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1 Bandwdth Packng E. G. Coman, Jr. and A. L. Stolyar Bell Labs, Lucent Technologes Murray Hll, NJ Abstract We model a server that allocates varyng amounts of bandwdth to \customers" durng servce. Customers could be computer jobs wth demands for storage bandwdth or they could be calls wth demands for transmsson bandwdth on a network lnk. Servce tmes are constants, each normalzed to 1 tme unt, and the system operates n dscrete tme, wth packng (schedulng) decsons made only at nteger tmes. Demands for bandwdths are for fractons of the total avalable and are lmted to the dscrete set f1=k; 2=k;:::; 1g where k s a gven parameter. More than one customer can be served at a tme, but the total bandwdth allocated to the customers n servce must be at most the total avalable. Customers arrve nk ows and jon a queue. The jth ow has rate j and contans just those customers wth bandwdth demands j=k. We study the performance of the two packng algorthms Frst Ft and Best Ft, both allocatng bandwdth by a greedy rule, the rst scannng the queue n arrval order and the second scannng the queue n decreasng order of bandwdth demand. We determne necessary and sucent condtons for stablty of the system under the two packng rules. The average total bandwdth demand of the arrvals n a tme slot must be less than 1 for stablty under any packng rule,.e., the condton := (=k) < 1 must hold. We prove that f the arrval rates 1 ;:::; are symmetrc,.e., = k, for all ; 1 k, 1, then < 1 s also sucent for stablty under both rules. Our Best Ft result strengthens an earler result conned to Posson ows and equal rates 1 = = ; and does so usng a far smper proof. Our Frst Ft result s completely new. The work here extends earler results on bandwdth packng n multmeda communcaton systems, on storage allocaton n computer systems, and on message transmsson along slotted communcaton channels. It s not surprsng that < 1 s sucent under Best Ft, snce n a congested system, Best Ft tends to serve two complementary (matched) customers n each tme slot, wth bandwdth demands beng =k and (k, )=k for some ; 1 k, 1. It s not so obvous, however, that <1 s also sucent under Frst Ft. Interestngly, when the system becomes congested, Frst Ft exhbts a "self-organzng" property whereby an orderng of the queue by tme of arrval becomes approxmately the same as an orderng by decreasng bandwdth demand. Part of the work of ths author was completed whle he was wth AT&T Labs{Research, Murray Hll, NJ (now located n Florham Park, NJ 07932). 1

2 1 Introducton We study a queueng model of storage and transmsson bandwdth allocaton n computer and communcaton systems. To dene the model, we use the termnology of queueng systems; later, we wll map ths termnology nto that of the applcatons. In our queueng model, customers are allocated avalable bandwdth accordng to ther demands, each customer holdng ts allocaton whle t s beng served, then releasng ts allocaton when t departs from the system. More than one customer can be served at a tme, but the total bandwdth allocated to customers n servce at any tme must be less than the total avalable. Bandwdth demands are dscretzed and speced n fractons; for some gven nteger k>0, a demand can be any multple of 1=k up to k=k =1. The dscretzaton loses no generalty n practce and expands the potental applcatons, as we shall see below. Customers arrve n k ows to a sngle queue, the th ow havng rate and just those customers wth bandwdth demands =k. Each customer servce tme s a constant, whch we take to be the unt of tme. In addton, the system operates n dscrete tme; packng decsons are made, and customer servces begn, only at nteger tmes. Unt tme ntervals begnnng at nteger tmes wll also be called tme slots. Note that our model s a stochastc verson of one dmensonal bn packng [5], where a bn corresponds to the total bandwdth avalable over a tme slot. We study the performance of both the Frst Ft and Best Ft packng rules. At servce completons, both rules scan the queue and pack customer bandwdth demands by a greedy rule: the demand beng consdered s packed f and only f t s for at most the bandwdth stll remanng from the demands already packed for customers to be served n the next tme slot. The derence between the two polces s that Frst-Ft scans the queue n arrval order, whle Best Ft scans the queue n decreasng order of bandwdth demand. Our analyss addresses stablty problems: determne necessary and sucent condtons on the arrval rates such that the system s stable under Frst Ft and Best Ft,.e., the underlyng Markov queueng processes are ergodc. We prove that f the bandwdth-demand rates 1 ;:::; are symmetrc,.e., = k, ; 1 k, 1, then under a very general class of arrval processes, := k (=k) < 1 s necessary and sucent for stablty for both Frst Ft and Best Ft. Snce s the average total bandwdth demand n each tme unt, < 1 s clearly a necessary condton, so our proofs focus entrely on showng that the condton s sucent. In what follows we wll take k =0. It would be trval for us to generalze our results to the case k > 0, but we have chosen not to do so as t creates a lack of symmetry and clutters the analyss. Models smlar to ours were studed n [3], where applcatons to multmeda communcatons were emphaszed. The term `bandwdth packng' was ntroduced n [3] as a name for the class of problems of nterest here. In the applcatons, bandwdth on a network lnk s beng 2

