Packet multiplexers with adversarial regulated traffic

Packet multiplexers with adversarial regulated traf®c

Srini Rajagopal

, Martin Reisslein

*, Keith W. Ross

aManugistics, 2115, East Jefferson Street, Rockville, MD 20852-4999, USA

bDepartment of Electrical Engineering, Arizona State University, Tempe, AZ 85287-7206, USA

cInstitut EureÂcom, BP 193-06904 Sophia-Antipolis, France Received 2 June 2000; revised 8 May 2001; accepted 29 May 2001

Abstract

We consider a ®nite-buffer packet multiplexer to which traf®c arrives from several independent sources. The traf®c from each of the sources isregulated, i.e. the amount of traf®c that can enter the multiplexer is constrained by known regulator constraints. The regulator constraints depend on the source and are more general than those resulting from cascaded leaky buckets. We assume that the traf®c is adversarial to the extent permitted by the regulators. For lossless multiplexing, we show that if the original multiplexer is lossless it is possible to allocate bandwidth and buffer to the sources so that the resulting segregated systems are lossless. For lossy multiplexing, we use our results for lossless multiplexing to estimate the loss probability of the multiplexer. Our estimate involves transforming the original system into two independent resource systems, and using adversarial sources for the two independent resources to obtain a bound on the loss probability of the transformed system. We show that the adversarial sources are not extremal on±off sources, even when the regulator consists of a peak rate controller in series with a leaky bucket. We explicitly characterize the form of the adversarial source for the transformed problem. We also provide numerical results for the case of the simple regulator.q2002 Elsevier Science B.V. All rights reserved.

Keywords: Buffer-bandwidth tradeoff curve; Call admission control; Leaky bucket; Quality-of-service guarantees; Resource allocation; Statistical multi- plexing; Worst-case sources

1. Introduction

Consider a ®nite-buffer packet multiplexer to which traf®c arrives from several independent sources. For the multiplexer to provide quality of service (QoS) guarantees, such as limits on packet loss probabilities, it must have some knowledge about the traf®c characteristics of the sources.

Because the reliability of statistical models of traf®c is ques- tionable for many source types, in recent years there have been several studies on the performance of packet-switched nodes that multiplex regulated traf®c, e.g. traf®c which conforms to known constraints imposed by leaky buckets.

These studies suppose that the traf®c from the sources is adversarial to the extent permitted by the regulators [1±

14,16,19±21,23±27]. Some of these studies assume that the multiplexer provides deterministic QoS guarantees (e.g. no packet loss) whereas others assume that multiplexer

provides less stringent probabilistic QoS guarantees (e.g. a limit on the packet loss probability).

In a recent paper, LoPresti et al. [16] examine a packet- switched node with regulated traf®c. Motivated by earlier work of Elwalid et al. [9], LoPresti et al. consider both deterministic QoS guarantees and probabilistic QoS guaran- tees. They assume that each source is regulated by asimple regulator, namely, a regulator that consists of a peak-rate controller in series with a leaky bucket. For deterministic QoS, LoPresti et al. show that if the multiplexer has suf®- cient link bandwidth and buffer capacity toprovide lossless multiplexing, then the multiplexer's buffer and bandwidth can be allocated among the sources so that the resulting segregated systems are lossless. For probabilistic QoS, they develop a new approach to estimate the loss probabil- ity. Speci®cally, they transform the two-resource (band- width and buffer) allocation problem into two independent single-resource allocation problems; they then analyze these simpler, independent resource problems, taking on±off periodic sources for their adversarial sources.

Although the simple regulator is a popular policing mechanism within several standards bodies, it has been observed that it is often a poor characterization of a source's

0140-3664/02/$ - see front matterq2002 Elsevier Science B.V. All rights reserved.

PII: S0140-3664(01)00359-0

www.elsevier.com/locate/comcom

qA shorter version of this paper has appeared in Proceedings of IEEE Infocom, San Francisco, CA, March 1998.

* Corresponding author. Tel.:11-480-965-8593; fax:11-480-965-8325.

E-mail address:reisslein@asu.edu (M. Reisslein).

1 www.eas.asu.edu/,mre.

worst-case traf®c. A tighter and more powerful characteri- zation is given by a more general regulator consisting of a cascade of multiple leaky buckets [10,15,25]. For example, when the sources are VBR video sources, it is often possible to admit signi®cantly more connections by replacing the simple regulator with cascaded-leaky-bucket regulators [25]. It is therefore desirable to extend the important work of LoPresti et al. [16] and Elwalid et al. [9] to the case of more general regulators.

In this paper, we reexamine the model of LoPresti et al.

[16] in the context ofgeneralized regulators, which are even more general than cascaded leaky buckets. We ®rst re- examine the lossless multiplexer of LoPresti et al., and extend their lossless results to generalized regulators.

Using elementary tools from calculus, we show that if the original multiplexer is lossless, then it is possible to allocate bandwidth and buffer to the sources so that the resulting segregated systems are also lossless. We determine the optimal resource allocations and show that the buffer-band- width tradeoff curve is convex for generalized regulators.

We also show that the segregation result does not necessa- rily hold for the delay-based QoS metric, even when the regulators are the simple regulators.

We then examine the multiplexer for probabilistic loss guarantees. We use our results for lossless multiplexing to estimate the loss probability of the multiplexer. As in Ref.

