Communcatons and Network, 2013, 5, 312-318 http://dx.do.org/10.4236/cn.2013.53b2058 Publshed Onlne September 2013 (http://www.scrp.org/journal/cn) Jont Power Control and Schedulng for Two-Cell Energy Effcent Broadcastng wth Network Codng Lnyu Huang 1, Ch Wan Sung 1, Seong-Lyun Km 2 1 Dept. of Electronc Engneerng, Cty Unversty of Hong Kong, Hong Kong SAR 2 School of Electrcal and Electronc Engneerng, Yonse Unversty, Seoul, Korea Emal: L.huang@my.ctyu.edu.hk, albert.sung@ctyu.edu.hk, slkm@ramo.yonse.ac.kr Receved July, 2013 ABSTRACT We consder the energy mnmzaton problem for a two-cell broadcastng system, where the focus s devsng energy effcent jont power control and schedulng algorthms. To mprove the retransmsson effcency, lnear network codng s appled to broadcast packets. Combned wth network codng, an optmal algorthm s proposed, whch s based on dynamc programmng. To reduce computatonal complexty, two sub-optmal algorthms are also proposed for large networks. Smulaton results show that the proposed schemes can reduce energy consumpton up to 57% compared wth the tradtonal Automatc Repeat-reQuest (ARQ). Keywords: Power Control; Network Codng; Broadcast; Schedulng; Energy Mnmzaton 1. Introducton Energy effcency s an mportant concern n wreless systems, both for envronmental and economcal reasons. The nformaton and communcaton technology (ICT) nfrastructure consumes about 3% of the world-wde energy and the CO 2 emssons are as many as one quarter of the CO 2 emssons by cars [1]. The CO 2 emssons from ICT are stll ncreasng at a rate of 6% per year [2]. Besdes, the energy bll accounts for a large proporton n the costs of runnng a network. The operatng costs can be sgnfcantly reduced by reducng energy consumpton [3]. Therefore, energy effcency s an ncreasngly mportant ssue. In ths paper, we focus on transmt energy mnmzaton for a wreless broadcastng system consstng of two cells. Lnear network codng [4] can be used to reduce energy consumpton by mprovng the retransmsson effcency. The tradtonal way to retransmt the lost packets s usng Automatc Repeat-request (ARQ). The retransmtted packets may not be useful to every user, and therefore the tradtonal ARQ s energy neffcent. The promsng network codng technque [4] has been proved to be an effcent approach for mult-recever broadcast. Each encoded packet s a lnear combnaton of orgnal packets, where the combnaton coeffcents are drawn from a certan fnte feld. These coeffcents for each packet are grouped together, whch s the encodng vector of that packet. The encodng vectors are broadcast together wth the correspondng packets to all the recevers. An encodng vector s sad to be nnovatve to a recever f t s not n the subspace spanned by the prevously receved encodng vectors of that user. An encodng vector s called nnovatve f t s nnovatve to all the recevers that have not receved enough packets for decodng. For easer readng, we say that a packet s nnovatve f ts correspondng encodng vector s nnovatve. Network codng can be appled nto broadcastng systems to mprove the retransmsson performance. There are some prevous works that apply network codng to wreless broadcastng systems. For example, XOR network codng and random lnear network codng (RLNC) are appled to wreless broadcast n [5] and [6], respectvely. However, XOR operates n the bnary feld and nnovatve vectors cannot be guaranteed when there are more than two recevers. Although RLNC can provde nnovatve vectors when the fnte feld s suffcently large, the large feld sze ncreases the computatonal complextes for encodng and decodng. In [7-8], Kwan et al. propose dfferent codng methods whch can guarantee nnovatve encodng vectors whle have a relatvely low requrement on feld sze. Ther methods only requre that the feld sze s no less the total number of recevers. We wll use the method of [8] to mprove the retransmsson effcency. Another way to reduce energy consumpton s to control the transmt power of base statons. By controllng the transmt power, we can control the receved Sgnal to Interference plus Nose Rato (SINR), so as to reduce energy consumpton. To our best knowledge, most of the
