A Robust Optimal Rate Allocation Algorithm and Pricing Policy for Hybrid Traffic in 4G-LTE

Transcription

1 03 IEEE 4th Internatonal Symposum on Personal, Indoor and Moble ado Communcatons: Moble and Wreless Networks A obust Optmal ate Allocaton Algorthm and Prcng Polcy for Hybrd Traffc n 4G-LTE Ahmed Abdel-Had and Charles Clancy ECE, Vrgna Tech, Arlngton, VA, 03, USA {aabdelhad, tcc}@vt.edu Abstract In ths paper, we consder resource allocaton optmzaton problem n the fourth generaton long-term evoluton (4G-LTE) wth elastc and nelastc real-tme traffc. Moble users are runnng ether delay-tolerant or real-tme applcatons. The users applcatons are approxmated by logarthmc or sgmodal-lke utlty functons. Our objectve s to allocate resources accordng to the utlty proportonal farness polcy. Pror utlty proportonal farness resource allocaton algorthms fal to converge for hgh-traffc stuatons. We present a robust algorthm that solves the drawbacks n pror algorthms for the utlty proportonal farness polcy. Our robust optmal algorthm allocates the optmal rates for both hgh-traffc and low-traffc stuatons. It prevents fluctuaton n the resource allocaton process. In addton, we show that our algorthm provdes traffcdependent prcng for network provders. Ths prcng could be used to flatten the network traffc and decrease the cost per bandwdth for the users. Fnally, numercal results are presented on the performance of the proposed algorthm. I. INTODUCTION In recent years, there has been a sgnfcant growth n the demand for hgher data rates to support the transton from voce-only communcatons to multmeda communcatons n moble systems, e.g. 4G-LTE. Ths demand motvates numerous research efforts to optmally allocate the avalable lmted bandwdth resources for users seekng better qualtyof-servce (QoS). One aspect of mprovng resource allocaton and achevng better QoS s to use network utlty optmzaton. Network utlty optmzaton was ntally used for rate allocaton n wred networks such as the Internet [], []. The utlty functons used n the Internet are delay-tolerant utlty functons that can be mathematcally represented by logarthmc functons. In recent research work, e.g. [3], [4], network utlty optmzaton s used to allocate resources n wreless networks for real-tme applcatons. eal-tme applcatons such as voce-over-ip (VoIP) and vdeo streamng are mathematcally represented by sgmodal-lke utlty functons wth dfferent parameters for dfferent real-tme applcatons [5]. The majorty of pror work done on wreless network utlty optmzaton for real-tme applcatons only provdes approxmatons of the optmal rate allocaton. In [6], the authors used resource allocaton optmzaton problem wth utlty proportonal farness objectve functon (.e the objectve s to provde far utlty percentage for all the users). The authors ncluded hybrd traffc n ther model where users utlty functons are ether logarthmc or sgmodal-lke functons (.e. correspond to delay-tolerant or real-tme applcatons). Despte the absence of concavty n the sgmodallke utlty functons, the authors have reformulated the optmzaton problem and proven that t s a convex optmzaton problem. Therefore, a tractable global optmal soluton exsts. Usng dualty, the authors presented a dstrbuted teratve algorthm for allocatng the optmal rates to users. Ths rate allocaton s far wth respect to utlty percentage resultng n allocaton prorty gven to users wth real-tme applcatons over users wth delay-tolerant applcatons. Meanwhle, the algorthm ensures that no user receves zero rate (.e. no user s dropped). Therefore, the algorthm ensures mnmum QoS for all network subscrbers and prorty to real-tme applcaton users who are payng more for better QoS. The dstrbuted rate allocaton algorthm presented n [6] converges to the optmal rate only when the maxmum avalable rate by the enodeb exceeds the sum of rates needed to acheve % utlty percentage for all the real-tme applcaton users. Therefore, the algorthm doesn t converge for an enodeb wth scarce bandwdth resources wth respect to the number of users and ther utltes. In ths paper, we analyze ths stuaton whch occurs frequently durng peak network usage perods of the day. Our algorthm presents a more robust algorthm that converges for both scarce and abundant bandwdth resources. In addton to allocatng the optmal rates, we show that our algorthm provdes a traffcdependent prcng approach that could be used by network provders. A. elated Work In [7], [8], the authors