Qualitative Properties of alpha-Weighted Scheduling Policies

Qualitative Pr oper ties of α -W eighted Sch eduling P olicies De va vrat Shah MIT , LIDS Cambridge, MA 02139 de va vrat@mit.edu John N. Tsitsiklis MIT , LIDS Cambridge, MA 02139 jnt@mit.edu Y ua n Z hong MIT , LIDS Cambridge, MA 02139 zh yu411 8@mit.edu ABSTRA CT W e consider a switched netw ork, a fairly general constrained queueing netw ork mo del that has been used successfully to mod el the detailed pack et-level d ynamics in comm unication netw orks, such as inp ut-qu eued switches and w ireless net- w orks. The main op erational issue in th is mo del is th at of deciding whic h queues to serve, sub ject t o certain con- strain ts. In this p aper, we study qualitative p erformance prop erties of the well kn o wn α -w eigh ted scheduling policies. The stability , in the sense of positive recurrence, of these p olicies has b een well understo od . W e establish expon ential upp er b ound s on the tail of the steady -state distribut ion of the backlo g. Along the wa y , w e p ro ve finiteness of t h e ex- p ected steady-state backlog when α < 1, a prop erty that w as known only for α ≥ 1. Finally , we analyze the excur- sions of the maximum backlog o ver a finite time horizon for α ≥ 1. As a consequence, for α ≥ 1, we establish t h e full state space collapse property [17, 18]. 1. INTR ODUCTION This paper studies v arious qualitativ e stabilit y an d perfor- mance prop erties of the so-called α -we ighted p olicies, as ap- plied to a switched netw ork model (cf.[22, 18]). This mo del is a sp ecial case of t h e “stochastic pro cessing netw ork mo del” (cf.[10]), whic h has b ecome the canonical framew ork for the study of a large class of netw orked queueing systems, in- cluding systems arising in communications, m anufacturing, transp ortation, fin ancial markets, etc. The primary reason for th e p opu larit y of the switched netw ork mo del is its abil- it y t o faithfully mo del the b ehavior of a broad sp ectrum of netw orks at a fine granularit y . Specifically , th e switc hed netw ork model is useful in describing pack et-level ( “micro” ) b ehavior of med ium access in a wireles s netw ork and of the input-qu eued switc hes th at reside in side Internet routers. This model has prov ed tractable enough t o allo w for sub- stanti al progress in understanding the stabilit y and p erfo r- mance prop erties of v arious control p olicies. At a h igh level, th e switched netw ork mo del in vo lves a col- Permission to m ake digital or hard copies of all or part of this work for personal or classroom use is granted without fee provi ded that copies are not m ade or distribu ted for profit or commercial advanta ge and that copies bear this notice and the full cita tion on the first page. T o cop y otherwise, to republi sh, to post on serve rs or to redistrib ute to l ists, requi res prior specific permission and/or a fee. Copyri ght 200X ACM X-XXXXX-XX-X/XX/XX ...$10.00. lection of q ueues. W ork arrives to these q ueues exogenously or from another queue and gets serviced; it th en either leav es the netw ork or gets re-routed to another queue. Service at the queues requ ires the use of some commonly shared con- strained resources. This leads to th e p roblem of sche duling the service of pack ets queued in th e switched netw ork. T o utilize t he n etw ork resources efficien tly , a prop erly designed sc heduling p olicy is required. Of particular interest are the p opular Maxim um W eight or MW- α p olicies, in trodu ced in [22]. They are th e only k n ow n simple and universally appli- cable policies with p erforma nce guaran tees. I n addition, th e MW- α p olicy has serv ed as an imp ortant guide for design- ing implementable algorithms for input-q ueued switc hes and wireless medium access (cf.[14, 21, 8, 7, 16]). This motiv ates the wo rk in this p ap er, whic h focuses on certain qu alitativ e prop erties of MW- α policies. Related Prior W ork. Because of the significance of the α - w eigh ted p olicies, there is a large bo dy of researc h on their prop erties. W e pro vide here a brief ov erview of the work that is most relev an t to our purp oses. The most basic performance question concerns through- put and stability . F ormally , we sa y th at an algorithm is thr oughput optimal or stable if t he underlying netw ork Marko v chai n is p ositive recurrent whenever th e system is under- lo ade d . F or the MW- α policy , under a general enough stochas- tic mo del, stabilit y has b een established for any α > 0 (cf. [22, 15, 6, 1]). A second, fin er, p erformance question concerns th e ev alu- ation of the av erage backlog in the system, in steady- state. Bounds on the av erage b ac klog are usually obtained by con- sidering th e same stochastic Lyapunov function that w as used to prov e stability , and by building on th e drift ineq ual- ities established in t he course of the stability p roof; see, e.g., [5]. Using this approach, it is known that t he av erage exp ected backlog under α -weigh ted p olicies is finite, when α ≥ 1 ([12]). How ever, such a result is n ot known when α ∈ (0 , 1). An important p erformance analysis meth od that has emerged o ver the past few d ecades fo cuses on the he avy tr affic regime, in whic h the system is loaded near capacity . F or the switched mod el, hea vy traffic analysis has revealed some intri guing relations b etw een the p olicy parameter α and the p erfor- mance of the system through a phen omenon known as state sp ac e c ol lapse . In particular, in the heavy traffic limit and for an appropriately scaled version of the system, the state evol ves in a muc h low er-dimensional space (the state space “collapses” ). The structure of the collapsed state space pro- vides imp ortant information ab out th e system b ehavior (cf. [11, 17, 18]). Un der certain somewhat sp ecific assumptions, a complete heavy traffic analysis of the switched netw ork mod el has b een carried out in [19, 4]. How ever, for the more general switc hed net w ork mo del, only a w eaker result is av ailable, inv olving a so-called multiplicativ e state space collapse prop erty [18, 17]. State space collapse results are related to understand ing certain tran sient prop erties of the netw ork, such as the evolution of the qu eues o ve r a fi nite time h orizon. T o the b est of our knowledge, a transient analysis of th e switc hed netw ork mod el is not av ailable. A somewhat different approach fo cuses on tail probab ili- ties of the steady-state backlog and the associated large de- viation p rinciple ( LDP). This app roac h provides imp ortant insigh ts ab out the ov erfl o w probability in t h e p resence of fi- nite buffers. There hav e b een notable w orks in this direction, for sp ecific instances of th e switc h ed netw ork mod el, e.g., [20]. In a similar setting, the reference [13] has also estab- lished a LDP for the MW-1 policy , using Garcia’s exten ded- contra ction principle for quasi-contin u ous mappings. More recentl y , [23, 24] h as announced a characteriza tion of t h e precise tail b ehavior of th e (1 + α )-norm of the backlog, under the MW- α p olicy . How ev er, in these w orks, the LDP exp onent is only giv en implicitly , as th e solution of a compli- cated, p ossibly infi nite dimensional optimization problem. Our Contributions. W e establish v arious qu alitativ e p erfo rmance b ounds for α -w eigh ted p olicies, un