Multiobjective flow control in telecommunication networks - CiteSeerX

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The a.pproa.cli is via multiobjec- t,ive optimiza.t,ion. The queueing model used in this chapter is similar t,o the one t1ia.t 1ia.s been amlyzed in the context.


Department of Electrical a d Computer Engineering University of A4iami, Coral Gables, F L 93156, USA


by Yemini and Kleinrock in [29] that previously known optsima.lalloca.tions of tra.nsmitting probabilities in broadcast In this paper, Pareto oplima1it.y is studied i n the context networks are Pareto optimal. They also proposed [30] Pareof multiclass telecon~.naui~ica~ions networks v i t h constraints to optimality as the right notion of optimal flow control. on the maxin~.untalloulahle user delays. T h e perforn~~a.i~~ce Kurose, Schwartz a a d Yemini [ l G ] presented an algorithof objective of each individual class is th.e naaxin~~izatioi~~ m based on competition and pricing for choosing optimal throughput under a delay constraiii.t. Users share the for multiple access protocols in broadcast network on a processor sharing basis. Th.e i~?~altiohjective probworks. Microeconomic concepts that lead to Pareto optilem that arises is solved via a traiasformaf~ioii.f l t a i 1ea.ds t o mality were a.pplied to CPU bala.ncing and file allocation in a multicriteria lilt ear progro~in. T h e solutioi~.inethod is out[17]. Sharing of resources by several processes in the Pareto h i e d by a th~0~07lgh s t u d y of a simple e u m p l e . Pareto opopt,ima.lsense Ins been studied by Courcoubetis [3]. Pareto timal solutiolis are compared 011 t h e basis of f n i ~ n e s sgaveit. opt,imalit.ywa.s defined as the right, notion of optimality for difle re nt co i t f i g ii ra t i o lis a ii d t r a f i c ch a ra ct e ris t.zcs . 1oa.d balancing in computer systems [lo], and flow control in computer networks [ll]as well. I n [1], [5] and [GI, Pareto optimality wa.s defined as the 1 INTRODUCTION main crikrion for the definit,ion of an optimal flow cont,rol st.ra.tegy when multiple classes of users share the resources of a. single queue. The set of Pareto optimal points \vas found when power - defined as the weighted ratio of Resources in a telecomm~inica.t.iotisnetwork are shared dythroughput over average delay - was used a.s the perfornamically by several users. To avoid excessive delays a.nd mance crit,erion. Pareto optimality was also treated as a assure a performa.nce level tlie demands of the users basic requirement for t,he definit>ioiiof a fair solution in a. are not always satisfied complet,ely. This is acconiplislied by .Jacksoii wit.11 no loop free routing in [22]. the use of flow cont,rol. An esbensive p i t , of tlie lit,crn.t.ure on flow control has dealt with tlie cha.racteriza.tion of the I n this paper, Pareto opt,imalit.yis studied in the context stricture of optimal policies given a. performa.nce criterion of mult,icla.sst,elecotnmunica.tionsnetworks with constraints [13][18]. A good overview of t,he suhject, ca.ti be found in on t,lie user delays. The a.pproa.cli is via multiobject,ive optimiza.t,ion. T h e queueing model used in this chapter (141. is similar t,o the one t1ia.t 1ia.s been amlyzed in the context When multiple cla.sses of t,raffic compek for t,hc reof decentralized flow control in [12] and in load sources of an integra.ted services iiet,work, t,liere a.r.ises a balancing in [l]. T h e multiobjective problem is solved via wide variety of t.rallic cla.sses like voice, video, data and a (.ra.nsformat,iont,ha.t leads to mult,icriteria linear program. traffic carrying control. This lea.ds to a.n environIIietit. where Solut,ions coiiipred on the ba.sis of fa.irness given difmultiple classes of traflic a.nd ma.ny users compete for fixed ferent configura.tions a.nd tra.ffic cha.ra.cteristics. A software network resources and depending 011 t,he ol,ject,ive one may pa.cka.gehas been developed that is based on existling 1inea.r apply flow control on a. by class or per user Insis. iniilt,icrit.eria programs tha,t provides the set of all Pareto However, since a.11 classes might not, have a commoii perforo p h i a l solutions. Ba.sed on the results of [22], a study of mance objective the problem is now one of mnltiple criteria. hiriless is a.lso performed for a wide va.riation of the control optimization. This paper a.ddresses t,he multicrit.eria, probpa.ramet,ers. lem in a Pareto optimal sense following a. game theoretic approach. 'The paper is orga.nized a.s follows. In section 2 an The introduction of mult,iple olijective opt~iriiiza.t.ion overview of the modeling, objectives and performance crimodels in control problems da.tes back to Za.de1i [:Ill. St,eria used i n t8elecommunica.tionsnetworks is presented. In ince then, these ideas have been used extensively i n tlie secbion 3 t,he problem is stated. Section 4 presents the solnoperations research [33][2][25][2G]but. liave found tion met,liod and the forrnula.lion of the problem as a mulfew applications in t,he engineering field [20][21][19]. t,icriteria linear progra.m. Section 5 presents several of the propertlies of (.lie Pa.reto optimal set. Section G deals with One of the first works i u the comput,er net.worlts area t,o the det,ailetl presenta.tion of a. single exa.mple t1ia.t shows all use multiobjective optiinizat(ioi1 ideas wa.s (.he realizat,ion

