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Communications and Network, 2013, 5, 312-318 http://dx.doi.org/10.4236/cn.2013.53B2058 Published Online September 2013 (http://www.scirp.org/journal/cn) Copyright © 2013 SciRes. CN Joint Power Control and Scheduling for Two-Cell Energy Efficient Broadcasting with Network Coding Linyu Huang1, Chi Wan Sung1, Seong-Lyun Kim2 1Dept. of Electronic Engineering, City University of Hong Kong, Hong Kong SAR 2School of Electrical and Electronic Engineering, Yonsei University, Seoul, Korea Email: L.huang@my.cityu.edu.hk, albert.sung@cityu.edu.hk, slkim@ramo.yonsei.ac.kr Received July, 2013 ABSTRACT We consider the energy minimization problem for a two-cell broadcasting system, where the focus is devising energy efficient joint power control and scheduling algorithms. To improve the retransmission efficiency, linear network cod- ing is applied to broadcast packets. Combined with network coding, an optimal algorithm is proposed, which is based on dynamic programming. To reduce computational complexity, two sub-optimal algorithms are also proposed for large networks. Simulation results show that the proposed schemes can reduce energy consumption up to 57% compared with the traditional Automatic Repeat-reQuest (ARQ). Keywords: Power Control; Network Coding; Broadcast; Scheduling; Energy Minimization 1. Introduction Energy efficiency is an important concern in wireless systems, both for environmental and economical reasons. The information and communication technology (ICT) infrastructure consumes about 3% of the world-wide en- ergy and the CO2 emissions are as many as one quarter of the CO2 emissions by cars [1]. The CO2 emissions from ICT are still increasing at a rate of 6% per year [2]. Be- sides, the energy bill accounts for a large proportion in the costs of running a network. The operating costs can be significantly reduced by reducing energy consumption [3]. Therefore, energy efficiency is an increasingly im- portant issue. In this paper, we focus on transmit energy minimization for a wireless broadcasting system consist- ing of two cells. Linear network coding [4] can be used to reduce en- ergy consumption by improving the retransmission effi- ciency. The traditional way to retransmit the lost packets is using Automatic Repeat-request (ARQ). The retrans- mitted packets may not be useful to every user, and therefore the traditional ARQ is energy inefficient. The promising network coding technique [4] has been proved to be an efficient approach for multi-receiver broadcast. Each encoded packet is a linear combination of original packets, where the combination coefficients are drawn from a certain finite field. These coefficients for each packet are grouped together, which is the encoding vec- tor of that packet. The encoding vectors are broadcast together with the corresponding packets to all the receiv- ers. An encoding vector is said to be innovative to a re- ceiver if it is not in the subspace spanned by the previ- ously received encoding vectors of that user. An encod- ing vector is called innovative if it is innovative to all the receivers that have not received enough packets for de- coding. For easier reading, we say that a packet is inno- vative if its corresponding encoding vector is innovative. Network coding can be applied into broadcasting systems to improve the retransmission performance. There are some previous works that apply network coding to wireless broadcasting systems. For example, XOR network coding and random linear network coding (RLNC) are applied to wireless broadcast in [5] and [6], respectively. However, XOR operates in the binary field and innovative vectors cannot be guaranteed when there are more than two receivers. Although RLNC can pro- vide innovative vectors when the finite field is suffi- ciently large, the large field size increases the computa- tional complexities for encoding and decoding. In [7-8], Kwan et al. propose different coding methods which can guarantee innovative encoding vectors while have a rela- tively low requirement on field size. Their methods only require that the field size is no less the total number of receivers. We will use the method of [8] to improve the retransmission efficiency. Another way to reduce energy consumption is to con- trol the transmit power of base stations. By controlling the transmit power, we can control the received Signal to Interference plus Noise Ratio (SINR), so as to reduce energy consumption. To our best knowledge, most of the L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 313 previous works on energy minimization for broadcasting are based on power control only [9] (and the references therein) [10], without using network coding. Although Tran et al. consider both power control and network coding in [5], their network coding method works over GF(2) which cannot guarantee that all the encoding vec- tors are innovative. Note that both our power control scheme and network coding method are different from those in [5]. In this paper, we combine power control together with network coding for broadcasting networks. We focus on the joint power control and scheduling problem