Resource Control for Loss-Sensitive Traffic in CDMA Networks

Transcription

1 Resource Control for Loss-Senstve Traffc n CDMA Networks Vaslos A. Srs Insttute of Computer Scence, FORTH and Department of Computer Scence, Unversty of Crete P. O. Box 1385, GR 711 1, Heraklon, Crete, Greece Emal: vsrs@cs.forth.gr Costas Courcoubets Department of Informatcs Athens Unversty of Economcs and Busness 47A Evelpdon & 33 Lefkados, GR , Athens, Greece Emal: courcou@aueb.gr Abstract We nvestgate the problem of effcent resource control for loss-senstve traffc n CDMA networks, usng an economc modellng framework that takes nto account the jont control of the transmsson rate and the sgnal qualty. Although the correspondng global optmzaton problem has a non-trval structure, and we cannot n general guarantee that a soluton can be found usng the Lagrangan method, we have strong expermental evdence that ths s possble for a wde range of user utltes. In ths case the global optmum can be acheved n a decentralzed manner, usng shadow prces to nfluence the ndvdual user resource requests. Based on ths evdence, the man contrbuton of the paper s to dscuss how exstng rate control and outer-loop power control procedures can obtan a smple and attractve form that takes nto account, through shadow prces, the level of demand and supply n order to acheve effcent resource utlzaton. Moreover, we descrbe and evaluate approxmatons of the proposed resource control model that can smplfy ts applcaton, and we present extensons of the model for the case where the packet success rato depends on the transmsson rate n addton to the sgnal qualty, and for network paths contanng multple wreless lnks. I. INTRODUCTION Flexble and effcent resource control n wreless networks s an ncreasngly mportant ssue, due to the lmted capacty of wreless networks, and ther ncreasng use for delay and loss-senstve applcatons, n both prvate and commercal envronments. In ths paper we nvestgate the applcaton of economc modellng for resource control n Code Dvson Multple Access (CDMA) wreless networks carryng losssenstve traffc. Although our approach s generally applcable to CDMA-based systems, we focus our dscusson on Wdeband CDMA (WCDMA), whch s the most wdely adopted thrd generaton (3G) ar nterface technology [1], and dentfy how the proposed model affects exstng control procedures n WCDMA systems. In WCDMA, data bts are spread over the entre spectrum that s avalable for transmsson, and unque dgtal codes are used to separate the sgnals from dfferent mobles. WCDMA supports varable bt rate transmsson wth the use of varable spreadng factors, whch determne how much a data bt s spread n tme, and wth the use of multple codes. In addton to ts use n cellular systems, CDMA s recevng ncreasng attenton for use n wreless LANs. Ths work has been supported n part by Brtsh Telecommuncatons, UK. A unque property of wreless networks, and CDMA networks n partcular, s that resource usage s determned by the transmsson rate and the transmsson power, whch can be dfferent for dfferent moble users. In WCDMA, resource control procedures nclude rate control, outer-loop power control, and fast closed-loop power control. Rate control s responsble for adjustng the varable bt spreadng factor and the number of dgtal codes. Outer-loop power control s responsble for adjustng the target sgnal qualty, determned by the bt-energy-to-nose-densty rato, n order to acheve a gven packet error rate; ths target sgnal qualty s then gven to fast closed-loop power control, whch s responsble for adjustng the transmsson power n order to acheve the target. Fast closed-loop power control operates on a faster tme-scale, approxmately 15 Hz, compared to rate control and outerloop power control, whch operate at a frequency typcally less than 1 Hz. Economc modellng has been dentfed as a flexble framework for effcent resource control n wred networks; e.g., see [2], [3], [4] and the references theren. Recently, economc modellng has also been appled for resource control n wreless networks [5], [6], [7], [8], [9], [1], [11], [12], [13]; we present a bref overvew of ths work later. Our work dffers from ths work n one or more of the followng: Frst, we consder the jont optmzaton of the transmsson rate and the bt-energy-to-nose-densty rato (sgnal qualty), and nvestgate how the two control procedures for adjustng these quanttes need to be modfed n order to acheve effcent resource utlzaton. It s mportant to note that our model does not affect closed-loop power control, whch can be performed as currently done n exstng systems. Moreover, our approach results n moble users selectng target bt-energy-to-nosedensty ratos that lead to feasble power allocatons, hence protects fast closed-loop power control from dvergng. Second, our work s based on the economc modellng framework of socal welfare maxmzaton, for whch we consder the partcular network constrants of CDMA networks. Thrd, our work consders partcular expressons for the user utlty that capture ts dependence on the average throughput and the data loss rate. In ths drecton we defne loss-nsenstve traffc to be traffc whose utlty depends solely on the average throughput, and s ndependent of the loss rate. An example of