[16], our estimate involves the following three steps: (i) choose a point on the buffer-bandwidth tradeoff curve and transform the original system into two independent resource systems; (ii) use adversarial sources for the two independent resources to obtain a bound on the loss probability of the transformed system; and (iii) minimize the bound by search- ing over all points on the buffer-bandwidth tradeoff curve.

Our principle contribution for probabilistic loss guarantees is an explicit characterization of the adversarial source for the transformed problem in Step (ii). Importantly,the most adversarial source is not a periodic on±off source for the transformed problem consisting of two independent resources. In fact, even in the case of simple regulated sources as studied in Ref. [16], the most adversarial source is not a periodic on±off source. Thus, in addition to general- izing the theory in Ref. [16] to the case of general regulated sources, we provide the true adversarial source for the case of the simple regulator. We also provide an algorithm to calculate the estimate of the loss probability, assuming the truly adversarial sources.

We mention here that in Ref. [9] the original multiplexer problem is transformed into a bufferless multiplexer problem, and then the loss probability is bounded with the Chernoff bound. In this case, the worst-case adversarial sources are indeed on±off periodic sources. But when the original problem is transformed into a problem consisting of two independent resources, one bufferless resource and one buffered resource, the worst-case sources are no longer on±

off periodic sources, even for simple regulators.

This paper is organized as follows. In Section 2, we de®ne the model and the generalized regulators. In Section 3, we address lossless multiplexing. In Section 4, we address lossy multiplexing. In Section 5, we provide numerical results for lossy multiplexing of simple regulators, i.e. regulators consisting of a peak rate controller in series with a leaky bucket. We summarize our contributions in Section 6.

2. Regulated traf®c

We consider a link of rateCwhich is preceded by a ®nite buffer of capacityB. Let Jbe the number of sources that send traf®c to the buffer, and let jˆ1;¼;J index the sources. Each sourcejhas an associated regulator function, denoted bye_j…t†;t$0:The regulator function constrains the amount of traf®c that thejth source can send over an time interval of lengtht toe_j…t†:More explicitly, if A_j…t† is the amount of traf®c that thejth source sends to the buffer over the interval‰0;tŠ;thenA_j…´†is required to satisfy

A_j…t1t†2A_j…t†#e_j…t† for allt$0; t$0:

Fig. 1 shows a multiplexer consisting of a link of rateC, a buffer of capacityB, andJsources with regulator functions, ej…t†;jˆ1;¼J:

A popular regulator is the simple regulator, which consists of a peak-rate controller in series with a leaky bucket; for the simple regulator, the regulator function takes the following form:

e_j…t† ˆminnr¹_jt;s_j²1r²_jto :

For a given source type, the bound on the traf®c provided by the simple regulator may be loose and lead to overly conservative admission control decisions. For many source types (e.g. for VBR video [25]), it is possible to get a tighter bound on the traf®c and dramatically increase the admission region. In particular, piece-wise linear regulator functions of the form

e_j…t† ˆminnr¹_jt;s_j²1r²_jt;¼;s_j^Lj1r^L_j^jto

are easily implemented with cascaded leaky buckets [1,5]

and can lead to improved admission regions [25].

In this paper, we shall consider extremely general regu- lator functions, which include as special cases the forms mentioned above. To avoid certain trivialities, however, we shall always assume thate_j…0† ˆ0;e_j…t†is non-decreasing

Fig. 1. Link of capacityC, buffer of capacityB, andJregulators.

in t, and that e_j…t† is subadditive in t (i.e. e_j…t₁1t₂†# e_j…t₁†1e_j…t₂† for all t₁ and t₂). Also, unless explicitly mentioned otherwise, we shall assume that each e_j…t† is concave int. Let

e…t† ˆX^J

jˆ1

e_j…t†

be the aggregate regulator function. Due to the concavity of the ej…t†⁰s; the aggregate regulator function e…t† is also concave.

Before preceding with our analysis of the lossless systems, it is convenient at this point to introduce some notation and state a few technical facts. Let e¹j …t† denote the right derivative forej…t†ande²j …t†denote the left deri- vative forej…t†:Lete⁰j…t†denote the derivative ofej…t†when- ever the derivative exists att. Similarly de®nee¹…t†;e²…t†

ande⁰…t†:We will make use of the following fact: Ife…t†is differentiable att^p, then all of thee_j…t†⁰s are differentiable at t^p(due to the concavity of thee_j…t†⁰s).

3. Guaranteed lossless service and optimal segregation It is well known [4] that the amount of traf®c in the buffer does not exceedBmin, where

B_minˆmax

t$0 {e…t†2Ct}: …1†

(To avoid trivialities we assume that the maximum is attained in Eq. (1).) Furthermore, due to sub-additivity, it is possible to de®ne traf®c functionsA_j…t†;jˆ1;¼;J;such that the buffer contents will attainBmin. Thus the minimum buffer size that will guarantee lossless operation is Bmin. Throughout the remainder of this section we assume that the multiplexer is lossless, i.e. we assume that the multi- plexer bufferBsatis®esB$B_min:It will be useful to write Eq. (1) in a more convenient form. Ife…t†is differentiable then from Eq. (1) we have

B_minˆe…t_max†2Ct_max; …2†

wheret_maxis any solution toe⁰…t† ˆC:More generally, there exists at_maxsuch that

e¹…t_max†#C#e²…t_max†; …3†

and anytmaxwhich satis®es Eq. (2) and also satis®es Eq. (3).