presented a non-convex optmzaton problem for maxmzaton of utlty functons n wreless networks. They used both hybrd utlty functons and presented the algorthm to solve t optmally when the dualty gap s zero. They ncluded a far allocaton heurstc algorthm to ensure network stablty whch resulted n a hgh aggregated utlty. In [9], the authors proposed a utlty max-mn farness resource allocaton for users wth elastc and nelastc traffc. In [], the authors proposed a utlty proportonal far optmzaton problem for hgh-sin wreless networks usng utlty max-mn archtecture. They compared ther algorthm to the tradtonal bandwdth proportonal far algorthms and presented a closed form soluton that prevents fluctuaton. In [3], the authors presented a dstrbuted power allocaton algorthm for cellular systems. They used non-concave sgmodal-lke utlty functons. The proposed algorthm approxmates the global optmal soluton and can drop users to /3/$ IEEE 85

2 maxmze the overall system utltes, therefore, t does not guarantee mnmum QoS for all users. B. Our Contrbutons Our contrbutons n ths paper are summarzed as: We consder the utlty proportonal farness resource allocaton optmzaton problem for both elastc and nelastc traffc. We analyze the convergence of the dstrbuted rate allocaton algorthm that s presented n [6]. We show that t doesn t converge to the optmal rates n hgh-traffc perods (.e. resources are scarce wth respect to number of actve users). We present a robust dstrbuted rate allocaton algorthm that converges to the optmal rates for hgh-traffc and low-traffc perods. We present a prcng polcy for network provders that can flatten traffc load on the network and decrease the overall servce cost to subscrbers. The remander of ths paper s organzed as follows. Secton II presents the problem formulaton. Secton III analyzes the rate allocaton algorthm n [6] and dscusses ts convergence. In Secton IV, we present our dstrbuted rate allocaton algorthm. Secton V dscusses smulaton setup and provdes quanttatve results along wth dscusson. Secton VI concludes the paper. II. POBLEM FOMULATION We consder a system model that s smlar to [6] where we have a sngle cell 4G-LTE moble system consstng of a sngle evolved Node B (enodeb) and M user equpments (UE)s. The rate allocated by the enodeb to th UE s gven by r. Each UE has ts own utlty functon U (r ) that corresponds to the type of traffc beng handled by t. Our objectve s to determne the optmal rates the enodeb allocates to the UEs. We assume the utlty functon U (r ) of the th UE to be a strctly concave or a sgmodal-lke functon. The utlty functons U(r) have the followng propertes: U(0) = 0 and U(r) s an ncreasng functon of r. U(r) s twce contnuously dfferentable n r and bounded above. In our model, we use the normalzed sgmodal-lke utlty functon, as n [3], to represent real-tme applcatons runnng on the UEs that can be expressed as ( ) U(r) =c +e d () a(r b) where c = +eab and d =. So, t satsfes U(0) = 0 and e ab +e ab U( ) =. The nflecton pont of the normalzed sgmodallke functon s at r nf = b. In addton, we use the normalzed logarthmc utlty functon, as n [], to represent delaytolerant applcatons runnng on the UEs that can be expressed as log( + kr) U(r) = () log( + kr max ) where r max s the maxmum requred rate for the user to acheve 0% utlty percentage and k s the rate of ncrease of utlty percentage wth the allocated rate. So, t satsfes U(0) = 0 and U(r max ) =. The nflecton pont of normalzed logarthmc functon s at r nf =0. The basc formulaton of the resource allocaton problem s gven by the followng optmzaton problem: max r subject to M U (r ) = M r = r 0, =,,..., M. where s the maxmum achevable rate of the enodeb and r = {r,r,..., r M } are the rates allocated to the UEs. The exstence of a tractable global optmal soluton for the optmzaton problem (3) s proven n [6]. III. ATE ALLOCATION ALGOITHM AND ITS FLUCTUATION A. ate Allocaton Algorthm In [6], the authors proposed a rate allocaton algorthm to allocate the optmal rates for the optmzaton problem n equaton (3). The algorthm s an teratve dstrbuted algorthm, whch s a modfed verson of Frank Kelly algorthm n []. Algorthm UE Algorthm n [6] Send ntal bd w () to enodeb eceve shadow prce from enodeb f STOP from enodeb then Calculate allocated rate r opt Solve r (n) = arg max r ( = w(n) log U (r ) r ) Send new bd w (n) =r (n) to enodeb end The algorthm s dvded nto two parts, Algorthm () whch runs on the UE sde and Algorthm () whch runs on the enodeb sde. The th UE solves for ts bd w (n) and sends t to the enodeb. The enodeb calculates the shadow prce and sends t to all UEs. Each UE uses the shadow prce to recalculate ts new bd untl w (n) w (n ) s less than a pre-specfed threshold δ. Now, we show the fluctuaton n Algorthm () and () (.e. convergence analyss) when runnng on an enodeb wth scarce bandwdth resources wth respect to the number of users and the shape of ther utlty functons. B. Convergence Analyss In ths secton, we present the convergence analyss of Algorthm () and () for dfferent values of. Lemma III.. For sgmodal-lke utlty functon U (r ),the log U(r) slope curvature functon has an nflecton pont at r = r s b and s convex for r >r s. (3) 86

3 Algorthm enodeb Algorthm n [6] eceve bds w (n) from UEs {Let w (0) = 0 } f w (n) w (n ) <δ then STOP and allocate rates (.e r opt to user ) M = Calculate = w(n) Send new shadow prce to all UEs end ( Proof: For the) sgmodal-lke functon U (r ) = log U(r) c d +e a (r b ),lets (r )= be the slope curvature functon. Then, we have that S a = d e a(r b) ) r c ( a e a(r b) ) d ( + e a(r b) ) (+e a(r b) and S r = a3 d e a(r b) ( d ( e a(r b) )) ) 3 c ( d ( + e a(r b) ) + a3 e a(r b) ( e a(r b) ) (+e a(r b) ) 3. (4) We analyze the curvature of the slope of the natural logarthm of sgmodal-lke utlty functon. For the frst dervatve, we have S < 0 r.thefrstterms of S n equaton (4) r can be wrtten as S = a3 (e eab ab + e a(r b) ) (5) (e ab e a(r b) ) 3 and we have lm r 0 S =, and lm For second term S followng propertes of S r S =0 for b. (6) r b a n equaton (4), we have the S (b )=0, S (r >b ) > 0, and S (r <b ) < 0. (7) From equaton (6) and (7), S has an nflecton pont at r = r s b. In addton, we have the curvature of S changes from a convex functon close to orgn to a concave functon before the nflecton pont r = r s then to a convex functon after the nflecton pont. Corollary III.. If M then Algorthm n () and = rnf () converges to the global optmal rates whch correspond to amax dmax the steady state shadow prce p ss < d max + amax where max =argmax b. ( Proof: For the) sgmodal-lke functon U (r ) = c d +e a (r b ), the optmal soluton s acheved by solvng the optmzaton problem (3). In Algorthm (), an mportant step to reach to the optmal soluton ( s to solve the ) optmzaton problem r (n) = arg max log U (r ) r r for every UE. The soluton of ths problem can be wrtten, usng Lagrange multplers method, n the form log U (r ) p = S (r ) p =0. (8) From equaton (6) and (7) n Lemma III., we have the curvature of S (r ) s convex for r >r s b. The Algorthm n () and () s guaranteed to converges to the global optmal soluton when the slope S (r ) of all the utlty functons natural logarthm log U (r ) are n the convex regon of the functons, smlar to analyss of logarthmc functons n [] and []. Therefore, the natural logarthm of sgmodal-lke functons log U (r ) converge to the global optmal soluton for r >r s b. The nflecton pont of sgmodal-lke functon U (r ) s at r nf = b.for M = rnf, Algorthm n () and () allocates rates r >b for all users. Snce S (r ) s convex for r >r s b then the optmal soluton can be acheved by Algorthm () and (). We have from equaton (8) and as S (r ) s convex for r >r s b,thatp ss <S (r =maxb ) amax dmax where S (r =maxb )= d max + amax and max = arg max b. Corollary III.3. For M = rnf shadow prce p ss a a b d e d (+e a b ) >and the global optmal + ae ab (+e a b ), then the soluton by Algorthm n () and () fluctuates about the global optmal soluton. Proof: It follows from lemma III. that for M = rnf > such that the optmal rates r opt < b. Therefore, f p ss ade ab + ae ab s the optmal shadow prce d (+e a b ) (+e a b ) for optmzaton problem (3). Then, a small change n the shadow prce n the n th teraton can lead the rate r (n) (root of S (r ) =0) to fluctuate between the concave and convex curvature of the slope curve S (r ) for the th user. Therefore, t causes fluctuaton n the bd w (n) sent to the enodeb and fluctuaton n the shadow prce set by enodeb. Therefore, the teratve soluton of Algorthm n () and () fluctuates about the global optmal rates r opt. Theorem III.4. Algorthm n () and () does not converge to the global optmal soluton for all values of. Proof: It follows from Corollary III. and III.3 that Algorthm n () and () does not converge to the global optmal soluton for all values of. C. Fluctuaton Example We consder an example of four users where two users run applcatons wth sgmodal-lke utlty functons and the other two users run applcatons wth logarthmc utlty functons. The sgmodal-lke utlty functons parameters are a = {5, 0.5} and b = {, 0}, respectvely. The logarthmc utlty functons parameters are k = {5, 0.} and r max = 0. We assume that the enodeb maxmum rate s = 5, therefore 4 = rnf = > = 5, therefore we can t guarantee convergence wth Algorthm n () and (), as stated by Corollary III.3. In Fgure, we show that the shadow prce fluctuates between a concave and convex curvature 87