2 loss-nsenstve traffc s bulk data transfer, where delays due to retransmssons are not mportant. For loss-senstve traffc, on the other hand, the utlty depends on both the average throughput and the loss rate. Examples of loss-nsenstve traffc are real-tme and nteractve applcatons, whch are senstve to delay, hence to losses whch result n retransmssons. Also, TCP traffc can be consdered loss-senstve, snce ts achevable throughput depends on the packet loss rate. In ths paper we present a new model for resource control n CDMA networks carryng loss-senstve traffc, based on an economc modellng framework that takes nto account the wreless resource constrants and the jont control of the transmsson rate and the sgnal qualty. The correspondng socal welfare optmzaton problem has a non-trval structure, and we cannot n general guarantee that a soluton can be found usng the Lagrangan method. For ths reason we have resorted to numercal expermentaton, whch ndcate, to our ntal surprse, that for a wde range of user utltes the Lagrangan method can be appled, hence a shadow prce exsts and the socal welfare maxmum can be acheved n a decentralzed manner, by havng each user solve a local optmzaton problem. Based on the aforementoned expermental evdence, the man contrbuton of the paper s to dscuss how exstng control procedures, namely rate control and outer-loop power control, can obtan a smple and attractve form that takes nto account, through shadow prces, the level of demand and supply n order to acheve effcent resource utlzaton. In ths drecton, we show that for loss-senstve traffc where the utlty depends on the loss rate n addton to the average throughput, unlke the case of loss-nsenstve traffc where the utlty depends only on the average throughput, the optmal sgnal qualty s no longer ndependent of the level of resource demand and supply. Hence, both rate control and outer-loop power control need to take nto account shadow prces. A second contrbuton of the paper s to dscuss and evaluate an approxmaton that can smplfy the applcaton of the proposed resource control model to dscrete rate control and to outer-loop power control, and for supportng class-based servce dfferentaton n terms of both throughput and losssenstvty. Fnally, we extend the model to the case where the packet success rato depends on the transmsson rate, n addton to the sgnal qualty, and to networks contanng multple wreless lnks. The rest of the paper s structured as follows. In Secton II we summarze results on resource control n CDMA networks carryng loss-nsenstve traffc. In Secton III we present and nvestgate a model for resource control n the case of losssenstve traffc, and n Secton IV we dscuss varous ssues regardng the applcaton of the proposed model. In Secton V we present extensons to the proposed model. Fnally, n Secton VI we present a bref overvew of related work and n Secton VII we conclude the paper. II. RESOURCE CONTROL FOR LOSS-INSENSITIVE TRAFFIC In ths secton we summarze background work on resource control for loss-nsenstve traffc n CDMA networks, as developed n [14]; for such traffc a user s utlty depends solely on hs average throughput. Consder the uplnk of a sngle CDMA cell. Note, however, that the results can be extended to the multple cell case by consderng the ntercell nterference coeffcent [15]. Let W be the chp rate. The bt-energy-to-nose-densty rato at the base staton s gven by [15], [16] ( Eb N ) = W r g p j g jp j + η, (1) where r s the transmsson rate, p s the transmsson power, g s the path gan between the base staton and moble, and η s the power of the background nose at the base staton. The rato W/r s the spreadng factor or processng gan for moble. The value of the bt-energy-to-nose-densty rato (E b /N ) corresponds to the sgnal qualty, snce t determnes the bt error rate, BER [15], [16]. Under the assumpton of addtve whte Gaussan nose, BER s a non-decreasng functon of E b /N, that depends on the multpath characterstcs, and the modulaton and forward error correcton (FEC) algorthms. Let γ be the target bt-energy-to-nose-densty rato requred to acheve a target bt or packet error rate; the adjustment of γ s performed by outer-loop power control. The target γ s gven to closed-loop power control, whch adjusts the transmsson power n order to acheve t. If we assume perfect power control, then (E b /N ) = γ. Solvng the set of equatons (1) for each moble, and assumng that there s a large number of mobles, each usng a small porton of the avalable resources, one can show that the wreless resource constrant can be approxmated by [16], [17], [1] r γ <W. Hence, the amount of resources used by a moble s gven by the product of the transmsson rate and the target bt-energyto-nose-densty rato (sgnal qualty). In actual systems, due to the lmted transmsson power of the moble hosts, mperfect power control, shadowng, and nter-cell nterference, the total load must be well below one. Indeed, n rado network plannng [1], all the above factors are used to determne an nterference margn (or nose rse) I margn, based on whch the wreless resource constrant becomes r γ <ρ UL W where ρ UL = I margn 1. (2) I margn In the case of loss-nsenstve traffc, where users value only the average throughput of successful data transmsson, whch s gven by the product r P (γ ) of the transmsson rate and the probablty of successful packet transmsson, the utlty s gven by U (r,γ )=U (r P (γ )).