Throughout the remainder of this section, ®x a tmax that satis®es Eq. (3) (and therefore Eq. (2) as well).

We now address the following question: Is it possible to allocate bandwidth and buffer to theJsources so that each of the resulting segregated systems is also lossless? We shall see that the answer to this question is yes, but depends critically on the concavity of thee_j…t†⁰s:

To address this issue, consider a new system which consists of a link of ratecpreceded by a ®nite buffer. Suppose only the traf®c from source j is sent to this system. The

minimum buffer size that will ensure lossless operation is B_min…j;c† ˆmax

t$0 {e_j…t†2ct}: …4†

We say that a collection ofJpositive numbersc₁;¼;c_Jis a bandwidth allocation if c₁1¼1c_JˆC: For a given bandwidth allocation, we create J segregated systems, with thejth segregated system having link ratecjand receiv- ing traf®c only from sourcej.

Theorem 1.

1. For all allocations we may have B_min#P_J

jˆ1B_min…j;c_j†:

2. If one or more of theej…t†⁰s is not concave then we may have B_min #P_J

jˆ1B_min…j;c_j†for all allocations c₁;¼;c_J: 3. If each e_j…t† is concave then B_minˆP_J

jˆ1B_min…j;c^p_j† where c^p_j ˆe⁰_j…t_max† if e…t† is differentiable at tˆt_max and where

c^p_j ˆe¹_j …t_max†1a‰e²_j …t_max†2e¹_j …t_max†Š

with

aˆ C2e¹…t_max† e²…t_max†2e¹…t_max†

ife…t†is non-differentiable at tˆt_max:

Proof. The proof of the ®rst claim follows from Eqs. (1) and (4):

B_min ˆmax

t$0 {e…t†2Ct}ˆmax

t$0

X^J

jˆ1

{ej…t†2c_jt}

#X^J

jˆ1

maxt$0 {ej…t†2c_jt}ˆX^J

jˆ1

B_min…j;c_j†

For the second claim, we offer the following counter- example withJˆ2;Cˆ1:The envelope function for the

®rst source is:

e₁…t† ˆ

t if 0#t#1;

1 if 1#t#3;

11…t23† if 3#t#4;

2 ift$4:

8>

>>

>>

<

>>

>>

>:

The envelope function for the second source is:

e₂…t† ˆ 2t if 0#t#2;

4 ift$2:

(

It is easily seen that B_minˆ3 whereas B_min…1;c₁†1 B_min…2;c₂†$10=3 for all allocations. Note that both e₁…t†

and e₂…t† are non-decreasing and sub-additive. However, e₁…t†is not concave.

For the third claim, we ®rst show thatc^p₁;¼;c^p_Jis a feasi- ble allocation. Suppose thate…t†is differentiable att_max. Due to the concavity assumption, this implies that each of the ej…t†'s is differentiable attmax. Thus

X^J

jˆ1

c^p_j ˆX^J

jˆ1

e⁰_j…t_max† ˆe⁰…t_max† ˆC:

If e…t†is not differentiable at tˆt_max;then it is easy to show directly from the de®nition of the c^p_js thatc^p₁1¼1 c^p_JˆC:It remains to show thatB_minˆP_J

jˆ1B_min…j;c^p_j†:For a ®xed transmission rate c, the concavity of theej…t†'s and Eq. (4) imply

B_min…j;c† ˆej…t^p†2ct^p; wheret^pis anytthat satis®es

e¹j …t†#c#e²j …t†: …5†

By the de®nition ofc^p_j; e¹_j …t_max†#c^p_j #e²_j …t_max†:

Thus,tmaxis atthat satis®es Eq. (5) forcˆc^p_j:Therefore, B_min…j;c^p_j† ˆe_j…t_max†2c^p_jt_max;

which in turn implies X^J

jˆ1

B_min…j;c^p_j† ˆX^J

jˆ1

‰e_j…t_max†2c^p_jt_maxŠ ˆe…t_max†2Ct_max

ˆB_min: B

From Theorem 1 we know that it is possible to allocate bandwidth and buffer so that the resulting segregated systems are lossless, provided that the regulator functions are concave. This result generalizes a result in Ref. [16], in which all regulators were assumed to be simple regulators.

This result also provides a motivation for the approach we take in Section 4 when we study probabilistic QoS.

Theorem 1 also gives fairly explicit formulas for these optimal allocations. In Section 3.1, we outline an ef®cient algorithm for calculating the allocations.