der the switc hed netw ork model. I n the stationary regime, we establish fin iteness of the exp ected backlo g, and an exp onential u pp er b ound on the steady- state tail probabilities of the bac klog. In the transient regime, w e establish a maximal ineq ualit y on t he queue-size pro- cess, and the strong state space collapse prop erty under α - w eigh ted p olicies, when α ≥ 1. Our analysis is b ased on drift inequalities on suitable Lyapuno v fun ctions. Our metho ds, how ever, depart from prior work b ecause they rely on d if- feren t classes of Lyapuno v functions, and also inv olve some new techniques. In more detail, w e begin by establishing the finiteness of the steady-state exp ected bac klog under the MW- α p olicy , for any α ∈ (0 , 1). In stead of the traditional Lyapuno v func- tion k · k α +1 α +1 , we rely on a Lyapuno v function which is a suitably smo othed version of k · k 2 α +1 . W e con tinue by deriving a d rift inequality for a “norm” or “norm-like” Lyapuno v function, namely , k · k α +1 or a suit- ably smo othed version. Using the drift inequ alit y , we es- tablish exp onential tail b ound s for th e steady-state bac klog distribution u nder the MW- α p olicy , for any α ∈ (0 , ∞ ). Our metho d bu ilds on certain results from [3] t hat allo w us to translate drift inequalities into closed-form tail boun d s; it yields an explicit b ound on the tail exp onent, in terms of the system load and t he total number of queues. This is in con trast with the earlier wo rk in [20, 24]. That work provides an exact but implicit characterization of the tail exp onents, in terms of a complicated optimization p roblem, and provides n o immediate insights on the dep endence of the tail exp onen ts on the system parameters, such as the load and t h e number of queues. F urthermore, in contrast to the sophisticated math ematical techniques used in [20, 24], our explicit b ou n ds are obtained th rough elementary metho d s. F or some additional p ersp ective, we also consider a sp ecial case and compare our upp er b oun d with av ailable lo w er b ou n ds. Finally , we p ro vide a transient analysis under MW- α p oli- cies, for the case where α ≥ 1. W e use a Lya punov drift inequality to ob t ain a b oun d on the probability that the maximal bac klog ov er a given fin ite time interv al exceeds a certain threshold. This b ound leads to the resolution of the strong state space collapse conjecture for the switc hed net- w ork mo del when α ≥ 1. This strengthens th e multiplicativ e state space collapse results in [17, 18]. Or ganization of the P aper. The rest of the pap er is organized as follow s. In S ection 2, we define the notation we will emplo y , and describ e the switc hed netw ork mo del. In Section 3, w e provide formal statements of our main results. In Section 4, we establish a drift inequalit y for a suitable Lyapunov function, which will b e key to t he pro of of the exp onential upp er b ound on tail probabilities. In Section 5, we pro ve the fi niteness of steady-state exp ected bac klog when α ∈ (0 , 1). W e prove the exp onential upp er b ound in Section 6. F or a sp ecial instance, we compare this u pp er b ound with a v ailable lo w er b ounds in the App endix. The transient analysis is presented in Section 7. W e start with a general lemma, and sp ecialize it t o obt ain a maximal ineq ualit y u nder t h e MW- α p olicy , for α ≥ 1. W e then apply the latter inequalit y to prov e th e full state space collapse result for α ≥ 1. W e conclude the pap er with a b rief discussion in Section 8. 2. MODEL AND NO T A TION 2.1 Notation W e introd uce here th e notation that will b e employ ed throughout the pap er. W e denote the real vector sp ace of dimension M by R M and the set of nonnegative M -tuples by R M + . W e write R for R 1 , and R + for R 1 + . W e let Z b e the set of integers, Z + the set of nonnegative integers, and N the set of p ositiv e integers. F or any vector x ∈ R M , and any α > 0, we defin e k x k α = M X i =1 | x i | α ! 1 /α . F or an y tw o vectors x = ( x i ) M i =1 and y = ( y i ) M i =1 of the same dimension, w e let x · y = P M i =1 x i y i b e the dot prod uct of x and y . F or tw o real num b ers x and y , we let x ∨ y = max { x, y } . W e also let [ x ] + = x ∨ 0.W e introduce th e Kroneck er delta sy mbol δ ij , d efined as δ ij = 1 if i = j , and δ ij = 0 if i 6 = j . W e let e i = ( δ ij ) M j =1 b e the i -t h u nit vector in R M , and 1 the vector of all ones. F or a set S , we den ote its cardinalit y by | S | , and its indicator function by I S . F or a matrix A , we let A T denote its transp ose. W e will also use the abb reviations “RHS/LHS” for “rig ht/lef t-hand side,” and “iff ” for “if and only if.” 2.2 Switched Network Mode l The Model. W e adopt the mo del in [18], while restricting to the case of single-hop n etw orks, for ease of exp osition. Ho w ever, our re- sults natu rally extend to mul ti-hop mod els, under the “back- pressure” v ariant of th e MW- α p olicy . Consider a collection of M q ueues. Let time be discrete: timeslot τ ∈ { 0 , 1 , . . . } ru ns from t ime τ to τ + 1. Let Q i ( τ ) den ote t h e (nonnegative integ er) length of queue i ∈ { 1 , 2 , . . . , M } at the b eginning of timeslot τ , and let Q ( τ ) b e the vector ( Q i ( τ )) M i =1 . Let Q (0) b e the vector of initial queue lengths. During each timeslot τ , th e queue vector Q ( τ ) is offered service described by a vector σ ( τ ) = ( σ i ( τ )) M i =1 draw n from a give n finite set S ⊂ { 0 , 1 } M of fe asible sche dules . Each queue i ∈ { 1 , 2 , . . . , M } has a ded icated exogenous arriv al process ( A i ( τ )) τ ≥ 0 , where A i ( τ ) denotes the num b er of pack- ets that arrive to queue i up to the b eginnin g of timeslot τ , and A i (0) = 0 for all i . W e also let a i ( τ ) = A i ( τ + 1) − A i ( τ ), whic h is the number of pack ets that arrive to qu eue i du r- ing timeslot τ . F or simplicity , w e assume that the a i ( · ) are indep endent Bernoulli pro cesses with parameter λ i . W e call λ = ( λ i ) M i =1 the arrival r ate ve ctor . Give n the service schedule σ ( τ ) ∈ S c hosen at timeslot τ , the queues evolv e according to th e relation Q i ( τ + 1) =  Q i ( τ ) − σ i ( τ )  + + a i ( τ ) . In order to av oid t riv ialities, w e assume, throughout the pap er, the follo wing. Assumption 2.1. F or every queue i , ther e exists a σ ∈ S such that σ i = 1 . An example: Input Queued (I Q) Switches. The switc hed n etw ork mod el captures imp ortant instances of comm unication netw ork scenarios (see [18] for v arious ex- amples). Specifically , it faithfully models the pac ket-lev el operation of an inp ut-queu ed ( IQ) switch inside an Internet router. F or an m -p ort I Q switch, it has m input and m output p orts. It has a separate queue for each input-out p ut pair ( i, j ), denoted by Q ij , 1 for a total of M = m 2 queues. A sc hedule is required to match eac h in p ut to exactly one output, and eac h output to exactly one input. Therefo re, the set of sc hedules S is ( σ = ( σ ij ) ∈ { 0 , 1 } m × m : m X k =1 σ ik = m X k =1 σ kj = 1 , ∀ i, j ) . W e assume that t h e arriv al process at each q ueue Q ij is an indep endent Bernoulli pro cess with mean λ ij . Capacity Re gion. W e define the capacity region Λ of a switc hed netw ork as ( λ ∈ R M + : λ ≤ X σ ∈ S α σ σ , α σ ≥ 0 , ∀ σ ∈ S , X σ ∈ S α σ < 1 ) . It is called the capacity region b ecause there exists a policy for whic h the Marko v chain describing th e netw ork is p os- itive recurren t iff λ ∈ Λ . W e define the lo ad induced by λ ∈ Λ , den oted by ρ ( λ ), as ρ ( λ ) = inf ( X σ ∈S α σ : λ ≤ X σ ∈ S α σ σ , α σ ≥ 0 , ∀ σ ∈ S ) . Note that ρ ( λ ) < 1, for all λ ∈ Λ . 