3 allocated to several competng demands n varyng amounts such as those needed for vdeo, audo, and data transmsson. It was proved n [3] that <1was sucent for the stabltyof Best Ft when the nput ows were specalzed to the Posson process and when the demand dstrbuton was unform wth all 's equal. The stablty queston for Frst Ft was left as an open problem. The analyss n [3] appled the classcal potental (Lyapunov) functon approach. Our approach uses the relatvely recent ud-lmt technques descrbed n Secton 2. As a consequence, our proofs are more compact and more easly adapted to general arrval processes. In another mportant applcaton, the avalable bandwdth refers to the storage (memory) bandwdth of a multprocessor system, a model studed n [9]. Customers are jobs usng varyng amounts of storage whle runnng on a computer. The model n [9] ders from ours n requrng a strct FIFO servce dscplne. Ths s a substantal smplcaton of our model, but the analyss n [9] leads to further results, ncludng formulas nvarant measures. Our work solves the stablty problem n ths applcaton for much more ecent packng rules. Our work also contrbutes new results to the analyss of an equvalent model of slotted communcaton systems [6]. In ths new nterpretaton, customers are messages and bandwdth demands are message duratons (fractons of a tme slot); the avalable bandwdth n a tme unt of our model becomes a unt-duraton slot n whch subsets of messages are packed and transmtted. Wth arrvng messages modeled by a dscretzed Markov process, the analyss n [6] focuses on the Next Ft algorthm: When a message arrves and nds no messages watng, t s packed (wll be sent) n the next tme slot. If a message arrves and nds other watng messages, t s packed n the latest tme slot already allocated at least one message, f t ts n the remanng unallocated tme of that slot; otherwse, the message s packed n the next, as yet unused, tme slot (and hence eventually transmtted one tme unt later). Our analyss extends the earler work to the much more ecent Frst Ft and Best Ft packng algorthms. For example, wth equal arrval rates 1 = =, the message-rate capacty under Next Ft s only 3=2, whereas t s 2 under Frst Ft and Best Ft. As a nal applcaton, one that takes us away from the bandwdth nterpretaton, we menton classcal k-server queues. Our model generalzes these queues by allowng customers to requre more than one server durng ther servce. In the termnology of our model, a bandwdth demand of =k s smply a request for servers. The next secton formalzes our probablty model and ntroduces the ud-lmt approach to our stablty problems. Our man results appear n Sectons 3 and 4 as theorems gvng necessary and sucent condtons for stablty under the Frst Ft and Best Ft rules, respectvely. In Secton 5 we present a moment convergence result, whch complements the stablty results for both Frst Ft and Best Ft. Whle our man results are n the stochastc analyss of algorthms, our methods also yeld useful results n the asymptotc average-case analyss of algorthms. In the average-case (or xed-nput) model, a xed number n of customers wth..d. bandwdth demands s gven, and the objectve s the large-n behavor of the expected total bandwdth wasted whle servng the n demands. Secton 6 apples the ud-lmt approach to the average-case 3

4 model by gvng a smple proof that, under Frst Ft and symmetrc bandwdth-demand dstrbutons, the expected total wasted bandwdth s o(n), so the expected number of tme slots needed to serve the n customers exceeds the expected sum of bandwdth demands (n=2) by a o(n) term. Secton 7 concludes the paper wth a dscusson of open problems and the senstvty of the analyss to varous model assumptons. 2 Prelmnares Under Frst Ft, a state of the queue s denoted by an element of the set of all ntely termnated sequences on f1;:::;k, 1g. The length of the sequence s the queue length, and the th element of the sequence gves the bandwdth demand of the customer that was th to arrve among the customers currently watng. Under Best Ft, the arrval order s not needed; the state just needs, for each =1;:::;k, 1 the number of customers watng wth bandwdth demands =k. Hereafter, a type- demand s one for a fracton =k of the bandwdth; type- customers are those wth type- demands. We assume that the aggregate arrval process of the k, 1 customer types can be descrbed by a nte number of ndependent, dscrete-tme regeneratve processes wth nte-mean regeneraton cycles. Our proofs rely on two consequences of ths assumpton: The underlyng queueng process, whch we denote by = ((t); t = 1; 2;:::), s a countable Markov chan, and the functonal strong law of large numbers holds for the nput process. To avod trval complcatons, we also assume that s rreducble and aperodc. These assumptons allow for vrtually any process havng a regeneratve structure, e.g., dscretetme versons of Markov modulated Posson processes, the processes generated by on-o sources, etc. However, to avod complcated notaton, n the rest of the paper we vew the k, 1 nput ows as ndependent wth the th beng an..d. sequence of nteger-valued random varables whch gve the numbers of type- arrvals n [t, 1;t] and have a nte mean, the same for all t =1; 2;:::. Wth ths smplcaton, the underlyng process becomes the queue-content process. In what follows, the norm k(t)k denotes the number of customers watng at tme t. Let (n) denote a process wth an ntal condton such that k (n) (0)k = n. In the analyss to follow, all varables assocated wth a process (n) wll be suppled wth the upper ndex (n). The followng theorem s a corollary of a more general result of Malyshev and Menshkov [10]. Theorem 1 Suppose there exsts an nteger T > 0 such that for any sequence of processes (n), we have lm n!1 E[ 1 n k (n) (nt )k] =0 (1) 4