3.1. Algorithm to calculate allocations

In this section, suppose that each of the regulator func- tions takes the form of cascaded leaky buckets:

ej…t† ˆmin{r¹jt;sj²1r²jt;¼;s_j^Lj1r^L_j^jt}:

Without loss of generality, we may assume that

0ˆs_j¹,s_j²,¼,s_j^Lj …6†

and

r¹_j .r²_j .¼.r^L_j^j: …7†

Let

T_j^lˆ sj^l112sj^l

r^l_j2r^l11_j ^{; lˆ1;2;¼;Lj21:}

In order to avoid trivialities, we assume that

T_j¹ ,T_j²,¼,T_j^Lj21: …8†

With these assumptions, T_j¹ ,T_j²,¼,T_j^Lj21 are the breakpoints ofe_j…t†:

Here is an ef®cient algorithm for determining the optimal allocationsc^p₁;¼;c^p_Jde®ned in Theorem 1. First sortT_j¹;lˆ 1;¼;L_j; jˆ1;¼;J;in increasing order. Number them as T₁;T₂;¼;T_L:These points are the break points ofe…t†:Letk be the maximumlsuch thate²…T_l†#C:Note that to calcu- latee²…T_l†it suf®ces to calculatee²j …T_l†forjˆ1;¼;J;and to calculatee²j …T_l†;we can determine theljsuch thatT_j^lj # T_l,T_j^lj11 and then set e²_j …T_l† ˆr^l_j^j11 if T_j^lj,T_l and set e²_j …T_l† ˆr^l_j^j ifT_j^ljˆT_l:

Thetmax in Theorem 1 isTk. Once having determinedk,

®nd kj such that T_j^kj #T_k,T_j^kj11 and set c^p_j ˆr^k_j^j11 if T_j^kj ,T_k or set c^p_j ˆr^k_j^j1a…r^k_j^j112r^k_j^j† if T_j^kj ˆT_k; whereais de®ned in Theorem 1 and can also be determined directly from ther^l_js:

3.2. The buffer-bandwidth tradeoff curve

For a given link rateCletB_min…C†be the maximum buffer contents de®ned by Eq. (1).The functionB_min…C†is usually referred to as thebuffer-bandwidth tradeoff curve[16] (it is also known as burstiness curve [17]). For a probabilistic analysis in the Section 4, it will be useful to understand the behavior of the buffer-bandwidth tradeoff curve. To this end, for each ®xed C let t…C† be a value oftmax that satis®es Eq. (3). It is easily seen thatt…C†is non-increasing inC.

Theorem 2. B_min…C†is non-increasing and convex in C.

Proof. We ®rst show thatB_min…C†is non-increasing. Let h.0:From Eq. (2) we have

B_min…C†2B_min…C1h† ˆe…t…C††2e…t…C1h††

1t…C1h†…C1h†2t…C†C: …9†

From the concavity ofe…t†we have e²…t…C††# e…t…C††2e…t…C1h††

t…C†2t…C1h† : …10†

From Eq. (3) we have

c#e²…t…C††: …11†

Combining Eqs. (9)±(11) gives

B_min…C†2B_min…C1h†$e²…t…C††‰t…C†2t…C1h†Š

1t…C1h†…C1h†2t…C†C$C‰t…C†2t…C1h†Š

1t…C1h†…C1h†2t…C†Cˆt…C1h†h$0;

which proves the ®rst statement.

For the convexity ofB_min…C†;letC₁#C₂ and leth.0:

We must show

B_min…C₂1h†2B_min…C₂†$B_min…C₁1h†2B_min…C₁†: …12†

By Eq. (2) it is equivalent to show

e…t…C₂1h††2e…t…C₂††1e…t…C₁††2e…t…C₁1h††

$t…C₂1h†…C₂1h†2t…C₂†C₂2t…C₁1h†…C₁1h†

1t…C₁†C₁:

…13†

Using the arguments in the proof of monotonicity, we have

e…t…C₂††2e…t…C₂1h††

t…C₂†2t…C₂1h† #C₂1h …14†

and

e…t…C₁††2e…t…C₁1h††

t…C₁†2t…C₁1h† $C₁: …15†

Combining Eqs. (13)±(15), we obtain Eq. (12). B From Theorem 2 we know that B_min…C† is a decreasing convex function ofC. If each of the regulator functionsej…t†

is piecewise linear, then it is easily shown thatB_min…C†is a decreasing convex piecewise-linear function. Using the arguments in the proof of Theorem 2, it is straightforward to show that the optimal allocationc^p_j for thejth segregated system is increasing inCand that the buffer requirement for thejth segregated system,B_min…j;c^p_j†;is decreasing inC.

3.3. Delay metric

In Section 3.1, we showed how to allocate bandwidth so that Ð for lossless operation Ð the collective buffer requirement of the segregated system is equal to the buffer requirement of the multiplexed system. In other words, for the buffer metricwe can ®nd a bandwidth allocation such that the segregated system performs as well as the multi- plexed system. In this section, we brie¯y consider a natural delay metric. We show that it is not generally true that the segregated system performs as well as the multiplexed system for the delay metric.

For the multiplexed system, the maximum delay isd:ˆ

B_min=C:For thejth segregated system with bandwidthc_jthe

maximum delay isd…j;c_j†:ˆB_min…j;c_j†=c_j:For a given allo- cation, we de®ne the maximum delay of the collective segregated system to be the maximum of the maximum delays of the individual segregated systems, that is, d_seg:ˆ max

1#j#J d…j;c_j†:

The following theorem draws comparisons between the maximum delay of the multiplexed system,d, and the maxi- mum delay of the collective segregated system,dseg.

Theorem 3.