1 Here we d eviate from our conv entio n of indexin g queues by a single subscript. This will ease exp osition in the context of IQ switc hes, without causing confusion. The Maximum-W eight- α P olicy. W e now describe th e so-called Maximum-Weight- α (MW- α ) p olicy . F or α > 0, we use Q ( τ ) α to denote the v ector ( Q α i ( τ )) M i =1 . W e defin e the weight of schedule σ ∈ S to b e σ · Q ( τ ) α . The MW- α p olicy chooses, at each timeslot τ , a sc hedule with t h e largest weigh t (breaking ties arbitrarily). F ormally , during timeslot τ , the p olicy c ho oses a sc hedule σ ( τ ) that satisfies σ ( τ ) · Q ( τ ) α = max σ ∈ S σ · Q ( τ ) α . W e define t h e maximum α - w eigh t of t he queu e length ve ctor Q by w α ( Q ) = max σ ∈ S σ · Q α . When α = 1, the policy is simply called the MW p olicy , and w e use the notation w ( Q ) instead of w 1 ( Q ). W e take note of the fact that under the MW- α p olicy , the resulting Marko v chain is known to b e p ositiv e recurrent, for an y λ ∈ Λ (cf. [15]). 3. SUMMAR Y OF RESUL TS In this section, we summarize our main results for b oth the steady-state and the transient regime. The pro ofs are giv en in su b sequent sections. 3.1 Stationary r egi me The Mark o v chain Q ( · ) that describ es a switc hed netw ork operating under the MW- α p olicy is kn o wn to b e p ositive recurrent, as long as the system is un derloaded, i.e., if λ ∈ Λ or, equiv alen tly , ρ ( λ ) < 1. I t is n ot hard to verify that this Marko v c hain is irreducible and ap erio dic. Therefore, there exists a unique stationary distribution, which we will denote by π . W e use E π and P π to denote exp ectations and probabilities und er π . F initenes s of Expected Queue-Size. W e establish that under the MW- α p olicy , the steady- state exp ected queue- size is finite, for any α ∈ (0 , 1). (Recall that this result is already known when α ≥ 1.) Theorem 3.1. C onsider a switche d network op er ating un- der the MW- α p olicy wi th α ∈ (0 , 1) , and assume that ρ ( λ ) < 1 . Then, the ste ady-state exp e cte d queue-size is finite, i.e., E π [ k Q k 1 ] < ∞ . Expone ntial Boun d on T a il Pr obabilities. F or the MW- α policy , and for an y α ∈ (0 , ∞ ), w e obtain an ex p licit exp onential up p er b oun d on the tail probabilities of the q ueue-size, in steady-state. Our result invo lves tw o constants defi ned by ¯ ν = E  k a (1) k α +1  , γ = 1 − ρ 2 M α α +1 , where ρ = ρ ( λ ). Theorem 3.2. C onsider a switche d network op er ating un- der the MW- α p olicy, and assume that ρ = ρ ( λ ) < 1 . T her e exist p ositive c onstants B and B ′ such that for al l ℓ ∈ Z + : (a) if α ≥ 1 , then P π  k Q ( τ ) k α +1 > B + 2 M 1 α +1 ℓ  ≤  ¯ ν ¯ ν + γ  ℓ +1 ; (b) if α ∈ (0 , 1) , then P π  k Q ( τ ) k α +1 > B ′ + 10 M 1 α +1 ℓ  ≤  5 ¯ ν 5 ¯ ν + γ  ℓ +1 . Note that Theorem 3.1 could b e obtained as a simple corollary of Theorem 3.2. On the other hand , our pro of of Theorem 3.2 requires the finiteness of E π [ k Q k 1 ], and so Theorem 3.1 need s to b e established fi rst. In th e App endix, we comment on the tightness of our upp er b oun ds by comparing them with explicit low er b ou n ds that follo w from the recent large d ev iations results in [24]. 3.2 T ransient r egime Here w e p rovide a simple inequality on the maximal ex- cursion of t he queue- size o ver a finite time interv al, under the MW- α p olicy , with α ≥ 1. Theorem 3.3. C onsider a switche d network op er ating un- der the MW- α p olicy wi th α ≥ 1 , and assume that ρ ( λ ) < 1 . Supp ose that Q (0) = 0 . L et Q max ( τ ) = max i ∈{ 1 ,...,M } Q i ( τ ) , and Q ∗ max ( T ) = max τ ∈{ 0 , 1 ,...,T } Q max ( τ ) . Then, for any b > 0 , P ( Q ∗ max ( T ) ≥ b ) ≤ K ( α, M ) T (1 − ρ ) α − 1 b α +1 , (1) for some p ositive c onstant K ( α, M ) dep ending only on α and M . As an imp ortant app lication, we use Theorem 3.3 to prov e a full state space collapse result, 2 for α ≥ 1, in Section 7.3. The precise statement can b e found in Theorem 7.7. 4. MW - α PO LICIES: A USEFUL DRIFT IN- EQU ALITY The key to many of our results is a drift ine quality that holds for every α > 0 and λ ∈ Λ . In th is section, we shall state and prove this inequality . It will be used in Section 6 to prove Theorem 3.2. W e remark th at similar d rift inequal- ities, b ut for a differen t Lyapunov function, ha ve pla yed an imp ortant role in establishing p ositive recurrence (cf. [22]) and multiplicativ e state space collapse (cf. [18]). W e will b e making extensive use of a second-order mean v alue th eorem [2], which we state b elow for easy reference. Pr oposition 4.1. L et g : R M → R b e twic e c ontinuously differ entiable over an op en spher e S c enter e d at a ve ctor x . Then, for any y such that x + y ∈ S , ther e exists a θ ∈ [0 , 1] such that g ( x + y ) = g ( x ) + y T ∇ g ( x ) + 1 2 y T H ( x + θ y ) y , (2) wher e ∇ g ( x ) is the gr adient of g at x , and H ( x ) is the Hes- sian of the f unction g at x . W e now define th e Lyapunov function that w e will employ . F or α ≥ 1, it will b e simply t he ( α + 1)-norm k x k 1+ α of a vector x . Ho w eve r, when α ∈ (0 , 1), this function has unbound ed second deriv ativ es as we approac h the boun dary of R M + . F or this reason, our Lyapuno v function will b e a suitably smo othed version of k · k α +1 . 2 This is strong state space colla pse and not full diffusion approximatio n. Definition 4.2. Define f α : R + → R + to b e f α ( r ) = r α , when α ≥ 1 , and f α ( r ) =  r α , if r ≥ 1 , ( α − 1) r 3 + (1 − α ) r 2 + r, if r ≤ 1 , when α ∈ (0 , 1) . L et F α : R + → R + b e the antiderivative of f α , so that F α ( r ) = R r 0 f α ( s ) ds . The Lyapunov function L α : R M + → R + is define d to b e L α ( x ) = " ( α + 1) M X i =1 F α ( x i ) # 1 α +1 . W e will make heavy use of v arious prop erties of th e func- tions f α , F α , and L α , which we summarize in the follo wing lemma. The pro of is elemen tary and is omitted. Lemma 4.3. Le t α ∈ (0 , 1) . The function f α has the f ol- lowing pr op erties: (i) i t i s c ontinuously differ entiable with f α (0) = 0 , f α (1) = 1 , f ′ α (0) = 1 , and f ′ α (1) = α ; (ii) it is incr e asing and, in p articular, f α ( r ) ≥ 0 for al l r ≥ 0 ; (iii) we have r α − 1 ≤ f α ( r ) ≤ r α + 1 , for al l r ∈ [0 , 1] ; (iv) f ′ α ( r ) ≤ 2 , f or al l r ≥ 0 . F urthermor e, f r om (iii), we al so have the fol lowing pr op erty of F α : (iii’) r α +1 − 2 ≤ ( α + 1) F α ( r ) ≤ r α +1 + 2 for al l r ≥ 0 . W e are no w ready to state the drift inequ alit y . Theorem 4.4. C onsider a switche d network op er ating un- der the MW- α p ol icy, and assume that ρ = ρ ( λ ) < 1 . Then, ther e exists a c onstant B > 0 , such that if L α ( Q ( τ )) > B , then E [ L α ( Q ( τ + 1)) − L α ( Q ( τ )) | Q ( τ )] ≤ − 1 − ρ 2 M 1 α +1 − 1 . (3) The pro of of th is drift inequality is quite tedious when α 6 = 1. T o make the pro of more accessible an d t o provide intuition, w e first presen t the somewhat simpler pro of for α = 1. W e then p ro vide t h e pro of for th e case of general α , by consid- ering separately the tw o cases where α > 1 and α ∈ (0 , 1). W e wish to d ra w attentio n here to the main difference from related d rift inequalities in the literature. The u sual proof of stabilit y inv olve s the Lya punov funct ion k Q k α +1 α +1 ; for instance, for the stand ard MW p olicy , it inv olve s a quadratic Lyapuno v function. In contrast, we use k Q k α +1 (or its smoothed ve rsion), whic h scales linearly along radial direc- tions. I n this sense, our approach is similar in spirit to [3], whic h employ ed piecewise linear Lyapunov fun ctions t o de- rive drift inequ alities and then moment and tail b ounds. 