3 Consder the global problem of maxmzng the aggregate utlty (socal welfare) maxmze U (r P (γ )) over r,γ subject to r γ <ρ UL W. By settng z = r γ, and assumng γ >, the above problem can be wrtten as ( ) P (γ ) maxmze U z γ whch s equvalent to over z,γ > subject to z <ρ UL W, maxmze U (z max γ > over z subject to z <ρ UL W. ) P (γ ) γ Hence, the global problem of maxmzng the socal welfare can be decomposed nto two separate problems, the frst gves an optmal sgnal qualty γ for each user that satsfes P (γ )γ = P (γ ), (3) whch s ndependent of the user utlty, and the second problem s an optmzaton over {z } ( P (γ maxmze U z ) ) γ over z subject to z <ρ UL W. If U ( ) s dfferentable and strctly concave, then the Lagrangan method can be appled, hence the global optmzaton problem s equvalent to each ratonal user solvng the problem maxmze U (r P (γ )) λr γ (4) over r,γ, where λ s the shadow prce for the wreless resource constrant (2). In practse, the shadow prce can be teratvely adjusted based on a tatonnement process, so that the aggregate demand equals the avalable wreless resources. The above results allow the decouplng of the two problems of selectng the optmal γ and the optmal transmsson rate r. Moreover, the optmal γ s ndependent of the level of resource demand and supply (shadow prce); ths s not the case for loss-senstve traffc as we wll see n the next secton. III. RESOURCE CONTROL FOR LOSS-SENSITIVE TRAFFIC In ths secton we frst propose a utlty for loss-senstve traffc that takes nto account the dependence on the average throughput and the loss rate, followed by a model for effcent resource control n CDMA networks based on an economc modellng framework. Then we present and evaluate an approxmaton of the proposed resource control model that can smplfy ts applcaton. A. Utlty for loss-senstve traffc Consder a user whose utlty depends, n addton to the average throughput, on the loss rate he s experencng. If the latter affects the utlty n a subtractve way, we have the followng expresson for user s utlty U (r,γ )=U (r P (γ )) V (r (1 P (γ ))), (5) where U (x ) represents user s valuaton for average throughput x, and V (y ) represents the decrease of the user s utlty for data loss rate equal to y. The global problem of maxmzng the aggregate utlty (socal welfare) s maxmze [U (r P (γ )) V (r (1 P (γ )))] over r,γ (6) subject to r γ <ρ UL W. The above problem has a non-trval structure, and nvolves a non-convex constrant over the two varables r,γ. Hence, unlke the case for loss-nsenstve traffc, we cannot n general guarantee that a soluton can be found usng the Lagrangan method. Nevertheless, we have strong expermental evdence that ths s possble for a wde range of user utltes. Here we present a subset of these experments wth both concave exponental and logarthmc functons for U, and convex exponental functons for V. Fgs. 1 and 2 show, for varous mxes of up to three dfferent utlty types, the optmal socal welfare SW n (6) as a functon of the total resource constrant ρ UL W. The Lagrangan method can be appled when SW (ρ UL W ) s a concave functon of the total avalable resources ρ UL W ; the fgures show that ths s ndeed the case. B. Resource control model Consderng the utlty functon for loss-senstve users gven by (5), and assumng as our expermental evdence ndcates that the Lagrangan method can be appled, the user optmzaton problem s maxmze U (r P (γ )) V (r (1 P (γ )) λr γ (7) over r,γ, where as n the case of loss-nsenstve traffc, λ s the shadow prce for the wreless resource constrant (2).

4 Optmal socal welfare, SW Mx 1 Mx 2 Packet success rato, P (γ) P (γ) loss-nsenstve loss-senstve Wreless resource constrant, ρ UL W (Kbps) Fg. 1. Two user types =1, 2 wth U (x )=u log x,v (y )= d (e c y 1), N 1 = N 2 =5.Mx1:u 1 = u 2 =1,c 1 =1.4,c 2 =.5,d 1 = d 2 =.5. Mx2:u 1 =1,u 2 =.2,c 1 = c 2 =1.4,d 1 = d 2 =.5. Optmal socal welfare, SW Mx 3 Mx Wreless resource constrant, ρ UL W (Kbps) Fg. 2. Three user types = 1, 2, 3 wth U (x ) = 1 e b x,v (y )=d (e c y 1), N 1 = N 2 =3,N 3 =4.Mx3: b 1 =.1,b 2 =.2,b 3 =.3,c = 1.4,d =.5. Mx 4: b =.1,c 1 =1.4,c 2 =.5,c 3 =.25,d =.5. If there exsts r > and γ > that acheve the maxmum n (7), then from the frst order condtons we have λγ P (γ )γ = P (γ ) λγ + V (r (1 P (γ. (8) ))) The last equaton s a generalzaton of (3), snce for lossnsenstve traffc we have V ( ) =. Hence, whereas for lossnsenstve traffc the optmal sgnal qualty depends only on P (γ ), for loss-senstve traffc t also depends on the shadow prce and the user s senstvty to losses. It s nterestng to observe, nevertheless, that the optmal sgnal qualty does not depend on the user s valuaton for hs average throughput. In the case of loss-senstve traffc, as expected, the optmal sgnal qualty γ s hgher than n the case of loss-nsenstve traffc, snce the rato on the rght-hand sde of (8) s smaller than one, and the packet success rato as a functon of γ has a sgmod shape, Fg. 3. Indeed, the above rato s the rato of the margnal charge due to the wreless constrant, λγ, and the total margnal cost ncurred by the user, whch ncludes the margnal charge due to the wreless constrant and the margnal Bt-energy-to-nose-densty rato, γ Fg. 3. Packet success rato as a functon of γ for DPSK modulaton, wth no error correcton and packet length L =6bts. For