1. For all allocations d#max_1#j#Jd…j;c_j†:

2. There exist concavee_j…t†'s such that d,max_1#j#Jd…j;c_j† for all allocations.

3. Ife₁…t† ˆ¼ˆe_J…t†(homogeneous regulator functions), then dˆmax_1#j#Jd…j; c=J†:

Proof. From Theorem 1 we have

B_min #X^J

jˆ1

B_min…j;c_j†:

Dividing both sides of the above by Cˆc₁1¼1c_J and using the inequality

x₁1¼1x_J

y₁1¼1y_J # max

1#j#J

x_j y_j: we obtain

d# X^J

jˆ1

B_min…j;c_j† X^J

jˆ1

c_j

# max

1#j#J

B_min…j;c_j† c_j

( )

ˆ max

1#j#Jd…j;c_j†;

which establishes the ®rst claim.

For the second claim, we offer the following example:

Cˆ1;Jˆ2;e₁…t† ˆ10 for allt$0 and

e₂…t† ˆ 2t if 0#t#5;

10 if t$5:

(

From Eq. (1) we have dˆ15; d…1;c₁† ˆ10=c₁; and d…2;c₂† ˆ10=c₂25:It is easily seen that for all allocations max…10=c₁;10=c₂25†.15:

The third statement follows directly from Eq. (1) and the de®nitions ofdandd…j;C=J†: B

For the remainder of the paper, we will use the original buffer metric.

4. Statistical multiplexing with small loss probabilities For VBR sources the admission region can typically be made signi®cantly larger by allowing loss to occur with minute probabilities, e.g. loss probabilities on the order of 10²⁷. In this section, we use our results of Section 3 to derive the worst-case loss probabilities for the multiplexer with regulated traf®c.

We consider the same system as de®ned in Section 2: the multiplexer consists of a link of rateCwhich is preceded by a ®nite buffer. There areJsources and thejth source has an associated regulator function, denoted bye_j…t†;t$0:In this section, we suppose that the system resources are not suf®- cient to provide guaranteed lossless service. In other words, we assume B,B_min…C†; so that there exists arrival processes which meet the regulator constraints but cause the buffer to over¯ow. LetPlossdenote the expected fraction of time during which the buffer over¯ows. Our goal is to determine a bound forPlossthat holds forallcombinations of arrival processes which meet the regulator constraints. To this end, we follow the methodology in Ref. [16] (which in turn is inspired by the paper [9]).

Leta_j…t†be the rate at which sourcejtransmits traf®c at timet. We view {a_j…t†;t$0} as a stochastic process. Our goal is to ®nd independent rate processes {a_j…t†;t$0}jˆ 1;¼;J;which maximize the loss probability over the class of all rate processes that meet the regulator constraints. To simplify the analysis, however, we only consider rate processes of the form

a_j…t† ˆb_j…t1uj†;

where b_j…t† is a deterministic periodic function with some periodTj, andujis a random variable, uniformly distributed over‰0;T_jŠ:We assume that the phasesu₁;¼;u_J are inde- pendent, which implies that the rate processes {a_j…t†;t$0};

jˆ1;¼;J; are also independent. We refer to b_j…t† as a source-j rate function.

We say that a source-jrate functionb_j…t†isfeasibleif Zt1t

t b_j…s†ds#e_j…t† for allt$0; t$0: …16†

Note that for a given rate function b_j…t†and phase uj the amount of source-j traf®c sent to the multiplexer over the interval‰0;tŠis

A_j…t† ˆZt

0b_j…s1uj†ds:

Thus the regulator constraint

A_j…t1t†2A_j…t†#ej…t† for allt$0; t$0 is satis®ed if and only ifb_j…t†is a feasible rate function.

As in Ref. [16], our derivation of a bound for Ploss

involves the following three steps: (i) choose a point on the buffer-bandwidth tradeoff curve and transform the origi- nal system into two independent resource systems; (ii) use adversarial rate functions for the two independent resources

to obtain a bound on the loss probability for the transformed system; (iii) minimize the bound by searching over all points on the buffer-bandwidth tradeoff curve. LoPresti et al. use an on±off rate function for their worst case rate function. Our approach differs from that of LoPresti et al.

[16] in two respects. First, we allow for generalized regula- tors as opposed to simple regulators. Second, we derive the true adversarial rate functions, and employ these true adver- sarial rate functions in the bound for Plossfor both simple and generalized regulators.

4.1. The virtual segregated system

Fix a point …C_n;B_n† on the buffer-bandwidth tradeoff curve, and consider a lossless multiplexer with total amount of bandwidthCn and buffer spaceBn. Because the system resource pair…B;C†lies below the buffer-bandwidth tradeoff curve, we must have eitherC_n.CorB_n.Bor both. For this lossless system, we use Theorem 1 to allocate band- widths cⁿ₁;¼;cⁿ_J from Cn and buffers bⁿ₁;¼;bⁿ_J from Bn

such that each of the corresponding Jsegregated systems is lossless. This collection ofJsegregated systems is called the virtual segregated system [16].

For each jˆ1;¼;J; ®x a feasible rate function b_j…t†:

Each rate function generates a stochastic arrival process

A_j…t† ˆZt

0b_j…s1u_j†ds:

For this arrival process, letUjbe a random variable that corresponds to the steady-state utilization of thejth segre- gated system; similarly, letVjbe the random variable that corresponds to the steady-state buffer contents of the jth segregated system. Because theuj's are independent across the sources, U₁;¼;U_J are independent of each other and V₁;¼;V_J are independent of each other.