A [11], SE A s SE Acan SE A s SE A s 313 prevous works on energy mnmzaton for broadcastng are based on power control only [9] (and the references theren) [10], wthout usng network codng. Although Tran et al. consder both power control and network codng n [5], ther network codng method works over GF(2) whch cannot guarantee that all the encodng vectors are nnovatve. Note that both our power control scheme and network codng method are dfferent from those n [5]. In ths paper, we combne power control together wth network codng for broadcastng networks. We focus on the jont power control and schedulng problem for two base statons. The objectve s to mnmze the expected transmt energy consumpton. We use the network codng method proposed n [8] to generate nnovatve packets for transmssons. Combned wth network codng, we propose three jont power control and schedulng algorthms to mnmze energy consumpton. In the frst algorthm, the problem s formulated as a dynamc programmng problem, whch can provde an optmal soluton, n the sense of mnmzng the expected energy consumpton. To reduce the computatonal complexty, two heurstc algorthms are also proposed for large networks. Smulaton shows that they perform very well. 2. System Model As shown n Fgure 1, we consder a tme slotted wreless broadcast system consstng of two partally overlapped cells and K users. The two base statons are denoted by BS 1 and BS 2, respectvely. The transmsson range of each base staton s a crcle wth radus r. The dstance between the two base statons s denoted by D. We defne a parameter δ = D/r, where 0 < δ < 2, to characterze the locatons of the two cells. The K users are randomly located n the two cells and labeled as U, where {1, 2,, K}. The dstance from BS b, where b {1, 2}, to user U s denoted by d b. A broadcast fle s dvded nto N equal-sze packets whch are called uncoded packets. Both BS 1 and BS 2 have all the N uncoded packets. They want to broadcast these N packets to all the K users by cooperatng wth each other. Fgure 1. System model. Both BS 1 and BS 2 can transmt to the users n ther own transmsson range. To avod collsons, they are not allowed to transmt smultaneously. At each tme slot, both BS 1 and BS 2 can change ts transmt power to a value whch s chosen from a preset power level collecton ϴ = {θ 1, θ 2,, θ ϴ,} (n mw unts). All downlnk channels are modeled as mutually ndependent channels, across tme and across users, whch are affected by path loss and Raylegh fadng. A packet s consdered to be successfully receved f the receved SINR s greater than a predetermned threshold Γ (n db unts). Otherwse, that packet wll be consdered to be lost. We use R b (θ m ) to denote the relablty of the downlnk channel from BS b to user U wth transmt power θ m. The relablty s defned as the probablty that the receved SINR s greater than the threshold Γ. Snce the receved sgnal s subject to Raylegh fadng, the nstantaneous SINR s dstrbuted accordng to an exponental dstrbuton wth parameter 1/A SE where A the average SINR. Accordng to the cumulatve dstrbuton functon of exponental dstrbuton, the relablty R b (θ m ) can be computed by Rb m P{ Sr } exp( ), (1) S where S r s the nstantaneous receved SINR and A the average SINR (n db unts). Note that A determned by the correspondng dstance d b and transmt power θ m. Dependng on the envronment, A be obtaned by usng the correspondng rado propagaton model, such as the Okumura model and Hata model. Once a user successfully receved a packet, t wll send an acknowledgement to the sender of that packet. All the acknowledgement channels are assumed to be relable channels wthout error and delay. BS 1 and BS 2 are also assumed to be connected va a wred cable and they can share the acknowledgement nformaton wth each other. All the packets transmtted by the base statons are coded packets based on lnear network codng. We assume that the codng s operated over a fnte feld wth q elements, GF(q), where q K. We use the network codng method proposed n [8]. Snce q K, t s guaranteed that each broadcast packet s nnovatve to all users. In other words, a user can recover the orgnal fle as long as t has successfully receved any N broadcast packets. An example of network codng s gven n Example 1. For more detals about the network codng method, we refer the reader to [8]. We say that U s complete f t has receved N broadcast packets. Otherwse, we say that U s ncomplete. Example 1: Let q = 3, K = 3 and N = 3. We assume that each encodng vector s a 1 3 row vector. For example, an encodng vector [1 0 1] means that the correspondng encoded packet s obtaned by takng lnear combnatons of the frst and the thrd uncoded packets