5 Then s ergodc. It was shown by Rybko and Stolyar [11] that an ergodcty condton of the form (1) naturally leads to a ud-lmt approach to the stablty problem of queueng systems. Ths approach was further developed by Da [7], Chen [2], Stolyar [12], and Da and Meyn [8]. As the form of (1) suggests, the approach studes a ud process x(t) obtaned as a lmt of the sequence of scaled processes 1 n (n) (nt);t 0; at the heart of the approach n ts standard form s a proof that x(t) startng from any ntal state wth norm kx(0)k = 1 reaches 0 n nte tme T and stays there. (In most cases of nterest, ncludng ours, weaker condtons are sucent, e.g., t s enough to verfy that nf t0 kx(t)k < 1, as shown n [12].) In our settng we need to dene what the scalng 1 n (n) (nt) means. In order for ths scalng to make sense, we wll need an alternatve denton of the queueng process. To ths end, we rst adopt the conventon (t) = (btc); t 0, whch allows us to vew as a contnuous-tme process dened for all t 0, but wth new arrvals and servces stll begnnng only at nteger tmes t = 0; 1; 2;:::. Next, we dene the followng random functons assocated wth the process (n) (t): F (n) (t) s the total numberoftype- customers (n) that arrved by tme t 0, ncludng the customers present at tme 0; and ^F (t) s the (n) number of type- customers that were served by tme t 0. Obvously, ^F (0) = 0 for all. As n [11] and [12], we \encode" the ntal state of the system; n partcular, we extend the denton of F (n) (t) to the negatve nterval t 2 [,n; 0) by assumng that the customers present n the system n ts ntal state (n) (0) arrved n the past at tme nstants,(n, 1);,(n, 2);:::; 0, exactly one customer at each tme nstant. In the case of Frst Ft, we requre the order of these arrvals to be the same as n the state at tme 0. By ths conventon F (n) (,n) = 0 for all and n, and P F (n) (0) = n. It s clear that the process (n) = ( (n) (t);t 0) s a projecton of the process S (n) = (F (n) ; ^F (n) ), where and F (n) =(F (n) (t); t,n; =1; 2;:::;k, 1) ^F (n) =(^F (n) (t); t 0; =1; 2;:::;k, 1);.e., a sample path of S (n) unquely denes a sample path of (n). Now consder the scaled process s (n) =(f (n) ; ^f (n) ), where f (n) =(f (n) (t) = 1 n F (n) (nt); t,1; =1; 2;:::;k, 1) and ^f (n) =(^f (n) (t) = 1 (n) ^F (nt); t 0; =1; 2;:::;k, 1) n The followng lemma establshes convergence to a ud process and s a varant of Theorem 4.1 n [7]. 5

6 Lemma 1 The followng statements hold wth probablty 1. For any sequence of processes (n), there exsts a subsequence (m) ; fmg fng, such that for each ; 1 k, 1, (f (m) (t);t,1)! (f (t);t,1) u:o:c: (2) (m) ( ^f (t);t 0)! ( ^f (t);t 0) u:o:c: (3) where the functons f ; ^f ; are non-negatve non-decreasng Lpschtz-contnuous n the gven tme ntervals, and u.o.c. means that convergence s unform on compact sets as n! 1. The lmtng set of functons also satses and for all ; 1 k, 1, for any 0 t 1 t 2, s =(f; ^f) =f(f (t);t,1); ( ^f (t);t 0);; 2;:::;k, 1g f (0) = 1 (4) f (,1) = 0; ^f (0)=0; (5) f (t), f (0) = t; t 0; (6) ^f (t) f (t); t 0 ; (7) k ( ^f (t 2 ), ^f (t 1 )) 1 : (8) Proof. It follows from the strong law of large numbers that, wth probablty 1 for every, (f (n) (t), f (n) (0); t 0)! ( t; t 0) u:o:c: Also, for every n and, the functons (f (n) (n) (t);,1 t 0) and ( ^f (t);t 0) can ncrease by at most k(1=n) n any nterval of length 1=n. Ths mples (2) and (3). We get (6) as a byproduct. Equatons (4) and (5) follow from the constructon representng the ntal state. Equaton (7) follows mmedately from dentons, and the conservaton law (8) from the trval observaton that the total bandwdth of customers served n one tme slot s lmted to 1. 3 Frst Ft To prove that <1 s sucent for stablty under Frst Ft, we need two lemmas. Lemma 2 For any xed T 1 > 1, the followng holds wth probablty 1. A lmtng set of functons s = (f; ^f) dened n Lemma 1 has the followng addtonal property: For all ; 1 k, 1, ^f (T 1 ) >f (0): (9) 6