3A.1.1 INFOCOM '92

CH3133-6/92/0000-0303 $3.00

0 1992 IEEE


the necessary steps of the solution. Sect,ion 7 concludes tlie paper.



In a telecommunica.tions environment, the t,wo most releva.nt performance quantities the werage throughput and t.he average delay. T h e conflict betaween throughput and delay is obvious since as more traffic enters the network queues become large and delays may increase to una.ccept,able levels. Maximization of throughput and mininliza.tion of t.he delay are thus the main objectives of any flow control or congestion control scheme. Incorporation of these two measures in a single performance criterion has been proposed t,lirougll the use of power - defined as the weighted ratio of the throughput over the average delay. Even though power ta.kes int.0 accouttt both throughput a.nd delay, delays can be very high [GI. I n am integrated environment classes of customers very strict delay requireii1ent.s; if for exa.mple voice pa.ckets experience a delay greater t1ia.n a t,lireshold the qualit,y of the transmitted voice is affect.ed sribst.ant,ially. To address this problem, L a m r [Is] proposed a s a performance criterion the maxitniaatioti of throughput. under a n h i e delay const.raint.. T h e queueing syst.em of figure I is used t,o demonst,rai.e the performance criterion and the flow cont,rol strategy.

) Network


N circulating


pa.cket,s. A t o h l of N packets is allowed in the network and N - k of them are assumed t,o be in the lower queue. N ca.n be arbit,rarily large t o model cases where there is no preset. limit, on the number of packets allowed in the network. l'he lower queue has an exponential server with rate ( A k ) , k = 1 , . . , , N . If 0 5 A t 5 S , with S E R+ the maxiniuin speed of the channel, the control is called admissible. This maximum speed may depend on the transmission medium used or on other regulatory constraints. Let E ~ NETN , denote the average throughput and the average delay experienced in the upper part of the queueing system, when N packets are allowed to circulate in the network, and let 'T be the maximum acceptable average delay. Uefinilton 1. T h e control X = ( A k ) , 1 5 k 5 N , is said to be 0pt.ima.l over the class of a.dmissible controls for a given 'T) 'T E R + , if the ma.ximum maxETNet.t.eroff without deteriora.ting tlie performance of a t least onr of the other players. Such points called Pareto optimal or efficient or nondominated. UpJinition 2. A point U* E U is called Pareto optimal ill' for a.ny (U' + 610 E U either



packets 01'

for a t least

+ 6 u ) - hi(l1') /l;(u*

3 Figure 1. A flow controlled closed queueing network with one class of packets

There is a closed queueing nrlwork wiih one class ol


15 i

5 71


611) -

/li(U*) < 0


A queueing net,work consist,ing of A4 interconnected queues rnodeletl as E\II/R,l/l queueing systems is considered (see figure 2). This queueing network represents a. telecornmunicat.ions net.worli where tlie queues correspond to the nodes of

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O I I ~ i

the network where packets received processed and t.hen sent to the desirable dest,ina.tion. 72 classes of users the resources of this net,work. These cla.sses inay arise from different ca.tegories of pa.cket.s like voice and d a h pa.ckets or they may a.rise due t,o the different priorities of t.he decision process in the cont,rol of this network. Ea,ch class i, i = 1, 2, . . . , n,,tra,nsinit,s pa.cltets - the basic units of information - through the with a. maximum transmission rate of Si packet.s per unit of t,iiiie, has an upper limit - a window - of Ni 1xicltet.s t.liat. ca.n be siinrilbaneously in process in the network and there is a.n upper bound ri on the average delay that a class i pa.cket can tolemte. ~

Tile t,otal number of packets in queue j is thus Cy=lk!, The set E of feasible states consists of s-

of vect.or a,.

lkj) =

t.a.ks where t,he total number of packets of each individual

c,lass does not, exceed t,he ma.ximum allowable number.