For these ®xed rate functions it can be argued [16] that

P_loss#P_n X^J

jˆ1

U_j.C 0

@

1

A1P_n X^J

jˆ1

V_j.B 0

@

1

A: …17†

(The argument in Ref. [16] is for a simple regulator. It can be easily extended to our generalized regulators.) P_n…P_J

jˆ1U_j.C†is the probability that the sum of the band- width processesU₁;¼;U_Jexceeds the nodal bandwidthC.

The subscriptn indicates that the steady-state random vari- ables U₁;¼;U_J are computed based on the allocation of bandwidth Cn to the J segregated systems. Similarly, P_n…P_J

jˆ1V_j.B†is the probability that the sum of the buffer processes V₁;¼;V_J exceeds the nodal buffer capacity B.

Again, the subscriptn indicates that the steady-state random variablesV₁;¼;V_Jare computed according to the allocation of bufferBnto theJsegregated systems.

The Eq. (17) is the starting point of our own analysis.

Using the Chernoff bound we get

P_loss#min

a$0

Y^J

jˆ1

Mⁿ_Uj…a† e^aC 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

; 1min

a$0

Y^J

jˆ1

M_Vⁿ_j…a† e^aB 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

;

; …18†

whereM_Uⁿ_j…a†andM_Vⁿ_j…a†are the moment generating func- tions ofUjandVj, respectively, i.e. M_Uⁿ_j…a† ˆE‰e^aUjŠ and M_Vⁿ_j…a† ˆE‰e^aVjŠ: Since Eq. (18) is valid for all points …C_n;B_n†on the buffer-bandwidth tradeoff curve, we have

P_loss# min

…Cn;Bn† min

a$0

Y^J

jˆ1

M_Uⁿ_j…a† e^aC 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

; 1min

a$0

Y^J

jˆ1

M_Vⁿ_j…a† e^aB 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

; 2

66 66 64

3 77 77 75:

…19†

We emphasize that the right-hand side of Eq. (19) depends on the ®xed feasible rate functions. In order to give a bound that holds for all feasible rate functions we need to maximize the right-hand side of Eq. (19) over the set of all feasible rate functions. To this end, we introduce the notion of asource-j adversarial rate function.

Corresponding to each choice of …n;a†; we say that a source-j rate function is adversarial if (i) it is feasible, and (ii) it has the largest value of Mⁿ_Uj…a† and M_Vⁿ_j…a† among all feasible source-j rate functions. Now suppose that we can ®nd the source-jadversarial rate functions for each choice of…n;a†;letU_j^p;V_j^p;jˆ1;¼;J;be the corre- sponding steady-state random variables. We then have the following bound onPloss:

P_loss# min

…Cn;Bn† min

a$0

Y^J

jˆ1

M_Uⁿ_j^p…a† e^aC 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

; 1min

a$0

Y^J

jˆ1

M_Vⁿ_j^p…a† e^aB 8>

>>

><

>>

>>

:

9>

>>

>=

>>

>>

; 2

66 66 64

3 77 77 75:

…20†

Note that by usingM_U^np_j…a†andM_Vⁿ_j^p…a†;which corresponds to the source-jadversarial rate function, we have obtained in Eq. (20) a bound onPlossthat is valid for all combinations of feasible arrival functions. We now proceed to characterize the adversarial rate functions.

4.2. Adversarial sources

Throughout this section ®x an,aandj. We now focus on

determining a feasible rate function which maximizes both Mⁿ_Uj…a† andM_Vⁿ_j…a†over the set of feasible rate functions.

We assume that the regulator functions have the form ej…t† ˆminnr¹jt;sj²1r²jt;¼;s_j^Lj1r^L_j^jto

:

Note thate_j…t†is non-decreasing, concave, piecewise-linear and sub-additive. (The analysis that follows can easily be extended to the case of more general e_j…t† which are non- decreasing, concave and sub-additive.) Without loss of generality, we also assume that Eqs. (6)±(8) hold. Note that the manner in which the allocations …cⁿ₁;¼;cⁿ_J† are chosen (see Theorem 1) ensures thatr^L_j^j#cⁿ_j #r¹j for all jˆ1;2;¼;J:

For a given feasible rate functionb_j…t†with periodTj, the arrival rate at time t is a_j…t† ˆb_j…t1uj† where uj is uniformly distributed over ‰0;T_jŠ: Corresponding to this a_j…t† arrival rate process, let v_j…t† be the buffer contents and u_j…t† be the link utilization at time t. Note that v_j…t†

andu_j…t†are periodic with periodTj. Also the steady-state random variables corresponding tov_j…t†andu_j…t†have distri- butions

P…V_j#x† ˆ 1 T_j

ZTj

0 1…v_j…s†#x†ds and

P…U_j#x† ˆ 1 T_j

ZTj

0 1…u_j…s†#x†ds:

Note that these distributions do not depend on the phaseuj

and are completely determined by the rate functionb_j…t†and the link ratecⁿ_j:

Throughout the remainder of this section we treat the case cⁿ_j .r^L_j^j:In Section 4.3, we deal with the simpler casecⁿ_j ˆ r^L_j^j:Let

d_jˆmax t.0: e_j…t†

t $cⁿ_j

: …21†

Note that sincer^L_j^j ,cⁿ_j #r¹j and sinceej…´†is an increasing concave function,djis a uniquely de®ned, ®nite and strictly positive number. We now de®ne an important class of rate functions. LetToffbe such that 0,T_off#d_jand let T_jˆ e_j…T_off†

r^L_j^{j :}

Now consider a rate functionb_j…t†with periodTjde®ned as follows:

b_j…t† ˆ e¹_j …t† 0#t#T_off; 0 T_off #t#T_j: 8<

:

Such a rate function is pictured in Fig. 2.