7 Proof. Consder the sequence of sample paths of the scaled process s (n) convergng to the set of functons s dened n Lemma 1. For any xed > 0 and > 0, we have for all sucently large n, f (n) () < 1+( )(1 + ): Snce as long as the queue s non-empty at least one customer s served n every tme slot, we conclude that ^f (n) ( +1+( )(1 + )) f (n) (); whch mples (9) va a smple passage to the lmt n!1, and by the fact that and can be arbtrarly small. For a set s of functons as dened n Lemma 1, let us dene and let (t) = nff,1j f () > ^f (t)g (10) (t) = mn (t): (11) 1 The proof of the Frst Ft stablty result centers on the analyss of the tmes (t); because of ther useful propertes, one beng a smple relaton to the ud lmt of the queue-length processes k (n) (t)k. For ths reason, we gve below an nformal nterpretaton of these tmes dened n terms of the unscaled processes S (n) = f(f (n) ; ^F (n) )g. Accordng to (10), (t) s the earlest tme by whch the number of type- arrvals exceeds the number of type- departures by tme t. Under sutable condtons to be covered n the lemma below, (t) can be expressed as the nverse of f evaluated at ^f (t),.e., (t) =f,1 ( ^f (t)); (12) as llustrated n Fgure 1. We remark that (t) need not be a smooth functon. For example, the ntal state can be contrved so that f (t) s at n some subnterval of [,1; 0], thus creatng a dscontnutyn (t). On the other hand, as proved later, (t) has to be Lpschtzcontnuous n the nterval 1 < t < 1. Under Frst Ft, the queue of type- customers at tme t conssts of just those type- customers that arrved durng [ (t);t]. Then (t) =t s a type- empty-queue condton. Recallng our dscusson of Theorem 1, we want to show that (t) tends towards t,.e., 0 (t) > 1; and absorbs n the empty-queue condton (t) =t. These and related propertes of the (t) are formalzed n the followng result. Lemma 3 Let T 1 > 1 be xed. There exst xed constants T 2 and T, T 1 T 2 T < 1 such that wth probablty 1, a lmtng set of functons s = (f; ^f) dened n Lemma 1 has the followng addtonal propertes: () We have (T 1 ) > 0 for all ; 1 k, 1: (13) 7

8 f (t) 1 ^f (t),1 (t) t Fgure 1: The functons (t), ^f (t), and f (t) wth f (t) = t + f (0) for t 0. () In the nterval t T 1, every functon (t); 1 k, 1, s non-decreasng Lpschtzcontnuous, and therefore so s (t). () At any regular pont t T 1,.e., a pont where all the dervatves of each of the functons f ; ^f ; ; and exst for all, 1 k, 1, we have 0 (t) = ^f 0 (t)= (14) (t) <t ) 0 (t) 1= (15) ( (t) < j (t) ^ <j) ) 0 j(t) =0: (16) (v) For all t T 2 ; <j ) (t) j (t): (17) (v) If the nput ows are symmetrc,.e., f = k, for every, then we have at any regular pont t T 2, (t) <t ) 0 (t) 1= > 1 (18) (v) For symmetrc nput ows, for all t T; whch s equvalent to the asserton that, for all t T; (t) =t; (19) ^f (t) =f (t) for all ; 1 k, 1: (20) 8

9 Proof Property () follows from Lemmas 1 and 2. When t T 1 > 1, the eects of the ntal state have dsspated and we know that s postve (by property ()), f (t) = f (0) + t, and ^f (t) s nondecreasng Lpschtz-contnuous (by Lemma 1). It follows easly that (t) s nondecreasng Lpschtz-contnuous for t T 1, whch proves property (). For property (), we rst derentate (12) at regular ponts t and substtute f 0 (t) = ; ths gves (14). To prove (15), dene M(t) :=f j (t) =(t)g; so that, snce t s a regular pont, we can wrte for all 2 M(t), (t) =(t); 0 (t) = 0 (t); ^f 0 (t)= = 0 (t): (21) For the unscaled sample path, t s easy to see that for any sucently small > 0 and all sucently large n, at least one customer of a type 2 M(t) wll be served n each tme slot of the tme nterval [nt; n(t + )]. Ths means that P ^f 0 2M (t) (t) 1; whch together wth (21) mples 0 (t) 1= P, thus provng (15). To prove (16) and complete the proof of property (), consder an unscaled sample path at tme nt. If (t) < j (t), then for small > 0 and all sucently large n, there are at least [ j (t), (t)](1, )n type- customers n the queue ahead of any type-j customer. Ths means that, f <j, there exsts a small >0 such that no type-j customers wll be served n the nterval [nt; n(t + )]. Ths n turn mples that enough, and therefore that (16) holds. ^f (n) j (t + ) = ^f (n) j (t) for all n large Property (v) follows from (15) and (16) n property (). The constant T 2 can be chosen to be T 2 = T 1 + max (T 1 ), (T 1 ) 1= P : To prove property (v), we note rst that, by property (v), the set M(t) for t T 2 has the form M(t) =fk, 1;k, 2;:::;rg wth r k, 1. Then we can rewrte (21) as (t) = (t) =:::= r (t) < r,1 (t) (22) 0 (t) =(t) 0 =:::= r(t) 0 > 0 (23) ^f (t)= 0 = = ^f r(t)= 0 r = 0 (t); (24) for some r k, 1. Here, we need to show that, f (t) <t, then 0 (t) 1= (25) Let us assume that k s odd; the proof of (25) for k even s very smlar and left to the reader. Frst, we make an observaton smlar to the one we made n the proof of (16). Consder the unscaled sample path at tme nt. Equaton (22) mples that, for any small > 0 and 9