4 log U (r ) / Sgmod a =5,b= Sgmod a =0.5,b=0 Log k =0. for Δw =5e n for Δw = n r / Fg.. The log U (r ) curve of fluctuaton example n Secton III-C, the shadow prce from Algorthm n () and (), and the shadow prce of Algorthm n (3) and (4) wth Δw =5e n and Δw = for =5 n (.e. r nf >). log U(r) of the curve. The fluctuaton n the shadow prce causes fluctuaton n the allocated rates and hnders the convergence to the optmal rates. Therefore, the optmal rate allocaton s not achevable by Algorthm n () and (). Algorthm 3 Our UE Algorthm Send ntal bd w () to enodeb eceve shadow prce from enodeb f STOP from enodeb then Calculate allocated rate r opt = w(n) Calculate new bd w (n) =r (n) f w (n) w (n ) > Δw(n) then w (n) =w (n )+sgn(w (n) w (n ))Δw(n) {Δw = l e n l or Δw = l3 n } Send new bd w (n) to enodeb end Algorthm 4 Our enodeb Algorthm eceve bds w (n) from UEs {Let w (0) = 0 } f w (n) w (n ) <δ then STOP and calculate rates r opt = w(n) M = Calculate = w(n) Send new shadow prce to all UEs end IV. OU DISTIBUTED ALGOITHM In ths secton, we present our robust algorthm to ensure the rate allocaton algorthm n [6] converges for all values of the enodeb maxmum rate. Our algorthm allocate rates concde wth the Algorthm n () and () for r nf >. For r nf, our algorthm avods the fluctuaton n the non-convergent regon dscussed n the prevous secton. Ths s acheved by addng a convergence measure Δw(n) that senses the fluctuaton n the bds w. In case of fluctuaton, our algorthm decreases the step sze between the current and the prevous bd w (n) w (n ) for every user usng fluctuaton decay functon. The fluctuaton decay functon could be n the followng forms: Exponental functon: It takes the form Δw(n) =l e n l. atonal functon: It takes the form Δw(n) = l3 n. where l,l,l 3 can be adjusted to change the rate of decay of the bds w. The new algorthm wth the fluctuaton decay functon s n Algorthm (3) and (4). emark IV.. The fluctuaton decay functon can be ncluded n Algorthm (3) of the UE or Algorthm (4) of the enodeb. In our model, we add the decay part n Algorthm (3) of the UE. In Fgure, we show the new shadow prce of the fluctuaton example n Secton III-C when usng Algorthm n (3) and (4). The shadow prce fluctuaton decreases wth every teraton n and converges to the optmal shadow prce that corresponds to the optmal rates. A detaled example s gven n the smulaton secton (Secton V). V. SIMULATION ESULTS Algorthm n (3) and (4) was appled to varous logarthmc and sgmodal-lke utlty functons wth dfferent parameters usng MATLAB. Our smulaton results showed convergence to the optmal global rates for all values of the enodeb rate. In ths secton, we use smulaton settng and parameters smlar to [6], we present the smulaton results of sx utlty functons correspondng to sx UEs shown n Fgure. We use three normalzed sgmodal-lke functons that are expressed by equaton () wth dfferent parameters, a = 5, b = whch s an approxmaton to a step functon at rate r = (e.g. VoIP), a = 3, b = 0 whch s an approxmaton of an adaptve real-tme applcaton wth nflecton pont at rate U(r) Sgmod a =5,b = Sgmod a =3,b =0 0.4 Sgmod a =,b = r Fg.. The users utlty functons U (r ) used n the smulaton (three sgmodal-lke functons and three logarthmc functons). 88

5 r(n) Sgmod a =5,b= Sgmod a =3,b=0 Sgmod a =,b= Fg. 3. The rates convergence r (n) of Algorthm n () and () wth number of teratons n for dfferent users and =. r(n) Sgmod a =5,b= Sgmod a =3,b=0 Sgmod a =,b= Fg. 4. The rates convergence r (n) of Algorthm n (3) and (4) wth number of teratons n for dfferent users and =. r =0(e.g. standard defnton vdeo streamng), and a =, b = also s an approxmaton of an adaptve real-tme applcaton wth nflecton pont at rate r = (e.g. hgh defnton vdeo streamng). We use three logarthmc functons that are expressed by equaton () wth r max = 0 and dfferent k parameters whch are approxmaton for delaytolerant applcatons (e.g. FTP). We use