When the network is in state k,an a.rriving class i packet is routed with proba,bility kif to queue j , j = 1 , .. . , M and there is no class i ar!iving pa.cket accepted with probability k?’: = 1 k?:. Since processor sha.ring is assumed a t server j, j = I , 2,. , . , A[, the capacity p! of server j available for class i


To complete our model, when a class i packet in the net,worlc is served ,at queue j , it then joins queue 1 with proba,bility 0 5 k le del ay s et c , 0 T h e problem as posed is a flow cont,rol problem. The rout,ing probabilit.ies bet.ween the queues were assumed constant.. I f w r are allowed 1.0 cIia.nge t,lieni as well, a multiobjective 1oa.d ba.lancing problem similar to the one presented i n [I] for the single objective case results with a solution similar t.0 tlie one presented a.bove.


l i t t.he oiicrat,ioiis resea.rch literat,ure, problem ( 3 ) has been treat,etl est,ensively and several of its key properties have heen['24][33][9]. We present some of these propert.ies to shed some liglit, on t,he structure of the solution and int,errelate t,his problem with other equivalent formulations. Unless the constra.ints satisfy certain conditions the soIiit,ion 1.0 t . l i c inilltiobjective program is not a iinique point. 'l'he set. of eliic.ient solutions includes the cases where each is alone i n the net,work and the objective is the maximizat ion of t.hroughput. under a time delay constraint. The set, of nondominat,ed ext,reme points is a. connected set.. a i i d t.hc srt. of elfic,ient. solutiolis includes the solihotis to all (.lie p r o l h n s of the form

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under const,raints ( l b ) ,

if these problems have a unique solution [32]. Note t,liorigli that arbitrary selection of a. weighting fact.or might. lead to pathological situations and not t,o Paret,o opt.ima1 point,s. Not only (4)is true, but. one can find the range of wrigliing factors wj that give a. extreme efficient. point a.s the solution as well. An extreme nondomina.ted point is t,he solution of the set of problems inax w iE - , ~ (5)

a n t l t,he global halance equations are given by:


under constra.ints ( l b ) where wj E [aj,bj] with 0 5 a.i 5 1, 0 5 6; 5 I , ai I b i , and Cy=lwj = 1. Several techniques to create the set of all the eflicient extreme points and then the set of a.11 the Paret.0 optima.1 points have been presented [7] [SI [24]. These algorit~l~ms usually follow the 3 pliases: Phase I: Starting from a.n iiiitia.1 ba.sis wliicli contains vectors associa.ted with artificial va.riables proceed to a. feasible basis if one exists or termii1a.t.e if problem incoiisist,ent,. Phase 2: From a feasihle basis proceed to an eficient, basis if one exists or tlet,ect, unboundedness. Phase 3: From a n cflicient, Ixisis proceed t.o enumernt,e the list of efficient basic feasible solutions.

A method present,ed by Evans and St,erier [D] aiitl tlocumented in the ADBASE package [2S] is modified antl used


as a subroutine in our soft,ware I d i a g e . Ot.lier sul,roiitines include the development. of t.he model i n a. forn1a.t. it. can be understood by ADBASE. The t.opology of t.lie net,worl;, tlie constralnts oii delays, t.he masinium nuniber of p a c k e k allowed and the values of the masiniuiii speeds a i i d service rates constit,ut,e the inpiit,s t,o t,liese subrout.incs. ' I ' l i ~ r ~is a. subroutine t1ia.t gets t,lie solut,ion of t.he iiiult,iob.jcct.ivcproblem a.nd assigns ranges of weights for tlie inclividiial criberia aiid a subroutine t,liat. creaks t.he set. of all 1'a.ret.o opt,imal points. This last, subrou(.iiie is Inset1 on an algorit.lim presented in [2'i]; the algorit.lim is presenktl iii sufficient. tlet.ail in the next section. The nest. sect,ion provides a ctetailetl solnt.ion of (.lie simplest possible exa.mple i n order t.0 shrtl light. t.o t.he s k p s of the procedure.