10 all sucently large n, there are at least ( r,1 (t), (t))(1, )n customers of each type = k, 1;k, 2;:::;r n the queue ahead of any customer of type j = r, 1;:::;1. Ths means that there exsts a small > 0 such that n the nterval [nt; n(t + )] the customers of types r, 1;:::;1 have lower prorty than customers of types k, 1;k, 2;:::;r. More precsely, no customer of type j < r wll be packed n a tme slot as long as a customer of type r can be packed nto that tme slot nstead. Therefore, as far as the behavor of (n) the functons ^f ; r, n the nterval [t; t + ] s concerned, we can gnore the customers of types j <r. In the remander of the proof of property (v), we x wth the above observaton n mnd; we conne ourselves to the behavor of scaled processes n the nterval [t; t + ], and the behavor of the correspondng unscaled processes n the nterval [nt; n(t + )], wth n sucently large. Let p =(k, 1)=2, and note that the symmetry condton = k, ; 1 p; mples " # p := k = k + k, k, k = p = =p+1 : (26) If r p +1,then 0 (t) = 1 P =r 1 P =p+1 = 1 : (27) To see ths, note that exactly one customer of some type r wll be served n each tme slot. For, snce 2r k + 1, two such customers have demands exceedng the total avalable bandwdth. Ths mmedately mples that =r ^f 0 (t) =1 whch means that 0 (t) P =r = 1, and hence that (27) holds. To nsh the proof of property (v), t remans to dspose of the case r p. We wll show that 0 (t) =1=. Frst, we observe that 0 (t) 1=. Ths s because, by an argument smlar to the one used for the case r p + 1 above, we have =p+1 ^f 0 (t) 1 and so 0 (t) 1 P =p+1 = 1 : Now assume that strct nequalty holds, 0 (t) < 1=: (28) 10

11 We wll prove that ths mples that ^f 0 r(t) 1=, whch s a contradcton, snce ^f 0 r(t) = 0 (t) by denton of r. If (28) were to hold, then for any >0, any sucently small ; 0 <<;(wth dependng on ), and all sucently large n (dependng on and ), the followng three observatons would hold for an unscaled sample path n the nterval [nt; n(t + )]: (a) The number of tme slots not servng any customers of types k, 1;:::;k, r + 1, whch do not t together wth type-r customers, s at least 2 4 1, ( 0 (t)+) =k,r n: (b) Consder a tme slot descrbed n (a). If ths slot does not serve any customers of types p; p, 1;:::;r+1,then t must serve at least one type-r customer. (c) The total number of served customers of types = p;:::;r+ 1 does not exceed [ 0 (t)+] p =r+1 1 A n: Snce the number of slots occuped by customers of types p; p, 1;:::;r+1s at most the number of such customers, observatons (a)-(c) mply that the lmtng type-r servce rate has the lower bound ^f 0 r(t) 2 4 1, ( 0 (t)+) Snce >0 can be arbtrarly small, we get ^f 0 r(t) 1, 0 (t) =k,r =k,r+1 3 5, [ 0 (t)+] + By the symmetry condton, P =k,r+1 = P r,1,so p ^f 0 r(t) 1, 0 (t) p =r (t) r = 1, 0 (t) + 0 (t) r > 0 (t) r ; 3 5 : p =r+1 the desred contradcton. Thus, (28) can not hold, we can conclude that 0 (t) = 1=, and property (v) s proved. It follows from property (v) that nfftj (t) =tg T 2 + T 2, (T 2 ) 1=, 1 T 2 + T 2 1=, 1 11

12 Let us choose the constant T to be T = T 2 + T 2 1=, 1 : Snce we know that d (t, (t)) 1, 1= < 0 at any regular pont t T dt 2 such that t, (t) > 0, we conclude that (t) = t for all t T. Ths proves property (v) and hence the lemma. Theorem 2 Suppose the nput ow ntenstes are symmetrc, and <1. Then under Frst Ft s ergodc. Proof The proof s a slght modcaton of the proof of Theorem 4.2 n [7]. In partcular, Lemmas 1 and 3 mply that there exsts a T > 0, whch can be chosen to be an nteger, such that for any sequence of processes f (n) g we have 1 lm n!1 n k (n) (nt )k = n!1 lm (f (n) (T ), ^f (n) (T )) = 0; (29) wth probablty 1. The unform ntegrablty of the sequence f (n) g can be proved n ways smlar to those n [11] and [7]. Unform ntegrablty and the convergence n (29) mply that lm E[ 1 n!1 n k (n) (nt )k] =0: Then the condton n (1) of Theorem 1 holds, and we are done. We can also make strong statements about convergence propertes and the exstence of moments. These apply to Best Ft as well, so we defer these results to Secton 5. 4 Best Ft Dscplne In ths secton, we prove that the analog of Theorem 2 for Best Ft also holds. We use the same general ud-lmt approach, but the arguments wll be smpler. We agan need Lemma 1, but we wll create a new verson of Lemma 3, a smpler verson n that there wll be no need to deal wth the tmes (t) or the encodng of the ntal state; nstead of analyzng t, (t), we wll analyze the derence q (t) :=f (t), ^f (t), provng that t reaches zero n nte tme and stays there. Theorem 3 Suppose the nput ow ntenstes are symmetrc, and <1. Then under Best Ft s ergodc. 12