k = {5, 3, 0.5}. w(n) Sgmod a =5,b= Sgmod a =3,b=0 Sgmod a =,b= Fg. 5. The bds convergence w (n) of Algorthm n () and () wth number of teratons n for dfferent users and =. w(n) Sgmod a =5,b= Sgmod a =3,b=0 Sgmod a =,b= Fg. 6. The bds convergence w (n) of Algorthm n (3) and (4) wth number of teratons n for dfferent users and = Fg. 7. n The shadow prce convergence wth the number of teratons of Algorthm n () and () of Algorthm n (3) and (4) A. Convergence Dynamcs for = In the followng smulatons, we set = and number of teratons n =. Here, we choose the total enodeb rate to be less than the sum of real-tme applcaton users nflecton ponts b. Therefore, Algorthm n () and () does not converge n ths regon. In Fgure 3, we show the r Sg a =5,b= Sg a =3,b=0 Sg a =,b= Fg. 8. The allocated rates r for dfferent values of and δ = 3 for Algorthm n (3) and (4). 89

6 w 60 0 Sgmod a =5,b= Sgmod a =3,b=0 Sgmod a =,b= Fg. 9. The fnal bds w for dfferent values of and δ = 3 for Algorthm n (3) and (4). p rates r (n) of dfferent users wth the number of teratons n for Algorthm n () and (). It s shown that the rates fluctuate around the optmal rates and so the optmal rates s not acheved and the ext condton s not satsfed (.e. endless teratons). Smlar behavor for bds w (n) wth the number of teratons n s shown n Fgure 5. Algorthm n (3) and (4) behavor s more robust due to the fluctuaton decay functon. It damps the fluctuaton wth every teraton so the network reaches the optmal rates of the optmzaton problem (3). The rates r (n) and bds w (n) of Algorthm n (3) and (4) are shown n Fgures 4 and 6, respectvely. Fgure 7 shows the fluctuatng shadow prce of Algorthm n () and () and the dampng shadow prce of Algorthm n (3) and (4). B. ate Allocaton and Prcng for 5 0 In the followng smulatons, we set δ = 3 and the total rate of the enodeb takes values between 5 and 0 wth step of 5. In Fgure 8, we show the fnal rates of dfferent users wth dfferent enodeb rate. Our dstrbuted algorthm s set to avod the stuaton of allocatng zero rate to any user (.e. no user s dropped). However, the enodeb allocates the majorty of the resources to the UEs runnng adaptve real-tme applcatons untl they reach the nflecton rate r = b.when the total rate exceeds the sum of the nflecton rates b of all the adaptve real-tme applcatons, enodeb allocates more resources to the UEs wth delay-tolerant applcatons, as shown n Fgure 8, when enodeb rate exceeds =65.Ths behavor s smlar to that n [6] but wth ncludng enodeb rate <60 where the bandwdth resources are scarce wth respect to the users utltes. In Fgure 9, we show the fnal bds of dfferent users wth dfferent enodeb total rate. The hgher the user bds the hgher the allocated rate. The realtme applcaton users bd hgh when the resources are scarce and ther bds decrease as ncreases. Therefore, the prcng whch s proportonal to the bds s traffc-dependent. Ths gves the servce provders the opton to ncrease the servce prce for subscrbers when the traffc load on the system s hgh. Therefore, servce provders can motvate subscrbers to use the network when the traffc load on the network s low as they pay less for the same servce. The shadow prce represents the total prce per unt bandwdth for all users. In Fgure, we show the shadow prce wth enodeb rate. The prce s hgh for hgh-traffc (.e. fxed number of users but less resources, s small) whch decreases for low-traffc (.e. same number of users but more resources, s large). A large decrease n the prce s apparent after = {,, 60} whch are the ponts where one of the users utlty exceed the nflecton pont. Ths large decrease occurs at the sum of δ = 3.5 nflecton ponts k = rnf,wherek = {,,..., M} s the users ndex and M s the number of users VI. CONCLUSION In ths paper, we presented the convergence analyss of resource allocaton problem of hybrd traffc n 4G-LTE. We Fg.. The fnal shadow prce p for dfferent values of and δ = 3 showed that pror methods are not convergent for dfferent for Algorthm n (3) and (4). network traffc condtons. We proposed a robust optmal algorthm that converges for hgh-traffc and low-traffc loads occurrng durng the day. Our robust algorthm damps the fluctuaton that could occur n low-traffc stuatons and converges to the optmal rates. In addton, we llustrated that our algorthm provdes a prcng approach for network provders that could be used to flatten the traffc loads durng peak traffc hours. 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