is Borel measurable under the Skorohod J1 topology (includingthe subtraction operation) and continuous under the local uniform topology (see Billingsley [4], Ethier andKurtz [17], or Whitt [40, Chapter 13]) on *"+$ (and hence -"+$-continuous), by an invocation of the con-traction principle (see Puhalskii [27, Corollary 3.1.22, p. 264]) we get that every limit point should satisfy thefollowing:

<k"t$ = max#

0# sup0's't

"-k"s$& ak"s$$&wk"0$$

= max#

wk"0$# sup0's't

"-k"s$& ak"s$$

$

&wk"0$2

Since wk"t$=wk"0$+ ak"t$&-k"t$+<k"t$, we also obtain

wk"t$=max#

ak"t$&-k"t$+wk"0$#ak"t$&-k"t$& inf0's't

"ak"s$&-k"s$$

$

2

Now, it is clear that <k"t$ is nonnegative and nondecreasing such that

<k"t$=

-

/

0

"-k"t$& ak"t$$+ if -k"t$& ak"t$= sup0's't

"-k"s$& ak"s$$ and -k"t$& ak"t$(wk"0$1

0 otherwise.

Note that the condition for nontrivial derivative of <k"t$ is equivalent to wk"t$ = 0; in other words, <k"t$can only increase when wk"t$= 0. Also note that -k"t$& ak"t$= sup0's't"-

k"s$& ak"s$$ directly implies that-k"t$( ak"t$, but nevertheless, we choose to emphasise the positive part in the formula above.Thus every limit point is an absolutely continuous solution of the following differential inclusion for all

t # &0#1':w"t$& &"t$ # a"t$& /H"w"t$$#

with w"0$ the initial condition such that &"t$( 0 and <k"t$wk"t$= 0 for all k= 1#2# 7 7 7 #K. !

Acknowledgments. This work was supported by Science Foundation of Ireland Grants 03/IN3/I396 and07/IN.1/I901. A preliminary version of this work was presented at the 2008 Information Theory and ApplicationsWorkshop at the University of California, San Diego, January 27–February 1, 2008. The author expresses hisgratitude to Anatolii Puhalskii for his detailed comments and suggestions to improve the presentation of aprevious version of this paper and for his encouragement. The author is indebted to Ken Duffy for detailedsuggestions and comments on a previous version of this paper that were critical in expanding its scope andimproving its presentation. The author also thanks the anonymous reviewers and editor for pointing out errors,omissions, and stylistic faults in an earlier version of this paper. Thanks are also due to Somsak Kittipiyakul fora detailed reading of this paper and for pointing out important typographical errors. Finally, many thanks areowed to David Malone for help with the simulations and to Helena Smigoc and Barak Pearlmutter for aidingthe author in obtaining a copy of Brézis[7].

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