13 Proof: We need only prove Lemma 4 below; wth Lemma 4 replacng Lemmas 2 and 3, the proof of Theorem 2 wll apply to Best Ft. Lemma 4 Suppose that the nput ows are symmetrc. Then there exst constants 0=T k < T <:::<T 1 = T < 1 such that, wth probablty 1, a lmtng set of functons s dened n Lemma 1 has the followng addtonal property for every =1; 2;:::;: At any regular pont t T +1, ^f (t) <f (t) ) ^f 0 (t) +(1, ); (30) and for any t T, Thus, for all t T, and for all ; 1 k, 1, ^f (t) =f (t) and therefore ^f 0 (t) = : (31) ^f (t) =f (t) (32) Proof All the conventons ntroduced n the proof of Lemma 3 are stll n force. Thus, s (n) ;n =1; 2;::: s the sequence of sample paths of the scaled process s (n) whch converges to s. And when we refer to the unscaled sample path, we mean the correspondng sample path of the process S (n). We consder only the case when k s odd; the proof for even k s analogous.) Dene p := (k, 1)=2, as before, and recall that q (t) := f (t), ^f (t), so that (30) and (31) become q (t) > 0 ) q(t) 0,(1, ) (33) and q (t) = 0 for every t T : (34) We need a couple of key observatons, the rst followng from the fact that, the hgher the bandwdth demand, the hgher the packng prorty under Best Ft. (a) For any xed r, the servce of customers of types ;k,2;:::;rs completely unaected by the servce of customers of types j <r. (b) The condton q (t) > 0 for the lmtng set of functons mples that, for a sucently small, xed > 0, and all n sucently large, the correspondng unscaled sample path s such that n the nterval [tn; (t + )n]: (b 1 ) there are always type- customers avalable for servce; (b 2 ) f p, then every tme slot servng a type-(k, ) customer must serve a type- customer; every tme slot not servng a type- customer, or a type-(k, ) or larger customer must serve one or more customers of types p; p, 1;:::;+1. The proof s by nducton on decreasng from k, 1to1.If = k, 1, t follows easly from observatons (a) and (b 1 ) that at any regular pont t 0 = T k, the condton q (t) > 0 mples that ^f 0 (t) =1 +(1, ), and hence that (33) holds. 13

14 Notce that q (0) 1, so f we choose T = T k +1=(1, ); (35) then (34) follows from (33). Ths establshes the bass of the nducton. For the nducton step, suppose (33) and (34) hold for = k, 1;k, 2;:::;r+1. We wll now prove that (33) and (34) also hold for = r. Consder a regular pont t T r+1. If r p + 1, then the condton q r (t) > 0must mply ^f 0 r(t) =1, =r+1 > r +(1, ); (36) whch gves q 0 r,(1, ). To see ths note that, n an unscaled sample path, one and only one customer of types k, 1;k, 2;:::;r can be served n a tme slot. Thus, (36) follows from observatons (a) and (b 1 ) and the nductve hypothess, whch asserts that the customers of each of the types = k, 1;:::;r+1are served at exactly the correspondng rate for all t>t. (We omt routne ; -techncaltes smlar to those used n the proof of Lemma 3.) If r p, then by applyng observatons (a) and (b 2 ), we get =1, so q 0 r,(1, ) agan holds. r,1 ^f 0 r(t) 1,, p =r+1 =k,r+1, p =r+1, r + r = r +(1, ) Now f we observe that q r (T r+1 ) 1+ r T r+1, and set, n analogy wth (35), T r =(1+ r T r+1 )=(1, ); then we see that (34) follows from (33). The nductve step and hence the proof of Lemma 4 and Theorem 3 s complete. 5 Moment Convergence It s shown n [8] that condton (1) mples not only stablty, but also very strong momentexstence and convergence propertes. For example, Theorem 4 below follows drectly from (1) and Theorem 6.2 n [8] (whch can easly be adjusted for our dscrete-tme case). 14

15 Theorem 4 Suppose the are symmetrc, < 1 holds, and the nput processes are..d. sequences wth nte (p+1)-st moments (p 1 s an nteger). Let (1) have the statonary dstrbuton of the Markov chan under ether Frst Ft or Best Ft. Then and for any ntal state (0), Ek(1)k p < 1 lm t!1 Ek(t)kp = Ek(1)k p 6 Connecton to Average-Case Analyss Our Frst Ft stablty analyss s closely related to the average-case analyss of Frst Ft bn packng under dscrete dstrbutons [4]. We can recast the average-case model nto our settng as follows. Suppose the ntal state conssts of a queue of n customers wth customer types beng a sequence of ndependent samples from a gven dstrbuton on f1;:::;k, 1g; and assume there are no new arrvals. A formula for the expected total wasted bandwdth n the packng of n customers s the objectve of the average-case analyss. The problem s dcult, so vrtually all of the results to-date descrbe large-n asymptotc behavor. In partcular, t has been shown that for certan customer-type dstrbutons, the expected total wasted bandwdth s o(n). Ths secton demonstrates how the Frst Ft average-case result can easly be obtaned from the propertes of the ud lmt of a stochastc system lke the one consdered n prevous sectons. Further results of ths type are dscussed n the next secton. Consder an nnte sequence of..d. customer types 1 ; 2 ;:::, takng values n the set f1;:::;g, wth the dstrbuton f ;;:::;g, P =1. The reason for adoptng our arrval rate notaton for the customer-type dstrbuton n an average-case model wll be clear when the theorem below lnks up the average-case and stochastc analyss. Dene a sequence of systems (just lke those analyzed n prevous sectons) ndexed by n = 1; 2;:::. The n th system has an ntal state consstng of customers of types 1 ;:::; n watng n queue n the order lsted. Suppose there are no new arrvals after tme 0. Note that n ths settng the ntal state s random. For the n th system, dene U (n) to be the tme slot n whch the last customer of the ntal state s served under Frst Ft, and dene W (n) = U (n), n =k ; W (n) s the total bandwdth (or server capacty) wasted by the Frst Ft packng process. Theorem 5 Under Frst Ft and a symmetrc dstrbuton f g = ; 1 k, 1; 15

16 the followng holds wth probablty 1: lm W (n) =n =0; (37) n!1 lm U (n) =n =1=2 : (38) n!1 Remark. Snce the random varables W (n) =n and U (n) =n are bounded above unformly n n, the probablty 1 convergence mples convergence of mean values. Proof Frst, t s clear that the propertes (37) and (38) are equvalent, snce 1 lm n!1 n n k = 1 2 ; wth probablty 1.Also, W (n) 0 obvously holds, and therefore lm nf n!1 U (n) =n 1=2; so t wll suce to show that lm sup n!1 U (n) =n 1=2 : (39) For every ndex n, consder a moded system n whch new arrvals after tme 0 do occur; the nput (say Posson) ow oftype arrvals has ntensty. By the denton of Frst Ft, such a modcaton can not change the random varables W (n) and U (n), because t has no eect on the servce of ntal customers. The prmary reason for consderng the moded system s to comply wth the formulatons of the results n prevous sectons, to ease the `reuse' of those results. We observe that our sequence of processes, wth ndex n, satses the condtons of Lemmas 1 and 3, except for the fact that the ntal state s now random. But snce the ntal states are drawn from a sequence of..d. random varables, the functonal strong law of large numbers apples to the sequence of ntal states. We can conclude that: Except for property (13), Lemma 1 and Lemma 3 are vald for our sequence of moded processes. Moreover, they are vald wth T 1 = T 2 =0and a lmtng set of functons (a ud process) s such that f (t) = (t, (,1));, 1 t 0; 8 (40) Indeed, t follows that f (t) = (t, (,1)); t,1; 8 (41) and hence that (0) =,1 for any. The only property of the constant T 1 requred n earler proofs was that each functon f () be strctly lnear wth slope n the nterval [ (T 1 ); 1); wth T 1 = 0, we stll have ths property. The only property of the constant T 2 requred n 16

17 the earler proofs was that T 2 T 1 and (T 2 ) ::: 1 (T 2 ). Agan, wth T 2 =0,we stll have ths property. Applyng the results of Lemma 1 and Lemma 3, we get 0 (t) 1= = 2 at any regular pont t 0. In fact, from the conservaton law (8), we see that equaltymust hold: 0 (t) =1= =2, at any regular pont t 0. Then the followng must also hold: (t) = (t) =:::= 1 (t) =,1+2t; t 0; (42) because an nequalty (t) < (t) for any xed t and would contradct property (8). We are now n poston to prove (39). Let us x a small >0. It follows from (42) that any lmtng set of functons s s such that, for all, ^f ((1, )=2) = (1, ) < = f (0): Ths means that, wth probablty 1 for any, the sequence of scaled processes that for all n, except perhaps for values n some nte set, (n) ^f () ssuch (1, 2) Ths n turn means that, n the unscaled systems: (n) ^f ((1, )=2) <f (n) (0) < (1 + ) : (a) n the rst b(n=2)(1, )c tme slots, no server bandwdth was wasted and only ntal customers were served; (b) U (n) (n=2)(1, )+nk 3. Therefore, wth probablty 1, lm sup n!1 U (n) =n (1, )=2+3k : Snce >0 can be chosen arbtrarly small, we get (39), whch concludes the proof. 7 Dscusson Our proofs of Theorems 2and3verfy that, for arrval processes wthn the broad framework gven n Secton 2, the stablty of the system under consderaton depends essentally on the nput ow ntenstes and s nsenstve to the precse probablstc structure of the nput ows. Other specal cases of nterest to whch the result of Theorem 2 s easly extended, are sets of dvsble bandwdth demands. If h=k and j=k; j >hare any two demands n a dvsble subset of f1=k; 2=k;:::; 1g, then h dvdes j and j n turn dvdes k. The specal case f1=2 a ; 1=2 a,1 ;:::;1=2; 1g for some postve nteger a s of nterest n computer applcatons. We leave to the nterested reader an easy adaptaton of the ud approach to a proof that <1 s sucent for stablty under Frst Ft and Best Ft ndependent of the relatve szes of the arrval rates. 17