for two base stations. The objective is to minimize the expected transmit energy consumption. We use the network cod- ing method proposed in [8] to generate innovative pack- ets for transmissions. Combined with network coding, we propose three joint power control and scheduling algo- rithms to minimize energy consumption. In the first algo- rithm, the problem is formulated as a dynamic program- ming problem, which can provide an optimal solution, in the sense of minimizing the expected energy consump- tion. To reduce the computational complexity, two heu- ristic algorithms are also proposed for large networks. Simulation shows that they perform very well. 2. System Model As shown in Figure 1, we consider a time slotted wire- less broadcast system consisting of two partially over- lapped cells and K users. The two base stations are de- noted by BS1 and BS2, respectively. The transmission range of each base station is a circle with radius r. The distance between the two base stations is denoted by D. We define a parameter δ = D/r, where 0 < δ < 2, to char- acterize the locations of the two cells. The K users are randomly located in the two cells and labeled as Ui, where i∈{1, 2, …, K}. The distance from BSb, where b ∈{1, 2}, to user Ui is denoted by dbi. A broadcast file is divided into N equal-size packets which are called un- coded packets. Both BS1 and BS2 have all the N uncoded packets. They want to broadcast these N packets to all the K users by cooperating with each other. Figure 1. System model. Both BS1 and BS2 can transmit to the users in their own transmission range. To avoid collisions, they are not al- lowed to transmit simultaneously. At each time slot, both BS1 and BS2 can change its transmit power to a value which is chosen from a preset power level collection ϴ = {θ1, θ2, …, θ| ϴ |,} (in mW units). All downlink channels are modeled as mutually independent channels, across time and across users, which are affected by path loss and Rayleigh fading. A packet is considered to be suc- cessfully received if the received SINR is greater than a predetermined threshold Γ (in dB units). Otherwise, that packet will be considered to be lost. We use Rbi(θm) to denote the reliability of the downlink channel from BSb to user Ui with transmit power θm. The reliability is de- fined as the probability that the received SINR is greater than the threshold Γ. Since the received signal is subject to Rayleigh fading, the instantaneous SINR is distributed according to an exponential distribution with parameter 1/ASE A [11], where ASE A is the average SINR. According to the cumulative distribution function of exponential dis- tribution, the reliability Rbi(θm) can be computed by {} exp(), bi mr RPS S (1) where Sr is the instantaneous received SINR and ASE A is the average SINR (in dB units). Note that ASE A is deter- mined by the corresponding distance dbi and transmit power θm. Depending on the environment, ASE Acan be ob- tained by using the corresponding radio propagation model, such as the Okumura model and Hata model. Once a user successfully received a packet, it will send an acknowledgement to the sender of that packet. All the acknowledgement channels are assumed to be reliable channels without error and delay. BS1 and BS2 are also assumed to be connected via a wired cable and they can share the acknowledgement information with each other. All the packets transmitted by the base stations are coded packets based on linear network coding. We as- sume that the coding is operated over a finite field with q elements, GF(q), where q ≥ K. We use the network cod- ing method proposed in [8]. Since q ≥ K, it is guaranteed that each broadcast packet is innovative to all users. In other words, a user can recover the original file as long as it has successfully received any N broadcast packets. An example of network coding is given in Example 1. For more details about the network coding method, we refer the reader to [8]. We say that Ui is complete if it has received N broadcast packets. Otherwise, we say that Ui is incomplete. Example 1: Let q = 3, K = 3 and N = 3. We assume that each encoding vector is a 1×3 row vector. For exam- ple, an encoding vector [1 0 1] means that the corre- sponding encoded packet is obtained by taking linear combinations of the first and the third uncoded packets L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 314 with coefficients being 1. Let Ci be the encoding matrix of user Ui, where its rows represent the encoding vectors Ui has received so far. At a particular time slot, consider 12 3 010 001100 , 101 110011 CCandC Each user needs to receive at least one more packet for successfully decoding the original file. If the network coding method proposed in [8] is used, we can broadcast an encoding vector v* = [0 1 2] to all users in one time slot. For each Ui, v* is not in the row space of Ci, i.e. v* cannot be obtained by taking linear combinations of row vectors in Ci over GF(3). Therefore, v* is innovative to all the three users. As long as Ui can successfully receive v*, it can get a full rank matrix by appending v* to the end of Ci. Then, Ui can successfully decode the original file