6 6.1


3y' ( 0 ) 3y?(O)

+ 3y1(1) + 3y2(1)

untlcr tlie constraints

ANEXAMPLE The Formulatioil of the Model

A single server queue wit,li t,wo classes of ciist.oiners is coilsidered. Let N1 = 1, N? = 1, S1 = 3, S? = 3, /' = 1, 71 = rz=l.l.Tliestat.espaceis {(O,O), ( 0 , 1 ) , (1,O), ( ] , I ) }

To solve t,liis ninlt,icrit.eria.1inea.r proglain a. simplex tableau is formed siini1a.r to t.he one usually formed in the single crit,erion case. Jnst,ea.d of a. single objective row there are 2 object.ive rows. 1nt.roclucing the two slack va.riables ylo a.nd 1/11 I.he siinplrs t,al)leau of 'lkble I is obta.ined. It can be oliservetl, is similar t,o the simples ta.bleau in the one ol).ject,ive case. 1nstea.d of the single row for the objective liere tlwre n.rc t,wo criteria. rows.

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The Extreme Points



- 1 . 1 4 8 ~ ~ 1 1.579t02 2 0 + w1 5 1.3333W2 -2.763111~+ 2 . 6 8 4 ~ 22 0 3 ?U1 5 0 . 9 7 ~ 2 1 .5i9t01 - 1 . 1 0 5 ~ ~0 3 wl >_ 0 . i w 2 -0.789~1 1 . 0 5 ~ 2 2 0 + ~1 5 1 . 3 3 3 3 ~ 2

Using the ADBASE package [2S], t.he following fivc ext.reine nondominated points are obta.inec1. E71 = 0.434 E72 = 0.434

x1 :



E71 = 0.394 E72 = 0.473' %5





Relationship with Weighted Criteria

p(0,O) = ? / ' ( O , O ) + ? / o ( o , o )


p ( 0 , l ) = ?/"(U, 1 ) +?/'(U, 1) = 0.Q55 p ( i , o ) = y 3 ( 1 , ~ ) + 1 / ? ( 1 , =o.4:34 0) y( I .

I ) = O.Oi9

a.nd the nonzero prolia.hilit.iesare

5 0.97~2)

Si n-Ii I a rl y \;V(cl) =

{ ( 7 U 1 , 7 0 2 ) : 0.9iw2 5 w1 5 1 . 0 2 9 ~ 2 1 W ( x 2 )= { ( ~ l ~ l , 1:~12. 0) 2 9 ~ 25 W1 5 1 . 4 2 ~ 2 ) CV(c4) = ( ( ~ 1 ~ : ~1 .24 2) ~ 25 wl}

E71 = 0.00 E72 = 0.75

To a.nalyze severa.1 of the properties of tlir solirt.ioii pick point x3. T h e equilibrium proba.bilities at. t,liis point, me:


IV(c3) = { ( W ~ , W:?0). 7 ~ 25 wl

Eyl = 0.75 E72 = O . O O

The last two extreme points correspond t o the cases where user 1 or user 2 ma.xiniizes its throughput respectively. Point z1 provides a completely symmetrical solution in the two users, while t2and c3 are symmetric to each ot,Iier with respect t o the two users. Note also that z1 ma.simizes the product of throughputs. This result connects with the results presented in [22] where a fair point was defined as the one t,lia.t.opt,imizes the procluct of throughputs.



E71 = 0.473 E72 = 0.394

e2 :




=5 { () w l rw 2 ): wl

5 0.7 w 2 }

Thus the set of all weighting factors has been decomposed in five subsets a.nd the solution of each weighted crit,erion problem in t,he interior of these subsets gives one ext.reme efTicient. point.. Points in the boundary give more than one ext,reme points.

An Algorithm to Find the Set of all Efficient Points


To con1put.e the set of all efficient, points a method proposed by S l i e n a.nd Lilt [2i] is used. T h e followiiig definitions are nrcessa.ry. DcJi71itioit 3 . A vertex L." = ( c y , . , . , ):e is called a brancl~ingvertex if il has more than one adjacent vertex 1.liat. have valucs of D or equal to t,he D value a t zo, wliere

D(xo) = Cy=,.cy.

A vertex x i is called a merging vertex if i t is t.o two or more distinct vertices and the D valite a.t, ea is grea.ter t1ia.n or equa.1 t,o the D value a t these adja.cent. vert ices. DcJiitition 3 .

l l r J i n i t i o n 5 . A path is a. series of distinct adjacent vert,ices arrilngetl in t,lie direction of increasing D value. I ) ~ f i n i f . i o n6. Paths are sa.itl to be alteriiative if they hranch oiit. froni t.hc sanle merging vertex with dist,inct. int . r r i i led i at~ever t,ices.