18 In heavy congeston, Frst Ft typcally matches demands =k wth ther complements (k, )=k, thus wastng no bandwdth and allowng < 1 to be sucent as well as necessary for stablty. For ths matchng to occur, the queue must reorder tself dynamcally nto decreasng order by sze. In an attempt to nd other examples where Frst Ft has a smlar self-organzng property, consder any dstrbuton that yelds perfect packngs respectng the arrval rates,.e., examples for whch there are ntegers n; n 1 ;:::;n such that, for some ; 0 < < 1, we have = n =n; 1 k, 1; and we can pack n 1 type-1 customers, n 2 type-2 customers,...,and n type-(k, 1) customers nto n tme unts wth no avalable bandwdth left over. It s easy to dene an algorthm that sutably restrcts the demand-type conguratons allowed n each tme unt so as to guarantee that < 1 s sucent for stablty. (What makes ths algorthmc technque mpractcal, of course, s that t requres advance knowledge of the arrval rates.) Typcally, however, t s not possble to make the same stablty clam about Frst Ft. As a smple example, take k = 7 and let the only nonzero arrval rates be 2 =2 3. A greedy algorthm that, whenever possble, packs 2 type-2 customers and 1 type-3 customer n a tme slot wll be stable as long as < 1. However, t can be shown that, durng perods of congeston under Frst Ft, a postve fracton of the tme slots wll be packed wth 2 type-3 customers or 3 type-2 customers, wastng 1/7 of the bandwdth n each case. A formal proof that Frst Ft s unstable for values of n an nterval [1, ; 1]; >0ssketched n the appendx. Examples lke those above suggest that the class of bandwdth-demand dstrbutons for whch < 1 s sucent for stablty under Frst Ft or Best Ft s lkely to be relatvely small. More dentve statements of ths knd present nterestng drectons for further research. We establshed n Secton 6 that the large-n packng process n the average-case model and the queueng process under heavy congeston n the stochastc model are descrbed by essentally the same ud process. Ths connecton makes the followng conjecture qute plausble. Consder an average-case model wth a xed customer-type dstrbuton fb g; and a famly of stochastc models wth ths same customer-type dstrbuton; then for each model n the famly = b ; where the total arrval rate = P ndexes the famly of stochastc models. We conjecture that, for the Frst Ft algorthm, the expected total wasted bandwdth s o(n) n the average-case model f and only f <1 s sucent for stablty n the famly of stochastc models. Expected total wasted bandwdth s known to be O( p nk) for Frst Ft packng [4] and customer types ndependently and unformly dstrbuted on f1;:::;k, 1g. Ths result and our Theorem 5 on the more general symmetrc dstrbutons lends support for the conjecture. Further support s provded by recent results of Albers and Mtzenmacher [1] who showed that, n the average-case model, the expected wasted bandwdth under Frst Ft s O(1) when b 1 = = b k,2 =1=(k, 2). The ud approach shows easly that Frst Ft s stable n the correspondng stochastc model wth ntenstes 1 = = k,2 and = 0, and wth <1. In fact, the proof of Theorem 2 s easly generalzed to prove the same result for any set of ntenstes satsfyng: () 1 s arbtrary, () for some gven nteger m; 2 m<k=2, we have the symmetry = k, for all ; m k, m, and () all other ntenstes are 18

19 0. Acknowledgments We gratefully acknowledge Anja Feldmann and Nabl Kahale for havng helped to ntate the research culmnatng n ths paper. Appendx Proposton 1 Consder Frst Ft bandwdth packng wth k =7and Posson nput ows of only types 2 and 3 customers, wth ntenstes satsfyng 2 =2 3 > 0. There exsts a >0 such that the system wth >1, s unstable. Proof sketch We consder rst a smpler, `saturated' verson of the system n whch new arrvals are generated f and only f there s room for more customers n the current tme slot. In other words, when packng the current tme slot, f the entre queue has been scanned and 2/7 of the tme slot s stll empty, the queue s mmedately extended by new arrvals, wth the numbers of new customers of types 2 and 3 beng Posson dstrbuted wth means 2 and 3. These arrvals are mmedately avalable for further packng. The process s repeated as necessary untl the slot s at most 1/7 empty. The queue state just after each nteger tme s a dscrete-tme countable Markov chan. Its ergodcty s easly vered. It s also easy to see that the average per-slot rates 2 and 3 at whch customers of types 2 and 3 are served are such that 3 = 2 =2 < 1. Also, the Markov chan can be vewed as a regeneratve process wth an empty queue beng a regeneraton state. All moments of the regeneraton cycle are nte. Let us now return to our orgnal system wth 2 and 3 such that 3 < 3 = 2 =2 < 1, whch means 3 <<1. Ths system s unstable. To prove ths, we consder an ntal state formed by arrvals wthout servce (packng s suspended) for M tme slots, wth M large. When the packng starts, the packng process s ndstngushable from the packng process n the saturated system, untl the tme slot when the last ntal customer s reached (.e., scanned for the rst tme). It wll take approxmately M tme slots, wth = 3 = 3 > 1; for the packng process to reach the last ntal customer. By that tme, the queue s longer than the ntal queue; t s extended by new arrvals durng approxmately M tme slots. The packng process wll then take approxmately 2 M slots to reach the end of that queue. Ths contnues, wth the maxmum queue length growng wthout bound. Usng routne large-devaton estmates, t s a smple matter to convert the above observatons nto a rgorous argument that, wth postve probablty, the queue length tends to nnty (see, for example, the nstablty example n [11]). References [1] S. Albers and M. Mtzenmacher. Average-Case Analyss of Frst Ft and Random Ft Bn Packng. Proc. Nnth Ann. ACM-SIAM Symp. Dsc. Alg., (1998), pp

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