314 wth coeffcents beng 1. Let C be the encodng matrx of user U, where ts rows represent the encodng vectors U has receved so far. At a partcular tme slot, consder 010 001 100 C1, C 2 andc 3 101 110 011 Each user needs to receve at least one more packet for successfully decodng the orgnal fle. If the network codng method proposed n [8] s used, we can broadcast an encodng vector v* = [0 1 2] to all users n one tme slot. For each U, v* s not n the row space of C,.e. v* cannot be obtaned by takng lnear combnatons of row vectors n C over GF(3). Therefore, v* s nnovatve to all the three users. As long as U can successfully receve v*, t can get a full rank matrx by appendng v* to the end of C. Then, U can successfully decode the orgnal fle by usng Gaussan elmnaton to solve the matrx equatons. However, f XOR network codng s used, t cannot fnd such an encodng vector that s nnovatve to all the three users smultaneously. It requres at least two transmssons. For example, the base staton may transmt the encodng vector [0 1 1] to U 1 and U 2 n the frst slot and the encodng vector [0 1 0] to U 3 n the second slot. The network codng method proposed n [8] outweghs XOR n ths case. Gven a schedulng scheme B, we use an nteger random varable T B to denote the requred number of transmssons for B to complete the broadcast. We defne the broadcast schedule of B as a bnary sequence, Φ B = {ϕ 1, ϕ 2,, ϕ TB }. If ϕ t = 0, BS 1 s selected to transmt at the t-th tme slot. Otherwse, f ϕ t = 1, BS 2 s selected. We defne the sequence of transmt power Λ B as{λ 1, λ 2,, λ TB }, where λ t ϴ s the transmt power level used at the t-th tme slot. At tme t, B specfes a functon f t to determne ϕ t and λ t : (, ) f (,, e,,, e,,,, e ) t t t 1 1 1 2 2 2 t1 t1 t1 where e t s the channel status at the t-th tme slot. It s a bnary random vector of length K. If e t () = 1, t means that the channel between the transmtter and U s too bad and U cannot receve the broadcast packet. Otherwse, U can receve that packet. Note that the probablty dstrbuton of e t depends on whether BS 1 or BS 2 s chosen as the transmtter at tme t. We call a realzaton of the sequence {e 1, e 2,, e TB } a channel realzaton under B, denoted by Ω B. For a transmsson wth transmt power λ t, the energy consumpton s λ t tmes the duraton of a tme slot. To smplfy notatons, we assume that the duraton s one unt and wll be gnored throughout ths paper. Therefore, the total energy consumpton of a partcular channel realzaton Ω B s denoted by X B and can be computed as X B TB t. t 1 The expected energy consumpton of B s denoted by E[X B ], where the expectaton s taken wth respect to the probablty dstrbuton of e t, whch depends on ϕ t. Our objectve s to mnmze E[X B ] by determnng an optmal schedulng scheme B whch requres to determne Φ B and Λ B as well. 3. Optmal Scheme The jont power control and schedulng problem can be formulated as a dynamc programmng problem wth the followng parameters. State We defne the state of all users as a 1 K state vector s = [s 1 s 2 s K ], where s, 0 < K, denotes the number of packets that U has receved. The state space s denoted by S = [ s 1 s 2 s K ], where 0 s N and 0 < K. The goal state s defned as s g = [N N N], whch means every user has receved N packets and can be consdered as a complete user. Acton At each tme step, we need to determne both the transmtter and the transmt power. We defne the acton set as A = {0, 1} ϴ. At each tme slot, an acton a = (b, θ) s selected from A. Acton a = (b, θ m ) means that BS b s selected to transmt wth transmt power θ m. State Transton Probablty Gven a state s, let p(s' s, a) denote the state transton probablty that transfers from s to s' under acton a. Now, we are gong to ntroduce how to compute the state transton probablty p(s' s, a). For a gven acton a = (b, θ m ) and gven states s = [s 1 s 2 s K ] and s' = [s 1 ' s 2 ' s K '], we use p(s ' s, a) to denote the probablty that the number of packets receved by U changes from s to s ' under acton a. Snce all the downlnk channels are assumed to be mutually ndependent across users, the state transton probablty p(s' s, a) s the product of p(s ' s, a) for all users. Hence, K p(', ssa) ps ( ' s, a) 1 We defne Q (a) as the success probablty of takng acton a to transmt to user U. Based on the prevous dscusson, Q (a) equals the relablty R b (θ m ) and Q (a) can be obtaned by Equaton (1). Based on the gven states s, s' and the number of packets that U has receved, the state transton for U can be classfed nto fve dfferent cases. The transton probablty of each case can be obtaned by the followng equaton.