lossnsenstve traffc the optmal γ that satsfes (3) s the value of γ at whch the lne passng through the orgn s tangent to P (γ). The lne passng the orgn whch s lower than the above lne corresponds to loss-senstve traffc, see (8), and the optmal γ s gven by ts rght-most ntersecton wth P (γ). Packet success rato, P (γ) DPSK, no FEC DPSK, FEC w/3 bt errors BPSK, QPSK Bt-energy-to-nose-densty rato, γ Fg. 4. Packet success rato as a functon of γ for DPSK wth forward error correcton (FEC) able to correct up to 3 bt errors, and for BPSK/QPSK modulaton. cost due to losses, expressed by V ( ). Fnally, observe that a hgher senstvty to losses corresponds to a smaller rato on the rght-hand sde of (8), hence to a hgher optmal sgnal qualty. Next we present an approxmaton of the resource control model presented above that can smplfy ts applcaton. The approxmaton s motvated by the fact that, for dfferent modulaton schemes, see Fg. 4, the optmal target sgnal qualty s such that the packet success rato obtans large values, typcally larger than 8%. Moreover, as dscussed above, loss-senstve traffc has a larger optmal target sgnal qualty compared to loss-nsenstve traffc. Based on the prevous dscusson, whch suggests that n practcal stuatons P (γ ) wll be close to one, we can consder the followng approxmaton for (8) that s expected to be accurate when r (1 P (γ )) s close to zero and the dervatve of V (y ) for loss rates y close to zero does not change

5 sgnfcantly, λγ P (γ )γ = P (γ ) λγ + V. (9) () The last equaton relates the optmal target bt-energy-to-nosedensty rato γ wth the shadow prce and the loss-senstvty at the neghborhood of loss rates around zero. If the optmal sgnal qualty s gven by (9), then the user optmzaton problem (7) can be wrtten as maxmze U (r P (γ (λ))) V (r (1 P (γ (λ)))) λr γ (λ) (1) over r. Observe that the optmzaton n the last equaton s over a sngle varable, the transmsson rate r, whle the functon γ (λ) hdes the effect of the loss-senstvty on the optmal sgnal qualty, and s mplctly performed by the outer-loop power control procedure, whch solves (9). C. The case of TCP It s nterestng to consder the applcaton of the model presented n the prevous subsectons to the case of TCP traffc. The resultng models can suggest the optmal target sgnal qualty when TCP traffc s sent over WCDMA networks. TCP traffc can be consdered loss-senstve, snce ts throughput depends on the packet loss rate. If we assume that a TCP segment fts n a sngle physcal layer packet, then we can defne the mplct utlty for TCP to be 2 U TCP (r, γ) = T 2 r(1 P (γ)), rp(γ) where T s the round trp tme. For the above mplct utlty, the correspondng average throughput has a behavor smlar to TCP s well-known nverse square root dependence on the packet loss rate [18]. Indeed, f we substtute the last expresson n (7), and fnd the optmal r,γ, then we obtan the followng average throughput r P (γ )= 1 2P (γ ) T 1 P (γ )+λγ, (11) wth γ satsfyng P (γ )γ = P (γ λγ ) λγ +1. (12) If P (γ ) n (11) s assumed close to one and s omtted from the numerator of the rght-hand sde, then the average throughput, r P (γ ), s nversely proportonal to the square root of the sum of the loss probablty, 1 P (γ ), and the congeston prce, λγ. If a sngle TCP segment s broken nto k physcal layer packets, then the loss rate s r(1 P (γ) k ), whch usng the Taylor expanson can be approxmated by rk(1 P (γ)). Hence, TCP s mplct utlty n ths case can be defned as 2 U TCP,k(r, γ) = T 2 rk(1 P (γ)). rp(γ) Loss-senstvty, V (y) c =1.4 c =.5 c = Loss rate, y (Kbps) Fg. 5. Loss-senstvty V (y) = d(e cy 1). The loss-senstvty ncreases from rght to left: a hgher value of c n (13) corresponds to a hgher loss-senstvty. d =.5. From the last equaton observe that the effect of losses s hgher when a TCP segment s broken nto multple physcal layer packets, compared to the case when a TCP segment fts n a sngle packet. In the former case, (12) becomes P (γ )γ = P (γ λγ ) λγ + k. D. Numercal nvestgatons In ths subsecton we nvestgate the resource control model and ts approxmaton that were presented n the prevous subsectons. The factor n (5) that gves the valuaton for average throughput x s assumed to be U(x) =1 e bx. Although the nvestgatons reported n ths paper consder the above throughput utlty factor, the qualtatve conclusons are smlar for other concave functons. The loss-senstvty factor n(5)sassumedtobe V (y) =d(e cy 1), (13) where y s the loss rate, d depcts the relatve mportance of losses compared to the average throughput, and c reflects the loss-senstvty, wth a larger value of c correspondng to a hgher loss-senstvty, Fg. 5. 1) Effect of loss-senstvty : The effect of the losssenstvty on the optmal γ s shown n Fg. 6. The optmal γ s gven by the ntersecton of the curve P (γ) wth the curve P (γ)/γ n the case of loss-nsenstve traffc (3), and wth the curve F (γ) = P (γ) γ λγ λγ + V (r(1 P (γ))), n the case of loss-senstve traffc (8). Observe that the optmal sgnal qualty γ n the case of loss-senstve traffc s larger than n the case of loss-nsenstve traffc. Furthermore, a larger loss-senstvty,.e. a larger value of c n (13), results n a hgher optmal sgnal qualty γ. Indeed, for the values consdered, see Fg. 3, the values of γ for loss-senstve traffc correspond to packet success ratos P (γ) larger than 9%.