Ilaving already foilnil all t,he 1'a.reto optimal vertices the met.liod proceeds a s follows:

St,ep 1 : Set, i = i o (where io is t.lie first siinplex iteration a I'itret,o opt.imaI verkx is found ) and start a path for ea,ch clist i n c l i n t,he i;" itmeration. S6rp 2: Set i = i


I f i > i , (where i , is t,lie last simplex iteration t , I i a t coiita.ins a.t. least one Pa.reto opbimal point ) go to step 6 else go t.0 st.cp 4 . Stel) 3:

Sbep 4 ; Skirt. new 1)at.h for vertices not, adjacent to any i n t,he ( i - I)'* iterat.ion. Step

5: Adjoin each vert.ex


t.lie i t h iteration to its

at1,iacent. vert ices i n t.hc ( i - I)"' it.era.tion thereby extending the esisting pa.t.lis.

St,el~6: Det.erinine all alternative pa.ths.

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T h e vertices contained in cach set, of alternative paths form a simplex. T h e union of t.hese simplexes and those edges and vertices that are not part. of any alternative patclis is the set of all Pareto optimal point,s.


al po~versin a .lackson network with loop-free routing and this point has a sct of properties that can define it as a “fair” point. T h e definition of a fair point was based on the N a 4 1 hrbit ration Scheme

The Set of Pareto Optiiiial Points

In the example, extreme point. t 1 is atIjacent, to est.reme points x 2 and z3.Ext,reme point. 1:’ is adjacent, t,o ext.reme points z1 and x4 and extreme point x3 is a.djacent t.o x1 and z 5 . Note that D ( z ’ ) = 1.13, D ( N ’ ) = 1.139, D(z3)l.139, 0 ( z 4 ) = 1.25 and D ( z 5 )= 1.25. Proceeding through t*hesteps of the a . l g o r i t h (see figure 3 ) the complete set, of Pareto optimal points is given by the convex hull of (z1,z2,z4} and {z1,x3,z5), i.c. edges (x’x’), (x2x4), (2’x3) a.nd (x3z5).

i $

lI 0.8.

2 0 3 0.6.








DELAY CONSTRAINT OF USER 211 [Er, = E n ] Figure 4. The individual throughputs and their product at the point that maximises the product (symmetric case)

Figure 3. The set of Pareto optimal points




0.8 -


To study the effect,s of t.lie delay coilstraitits. t,lic t.opology the network and t.he niimber of packet.s a.llowetl i n t.hc ne,work on the set of efficieiit. poirit.sseveral esaiiiplcs \wr(?riin. The number of efficient extreine point,s does not. cliaiigc significantly with the delay const.r; h t , is ii.fIbct.rtl IQ l . 1 ~ number of pa.ckets allo\ved in t,lie uetwork. I)ela.ys aIr(?ct the performance since Iiiglier delay const.raint.s allow t.lir existence of more packet,s i n the iietwork, iniproving i , I i ~ throughputs. T h e points t.1ia.tma.ximize t.he s u m of the po\rers and the points that maximize t.he product. of the powers were a.lso compared in o u r study. T h i s is done Im-arise a manager of the network would to choose a part.icn1a.r operat,ing point among the Pareto opt.inial ones. Maxiinizat.ioii of the sum of the indivitlira.l ol)jective fniict.ions is an intuitively appealing objective and is closely coiiiiect,etl with the equiva.lence between weighkd crit,eria aiid Inult.iol).jective programming t1ia.t we referred t.o in t.he previous section. It turns out. though t.1ia.t. there are cases where a user is penalized by a. zero t~lirougliput.when t.liis oIlject.ive is chosen [GI. In [22], the opt,inialit.y of t.hc p r o d i i c t ~fornis w a s shown. In part,icrilar. it, was provon t.liat I.lirrc~is n unique set of throughputs t.liat niaxiniizes t.hc prodiict of iiitlividu-





2 3




E 3,

0.2 -






PI= Eh1

Figure 5. The individual throughputs and their sum at the point that maximizes the sum (symmetric case)

In ligilres 4-7 we consider a single queue case with of pac1iet.s. T h e maximuin a.rrival rates are SI = = 3 packet,s/sec a.nd t.lie service ra.te is /I = 1 packet./sec. The need for flow cont,rol is We examine a symmet.ric case where the ma.ximum number of packets for class 1 is eqital t.o t.he maximum nuniher of packets for class ‘L a.nd a.n asymmet,ric case where class 2 is allowed 2 I’iiclict,s. We plot the p0int.s tl1a.t. maximize the sum and t.litr product. of throughputs in these two cases. In the symt.wo cliisses