315 0, f s ' s, 1, f s ' s N, ps ( ' s, a) 1 Q( a), f s ' s N, Q( a), f s ' s 1, 0, f s ' s 2. Now we are gong to dscuss how to fnd the optmal acton for each state. For s S, we defne V(s) as the expected energy consumpton for the system to evolve from s to s g gven that optmal actons are chosen n every state, and defne π(s) as the optmal acton at state s. It s clear that V(s) s non-zero except when s = s g. For s S \ {s g }, we defne V(s, a) as the expected energy consumpton for the system to evolve from s to s g gven that acton a s chosen for state s and optmal actons are chosen for the other states. For a gven acton a = (b, θ m ) A, V(s, a) can be computed by solvng the followng equaton: V(, s a) m p(' s s, a) V(') s p( s s, a) V(, s a), (2) s' S\ { s} where θ m means that each state transton takes one transmsson and θ m s the energy consumpton of that transmsson. Note that the summaton only nvolves those states that can be mmedately reached from s. In other words, state s' s nvolved only when p(s' s, a) s non-zero. Equaton (2) can be wrtten as: V(, s a) [ m p(' s s, a) V(')]/[1 s p( s s, a)] (3) s' S\ { s} Note that our problem has the property that the state-transton graph s a drected acyclc graph. It mea-ns that gven two dstnct states, s and s', f s' can be reached from s after a certan number of state transtons, then s cannot be reached from s'. Fgure 2 shows an example of the state transton graph for K = 2 and N = 2. For drected acyclc graph, t s well known that there s a topologcal orderng of vertces, whch means that f there s a state transton from s to s', then s comes before s' n the orderng. Due to ths property, V(s) can be computed backward from V(s g ). To compute V(s), we apply the formula n (3) to fnd V(s, a). V(s) and the optmal acton at s can then be obtaned by Fgure 2. State-transton graph for K=2 and N=2. Algorthm 1: Dynamc Programmng (DP) Algorthm Input: The ntal state s=0 1 K Output: V(s) and π(s) for all s S set V(s):=0 and π(s):= 0 for all s S //global varables do OptmalActon(s) return V(s) and π(s) for all s S Functon: OptmalActon(s) for s' S \ {s g, s} do f p(s' s, a) > 0 for any a A and V(s')=0 do OptmalActon(s') end f end for for a = (b,θ m ) A do V(, s a): [ p(' s s, a) V(')]/[1 s p( s s, a)] m s' S\ { s} end for V(): s mn V(, s a) aa π( s): argmn V(, s a) end functon aa V() s mn V(, s a) and π() s argmn V(, s a), aa aa respectvely. To mplement ths dea, recurson may be used, and we state the recursve algorthm n Algorthm (1). We call t Dynamc Programmng (DP) scheme. By the prncple of optmalty [12], DP s optmal n terms of mnmzng expected energy consumpton. 4. Heurstc Schemes Whle DP s optmal, ts tme complexty grows exponentally wth K. To reduce complexty, we propose two fast heurstc algorthms for large networks. When the number of users s relatvely small, we propose a greedy algorthm whch dynamcally changes the transmt power. When the number of users s very large, we propose an algorthm whch uses fxed transmt power. 4.1. Greedy Algorthm The greedy algorthm has the followng three steps. Step 1: At the begnnng of a tme slot, let s j, where 0 < j K, denote the number of packets that U j has receved. We frst choose the user U whch has receved the least number of packets. If there are multple qualfed users, we wll randomly pck one of them. Step 2: Select the base staton BS b * that s closer to U as the sender. If both BS 1 and BS 2 have the same dstance from U, randomly pck one of them as the sender n the current slot. Step 3: We defne a parameter E b (θ m ) for each trans-