by using Gaussian elimination to solve the matrix equations. However, if XOR network coding is used, it cannot find such an encoding vector that is innovative to all the three users simultaneously. It requires at least two trans- missions. For example, the base station may transmit the encoding vector [0 1 1] to U1 and U2 in the first slot and the encoding vector [0 1 0] to U3 in the second slot. The network coding method proposed in [8] outweighs XOR in this case. Given a scheduling scheme B, we use an integer ran- dom variable TB to denote the required number of trans- missions for B to complete the broadcast. We define the broadcast schedule of B as a binary sequence, ΦB = { ϕ 1, ϕ 2, …, ϕ TB }. If ϕ t = 0, BS1 is selected to transmit at the t-th time slot. Otherwise, if ϕ t = 1, BS2 is selected. We define the sequence of transmit power ΛB as{λ1, λ2, …, λTB}, where λt∈ϴ is the transmit power level used at the t-th time slot. At time t, B specifies a function ft to determine ϕ t and λt: 1112 22111 (,)(,, ,,,,,,,) ttttt t f ee e where et is the channel status at the t-th time slot. It is a binary random vector of length K. If et(i) = 1, it means that the channel between the transmitter and Ui is too bad and Ui cannot receive the broadcast packet. Otherwise, Ui can receive that packet. Note that the probability distri- bution of et depends on whether BS1 or BS2 is chosen as the transmitter at time t. We call a realization of the se- quence {e1, e2, …, eTB} a channel realization under B, denoted by ΩB. For a transmission with transmit power λt, the energy consumption is λt times the duration of a time slot. To simplify notations, we assume that the duration is one unit and will be ignored throughout this paper. Therefore, the total energy consumption of a particular channel re- alization ΩB is denoted by XB and can be computed as 1 . T t t X B B The expected energy consumption of B is denoted by E[XB], where the expectation is taken with respect to the probability distribution of et, which depends on ϕ t. Our objective is to minimize E[XB] by determining an optimal scheduling scheme B which requires to determine ΦB and ΛB as well. 3. Optimal Scheme The joint power control and scheduling problem can be formulated as a dynamic programming problem with the following parameters. State We define the state of all users as a 1×K state vector s = [s1 s2 … sK], where si, 0 < i ≤ K, denotes the number of packets that Ui has received. The state space is denoted by S = [ s1 s2 … sK], where 0 ≤ si ≤ N and 0 < i ≤ K. The goal state is defined as sg = [N N … N], which means every user has received N packets and can be con- sidered as a complete user. Action At each time step, we need to determine both the transmitter and the transmit power. We define the action set as A = {0, 1} × ϴ. At each time slot, an action a = (b, θ) is selected from A. Action a = (b, θm) means that BSb is selected to transmit with transmit power θm. State Transition Probability Given a state s, let p(s'|s, a) denote the state transition probability that trans- fers from s to s' under action a. Now, we are going to introduce how to compute the state transition probability p(s'|s, a). For a given action a = (b, θm) and given states s = [s1 s2 … sK] and s' = [s1' s2' … sK'], we use p(si'|si, a) to de- note the probability that the number of packets received by Ui changes from si to si' under action a. Since all the downlink channels are assumed to be mutually inde- pendent across users, the state transition probability p(s'|s, a) is the product of p(si'|si, a) for all users. Hence, 1 ('|, )( '|,) K ii i p apssass We define Qi(a) as the success probability of taking action a to transmit to user Ui. Based on the previous dis- cussion, Qi(a) equals the reliability Rbi(θm) and Qi(a) can be obtained by Equation (1). Based on the given states s, s' and the number of packets that Ui has received, the state transition for Ui can be classified into five different cases. The transition probability of each case can be ob- tained by the following equation. L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 315 0,' , 1,' , ('|,) 1(),', (),' 1, 0,' 2. ii ii iiiii iii ii ss s sN pssaQ assN Qas s ss if if if if if Now we are going to discuss how to find the optimal action for each state. For s∈S, we define V(s) as the ex- pected energy consumption for the system to evolve from s to sg given that optimal actions are chosen in every state, and define π(s) as the optimal action at state s. It is clear that V(s) is non-zero except when s = sg. For s∈S \ {sg}, we define V(s, a) as the expected energy consumption for the system to evolve from s to sg given that action a is chosen for state s and optimal actions are chosen for the other states. For a given action a = (b, θm) ∈A, V(s, a) can be computed by solving the following equation: '{} (,)('| ,)(')( | ,)(,), m VapaVpaVa sS\s sssssss (2) where θm means that each state transition takes one transmission and θm is the energy consumption of that transmission. Note that the summation only involves those states that can be immediately reached from s. In other words, state s' is involved only when p(s'|s, a) is