6 P (γ) loss-nsenstve, P (γ)/γ loss-senstve, F (γ),c =.25 loss-senstve, F (γ),c =.5 loss-senstve, F (γ),c = Bt-energy-to-nose-densty rato, γ Optmal bt-energy-to-nose-densty rato, γ loss-nsenstve, γ =5 exact, loss-senstve approx., loss-senstve 1e Shadow prce, λ Fg. 6. The optmal γ s the ntersecton of P (γ) the curve F (γ), whch depends on the loss-senstvty through c. A hgher losssenstvty results n a hgher optmal sgnal qualty γ. r =2Kbps, d =.5,λ= P (γ) loss-nsenstve, P (γ)/γ exact, loss-senstve, c =.25 approx., loss-senstve, c =.25 exact, loss-senstve, c =1.4 approx., loss-senstve, c = Bt-energy-to-nose-densty rato, γ Fg. 7. Effect of the loss-senstvty on the accuracy of (9). r = 2 Kbps, d =.5,λ=.2. 2) Accuracy of the approxmate model: Fg. 7 shows, for dfferent loss-senstvtes, the accuracy of (9), whch s an approxmaton of (8). Observe that the approxmaton s more accurate when the dervatve of the loss-senstvty factor does not change sgnfcantly n the neghborhood of loss rates close to zero, whch s the case for small values of c. The prevous results were for a partcular value of the shadow prce λ. Fg. 8 shows the accuracy of approxmaton (9) for dfferent values of λ. Observe that the approxmaton s very accurate for small and large values of λ, the latter correspondng to hgh demand. Moreover, observe that for large values of the shadow prce, the optmal sgnal qualty for loss-senstve traffc approaches the optmal sgnal qualty for loss-nsenstve traffc; ths s expected snce for large values of the shadow prce the rato on the rght-hand sde of (8) approaches one, hence (8) approaches the correspondng equaton for loss-nsenstve traffc (3). We end ths dscusson by notng that a heurstc for (1) s maxmze U (r ) λr γ (λ) over r, Fg. 8. Effect of the shadow prce on the accuracy of the approxmaton. d =.5,c =1.4. whch s expected to be accurate when P (γ ) s close to one, n whch case r P (γ ) and r (1 P (γ )) are close to r and zero respectvely, and V (y ) for loss rates y close to zero s small compared to U (r ). The last expresson for the user optmzaton problem s attractve snce t has the same form wth the correspondng problem n the case of loss-nsenstve traffc (4). The only dfference s that now the bt-energy-tonose-densty rato s not constant, but depends on the shadow prce; ths dependence s captured through the functon γ (λ). IV. APPLICATION Unlke the case of loss-nsenstve traffc, for loss-senstve traffc the optmal target bt-energy-to-nose densty rato should depend on the shadow prce. Hence, n order to acheve effcency both the outer-loop power control and the rate control procedures need to take nto account the shadow prce, Fg. 9, n order to solve (9) and (1), respectvely. Observe that the optmal target bt-energy-to-nose-densty rato s used as nput to both the rate control and the closed-loop power control procedures. For the uplnk, outer-loop power control s performed at the Rado Network Controller (RNC) of a WCDMA cell, and rate control can be mplemented at the moble host or at the RNC. An example of the latter case s a partcular nstantaton of the proposed resource control model, whch we descrbe later n ths secton, that supports class-based servce dfferentaton and can be mplemented solely at the RNC. Note that t s not our objectve to dscuss n depth all possble mplementaton detals, but rather to hghlght some mportant mplementaton aspects argung that the proposed resource control model does not requre radcally dfferent procedures than those already mplemented n exstng systems. A. Outer-loop power control In current systems, outer-loop power control adjusts the target sgnal qualty n order to acheve a predefned packet error rate, whch s typcally 1-2% for non-real-tme servces, and 1% for real-tme servces. As ndcated by the models

7 λ Dscrete rate control BS BS γ RNC RNC λ Outer loop power control r u,v γ feedback (shadow prce) r = R(u,v ) Closed loop power control Fg. 9. The proposed resource control model affects only the rate control and the outer-loop power control procedures (shaded boxes), whch must take nto account the shadow prce λ. The target sgnal qualty γ s used as nput to both rate control and closed-loop power control. presented n Sectons II and III, ths does not lead to effcent resource utlzaton. Current systems mplement outer-loop power control by ncreasng (or decreasng) the target bt-energy-to-nose-densty rato dependng on whether the measured packet error rate s larger (or smaller) than the target packet error rate [1]. Due to the sgmod shape of P (γ), a smlar procedure can be used to acheve (9). In partcular, the sgnal qualty γ s ncreased wth step γ f P (γ) P (γ + γ) P (γ) γ > P (γ) γ and s decreased wth step γ otherwse. λγ λγ + V (), B. Dscrete rate control In WCDMA, rates can obtan only dscrete values, correspondng to spreadng factors that are powers of 2. In the uplnk, the spreadng factor can take values from 256, gvng a channel bt rate of 15 Kbps, to 4, gvng a channel bt rate of 96 Kbps; hgher bt rates are acheved by usng up to 6 parallel codes wth spreadng factor 4 (gvng a channel bt rate of 574 Kbps). Hence, n the uplnk, assumng 1/2 rate codng, the user data rates that can be acheved usng a sngle code are 7.5, 15, 3, 6, 12, 24, and 48 Kbps [1]. Assume that there exsts drect feedback of the shadow prce from the Base Staton/Rado Network Controller (BS/RNC) to the moble statons, Fg. 1(a). Ths feedback can be provded over an exstng rado control channel or over a hgher layer sgnallng mechansms, such as Explct Congeston Notfcaton (ECN) n the IP layer. Based on ths feedback, the moble statons can adjust ther transmsson rate as follows: An ntal selecton