4.1 Pr oof of Theor em 4.4: α = 1 In th is section, w e assume t h at α = 1. A s remarked ear- lier, we h av e L α ( x ) = k x k 2 . Supp ose that k Q ( τ ) k 2 > 0. W e claim that on every sam- ple path, we hav e k Q ( τ + 1) k 2 − k Q ( τ ) k 2 ≤ Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2 k Q ( τ ) k 2 , (4) where δ ( τ ) = Q ( τ + 1) − Q ( τ ). T o see this, we pro ceed as follo ws. W e hav e  k Q ( τ ) k 2 + Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2 k Q ( τ ) k 2  2 ≥ k Q ( τ ) k 2 2 + 2  Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2  ≥ k Q ( τ ) k 2 2 + 2 Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2 = k Q ( τ ) + δ ( τ ) k 2 2 = k Q ( τ + 1) k 2 2 . (5) Note that k Q ( τ ) k 2 2 + Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2 =     Q ( τ ) + δ ( τ ) 2     2 2 + 3 4 k δ ( τ ) k 2 2 ≥ 0 . W e divide by k Q ( τ ) k 2 , to obt ain k Q ( τ ) k 2 + Q ( τ ) · δ ( τ ) + k δ ( τ ) k 2 2 k Q ( τ ) k 2 ≥ 0 . Therefore, w e can take square ro ots of b oth sides of (5), without reversi ng the direction of the ineq uality , and the claimed inequ alit y (4) follow s. Recall that | δ i ( τ ) | ≤ 1, b ecause of the Bernoulli arriv al assumption. It follo ws that k δ ( τ ) k 2 ≤ M 1 / 2 . W e now take the conditional expectation of b oth sides of (4). W e hav e E h k Q ( τ + 1) k 2 − k Q ( τ ) k 2    Q ( τ ) i ≤ E  Q ( τ ) · a ( τ ) − Q ( τ ) · σ ( τ ) + M k Q ( τ ) k 2    Q ( τ )  = P M i =1 Q i ( τ ) E [ a i ( τ )] − Q ( τ ) · σ ( τ ) + M k Q ( τ ) k 2 = P M i =1 Q i ( τ ) λ i − w ( Q ( τ )) + M k Q ( τ ) k 2 ≤ M − (1 − ρ ) w ( Q ( τ )) k Q ( τ ) k 2 . (6) The last inequalit y above is justified as follo ws. F rom the definition of ρ = ρ ( λ ), there exist constan ts α σ ≥ 0 such that P σ ∈S α σ ≤ ρ , and λ ≤ X σ ∈ S α σ σ . (7) Therefore, X i Q i ( τ ) λ i = Q ( τ ) · λ ≤ X σ ∈ S α σ Q ( τ ) · σ ≤ X σ ∈ S α σ w ( Q ( τ )) ≤ ρw ( Q ( τ )) . (8) Let Q max ( τ ) = max M i =1 Q i ( τ ). Then , k Q ( τ ) k 2 ≤ ( M Q 2 max ( τ )) 1 2 = M 1 2 Q max ( τ ) . F rom Assumption 2.1, w e ha ve w ( Q ( τ )) ≥ Q max ( τ ) . Therefore, th e R HS of (6) can be u pp er b ounded by − (1 − ρ ) M − 1 / 2 + M k Q ( τ ) k 2 ≤ − 1 2 (1 − ρ ) M − 1 / 2 , when k Q ( τ ) k 2 is sufficiently large. 4.2 Pr oof of Theor em 4.4: α > 1 W e wish to obtain an inequality similar to (6) for L α ( Q ( · )) = k Q ( · ) k 1+ α under the MW- α p olicy , and we accomplish this using th e second-order mean v alue theorem (cf. Prop osition 4.1). Throughout this pro of, we will drop the subscript α + 1 and use th e notation k · k instead of k · k α +1 . Consider the norm function g ( x ) = k x k = ( x α +1 1 + . . . + x α +1 M ) 1 α +1 . The first deriv ative is ∇ g ( x ) = k x k − α ( x α 1 , . . . , x α M ) = x α k x k α . Let H ( x ) = [ H ij ( x )] M i,j =1 b e the second deriv ative (Hessian) matrix of g . Then, H ij ( x ) = ∂ 2 g ∂ x i ∂ x j ( x ) = δ ij αx α − 1 i k x k α − αx α i x α j k x k 2 α +1 , where δ ij is the Kroneck er delta. By Prop osition 4.1, for any x , y ∈ R M + , and with δ = y − x , there exists a θ ∈ [0 , 1] for whic h g ( y ) = g ( x ) + δ T ∇ g ( x ) + 1 2 δ T H ( x + θ δ ) δ = g ( x ) + k x k − α X i δ i x α i ! + α 2 k x + θ δ k − α X i ( x i + θδ i ) α − 1 δ 2 i ! − α 2 k x + θ δ k − 1 − 2 α X i,j ( x i + θ δ i ) α ( x j + θ δ j ) α δ i δ j ! = g ( x ) + k x k − α X i δ i x α i ! + α 2 k x + θ δ k − α X i ( x i + θδ i ) α − 1 δ 2 i ! − α 2 k x + θ δ k − 1 − 2 α X i ( x i + θ δ i ) α δ i ! 2 . Using x = Q ( τ ), y = Q ( τ + 1) and δ ( τ ) = Q ( τ + 1) − Q ( τ ), w e hav e k Q ( τ + 1) k = k Q ( τ ) k +  P i δ i ( τ ) Q α i ( τ ) k Q ( τ ) k α  + α 2  P i ( Q i ( τ ) + θ δ i ( τ )) α − 1 δ 2 i ( τ ) k Q ( τ ) + θ δ ( τ ) k α  − α 2 "  P i ( Q i ( τ ) + θ δ i ( τ )) α δ i ( τ )  2 k Q ( τ ) + θ δ ( τ ) k 1+2 α # . (9) Therefore, using th e fact that δ i ( τ ) ∈ {− 1 , 0 , 1 } , we h a ve k Q ( τ + 1) k − k Q ( τ ) k ≤  P i δ i ( τ ) Q α i ( τ ) k Q ( τ ) k α  + α 2  P i ( Q i ( τ ) + θ δ i ( τ )) α − 1 k Q ( τ ) + θ δ ( τ ) k α  . (10) W e take conditional exp ectations of b oth sides, give n Q ( τ ). T o b ound the first term on the R HS, w e use t he definition of th e MW- α p olicy , the b ou n d (7) on λ , and t he argument used to establish (8) in th e pro of of Theorem 4.4 for α = 1 (with w ( Q ( τ )) replaced by w α ( Q ( τ ))). W e obtain E  P i δ i ( τ ) Q α i ( τ ) k Q ( τ ) k α    Q ( τ )  ≤ − (1 − ρ ) w α ( Q ( τ )) k Q ( τ ) k α . (11) Note that k Q ( τ ) k α ≤  M Q max ( τ ) α +1  α α +1 = M α α +1 Q α max ( τ ) , (12) and w α ( Q ( τ )) ≥ Q α max ( τ ) . Therefore, E  P i δ i ( τ ) Q α i ( τ ) k Q ( τ ) k α    Q ( τ )  ≤ − (1 − ρ ) M − α 1+ α . (13) Consider now the second term of the conditional exp ec- tation of the RHS of I nequality (10). Since α > 1, and δ i ( τ ) ∈ {− 1 , 0 , 1 } , the numerator of the expression inside the brack et satisfies X i ( Q i ( τ ) + θ δ i ( τ )) α − 1 ≤ M ( Q max ( τ ) + 1) α − 1 , and the denominator satisfies k Q ( τ ) + θ δ ( τ ) k α ≥  [ Q max ( τ ) − 1] +  α , where we u se the notation [ c ] + = 0 ∨ c . Th us, α 2  P i ( Q i ( τ ) + θ δ i ( τ )) α − 1 k Q ( τ ) + θ δ ( τ ) k α  ≤ α 2 · M ( Q max + 1) α − 1 ([ Q max ( τ ) − 1] + ) α . Now if k Q ( τ ) k is large enough, Q max ( τ ) is large enough, and α 2 · M ( Q max +1) α − 1 ([ Q max ( τ ) − 1] + ) α can b e made arbitrarily small. Thus, the conditional exp ectation of the second term on th e RHS of (10) can b e made arbitrarily small for large enough k Q ( τ ) k . This fact, together with Inequality (13), implies th at there exists B > 0 such th at if k Q ( τ ) k > B , then E h k Q ( τ + 1) k − k Q ( τ ) k    Q ( τ ) i ≤ − 1 − ρ 2 M − α 1+ α . 4.3 Proof of Theor em 4.4: α ∈ (0 , 1) The pro of in this section is similar to that for th e case α > 1. W e in vo ke Prop osition 4.1 to write the drift term as a sum of terms, which w e b ound separately . N ote that to use Prop osition 4.1, w e need L α to b e t wice contin uously differentiable. Ind eed, by L emma 4.3 (i), f α is contin uously differentiable, so its antideriv ative F α is t wice con tinuously differentiable, and so is L α . Thus, by the second order mean v alue theorem, w e obtain an equation similar to Eq uation (9): L α ( Q ( τ + 1)) − L α ( Q ( τ )) =  P i δ i ( τ ) f α ( Q i ( τ )) L α α ( Q ( τ ))  + 1 2  P i f ′ α ( Q i ( τ ) + θ δ i ( τ )) δ 2 i ( τ ) L α α ( Q ( τ ) + θ δ ( τ ))  − α 2  ( P i δ i ( τ ) f α ( Q i ( τ ) + θ δ i ( τ ))) 2 L 2 α +1 α ( Q ( τ ) + θ δ ( τ ))  . (14) Again, using th e fact δ i ( τ ) ∈ {− 1 , 0 , 1 } , L α ( Q ( τ + 1)) − L α ( Q ( τ )) ≤ T 1 + T 2 , where T 1 = P i δ i ( τ ) f α ( Q i ( τ )) L α α ( Q ( τ )) , and T 2 = 1 2  P i f ′ α ( Q i ( τ ) + θ δ i ( τ )) L α α ( Q ( τ ) + θ δ ( τ ))  . Let us consider T 2 first. F or α ∈ (0 , 1), by Lemma 4.3 (iv), f ′ α ( r ) ≤ 2 for all r ≥ 0. Thus T 2 ≤ 1 2  2 M L α α ( Q ( τ ) + θ δ ( τ ))  = M L α α ( Q ( τ ) + θ δ ( τ )) . whic h b ecomes arbitrarily small when L α ( Q ( τ )) is large enough. W e now consider T 1 . Since f α ( r ) ≤ r α + 1 for all r ≥ 0 (cf. Lemma 4.3 (iii)), and δ i ( τ ) ∈ {− 1 , 0 , 1 } , T 1 ≤ P i δ i ( τ ) Q α i ( τ ) L α α ( Q ( τ )) + M L α α ( Q ( τ )) . When we take the conditional exp ectation, an argument sim- ilar to the one for the case α > 1 yields E  P i δ i ( τ ) Q α i ( τ ) L α α ( Q ( τ ))    Q ( τ )  ≤ − (1 − ρ ) w α ( Q ( τ )) L α α ( Q ( τ )) . (15) Again, as b efore, w α ( Q ( τ )) ≥ Q α max ( τ ). F or the denomi- nator, by Lemma 4.3 (iii’), for any r ≥ 0, we hav e ( α + 1) F α ( r ) ≤ r α +1 + 2. Thus L α ( Q ( τ )) ≤ " X i ( Q i ( τ ) + 2) α +1 # 1 α +1 ≤  M ( Q max ( τ ) + 2) α +1  1 α +1 = M 1 α +1 ( Q max ( τ ) + 2) . Therefore, E  P i δ i ( τ ) Q α i ( τ ) L α α Q ( τ )    Q ( τ )  ≤ − (1 − ρ ) M − α α +1 Q α max ( τ ) ( Q max + 2) α . If Q max ( τ ) is large en ough, we can further upp er b oun d the RHS by , say , − 3 4 (1 − ρ ) M − α α +1 . Putting everything together, we hav e E h L α ( Q ( τ + 1)) − L α ( Q ( τ ))    Q ( τ ) i ≤ − 3 4 (1 − ρ ) M − α 1+ α + M L α α ( Q ( τ )) + E [ T 2 | Q ( τ )] , (16) if Q max ( τ ) is large enough. As b efore, if L α ( Q ( τ )) is large enough, th en Q max ( τ ) is large enough, and T 2 and M L α α ( Q ( τ )) can b e made arbitrarily small. Thus, there exists B > 0 such that if L α ( Q ( τ )) > B , then E h L α ( Q ( τ + 1)) − L α ( Q ( τ ))    Q ( τ ) i ≤ − 1 2 (1 − ρ ) M − α 1+ α . 5. PROOF OF THEOREM 3.1 In this section, we fix some α ∈ (0 , 1) and prov e that the MW- α p olicy induces finite steady-state expected queue lengths. The key to our p ro of is the use of the Ly apunov function Φ( x ) = L 2 α ( x ). This is to b e contrasted with the use of t h e standard Lyapuno v function, P i x 1+ α i , in th e lit- erature, or the “norm” -Lyapunov function L α ( x ) that w e used in establishing the drift inequality of Theorem 4.4. Throughout the pro of, w e drop the subscript α from L α , F α , and f α , as th ey are clear from the context. W e also use k x k to denote the ( α + 1)-norm of th e vector x , again dropping the subscript. As u su al, w e consider the conditional ex p ected drift at time τ , D ( Q ( τ )) = E h Φ( Q ( τ + 1)) − Φ( Q ( τ ) )    Q ( τ ) i . Recall the n otation Q max ( τ ) = max { Q 1 ( τ ) , . . . , Q M ( τ ) } . Since for Q max < 2, D ( Q ( τ )) is b ound ed by a constant, w e assume throughout the proof th at Q max ( τ ) ≥ 2. As in the proof of Theorem 4.4 for the case α ∈ (0 , 1), we shall use the second order mean val ue theorem to obtain a bou n d on D ( Q ( τ )). Using the definition Φ( x ) = L 2 ( x ), we have [ ∇ Φ( x )] i = 2 L ( x ) ∂ L ( x ) ∂ x i = 2 f ( x i ) L 1 − α ( x ) , (17) and ∂ 2 Φ ∂ x i ∂ x j ( x ) = 2 ∂ L ( x ) ∂ x i · ∂ L ( x ) ∂ x j + 2 L ( x ) ∂ 2 L ( x ) ∂ x i ∂ x j = 2 f ( x i ) f ( x j ) L 2 α ( x ) + 2 L ( x )  δ ij f ′ ( x i ) L α ( x ) − αf ( x i ) f ( x j ) L 2 α +1 ( x )  = 2(1 − α ) f ( x i ) f ( x j ) L 2 α ( x ) + 2 δ ij f ′ ( x i ) L 1 − α ( x ) . (18) Using the second order mean v alue theorem and th e notation Q ( τ + 1) = Q ( τ ) + δ ( τ ), we h a ve, for some θ ∈ [0 , 1], Φ( Q ( τ + 1)) − Φ( Q ( τ ) ) ≤ 2 L 1 − α ( Q ( τ )) X i f ( Q i ( τ )) δ i ( τ ) ! + L 1 − α ( Q ( τ ) + θ δ ( τ )) X i f ′ ( Q i ( τ ) + θ δ i ( τ )) ! + (1 − α )  P i f ( Q i ( τ ) + θ δ i ( τ )) δ i ( τ )  2 L 2 α ( Q ( τ ) + θ δ ( τ )) . (19) Let us denote the three terms on the RHS of (19) as ¯ T 1 , ¯ T 2 and ¯ T 3 respectively , so that ¯ T 1 = 2 L 1 − α ( Q ( τ )) X i f ( Q i ( τ )) δ i ( τ ) ! , ¯ T 2 = L 1 − α ( Q ( τ ) + θ δ ( τ )) X i f ′ ( Q i ( τ ) + θ δ i ( τ )) ! , and ¯ T 3 = ( 1 − α )  P i f ( Q i ( τ ) + θ δ i ( τ )) δ i ( τ )  2 L 2 α ( Q ( τ ) + θ δ ( τ )) . W e consider t hese terms one at a time. a) By Lemma 4.3 (iii), f ( r ) ≤ r α + 1. Using th e fact that δ i ( τ ) ∈ {− 1 , 0 , 1 } , we obtain ¯ T 1 ≤ 2 L 1 − α ( Q ( τ )) M + X i Q α i ( τ ) δ i ( τ ) ! . When we take a conditional expectation, an argument sim- ilar to the one in earlier sections yields E " X i Q α i ( τ ) δ i ( τ )    Q ( τ ) # ≤ − (1 − ρ ) w α ( Q ( τ )) . Thus, E h ¯ T 1    Q ( τ ) i ≤ − 2(1 − ρ ) w α ( Q ( τ )) L 1 − α ( Q ( τ )) +2 M L 1 − α ( Q ( τ )) . In general, for r , s ≥ 0 and β ∈ [0 , 1], ( r + s ) β ≤ r β + s β . (20) Now , by Lemma 4.3 (iii’), r α +1 − 2 ≤ ( α + 1) F ( r ) ≤ r α +1 + 2, so X i x α +1 i − 2 M ≤ ( α + 1) X i F ( x i ) ≤ X i x α +1 i + 2 M . W e use inequality (20), with r = x α +1 i , s = 2 M , and β = (1 − α ) / (1 + α ) ∈ (0 , 1), to obtain L 1 − α ( x ) = ( α + 1) X i F ( x i ) ! 1 − α 1+ α ≤ 2 M + X i x α +1 i ! 1 − α 1+ α ≤ (2 M ) 1 − α 1+ α + X i x α +1 i ! 1 − α 1+ α = (2 M ) 1 − α 1+ α + k x k 1 − α . A simila r argument, based on inequality (20 ), with r = ( α + 1) F ( x i ) and s = 2 M , yields k x k 1 − α − (2 M ) 1 − α 1+ α ≤ L 1 − α ( x ) . W e also know that w α ( Q ( τ )) ≥ Q α max ( τ ) ≥ M − α α +1 k Q ( τ ) k α . Putting all th ese facts together, we obtain E h ¯ T 1    Q ( τ ) i ≤ − 2(1 − ρ ) w α ( Q ( τ )) L 1 − α ( Q ( τ )) + 2 M L 1 − α ( Q ( τ )) ≤ − 2(1 − ρ ) M − α α +1 k Q ( τ ) k α  k Q ( τ ) k 1 − α − (2 M ) 1 − α 1+ α  + 2 M  (2 M ) 1 − α 1+ α + k Q ( τ ) k 1 − α  = − 2(1 − ρ ) M − α α +1 k Q ( τ ) k + 2 M k Q ( τ ) k 1 − α + 2 2 1+ α (1 − ρ ) M 1 − 2 α 1+ α k Q ( τ ) k α + (2 M ) 2 1+ α . (21) b) W e no w consider the term ¯ T 2 . S in ce α ∈ (0 , 1), we hav e f ′ ( r ) ≤ 2 for all r ≥ 0. Since w e also hav e θ ∈ [0 , 1] and δ i ( τ ) ∈ {− 1 , 0 , 1 } , and using the fact that L 1 − α ( x ) ≤ (2 M ) 1 − α 1+ α + k x k 1 − α , we h a ve ¯ T 2 ≤ 2 M L ( Q ( τ ) + θ δ ( τ )) 1 − α ≤ 2 M  (2 M ) 1 − α 1+ α + k Q ( τ ) + θ δ ( τ ) k 1 − α  = (2 M ) 2 1+ α + 2 M k Q ( τ ) + θ δ ( τ ) k 1 − α . Now k Q ( τ ) + θ δ ( τ ) k ≤ k Q ( τ ) + 1 k ≤ k Q ( τ ) k + k 1 k = k Q ( τ ) k + M 1 α +1 . Since α ∈ (0 , 1), we h av e 0 < 1 − α < 1, and so k Q ( τ ) + θ δ ( τ ) k 1 − α ≤  k Q ( τ ) k + M 1 α +1  1 − α ≤ k Q ( τ ) k 1 − α + M 1 − α α +1 . Putting everything together, w e hav e ¯ T 2 ≤ (2 M ) 2 1+ α + 2 M  k Q ( τ ) k 1 − α + M 1 − α α +1  = (2 + 2 2 1+ α ) αM 2 1+ α + 2 M k Q ( τ ) k 1 − α . (22) c) W e fi n ally consider ¯ T 3 . F or notational convenience, w e write x = Q ( τ ) + θ δ ( τ ), and let x max = max { x 1 , . . . , x M } . Note t hat since δ i ( τ ) ∈ {− 1 , 0 , 1 } , θ ∈ [0 , 1], and we assumed that Q max ≥ 2, w e alw a ys ha ve x max ≥ 1. W e consider th e numerator and the denominator separately . First u se the facts that f ( r ) ≥ 0 for all r ≥ 0 (cf. Lemma 4.3 (ii)), and δ i ( τ ) ∈ {− 1 , 0 , 1 } , to obtain X i f ( x i ) δ i ( τ ) ! 2 ≤ X i f ( x i ) ! 2 . Since f is increasing in r (cf. Lemma 4.3 (ii)), X i f ( x i ) ! 2 ≤ ( M f ( x max )) 2 = M 2 f 2 ( x max ) . Thus, X i f ( x i ) δ i ( τ ) ! 2 ≤ M 2 f 2 ( x max ) . Next, since F ( r ) = R r 0 f ( s ) ds and f ≥ 0, w e h ave F ≥ 0 as w ell. Thus, L 2 α ( x ) = ( α + 1) X i F ( x i ) ! 