316 mt power level θ m. At a partcular slot, E b (θ m ) s defned as the expected energy consumpton requred to make user U successfully receve a packet, assumng that BS b s selected to transmt wth power θ m. E b (θ m ) can be computed as follows. 1 Eb m m, (4) Rb ( m ) where R b (θ m ) s the relablty of the channel from BS b to U wth transmt power θ m. After computng the E b (θ m ) value for each θ m, choose the transmt power θ m * that corresponds to the mnmal E b (θ m ) as the transmt power of current slot. If there exsts more than one E b (θ m ) whch has the mnmal value, just randomly pck one of them. Then BS b * wll broadcast a packet to all users wth transmt power θ m *. Then go back to Step 1 and repeat the procedure untl all the users are complete. The greedy algorthm can be summarzed as Algorthm (2), whch s called Greedy Transmsson (GT) algorthm. 4.2. Fxed Power Algorthm When the number of users s very large, the users wth worst channels locate at the boundares of the cells wth hgh probabltes. For such scenaros, we propose the followng algorthm wth fxed transmt power. BS 1 transmts at odd tme slots and BS 2 transmts at even slots. The transmt power s always set to a fxed power level θ*. BS 1 and BS 2 take turns to transmt untl all users are complete. The transmt power θ* s determned as follows. Consder a wreless system whch conssts of BS 1 and only one user U 1. Suppose U 1 s located at the border of the cell,.e. the dstance between U 1 and BS 1 s the radus r. For each θ m, we calculate the expected energy consumpton E 11 (θ m ) defned by Equaton (4). Then select the power level wth mnmal E 11 (θ m ) as θ*, whch means θ* s the optmal power level for U 1, a user locatng at the boundary. Algorthm 2: Greedy Transmsson (GT) Algorthm whle s s g do Set arg mn{ s}, 0 K Set b arg mn{ d }, b{1,2} b for m=1 to ϴ do Compute E b (θ m ) by usng Equaton (4). end for Set m arg mn { E ( )}. 0 m b Let BS b transmt a coded packet wth power θ m. Update s. end whle m Note that ths algorthm has such a feature that t does not requre any channel state nformaton of users. We call ths algorthm the Fxed Power (FP) algorthm. 5. Numercal Results We evaluate the performance of proposed algorthms va smulatons. The downlnk channels are modeled contanng combned effects of path loss and Raylegh fadng. The mnmal transmt power θ mn and maxmal transmt power θ max are set to 100mW (20 dbm) and 20W (about 43 dbm), respectvely. ϴ ncludes 20 ordered power levels whch are lnearly spaced between and ncludng θ mn and θ max. The SINR threshold Γ s set to 3 db. δ s set to dfferent values to evaluate the performances under dfferent cell locatons. Other parameters are shown n Table 1 [13]. For each set of parameters, the results are averaged over 10,000 repettons. We compare the proposed algorthms wth the scheme named NCPC n [5] and an ARQ scheme. For NCPC, we set the parameter, the extenson radus of the center cell, to 1.15r, whch s the same as the smulaton n [5]. We refer the reader to [5] for detals of NCPC. In the ARQ scheme, BS 1 transmts at odd slots and BS 2 transmts at even slots. Note that f all the users n a cell are complete, only the base staton of the other cell wll transmt at the remanng tme slots. In each cell, the base staton retransmts each uncoded packet untl all the users n ts cell has receved t. The transmt power s determned as follows. At a partcular slot, the base staton BS b frst selects a target user U whch requres the packet to be broadcast whle has the longest dstance from the base staton. Then, use Equaton (4) to calculate the expected energy consumpton E b (θ m ) for each θ m. Fnally, select the power level wth mnmal E b (θ m ) as the transmt power n that slot. For