of r s made. Snce there are only seven possble rate values, a lnear search for the rate that maxmzes (1) can be suffcent. Neghborng rates are examned to see whether they yeld a larger user beneft (utlty mnus charge) n (1). If a (a) drect prce feedback (b) class-based Fg. 1. Wth drect prce communcaton (left fgure), a moble selects ts transmsson rate based on the shadow prce usng (1). Wth the class-based approach approach (rght fgure), the transmsson rate s selected by the RNC based on the user s declared throughput u and loss-senstvty v class usng (15). neghborng rate s found to gve a hgher beneft, then the next rate n the same drecton s examned; ths procedure s repeated throughout the duraton of the connecton. C. Class-based servce dfferentaton In ths secton we present a partcular nstantaton of the model dscussed n Secton III-B that supports servce dfferentaton based on weghts, where dfferent weghts can correspond to dfferent classes. Assume that user s valuaton for hs average throughput s gven by U (r P (γ )) = u log(r P (γ )), and hs loss-senstvty factor s V (r (1 P (γ ))) = v r (1 P (γ )). From the frst order condtons of (7) we get u = v r (1 P (γ )) + λr γ. (14) The rght-hand sde n the last equaton can be nterpreted as the cost user s wllng to ncur, whch ncludes the cost n terms of lost data and the charge due to the use of wreless resources. For the utlty correspondng to the average throughput and loss-senstvty factors dentfed above, the outer-loop power control objectve s gven by (9) wth V () = v. Based on (14), the rate for user can be computed from r = v (1 P (γ )) +. (15) λγ In practse, a network provder can offer a small set of possble values for u, each correspondng to a dfferent throughput class, and a small set of values for v, each correspondng to a dfferent loss-senstvty class. Note that the above rate selecton can be performed at the Rado Network Controller (RNC), rather than at the moble statons. After the rate selecton, the RNC communcates the rate values to the moble statons, Fg. 1(b). Such an approach u

8 places more control at the RNC, whch n current systems has the ntellgence for supportng flexble packet schedulng and load control [1]. Moreover, t does not requre the communcaton of shadow prces to the moble statons, snce the selecton of both the target bt-energy-to-nose-densty rato and the transmsson rate s performed at the RNC, based on (9), wth V () = v, and (15), respectvely. On the other hand, t does requre the communcaton of the transmsson rates from the RNC to the moble statons, whch however s already supported n current systems. Fnally, note that the approach for adjustng dscrete rates that was presented n the prevous subsecton can be combned wth the approach for class-based servce dfferentaton that was dscussed n ths subsecton. V. EXTENSIONS Next we present two extensons of the resource control model dscussed n Secton III. Frst, we extend the model to the case where the packet success rato depends on the transmsson rate, n addton to the bt-energy-to-nose-densty rato. Second, we extend the model to the case of a network contanng multple wreless lnks. A. Dependence of packet success rato on rate Up to now we have assumed that the packet success rato P (γ ) s a functon of only the target bt-energy-to-nosedensty rato. In practse the packet success rato also depends on the transmsson rate, hence we can wrte P (γ,r ). Indeed, the requred sgnal qualty n order to acheve the same packet success rato s smaller for hgher bt rates. Ths s because the performance of closed-loop power control depends on the accuracy of the channel and bt-energy-to-nose-densty rato estmaton algorthms, whch are based on reference symbols carred over the physcal control channels. The more power allocated to the control channels, the more accurate the estmaton procedure. The power levels for the control channels s typcally hgher for hgher bt rates, hence hgher bt rates yeld better performance n terms of the packet success rato, for the same target bt-energy-to-nose-densty rato. 1) Loss-nsenstve traffc: Frst consder the case of lossnsenstve traffc, where the utlty depends solely on the average throughput. If the packet success rato s of the form P (γ,r ), then the user optmzaton problem s maxmze U (r P (γ,r )) λr γ (16) over r,γ, where λ s the shadow prce for the wreless resource constrant (2). A necessary condton for achevng the optmal n (16), assumng that r,γ >, s the followng ϑp ϑγ γ,r γ = P (γ,r )+ ϑp ϑr r. (17) γ,r From the above dscusson on the effect of the transmsson rate on the packet success rato, we have ϑp ϑr >, hence from the last equaton the optmal γ s now lower than when there s no dependence of the packet success rato on the transmsson rate. 2) Loss-senstve traffc: In the case of loss-senstve traffc, where the user utlty s gven by (5), the user optmzaton problem s maxmze U (r P (γ,r )) V (r (1 P (γ,r )) λr γ (18) over r,γ, where λ s the shadow prce for the wreless resource constrant (2). From the frst order condtons we have ( ϑp ϑγ γ = P (γ,r )+ ϑp ) γ,r ϑr r γ,r λγ λγ + V (r (1 P (γ,r ))), whch s a combnaton of (8) and (17). B. Extenson to networks wth multple wreless hops Next we present the extenson of the proposed resource control model to networks contanng multple wreless hops. Ths would be the case of two moble statons both communcatng through WCDMA access lnks. As we wll see, the optmal sgnal qualtes on the wreless lnks are no longer ndependent. Although the models we present n ths secton consder only two wreless lnks, the extenson to more than two lnks can be done n a straghtforward way, although the equatons become more lengthy. Note that, for loss-nsenstve traffc, the extenson to networks contanng fxed lnks was presented n [14]. For the uplnk, the wreless resource constrant s gven by (2). In the downlnk, the constraned resource s the total transmsson power at the base staton, whch can lead to a resource control model that s dfferent from the one n the uplnk. Nevertheless, one can show, see [1], that for resource dmensonng and network plannng the downlnk constrant can be approxmated by an nequalty smlar to (2) r γ <ρ DL W, where ρ DL < 1 depends on the total base staton power, the nose, the average attenuaton from the base staton to the mobles, the average downlnk orthogonalty factor, and the average nter-cell nterference. 