2 α α +1 ≥ ( ( α + 1) F ( x max )) 2 α α +1 , and so ¯ T 3 ≤ (1 − α ) M 2 f 2 ( x max ) (( α + 1) F ( x max )) 2 α α +1 . W e will show th at ¯ T 3 is b ounded above by a p ositive con- stant, when ever x max ≥ 1. I ndeed, by Lemma 4.3 (iii) and (iii’), as x max → ∞ , f 2 ( x max ) x 2 α max → 1 and (( α + 1) F ( x max )) 2 α α +1 x 2 α max → 1 , so (1 − α ) M 2 f 2 ( x max ) (( α + 1) F ( x max )) 2 α α +1 → (1 − α ) M 2 as x max → ∞ . Using the contin uity of f and F for x max ≥ 1, it follo ws that there exists a constan t ˜ K > 0 su ch that ¯ T 3 ≤ ( 1 − α ) M 2 f 2 ( x max ) (( α + 1) F ( x max )) 2 α α +1 ≤ ˜ K , (23) whenever x max ≥ 1. Putting together the b ounds (21), (22 ), and (23) for ¯ T 1 , ¯ T 2 , and ¯ T 3 , resp ectively , w e conclude th at, for x max ≥ 1, D ( Q ( τ )) ≤ − 2(1 − ρ ) M − α α +1 k Q ( τ ) k + 2 M k Q ( τ ) k 1 − α + 4(1 − ρ ) M 1 − 2 α 1+ α k Q ( τ ) k α + (2 M ) 2 1+ α + (2 + 2 2 1+ α ) αM 2 1+ α + 2 M k Q ( τ ) k 1 − α + ˜ K = − ¯ A k Q ( τ ) k + C 1 k Q ( τ ) k 1 − α + C 2 k Q ( τ ) k α + K, (24) for some positive constants ¯ A , C 1 , C 2 and K . S ince α ∈ (0 , 1), th e k Q ( τ ) k t erm dominates. In particular, there exist p ositiv e constan ts A and D such that as long as max i Q i ( τ ) ≥ D , w e hav e D ( Q ( τ )) ≤ − A k Q ( τ ) k + K. (25) On the other hand, on th e bou nded set where max i Q i ( τ ) ≤ D , the drift D ( Q ( τ )) is also b ounded by a constant. By suitably redefi ning the constant K , we conclude that Eq. (25) holds for all p ossi ble v alues of Q ( τ ). The drift condition (25) is the standard F oster-Lyapuno v criterion for the Lya punov function Φ and implies the p os- itive recurrence of the Marko v chain Q ( · ) under the MW- α p olicy , for α ∈ (0 , 1). The irreducibility and ap erio dicity of the un derlying Marko v chain implies the existence of a unique stationary d istribu t ion π as w ell as ergodicity . Let Q ∞ b e a rand om v ariable distributed according to π . Then Q ( τ ) conv erges to Q ∞ in distribu t ion. Using Skorohod’s representa tion th eorem, we can embed the random vectors Q ( τ ) in a suitable probability space so that they converg e to Q ∞ almost surely . With this emb edding, k Q ( τ ) k → k Q ∞ k , and ( P T − 1 τ =0 k Q ( τ ) k ) /T → k Q ∞ k , almost surely . Using F a- tou’s Lemma, we hav e E [ k Q ∞ k ] = E " lim inf T →∞ 1 T T − 1 X τ =0 k Q ( τ ) k # ≤ lim inf T E " 1 T T − 1 X τ =0 k Q ( τ ) k # . On th e other h and, the drift ineq ualit y (25) is w ell k nown to imp ly that t he RH S ab ov e is finite; see, e.g., Lemma 4.1 of [7]. This pro ves t h at E [ k Q ∞ k ] < ∞ . By the equiv alence of norms, the result for k Q k 1 follo ws as well. 6. EXPONENTIAL BOU ND UNDER MW - α In this section we derive an exp onential up p er boun d on the tail probability of t h e stationary queue-size distribution, under the MW- α p olicy . 6.1 Proof of Theor em 3.2: α ≥ 1 The proof of Theorem 3.2 relies on t he follow ing prop osi- tion, and the d rift inequality established in Theorem 4.4. Pr oposition 6.1. Consider a swi tche d network op er ating under the MW- α p olicy with α ≥ 1 , and arrival r ate ve ctor λ with ρ = ρ ( λ ) < 1 . L et π b e the L et unique stationary distribution of the Markov chain Q ( · ) . Supp ose that f or al l τ ,    k Q ( τ + 1) k α +1 − k Q ( τ ) k α +1    ≤ ν max . F urthermor e, supp ose that for some c onstant s B > 0 and γ > 0 , and whenever k Q ( τ ) k 1+ α > B , we have E [ k Q ( τ + 1) k α +1 − k Q ( τ ) k α +1    Q ( τ )] ≤ − γ . Then for any ℓ ∈ Z + , P π  k Q ( τ ) k α +1 > B + 2 ν max ℓ  ≤  ¯ ν ¯ ν + γ  ℓ +1 . Proposition 6.1 follo ws immediately from the follo wing Lemma, whic h is a minor adaptation of Lemma 1 of [3]. An interes ted reader ma y refer to the pro of of Theorem 1(a) in [3] to see how Lemma 6.2 leads to the b ound claimed in Prop osition 6.1. Lemma 6.2. Under the same assumptions in Pr op osition 6.1, and for any c > B − ν max , P π ( k Q ( τ ) k α +1 > c + ν max ) ≤  ¯ ν ¯ ν + γ  P π ( k Q ( τ ) k α +1 > c − ν max ) . (26) Pr oof. Since this Lemma is a minor adaptation of Lemma 1 in [3], we only indicate the c hanges to the proof of Lemma 1 in [3 ] that lead to our claimed result. First let u s point out that the pro of in [3] makes u se of th e finiteness of the exp ected v alue of the Lyapunov function under th e station- ary distribution π . In our case, th e Lyapuno v funct ion in question is k · k α +1 , and the finiteness follo ws from Theorem 3.1 by noticing that all norms are equiv alen t. As in the proof of Lemma 1 in [3], define ˆ Φ( x ) = max { c, k x k α +1 } . Note that the maximal change in Φ( x ) in one time step is at most ν max . As in [3], we consider all x satisfying c − ν max < k x k α +1 ≤ c + ν max . Then, E  ˆ Φ( Q ( τ + 1)) | Q ( τ ) = x  − ˆ Φ( x ) ≤ X x ′ : k x ′ k > k x k p ( x , x ′ )( k x ′ k − k x k ) ≤ E  k a ( τ ) k  = ¯ ν . The pro of of Lemma 1 in [3] esentially used ν max as an upp er b ound on ¯ ν . F or our result, w e keep ¯ ν and then pro ceed as in the p roof in [3]. Completing the Pr oof of Theor em 3.2 ( α ≥ 1 ). Now the pro of of Theorem 3.2 follo ws immediately from Proposition 6.1 by noticing t h at Theorem 4.4 pro vides th e desired drift inequality , and the maximal c hange in k Q ( τ ) k 1+ α in one time step is at most ν max = M 1 1+ α , b ecause each queue can receive at most one arriv al and hav e at most one departure p er t ime step. 6.2 Proof of Theor em 3.2: α ∈ (0 , 1) The pro of for the case α ∈ (0 , 1) is entirel y parallel to that in the p revious section and we do not repro duce it here. 7. TRANSIENT ANAL YSIS In this section, w e present a transient analysis of th e MW- α p olicy with α ≥ 1. First w e present a general maximal lemma, which is then sp ecialized to the switched netw ork. In particular, w e p rove a drift inequality for the Lyapunov func- tion ˜ L ( x ) = 1 α +1 P i x α +1 i . W e com bine the drift inequ alit y with the maximal lemma to obtain a maximal inequality for the switc hed netw ork. W e then apply the maximal inequal- it y to p ro ve full state space collapse for α ≥ 1. 7.1 The Key Lemma Our analysis relies on the follo wing lemma: Lemma 7.1. Le t ( F n ) n ∈ Z + b e a filtr ation on a pr ob ability sp ac e. L et ( X n ) n ∈ Z + b e a nonne gative F n -adapte d sto chastic pr o c ess that satisfies E [ X n +1 | F n ] ≤ X n + B n (27) wher e B n ’s ar e nonne gative r andom variables (not ne c essar- ily F n -adapte d) with finite m e ans. L et X ∗ n = max { X 0 , . . . , X n } and supp ose that X 0 = 0 . Then, for any a > 0 and any T ∈ Z + , P ( X ∗ T ≥ a ) ≤ P T − 1 n =0 E [ B n ] a . This lemma is a simple consequ ence of the follo wing stan- dard maximal inequalit y for n onnegative sup ermartingales (see for example, Ex ercise 4, S ection 12.4, of [9]): Theorem 7.2. Le t ( F n ) n ∈ Z + b e a fil tr ation on a pr ob- ability sp ac e. L et ( Y n ) n ∈ Z + b e a nonne gative F n -adapte d sup ermartingale, i. e., for al l n , E [ Y n +1 | F n ] ≤ Y n . L et Y ∗ T = max { Y 0 , . . . , Y T } . Then, P ( Y ∗ T ≥ a ) ≤ E [ Y 0 ] a . Pr oof of Lemma 7.1. First note that if we take the con- ditional exp ectation on b oth sides of (27), give n F n , we h a ve E [ X n +1 | F n ] ≤ E [ X n | F n ] + E [ B n | F n ] = X n + E [ B n | F n ] . Fix T ∈ Z + . F or any n ≤ T , define Y n = X n + E " T − 1 X k = n B k    F n # . Then E [ Y n +1 | F n ] = E [ X n +1 | F n ] + E " E " T − 1 X k = n +1 B k    F n +1 #    F n # ≤ X n + E [ B n | F n ] + E " T − 1 X k = n +1 B k    F n # = Y n . Thus, Y n is an F n -adapted sup ermartingale; furthermore, by definition, Y n is non-n egativ e for all n . Therefore, by Theorem 7.2, P ( Y ∗ T ≥ a ) ≤ E [ Y 0 ] a = E h P T − 1 k =0 B k i a . But Y n ≥ X n for all n , since the B k are nonnegative. Th us P ( X ∗ T ≥ a ) ≤ P ( Y ∗ T ≥ a ) ≤ E h P T − 1 k =0 B k i a . W e h a ve th e follo wing corollary of Lemma 7.1 in which w e take all t he B n equal to the same constant: Cor ollar y 7.3. L et F n , X n and X ∗ n b e as in L emma 7.1. Supp ose that E [ X n +1 | F n ] ≤ X n + B , for al l n ≥ 0 , wher e B is a nonne gative c onstant. Then, for any a > 0 and any T ∈ Z + , P ( X ∗ T ≥ a ) ≤ B T a . 