This rate function is completely characterized by the parameter T_off. Note that the average arrival rate for this rate function is simply r^L_j^j: Let S_j be the collection of all

Fig. 2. Example of a rate function in SetSj whenToffˆ3 and ej…t† ˆ min{3t;2:510:5t}:

rate functions of this form. Each rate function inSjis iden- ti®ed through itsT_offparameter.

We will show that the setS_jhas the following important properties:

1. Each member ofS_jis a feasible source-jrate function.

2. All members in S_j have identical M_Uⁿ_j…a†; and the members ofSjmaximizeMⁿ_Uj…a†over the set of all feasi- ble source-jrate functions.

3. The member inSjwhich has the largest Mⁿ_Vj…a†has, in fact, the largestM_Vⁿ_j…a†among all feasible source-j rate functions.

Hence, we will have shown that in order to ®nd the source-j adversarial rate function corresponding to each choice…n;a† we need only consider the rate functions in the set Sj. Further, since the rate functions inSjare characterized by a single parameter, Toff, this essentially involves a single- parameter optimization problem. We now proceed to formally state and prove the properties listed above.

Theorem 4. Every member of S_jis a feasible rate function.

Proof. Fix aToffand letb_j…t†be the corresponding member ofSj. It follows immediately from the de®nition ofb_j…t†that Zt

0b_j…s†ds#e_j for all 0#t#T_j: …22†

We can, in fact, show that Zt

0b_j…s†ds#ej…t† for allt$0: …23†

To see this consider any arbitrarytˆnT_j1s;wherenis some non-negative integer and 0#s#T_j:

Zt

0b_j…s†dsˆZTj

0 b_j…s†ds1¼1ZnTj

…n21†Tj

b_j…s†ds

1ZnTj1s nTj

b_j…s†ds#nT_jr^L_j^j1ej…s†

#…e_j…nT_j1s†2e_j…s††1e_j…s† ˆe_j…t†:

The ®rst inequality follows from Eq. (22) and from the fact that the average rate ofb_j…t†over any period isr^L_j^j:The second inequality follows because the slope ofej…t†is never less thanr^L_j^j:

Also, because b_j…t† is non-increasing over each of its periods, we have

Zt1t

t b_j…s†ds#Zt

0b_j…s†ds for allt$0; t$0: …24†

Combining Eqs. (23) and (24) gives the desired result. B

Theorem 5. Each member of Sj maximizes Mⁿ_Uj…a† over the set of all feasible rate functions.

Proof. Each rate function inSjleads to the following form foru_j…t†;the utilization of thejth segregated system:u_j…t†is periodic with periodTj; and

u_j…t† ˆ cⁿ_j 0#t#D_on 0 D_on#t#T_j; (

whereD_onˆe_j…T_off†=cⁿ_j ˆ …r^L_j^j=cⁿ_j†T_j:

The corresponding steady-state random variable is

U_jˆ

cⁿ_j with probability r^L_j^j cⁿ_j ; 0 with probability 12 r^L_j^j

cⁿ_j 0

@

1 A: 8>

>>

>>

<

>>

>>

>:

…25†

Note thatE‰U_jŠ ˆr^L_j^j:

For any feasible source, the steady state rate at which traf®c leaves the jth segregated system, U⁰_j (say), must have a peak value less than or equal tocⁿ_j:Further, because the segregated system is lossless, the long-run average rate at which traf®c departs thejth segregated system must equal the long-run average rate at which traf®c enters the system, which is at most r^L_j^j: Hence, we must have E‰U⁰_jŠ#r^L_j^j: Among all random variables which have a peak value less than or equal tocⁿ_j and a mean value less than or equal tor^L_j^j; U_jas de®ned in Eq. (25) has the highest moment generating function,M_Uⁿ_j…a†:This is shown in the following argument (adapted from Ref. [18]). Let U⁰_j be any non-negative random variable with distribution F_U⁰_j…x† with a peak valuec⁰#cⁿ_j and mean valuem⁰#r^L_j^j:Then, sincea$0;

MUⁿj…a†2MUⁿ⁰_j…a† ˆ r^L_j^j cⁿj

0

@ 1

Ae^acnj 2 r^L_j^j

cⁿj 112Z^c0

0 e^axdFU⁰_j…x†

$ m⁰ cⁿ_j

!

e^acnj 2 m⁰ cⁿ_j 2Zc⁰

0 …e^ax21†dF_U⁰_j…x†

ˆ 1 cⁿ_j

Zc⁰

0 ‰x…e^acnj 21†2cⁿ_j…e^ax21†ŠdF_U⁰_j…x†$0:

B Let b^p_j…t† be a rate function in Sj that has the largest M_Vⁿ_j…a†:

Theorem 6. b^p_j…t† maximizes Mⁿ_Vj…a† among all feasible rate functions.