1) Loss-nsenstve traffc: Consder a network wth two wreless lnks, wth packet success rato P 1 (γ 1 ) and P 2 (γ 1 ), respectvely, where γ 1 and γ 2 are the target sgnal qualtes on the two lnks. The user optmzaton problem for lossnsenstve traffc s the followng (to smplfy the notaton, we drop the subscrpts dentfyng dfferent users) maxmze U(rP 1 (γ 1 )P 2 (γ 2 )) λ 1 rγ 1 λ 2 rp 1 (γ 1 )γ 2 over r,γ 1,γ 2,

9 where λ 1, λ 2 are the shadow prces for the constrants at the two wreless lnks. Observe that the rate on the second lnk s rp 1 (γ 1 ), snce some packets wll be lost on the frst wreless lnk. From the frst order condtons of the above optmzaton problem, and after some manpulaton one can show that the optmal γ1 and γ2, assumng that they exst and r,γ1,γ 2 >, satsfy P 1(γ 1)γ 1 = P 1 (γ1), P 2(γ 2)γ 2 λ 2 P 1 (γ = 1)γ2 λ 1 γ1 + λ 2P 1 (γ1 )γ P 2 (γ2). 2 Hence we observe that n the case of two wreless lnks, for loss-nsenstve traffc, the optmal target bt-energy-to-nosedensty rato for the frst lnk s the same as that n the sngle lnk case. Observe that the rato on the rght-hand sde of the last equaton for γ2 denotes the percentage of the total charge due to the second wreless lnk. Hence, the optmal target btenergy-to-nose-densty rato for the second lnk s larger than what t would be f t where the only lnk. Moreover, f the percentage of the charge due to the second lnk decreases, then γ2 ncreases. 2) Loss-senstve traffc: In the case of loss-senstve traffc, the user optmzaton problem s maxmze U(rP 1 (γ 1 )P 2 (γ 2 )) V (r(1 P 1 (γ 1 )P 2 (γ 2 ))) λ 1 rγ 1 λ 2 rp 1 (γ 1 )γ 2 over r 1,γ 1,γ 2, where λ 1, λ 2 are the shadow prces for the constrants at the two wreless lnks. From the frst order condtons of the above optmzaton problem, and after some manpulaton one can show that the optmal r,γ 1, and γ 2, assumng that they exst and are postve, satsfy P 1(γ 1)γ 1 λ 1 γ1 = λ 1 γ1 + V (r (1 P 1 (γ1 )P 2(γ2 )))P 1(γ1), whch s the same as the case for a sngle wreless lnk, (8). For the second wreless lnk we have P 2(γ 2 )γ2 = λ 2P 1(γ1 )γ2 λ 1γ1 + λ2p1(γ 1 )γ 2 + V (r (1 P 1(γ1 )P2(γ 2 ))) P2(γ 2 ), where the rato on the rght-hand sde s the percentage of the total charge that s due to the second wreless lnk, assumng that the total charge s the sum of the margnal charge of the two lnks and the cost due to losses V ( ). The results of ths secton dentfy how the optmal target sgnal qualtes on the two wreless lnks are related. An nterestng open ssue s how the two lnks can communcate n order to coordnate ther selectons. VI. RELATED WORK Next we present a bref overvew of related research, dentfyng ts dfferences wth the work presented n ths paper. The work n [5] consders a utlty that s nterpreted as the number of nformaton bts transmtted per unt of energy. It s shown that the non-cooperatve game, where mobles adjust ther power to maxmze ther utlty, has a unque Nash equlbrum, whch however s neffcent. Wth the ntroducton of prces [6], Pareto mprovements are acheved, but not the socal welfare optmal. The work n [1] consders a utlty that s a monotoncally ncreasng concave functon of the bt-energyto-nose-densty rato and a monotoncally decreasng concave functon of the moble s power, and prove the exstence of a Nash equlbrum. The work n [13] consders a weghted logarthmc utlty functon of the sgnal qualty and a prce proportonal to the transmtted uplnk power, and prove the exstence of a Nash equlbrum. The work n [7] consders a utlty that s a functon of the transmsson rate, and nvestgate the problem of maxmzng the sum of all utltes n the forward lnk (downlnk), under constrants on the total transmsson power at the base staton, and constrants on the maxmum error rate for each user. Our work consders user utltes that depend on the loss rate, n addton to the average throughput. The work n [9], [11], [12] also consders the downlnk power control takng nto account the constrant on the total transmsson power. The work n [9] consders a user utlty that s a step functon of the bt-energy-to-nose-densty rato, and a moble s charge contans a constant term (prce per code) and a term lnear n the transmtted power from the base staton. The work n [11] focuses on revenue maxmzaton through prcng, where prces are per unt of tme. The work n [12] nvestgates socal welfare maxmzaton n a mult-class CDMA system, and proposes an algorthm that can obtan a Pareto optmal allocaton, whch s a good approxmaton of the socal optmal power allocaton. The work n [8] consders a utlty that s a functon of the bt-energy-to-nose-densty rato, whch can have a sgmod shape, and formulate a utlty-based dstrbuted power control algorthm where each user seeks to maxmze hs net utlty, and charges are proportonal to the power. For a constant prce per unt of power, t s proved that the power update algorthm converges. Our work dffers from the above n that t consders the jont optmzaton of the sgnal qualty and transmsson rate, and takes nto account the partcular resource constrants n the uplnk. Moreover, we dscuss the form of the utlty functon for loss-senstve traffc, takng nto account both the user s valuaton for hs average data throughput and hs senstvty to losses. The aforementoned work s geared towards mechansms for power control; on the other hand, our work deals wth control mechansms, namely rate control and outer-loop power control, that operate on a slower tmescale, hence on top of fast closed-loop power control, wthout modfyng t. Fnally, n the proposed resource control model there s no dfferentaton of moble users based on ther poston. On the other hand, n the approaches of [5], [6], [8], [13], [11], moble users far from the base staton that encounter hgh path loss face a hgher charge and receve less resources, compared