7.2 The Maxima l Inequality for Switched Net- works W e employ the Lyapunov function ˜ L ( x ) = 1 α + 1 M X i =1 x α +1 i , (28) to stu dy the MW- α p olicy . This is the Lyapuno v function that was u sed in [15] to establish p ositiv e recurrence of the chai n Q ( · ) under the MW- α policy . Below we fin e-tune the proof in [15] to obtain a more precise b ou n d. Lemma 7.4. Le t α ≥ 1 . F or a switche d network mo del op er ating under the MW- α p olicy with ρ = ρ ( λ ) < 1 , we have: E  ˜ L ( Q ( τ + 1)) − ˜ L ( Q ( τ ))   Q ( τ )  ≤ ¯ K ( α, M ) (1 − ρ ) α − 1 , (29) wher e ¯ K ( α, M ) is a c onstant dep ending only on α and M . Pr oof. W e employ the same strategy as in previous sec- tions. By th e second-order mean v alue th eorem, there exists θ ∈ [0 , 1] su ch that ˜ L ( Q ( τ + 1)) − ˜ L ( Q ( τ )) = 1 α + 1 M X i =1 (( Q i ( τ ) + δ i ( τ )) α +1 − Q α +1 i ( τ )) = M X i =1 Q α i ( τ ) δ i ( τ ) + M X i =1 α ( Q i ( τ ) + θ δ i ( τ )) α − 1 δ 2 i ( τ ) . Let us b oun d the second term on the RHS. W e hav e M X i =1 α ( Q i ( τ ) + θ δ i ( τ )) α − 1 δ 2 i ( τ ) ≤ M X i =1 α ( Q i ( τ ) + θ ) α − 1 ≤ M X i =1 α ( Q i ( τ ) + 1) α − 1 ≤ α M X i =1 (2 α − 1 Q α − 1 i ( τ ) + 1) = α 2 α − 1 M X i =1 Q α − 1 i ( τ ) + αM ≤ α 2 α − 1 M Q α − 1 max ( τ ) + αM . The third inequality follo ws b ecause when Q i ( τ ) ≥ 1, ( Q i ( τ )+ 1) α − 1 ≤ ( 2 Q i ( τ )) α − 1 = 2 α − 1 Q α − 1 i ( τ ), an d when Q i ( τ ) = 0, ( Q i ( τ ) + 1) α − 1 = 1. Let us now take conditional exp ectations. F rom Section 4 , w e know that E " M X i =1 Q α i ( τ ) δ i ( τ )   Q ( τ ) # ≤ − (1 − ρ ) w α ( Q ( τ )) ≤ − (1 − ρ ) Q α max ( τ ) . Thus, if we combine t h e inequalities ab ov e, we hav e E  ˜ L ( Q ( τ + 1)) − ˜ L ( Q ( τ ))   Q ( τ )  ≤ − (1 − ρ ) Q α max ( τ ) + α 2 α − 1 M Q α − 1 max ( τ ) + αM . (30) It is a simple exercise in calculus to see that the RH S of (30 ) is maximized at Q max ( τ ) = ( α − 1)2 α − 1 M / (1 − ρ ), giving the maximum va lue ( α − 1) α − 1 2 α ( α − 1) M α (1 − ρ ) α − 1 + αM = O ((1 − ρ ) 1 − α ) . Pr o of of Theorem 3. 3. Let b > 0. Then P ( Q ∗ max ( T ) ≥ b ) = P  1 α + 1  Q ∗ max ( T )  α +1 ≥ 1 α + 1 b α +1  ≤ P  max τ ∈{ 0 ,...,T } ˜ L ( Q ( τ )) ≥ 1 α + 1 b α +1  . Now , by Lemma 7.4 and Corollary 7.3, P  max τ ∈{ 0 ,...,T } ˜ L ( Q ( τ )) ≥ 1 α + 1 b α +1  ≤ ( α + 1) ¯ K ( α, M ) T (1 − ρ ) α − 1 b α +1 = K ( α, M ) T (1 − ρ ) α − 1 b α +1 , where K ( α, M ) = ( α + 1) ¯ K ( α, M ). 7.3 Full State Space Collapse f or α ≥ 1 Throughout this section, w e assume that we are giv en α ≥ 1, and correspondingly , th e Lyapunov function ˜ L ( x ) = 1 α +1 P M i =1 x α +1 i . T o state the full state space collapse result for α ≥ 1, we need some p reliminary definitions and the statement of t he multiplicati ve state sp ace collapse result. Let Σ b e the conv ex hull of S (the set of feasible schedules), and let ¯ Λ b e defi ned by ¯ Λ = n λ ∈ R M + : λ ≤ σ ′ compon entwi se, for some σ ′ ∈ Σ o . Note th at this is the closure of the capacity region Λ defined earlier. Recall the defin ition of the lo ad ρ ( λ ) of an arriv al rate vector λ . It is clear th at λ ∈ ¯ Λ iff ρ ( λ ) ≤ 1. Define ∂ Λ the set of critic al arriv al rate vectors: ∂ Λ = ¯ Λ − Λ = n λ ∈ ¯ Λ : ρ ( λ ) = 1 o . Now consider the linear optimization problem, named DUAL( λ ) in [18]: maximize ξ · λ sub ject to max σ ∈ S ξ · σ ≤ 1 , ξ ∈ R M + . F or λ ∈ ∂ Λ , the optimal v alue of the ob jectiv e in DUA L( λ ) is 1 (cf.[18]). The set of optimal solutions to DUA L( λ ) is a b ounded p olyh ed ron, and we let S ∗ = S ∗ ( λ ) b e the set of its extreme p oints. Fix λ ∈ ∂ Λ . W e then consider the opt imization problem ALGD( w ): minimize ˜ L ( x ) sub ject to ξ · x ≥ w ξ for all ξ ∈ S ∗ ( λ ) , x ∈ R M + . W e know from [18] that ALGD ( w ) h as a u nique solution. W e no w define the li fting map : Definition 7.5. Fix some λ ∈ ∂ Λ . The lifting map ∆ λ : R |S ∗ ( λ ) | + → R M + maps w to the unique solution to ALGD( w ). We also define the worklo ad map W λ : R M + → R |S ∗ ( λ ) | + by W λ ( q ) = ( ξ · q ) ξ ∈S ∗ ( λ ) . Fix λ ∈ ∂ Λ . Consider a sequence of switched netw orks indexed by r ∈ N , op erating under the MW- α p olicy (recall that α ≥ 1 h ere), all with th e same num b er M of queues and feasible sc hedules. Supp ose that λ r ∈ Λ for all r , and that λ r = λ − Γ /r , for some Γ ∈ R M + . F or simplicity , su p p ose that all n etw orks start with empty queues. Consider the follo wing central limit scaling, ˆ q r ( t ) = Q r ( r 2 t ) /r, (31) where Q r ( τ ) is the queue size vector of the r th netw ork at time τ , and where we extend the domain of Q r ( · ) to R + by linear interpolation in each interv al ( τ − 1 , τ ). W e are finally ready to state th e m ultiplicativ e state space collapse result (Theorem 8.2 in [18]): Theorem 7.6. Fi x T > 0 , and let k x ( · ) k = sup i ∈{ 1 ,...,M } , 0 ≤ t ≤ T | x i ( t ) | . Under the ab ove assumptions, for any ε > 0 , lim r →∞ P  k ˆ q r ( · ) − ∆ λ ( W λ ( ˆ q r ( · ))) k k ˆ q r ( · ) k ∨ 1 < ε  = 1 . W e no w state and pro ve the full state space collapse result. Theorem 7.7. Under the same assumptions in T he or em 7.6, and for any ε > 0 , lim r →∞ P  k ˆ q r ( · ) − ∆ λ ( W λ ( ˆ q r ( · ))) k < ε  = 1 . Pr oof. First note that since λ r = λ − Γ /r , the cor- respond ing loads satisfy ρ r ≤ 1 − C /r , for some p ositive constant C > 0. By Theorem 3.3, for any b > 0, P  max τ ∈{ 0 , 1 ,...,r 2 T } Q r max ( τ ) ≥ b  ≤ K ( α, M ) r 2 T (1 − ρ ) α − 1 b α +1 ≤ K ( α, M ) r 1+ α T C α − 1 b α +1 . Then with a = b/r and under the scaling in (31), P  k ˆ q r ( · ) k ≥ a  ≤ K ( α, M ) C α − 1 T a α +1 , for any a > 0. F or notational con venience, we write D ( r ) = k ˆ q r ( · ) − ∆ λ ( W λ ( ˆ q r ( · ))) k . Then, for any a > 1, P  D ( r ) ≥ ε  ≤ P  D ( r ) k ˆ q r ( · ) k > ε a or k ˆ q r ( · ) k ≥ a  ≤ P  D ( r ) k ˆ q r ( · ) k > ε a  + P  k ˆ q r ( · ) k ≥ a  . Note that by Theorem 7.6, the first term on the RHS goes to 0 as r → ∞ , for an y a > 0. The second term on the RHS can b e made arbitrarily small by taking a sufficiently large. Th us, P ( D ( r ) ≥ ε ) → 0 as r → ∞ . This conclud es the pro of. 8. DISCUSSION The results in this p ap er can b e v iew ed from tw o differ- ent p erspectives. On th e one hand , they provide m uch new information on the qualitative b ehavior (e.g., finiteness of exp ected backlog, b oun ds on steady- state tail probabilities and finite- horizon maximum ex cursion probabilities, etc.) of MW- α p olicies for switched net w ork mo dels. On th e oth er hand, at a technical level, our results highligh t the imp or- tance of choosing a suitable Ly apunov function: even if a netw ork is shown to b e stable by using a particular Lya- punov function, different choices ma y lead to more p ow erful b ounds. The metho ds and results in this pap er ex tend in tw o di- rections. First, all of the results, suitably restated, remain v alid for multihop netw orks under bac kpressure- α p olicies. 