Proof. Consider any feasible source-j rate function b_j…t†

with periodTj. The actual arrival rate at time t is a_j…t† ˆ b_j…t1u_j† where uj is the random phase. Here, we are concerned only with the steady-state distribution of the buffer contents and the utilization rate of thejth segregated system which are independent of the phase. Hence, in the rest of the proof, we will, without loss of generality, set the phase to zero and considerb_j…t†to be the arrival rate at time t. The corresponding buffer contents process, v_j…t†; is also periodic with periodTj.

In general, both b_j…t† and v_j…t† can have rather compli- cated forms with several intervals within a period where each is non-zero. However, we will ®rst show the desired result for feasible rate functions that give a buffer content process of the formv_j…t†.0 for 0,t,t_j andn_j…t† ˆ0 fort_j#t#T_j;for some 0,t_j,T_j:For rate processes of this form we have

v_j…t† ˆ Zt

0b_j…s†ds2cⁿ_jt 0#t#t_j; 0 tj#t#T_j: 8>

<

>:

Note that, sincev_j…t†.0 for all 0,t,t_j;we must have

t_j#d_j: …26†

We show next thatM_Vⁿ_j…a†corresponding to such a feasi- ble rate function is smaller than that corresponding tob^p_j…t†:

We do this by showing that there is a rate function in the set S_j, b_j…t†; with steady-state buffer contents V_j which is stochastically larger thanVjand which, hence, has a larger MGF (moment generating function).

LetToffbe such that ej…T_off† ˆc_jtj:From Eqs. (21) and (26) we get,ej…T_off†,ej…tj†iftj,djandej…T_off† ˆej…dj†if tjˆdj: Hence, since ej…´† is non-decreasing and dj is uniquely de®ned, T_off #tj#dj: By de®nition, the rate function in Sj corresponding to this Toff is periodic with periodT_jˆe_j…T_off†=r^L_j^j and has the form

b_j…t† ˆ e¹j …t† 0#t#T_off; 0 T_off#t#T_j: 8<

:

The corresponding buffer contents at timet,yj…t†;is given as

y_j…t† ˆ

ej…t†2c^v_jt 0#t#T_off; ej…T_off†2c^v_jt T_off #t#tj; 0 tj#t#T_j: 8>

><

>>

:

Denote the corresponding steady-state random variable as V_j:

Clearly,v_j…t†#v_j…t†for all 0#t#T_off:Note, also, that we cannot havev_j…t†.v_j…t†for anyT_off #t#t_jsince that would requirev_j…t†to decrease at a rate strictly faster than cⁿ_j;in order for bothv_j…t†andv_j…t†to be zero attj. Hence, we get

n_j…t†#n_j…t† for all 0#t#t_j: …27†

Also, we can show that

T_j$T_j: …28†

To see this, note that the utilization rate of thejth segre- gated system with arrival rate b_j…t†is cⁿ_j whenever v_j…t†is non-zero. Hence,P…U_jˆcⁿ_j†$tj=T_j:Also, since the aver- age utilization rate must be equal to the average arrival rate, which in turn is smaller thanr^L_j^j;

r^L_j^j$E‰U_jŠ$c^v_jP…U_jˆc^v_j†$c^v_j tj

T_j and so,

T_j$ c^v_jtj

r^L_j^{j ˆ} ej…T_off†

r^L_j^{j ˆTj:}

Eqs. (28) and (27) imply that

P…V_j.x†#P…V_j.x† for allx$0:

We have thus shown thatV_jis stochastically smaller than V_jand hence has a smaller MGF. It is immediate from the de®nition of b^p_j…t†thatMⁿ_Vj…a†is smaller than the MGF of b^p_j…t†:

We now extend this argument to the case of a general feasible rate functionb_j…t†:Assume, without loss of gener- ality, that the corresponding buffer content process v_j…t†

has m(some positive integer) non-zero portions within a single period, identi®ed by v¹_j;v²_j;¼;v^m_j in the following manner:

v_j…t† ˆ

0 0#t#t_j¹; v¹_j…t2t_j¹† t_j¹#t#t¹_j 1t¹j; v²_j…t2t_j²† t_j²#t#t²_j 1t²j;

...

v^m_j…t2t_j^m† t^m_j #t#t^m_j 1t^m_j ; 0 t_j^m1t^m_j #t#T_j; 8>

>>

>>

>>

>>

>>

><

>>

>>

>>

>>

>>

>>

:

where tⁱj.0; iˆ1;2;¼;m; and tⁱ_j$tⁱ²¹_j 1tⁱ²¹j ; iˆ 2;¼;m: Here, tⁱ_j and tⁱ_j1tⁱ_j represent the end points of theith non-zero portion.

We can express each non-zero portionvⁱ_j…t†as a periodic function, with periodTj, of the following form:

vⁱ_j…t† ˆ Ztⁱj1t

tⁱ_j b_j…s†ds2c^v_jt 0#t#tⁱ_j; 0 tⁱj#t#T_j: 8>

><

>>

:

Let V_jⁱ denote the corresponding steady-state random