small scale networks, we compare the performances of DP, GT and FP wth NCPC and ARQ. In Fgure 3, we frst evaluate the performances of dfferent cell locatons wth N = 4 and K = 4 and δ vares from 0.3 to 1.9. When δ = 0.3, the two cells are almost totally overlapped wth each other. When δ = 1.9, the two cells are almost separated. We can fnd that both DP and GT always perform better than NCPC and ARQ. In partcular, Table 1. Smulaton parameters[13]. Parameter Value Cell radus r 1 Km Bandwdth 20 MHz No. of resource blocks 100 BS antenna gan 11 db MS antenna gan 0 db Mnmum couplng loss 53 db Path loss 128.1+37.6log 10 (d b ), d b n Km Nose densty -165 dbm/hz Nose fgure 9 db
L. Y. HUANG ET AL. 317 GT and FP can reduce energy consumpton by 18% and 14%, respectvely. Compared wth ARQ, when K = 100, GT and FP can reduce energy consumpton by 57% and 55%, respectvely. In both Fgure 5 and Fgure 6, the performance of FP s comparable to that of GT. In Fgure 6, the gap s less than 5% when K 40. As the number of users ncreases, the gap between GT and FP becomes smaller and smaller. The reason s that, when number of users s large, the users wth worst channels are located near the cell boundary. In that case, the transmt power selected by GT s close, or even equal, to that selected by FP. Therefore, ther performances are very close. 6. Conclusons Fgure 3. Energy consumpton for N=4, K=4 and varous δ. compared wth NCPC when δ = 0.3, DP and GT can reduce energy consumpton by 37% and 34%, respectvely. Compared wth ARQ, when δ = 0.3, DP and GT can reduce energy consumpton by 26% and 22%, respectvely. To nvestgate the effects of number of users, we run smulatons wth N = 4, δ = 1.0 and K vares from 2 to 4. The smulaton results are shown n Fgure 4. DP and GT also outwegh NCPC and ARQ n all the scenaros. Especally, compared wth NCPC, both DP and GT can reduce energy consumpton by about 50% when K = 2. Compared wth ARQ, when K = 4, DP and GT can reduce energy consumpton by 17% and 16%, respectvely. In both Fgure 3 and Fgure 4, the performance gap between DP and GT s less than 6%, whch means the GT performs very well and s close to the optmal scheme. In both Fgure 3 and Fgure 4, FP performs worse than other schemes. The reason s that, when the number of users s small, the worst user s not located near the cell boundary wth hgh probabltes, where the worst user means t has the longest dstance from base staton. That means the transmt power of FP s too large and FP s energy neffcent. For large scale networks, we evaluate the performance of GT, FT, NCPC and ARQ wth changng K and δ. Consderng the computatonal complexty of DP, t s not taken nto comparson. In Fgure 5, the parameters are set to N = 60, K = 50 and δ vares from 0.3 to 1.9. Both GT and FP perform better than NCPC and ARQ, no matter how the cells are located. In partcular, compared wth NCPC, when δ = 0.3, GT and FP can reduce energy consumpton by 19% and 15%, respectvely. Compared wth ARQ, when δ = 1.9, GT and FP can reduce energy consumpton by 50% and 47%, respectvely. We also evaluate the performances wth N = 60, δ = 1 and K vares from 20 to 100. The smulaton results are shown n Fgure 6. Both GT and FP also requre less transmt energy. In partcular, compared wth NCPC, when K = 100, In ths paper, we consder the problem of mnmzng the expected transmt energy consumpton for a two-cell broadcastng system. On one hand, we apply the nnovatve network codng to reduce the number of retransmssons. On the other hand, we use power control to control Fgure 4. Energy consumpton for N=4, δ=1 and varous K. Fgure 5. Energy consumpton for N= 60, K= 50 and varous δ.