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metric case there is no tlilrererlce bet,weelt t.he poii1t.s t,Iiat, maximize the product, and t.he poi1it.s t,liat. iiiasiiiiize t.lte suin of the tliroughpiit,s and t.liere is also 110tlifI’ereiice between the performance of the two classes as expected. 111 the asymmetric case t.he product of tliroughptit,s crit.erion gives symmetric performance t,o t,he two risers, providing a “fair” result. T h e sum of t,liroiighputs crit,erion allows more packets for user 2 t.0 the queue when the delay constraints not that. st.rict.

Class 2 packet,s sliare queue 1 and then are served by queue 3 wit.11 service rat.e 112 = 2 packets/sec. The maximum i t u i i i l w r or a.llowable 1mcket.s lor class 1 a.nd class 2 are N1 = I p.-ltet, and N? = 2 pacliets. As expected class 1 gets a smaller portion of the t,otal throughput,. T h e difference t.liough is not, as profound in the product, of throughputs case t.o the sum of throughputs.


I v

F PMETO: Nl=l, NZp2, Sl=S=J. sERw( ME-1






8. A t w o class, three queue example.

p = p i = l pack/sec,



2 packlsec.

l3y comparing t.lw Parct.0 opt.imal poiut,s t1ia.t. masiniize (.lie S I I I I Iof t.he t.liroiighput.s 1.0 the Pareto optimal points

t1ta.t. masimize t.lie product of t.he t,lirorigliputs we could see t.ltat. (.lie lilt.t,er provide a more bala,iicetl set of throughputs coniparetl t,o t.lie former. As t.he difference between the delays a.nd/or t,he numher of packets allowed in the network increases t.lie maximizat.ion of the product. does not penalize t.lie rest.ricted users a s much.

DELAY CONSTRAINT OF USER 2 / 1 [Er, = Er2] Figure 6. The individual throughputsnnd their product nt

7 the point that maximizes I

the product (nsymmetric cnre)



NZ=2. Sl=S2=3 SONER M l W



’l’ltv Ilow cont.rol problem i n a iniilt, t.elecommuiiicat.ions enviroiiinriit, where users try to maximize their t,liioiighput,s sril~ject.t.o delay coiistraints has been atltlressetl. i\ inult.icriteria fortiiiilat,ioii has led t,o the study of 1’arc.t.o opt~imunisolutions. T h e resulting nonlinear mult.iobjective opt,imization problem has been transformed to a 1inea.r iiiiilt.iol~,ject.ive pr0gra.m. A complete solution to this

DELAY CONSTRAINT OF USER 2 f 1 (ET, = Er21 Figure 7. The individual throughputs nnd their sum nt the point thnt mnximizes the sum (asymmetric case)

T h e performance of the inore geiieral net.\vork of ligiirc 8 is shown in tables 3 and 4. Class I pa.cket,s I>eing served by queue 1 wit.11 service rat.e / L = I l>ilckPt./scc are served by queue 2 with service rat,e = 1 I)a.cIi(!L/sec.

prohlem was presented. Connections wit,li weighted criteria opt,imization problems and fa.iriiess issues were discussed. \i package ltas been developed t,liat a.ccepts as in1),lie t.opology of the iiet,work, t,he routing configurations clii(l the delay coi1st.raiitt.s a.nd provides a s out,puts the set of Paret,o opt.imal points and t,he points t1ia.t ma.ximize the product ancl t,he sitin of t,lie t,lirougltputs. Even t,lioiigh a. closed form solution for the set of Pa.reto optimal point,s has not been obt.ained a s in [GI,our present st.utly allows t.he invest,igation of a wider class of networks t,Iiait (.lie siiigle queue studied in [GI. Ackiiowlt!tlgineiits

3A.1.8 0310

This work was Inrtia.lly support-

ed by the Florida Iligli Technology alrcl Iii(Iiis1ry (hnncil. IIel pful and insight f i I 1 d iscri ssion s wit,h 1) rs I t . 14 a z iim d ii r a.nd L. G. Mason of INRS Telecommiinicat.ioiis,Canada., are gratefully a.cknowledged. A n uptlat,etl vcrsioii of t,he software package was done by Jean Il>(Ai1r.

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