10 to users close to the base staton; ths s termed near-far unfarness n [8]. VII. CONCLUDING REMARKS We have presented a new model for resource control n CDMA networks carryng loss-senstve traffc, based on an economc modellng framework that takes nto account the wreless resource constrants and the jont control of the transmsson rate and the sgnal qualty. The correspondng socal welfare optmzaton problem has a non-trval structure, for whch we cannot n general guarantee that a soluton can be found usng the Lagrangan method. Nevertheless, we have strong expermental evdence that ths s possble for a wde range of user utltes, n whch case the socal welfare maxmum can be acheved n a decentralzed manner by havng each user solve a local optmzaton problem. Based on the above evdence, the man contrbuton of the paper s to dscuss how exstng control procedures, namely rate control and outer-loop power control, can obtan a smple and attractve form that takes nto account, through shadow prces, the level of demand and supply n order to acheve effcent resource utlzaton. Moreover, we descrbe and evaluate an approxmaton of the proposed resource control model that can smplfy ts applcaton, wthout requrng radcally dfferent procedures than those mplemented n exstng systems. One applcaton scenaro, whch can be mplemented solely at the Rado Network Controller (RNC) of a WCDMA network, can support servce dfferentaton based on throughput and losssenstvty classes. Ongong work ncludes nvestgatng n depth the partcular characterstcs of the socal welfare maxmzaton (6), to understand why and when the Lagrangan method can be appled for the varous user utltes that we have expermentally nvestgated. Our ntal fndngs suggest that ths s related to the fact that the optmal sgnal qualty for loss-senstve traffc s hgher than for loss-nsenstve traffc, and corresponds to packet success ratos close to one. Other related work ncludes the applcaton of economc modellng for resource control and servce dfferentaton n wreless LANs based on the IEEE standard. REFERENCES [1] H. Holma and A. Toskala, WCDMA for UMTS. New York: Wley, 2. [2] F. P. Kelly, Chargng and rate control for elastc traffc, European Transactons on Telecommuncatons, vol. 8, pp , January [3] R. J. Gbbens and F. P. Kelly, Resource prcng and the evoluton of congeston control, Automatca, vol. 35, pp , [4] F. P. Kelly, Mathematcal modellng of the Internet, n Mathematcs Unlmted - 21 and Beyond, B. Engqust and W. Schmd, Eds. Hedelberg: Spnger-Verlag, 2, avalable at: frank/. [5] D. J. Goodman and N. B. Mandayam, Power control for wreless data, IEEE Personal Commun., vol. 7, pp , Aprl 2. [6] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, Effcent power control va prcng n wreless data networks, IEEE Trans. Commun., vol. 5, no. 2, pp , February 22. [7] L. Song and N. B. Mandayam, Herarchcal SIR and rate control on the forward lnk for CDMA data users under delay and error constrants, IEEE J. Select. Areas Commun., vol. 19, no. 1, pp , October 21. [8] M. Xao, N. B. Shroff, and E. K. P. Chong, Utlty-based power control n cellular wreless systems, n Proc. of IEEE INFOCOM 1, Aprl 21. [9] P. Lu, M. L. Hong, and S. A. Jordan, Forward-lnk CDMA resource allocaton based on prcng, n Proc. of IEEE Wreless Communcatons and Networkng Conference (WCNC), September 2. [1] H. J and C.-Y. Huang, Non-cooperatve uplnk power control n cellular rado systems, ACM/Baltzer Wreless Networks Journal, vol. 4, pp , [11] P. Marbach and R. Berry, Downlnk resource allocaton and prcng for wreless networks, n Proc. of IEEE INFOCOM 2, June 22. [12] J. W. Lee, R. R. Mazumdar, and N. B. Shroff, Downlnk power allocaton for mult-class CDMA wreless networks, n Proc. of IEEE INFOCOM 2, June 22. [13] T. Alpcan, T. Basar, R. Srkant, and E. Altman, CDMA uplnk power control as a noncooperatve game, Wreless Networks, vol. 8, no. 6, pp , November 22. [14] V. A. Srs, Resource control for elastc traffc n CDMA networks, n Proc. of ACM Internatonal Conference on Moble Computng and Networkng (MOBICOM), September 22. [15] K. S. Glhousen, I. M. Jacobs, R. Padovan, A. J. Vterb, L. A. Weaver, and C. E. W. III, On the capacty of a cellular CDMA system, IEEE Trans. on Vehcular Technology, vol. 4, no. 2, pp , May [16] L. C. Yun and D. G. Messerschmtt, Power control for varable QoS on a CDMA channel, n Proc. of IEEE MILCOM 94, October [17] A. Sampath, P. S. Kumar, and J. M. Holtzman, Power control and resource management for a multmeda CDMA wreless system, n Proc. of IEEE Int. Symp. on Personal, Indoor, Moble Rado Commun. (PIMRC), September [18] M. Maths, J. Semke, J. Mahdav, and T. Ott, The macroscopc behavor of the TCP congeston avodance algorthm, ACM Computer Communcaton Revew, vol. 27, pp , ACKNOWLEDGMENT The authors would lke to thank Bob Brscoe and Dave Songhurst for helpful dscussons on parts of ths work.