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Ephremides. Stability prop erties of constrained qu eueing systems and scheduling p olicies for maximum t h roughput in multihop radio netw orks. I EEE T r ansactions on Automatic Contr ol , 37:1936 –1948, 1992. [23] V . J. V enk ataramanan and X . Lin. Struct u ral prop erties of LDP for queue-length based wireless sc heduling algorithms. In 45th Annual A l lerton Confer enc e on C ommunic ation, Contr ol, and Computing , September 2007. [24] V . J. V enk ataramanan and X . Lin. On th e large-deviations optimalit y of scheduling p olicies minimizing the drift of a lyapuno v function. In 47th Ann ual A l lerton Confer enc e on Communi c ation, Contr ol, and Computing , September 2009. APPENDIX A. TIGHTNESS OF THE UPPER BOUND Here we compare the exp onen tial upp er boun d in Theo- rem 3.2 with a lo w er b ound that we shall obtain from th e LDP results in [24]. T o b e able to ev aluate a useful low er b ound explicitly , we consider the case of an inpu t -queued (IQ) switc h . As discussed in Section 2.2, in an m - p ort IQ switc h, there are M = m 2 queues. Let λ = ( λ ij ) be the vector of arriv al rates. Then, the load ρ = ρ ( λ ) is ρ = max max j m X i =1 λ ij ! , max i m X j =1 λ ij !! . Upper bound for IQ switch. First, we sp ecialize Theorem 3.2 to the case of an IQ switc h. Pr oposition A .1. Consider an m -p ort IQ switch op er- ating under the MW- α p olicy, with arrival ve ctor λ so that ρ = ρ ( λ ) < 1 . L et π b e the stationary distribution of the queue-size. Then, for some lar ge enough c onstants B > 0 and B ′ > 0 , and f or every ℓ ∈ Z + : (i) if α ≥ 1 , then P π  k Q k α +1 > B + 2 m 2 α +1 ℓ  ≤ 1 1 + 1 − ρ 2 m ! ℓ +1 , (ii) and if α ∈ (0 , 1) , then P π  k Q k α +1 > B ′ + 10 m 2 α +1 ℓ  ≤ 1 1 + 1 − ρ 10 m ! ℓ +1 . Before p ro viding the pro of, we note that Prop osition A.1 suggests that for all α > 0, as ρ → 1, and using log(1 / (1 + r )) ≈ − r for small r > 0, we h a ve lim sup R →∞ 1 R log P π ( || Q ( τ ) || > R ) ≤ − 1 − ρ 100 m − 1 − 2 α +1 . (32) Pr oof of Proposition A.1. W e need to iden tify the max - imal change ν max , th e drift constan t γ , and ¯ ν . Since M = m 2 , k Q ( τ + 1) − Q ( τ ) k 1+ α ≤ m 2 1+ α F or α ≥ 1, m 2 α +1 can serve as ν max for k Q ( · ) k α +1 . F or α ∈ (0 , 1), 5 m 2 α +1 can serve as ν max for L α ( Q ( · )). The drift constant γ = 1 − ρ 2 m α/ (1+ α ) is obtained in Lemma A.2 stated b e- lo w. Finally , for ¯ ν , we hav e ¯ ν = E [ k a (1) k 1+ α ] ≤ ( ρm ) 1 1+ α ≤ m 1 1+ α . Thus, γ ¯ ν ≥ 1 − ρ 2 m . Putting ev erything together and applying Theorem 3.2, we obtain for α ≥ 1 and some B > 0, P π  k Q k α +1 > B + 2 m 2 α +1 ℓ  ≤ 1 1 + 1 − ρ 2 m ! ℓ +1 , and for α ∈ (0 , 1) and some B ′ > 0, P π  k Q k α +1 > B ′ + 10 m 2 α +1 ℓ  ≤ 1 1 + 1 − ρ 10 m ! ℓ +1 , This completes t he pro of of Prop osition A .1. Lemma A.2. Under the same assumptions of Pr op osition A.1, and for al l α > 0 , ther e exists a c onstant B > 0 such that E  L α ( Q ( τ + 1)) − L α ( Q ( τ ))   Q ( τ )  ≤ − 1 − ρ 2 m α/ (1+ α ) , (33) whenever L α ( Q ( τ )) > B . Pr oof. F or an IQ switch, and with a slight abuse of nota- tion, w e write Q = [ Q ij ]. R ecall th at a sc h edule can serv e m queues sim ultaneously sub ject to the matching constraints. Because of this stru ctural prop erty , w e claim that for any Q = [ Q ij ] ∈ Z m × m + , m X i,j =1 Q α +1 ij ≤ mw α ( Q ) α +1 α . (34) Equiv alen tly , w α ( Q ( τ )) / || Q ( τ ) || α α +1 ≥ 1 m α/ (1+ α ) . (35) By inspectin g the p roof of Theorem 4.4 for general switc hed netw orks, we realize that th e desired b ound (33) follows from (35). Therefore, to establish Lemma A .2, it is sufficient to verif y (34). Supp ose that σ 0 = [ σ 0 ij ] is a schedule with maximum α - w eigh t for a given Q . With some abuse of notation, let us denote σ 0 ( i ) = j if σ 0 ij = 1. Then, w α ( Q ) = P m i =1 Q α iσ 0 ( i ) . It can b e shown that there exist m − 1 other schedules (or matc hings) σ 1 , . . . , σ m − 1 so that all m 2 queues are served by the m schedules σ 0 , σ 1 , . . . , σ m − 1 . Since w α ( Q ) is th e maximum α -wei ght, it follo ws that for any k (0 ≤ k ≤ m − 1), w α ( Q ) ≥ m X i =1 Q α iσ k ( i ) ≥ m X i =1 Q α +1 iσ k ( i ) ! α 1+ α , (36) where the last inequality follo ws from the standard norm inequality k x k α ≥ k x k 1+ α . Raising to the p o w er (1 + α ) /α on b oth sides of (36), th en summing ove r all k , and using the prop erty t h at the schedules σ k co ver all of the m 2 queues, w e obtain mw α ( Q ) 1+ α α ≥ m X i,j =1 Q 1+ α ij . (37) This completes the verification of (34) and the pro of of Lemma A.2. Lower Bound for IQ Switch. W e n o w derive an exp onen tial low er b ound using the re- sults of [24] and assuming uniform arriv al rates, i.e., λ = [ ρ/m ] with ρ < 1. In [24], the authors establish an LDP for the switched netw ork mo del. They show that under the MW- α policy , with arriv al rates λ satisfying ρ ( λ ) < 1, there exists θ α so that lim R →∞ 1 R log P π ( k Q ( τ ) k α +1 > R ) = − θ α . The tail exp onent θ α is c haracterized as the solution of a v ariational problem: θ α = in f { H ( ˜ λ || λ ) /r ( ˜ λ ) : ˜ λ ∈ [0 , 1] m × m , ρ ( ˜ λ ) > 1 } , (38) where H ( ˜ λ || λ ) = m X i,j =1 ˜ λ ij log  ˜ λ ij λ ij  + (1 − ˜ λ ij ) log  1 − ˜ λ ij 1 − λ ij  . F urt h ermore, r ( ˜ λ ) is the solution to the optimization prob- lem: minimize k x k 1+ α sub ject to r ∈ R M + x ≥ ˜ λ − σ ′ for some σ ′ ∈ Σ , where Σ is the conv ex hull of the set of feasible sc hedules S . Clearly , an explicit form ula for θ α in terms of ρ ( λ ) and the switch size m seems imp ossible, even when λ is uniform, i.e., λ = [ ρ/m ]. How ever, as w e sho w n ext, it is possible to obtain a useful low er b ound . In order to obtain a low er b ound on the large d eviation probabilit y , or equ iv alently , an upp er b ound on θ α , it is suf- ficient to restrict to symmetric o v erload arriv al rates ˜ λ in the optimization problem (38). Th us, let u s assume that ˜ λ ij = (1 + ε ) /m for all i, j , where ε > 0. Then, it can b e chec ked th at r ( ˜ λ ) = m 2  1 + ε − 1 m  α +1 ! 1 α +1 = ε m 1 − α 1+ α . Therefore, the optimization in (38) reduces to minimizing m 2 εm 1 − α 1+ α  1 + ε m log 1 + ε ρ + (1 − 1 + ε m ) log 1 − 1+ ε m 1 − ρ m  , (39) o ver all ε > 0. A gain, a closed form solution seems im- p ossible, but we can look for an approximation. W e are interes ted in comparing the boun ds for large m and ρ near 1, and w e will deve lop a go od app ro ximation in that regime. W e exp ect the optimizing v alue of ε in (39) to b e small. Therefore, we shall use the T aylor series expansion for log up t o t h e first tw o terms, i.e. log (1 + x ) ≈ x − x 2 / 2. With these approximations, minimizing (39) boils do wn to solv- ing a q uadratic equation. This leads to an op t imal solution ε ∗ ≈ 1 − ρ . I ndeed, if ρ is near 1, ε ∗ is q uite small, thus justifying our app ro ximations. Using ε ∗ ≈ 1 − ρ , w e obtain θ α ≤ 2 m 2 α/ (1+ α ) (1 − ρ ) . (40) That is, for ρ near 1 and for m large enough, we hav e lim inf R →∞ 1 R log P π ( k Q ( τ ) k α +1 > R ) ≥ − 2 m 2 α/ (1+ α ) (1 − ρ ) . (41) Comparison . Putting the b ounds (32) and (41) together, w e obtain that − 2 m 2 α/ (1+ α ) (1 − ρ ) ≤ lim inf R →∞ 1 R log P π ( k Q ( τ ) k 1+ α > R ) ≤ lim sup R →∞ 1 R log P π ( k Q ( τ ) k 1+ α > R ) ≤ − 1 − ρ 100 m − 1 − 2 α +1 . F or an y α , ignoring small constants, the ratio b etw een the tw o tail exp onents is p recisely m 3 . F rom this, w e see that the dep endence of our upp er b ound exp onent on th e load ρ is t ight, when the sy stem is hea vily loaded. Ho w ever, th e dep endence on th e num ber of q ueues is not.