318 Fgure 6. Energy consumpton for N=60, δ=1 and varous K. the receved SINR such that the transmt effcency can be mproved. Besdes, we also consder the schedulng problem of the two base statons. We frst propose an optmal algorthm called DP, n the sense of mnmzng expected energy consumpton, whch uses dynamc programmng to choose the optmal acton for each state. To reduce complexty, we also propose two sub-optmal greedy algorthms, called GT and FP, respectvely, for large scale networks. Smulaton shows that GT has a very close performance to that of the optmal DP when the network s small. When the number of users s very large, both FP and GT outwegh NCPC and ARQ. Although the algorthms are desgned for two-cell systems n ths paper, they can be extended to multple-cell systems. 7. Acknowledgements The work descrbed n ths paper was partally supported by a grant from the Unversty Grants Commttee of the Hong Kong Specal Admnstratve Regon, Chna (Project No. AoE/E-02/08). Part of ths work was done when the frst author vsted Yonse Unversty n 2012 Fall. REFERENCES [1] S. Vadgama, Trends n Green Wreless Access, FU- JITSU Scentfc & Techncal Journal, Vol. 45, 2009, pp. 404-408. [2] S. Zeadally, S. U. Khan and N. Chlamkurt, Energy-Effcent Networkng: Past, Present, and Future, The Journal of Supercomputng, Vol. 62, No. 3, 2012, pp. 1093-1118. HUdo:10.1007/s11227-011-0632-2UH [3] Ercsson, Sustanable Energy Use n Moble Communcatons, Whte paper, Aug. 2007. [4] S. Y. R. L, R. W. Yeung and N. Ca, Lnear Network Codng, IEEE Transactons on Informaton Theory, Jun. 2003, pp. 371-381. HUdo:10.1109/TIT.2002.807285UH [5] T. Tran, D. Nguyen, T. Nguyen and D. Tran, Jont Network Codng and Power Control for Cellular Rado Networks, Proceedngs of the Second Internatonal Conference on Communcatons and Electroncs, Jun. 2008, pp. 109-114. [6] L. Lu, F. Sun, M. Xao and L. K. Rasmussen, Relay-Aded Mult-cell Broadcastng wth Random Network Codng, Proceedngs of Internatonal Symposum on Informaton Theory and ts Applcatons, Dec. 2010, pp. 957-962. [7] H. Y. Kwan, K. W. Shum and C. W. Sung, Generaton of Innovatve and Sparse Encodng Vectors for Broadcast Systems wth Feedback, Proceedngs of IEEE Internatonal Symposum on Informaton Theory, Sant Petersburg, Russa, Aug. 2011, pp. 1161-1165. [8] H. Y. Kwan, K. W. Shum and C. W. Sung, Lnear Network Code for Erasure Broadcast Channel wth Feedback: Complexty and Algorthms, arxv: 1205.5324v1, May 2012. [9] M. Chang, P. Hande, T. Lan and W. Tan, Power Control n Wreless Cellular Networks, Foundatons and Trends n Networkng, Vol. 2, No. 4, 2008, pp. 381-533. HUdo:10.1561/1300000009UH [10] A. Srdhar and A. Ephremdes, Energy Optmzaton n Wreless Broadcastng through Power Control, Ad Hoc Networks, Vol. 6, 2008, pp. 155-167. HUdo:10.1016/j.adhoc.2006.11.001UH [11] M. K. Smon and M. S. Aloun, Fadng Channel Characterzaton and Modelng, Dgtal Communcaton over Fadng Channels A Unfed Approach to Performance Analyss, 2nd ed., Wley, 2005. [12] D. P. Bertsekas, The Dynamc Programmng Algorthm, Dynamc Programmng and Optmal Control, 3nd ed., Mass.: Athena Scentfc, 2005. [13] 3GPP TR 25.942 v9.0, RF System Scenaros, Dec. 2009.