non-zero. Equation (2) can be written as: '{} (,)[( '|,)(')]/[1(|,)] m VapaVp a sS\s ssssss (3) Note that our problem has the property that the state-transition graph is a directed acyclic graph. It mea-ns that given two distinct states, s and s', if s' can be reached from s after a certain number of state transitions, then s cannot be reached from s'. Figure 2 shows an ex- ample of the state transition graph for K = 2 and N = 2. For directed acyclic graph, it is well known that there is a topological ordering of vertices, which means that if there is a state transition from s to s', then s comes before s' in the ordering. Due to this property, V(s) can be com- puted backward from V(sg). To compute V(s), we apply the formula in (3) to find V(s, a). V(s) and the optimal action at s can then be obtained by Figure 2. State-transition graph for K=2 and N=2. Algorithm 1: Dynamic Programming (DP) Algorithm Input: The initial state s=01×K Output: V(s) and π(s) for all s∈S set V(s):=0 and π(s):= 0 for all s∈S //global vari- ables do OptimalAction(s) return V(s) and π(s) for all s∈S Function: OptimalAction(s) for s'∈S \ {sg, s} do if p(s'|s, a) > 0 for any a∈A and V(s')=0 do OptimalAction(s') end if end for for a = (b,θm) ∈A do '{} (,):[('| ,)(')]/[1(| ,)] m VapaVp a sS\s ssssss end for (): min(, ) π(): argmin(,) a a VVa Va A A ss ss end function ()min(, )()argmin(, ), aa VVa Va AA ssandπss respectively. To implement this idea, recursion may be used, and we state the recursive algorithm in Algorithm (1). We call it Dynamic Programming (DP) scheme. By the principle of optimality [12], DP is optimal in terms of minimizing expected energy consumption. 4. Heuristic Schemes While DP is optimal, its time complexity grows expo- nentially with K. To reduce complexity, we propose two fast heuristic algorithms for large networks. When the number of users is relatively small, we propose a greedy algorithm which dynamically changes the transmit power. When the number of users is very large, we propose an algorithm which uses fixed transmit power. 4.1. Greedy Algorithm The greedy algorithm has the following three steps. Step 1: At the beginning of a time slot, let sj, where 0 < j ≤ K, denote the number of packets that Uj has re- ceived. We first choose the user Ui which has received the least number of packets. If there are multiple quali- fied users, we will randomly pick one of them. Step 2: Select the base station BSb* that is closer to Ui as the sender. If both BS1 and BS2 have the same distance from Ui, randomly pick one of them as the sender in the current slot. Step 3: We define a parameter Ebi(θm) for each trans- L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 316 mit power level θm. At a particular slot, Ebi(θm) is defined as the expected energy consumption required to make user Ui successfully receive a packet, assuming that BSb is selected to transmit with power θm. Ebi(θm) can be computed as follows. 1, () bi mm bi m ER (4) where Rbi(θm) is the reliability of the channel from BSb to Ui with transmit power θm. After computing the Ebi(θm) value for each θm, choose the transmit power θm* that corresponds to the minimal Ebi(θm) as the transmit power of current slot. If there ex- ists more than one Ebi(θm) which has the minimal value, just randomly pick one of them. Then BSb* will broad- cast a packet to all users with transmit power θm *. Then go back to Step 1 and repeat the procedure until all the users are complete. The greedy algorithm can be summarized as Algorithm (2), which is called Greedy Transmission (GT) algorithm. 4.2. Fixed Power Algorithm When the number of users is very large, the users with worst channels locate at the boundaries of the cells with high probabilities. For such scenarios, we propose the following algorithm with fixed transmit power. BS1 transmits at odd time slots and BS2 transmits at even slots. The transmit power is always set to a fixed power level θ*. BS 1 and BS2 take turns to transmit until all users are complete. The transmit power θ* is determined as follows. Con- sider a wireless system which consists of BS1 and only one user U1. Suppose U1 is located at the border of the cell, i.e. the distance between U1 and BS1 is the radius r. For each θm, we calculate the expected energy consump- tion E11(θm) defined by Equation (4). Then select the power level with minimal E11(θm) as θ*, which means θ* is the optimal power level for U1, a user locating at the boundary. Algorithm 2: Greedy Transmission (GT) Algorithm while s ≠ sg do 0 {1, 2 } Set argmin{}, Set argmin{}, i iK bi b is bd for m=1 to |ϴ| do Compute Ebi(θm) by using Equation (4). end for 0|| Set argmin{()}. bi m m mE Let BSb transmit a coded packet with power θm. Update s. end while Note that this algorithm has such a feature that it does not require any channel state information of users. We call this algorithm the Fixed Power (FP) algorithm. 5. Numerical Results We evaluate the performance of proposed algorithms via simulations. The downlink channels are modeled containing combined effects of path loss and Rayleigh fading. The minimal transmit power θmin and maximal transmit power θmax are set to 100mW (20 dBm) and 20W (about 43 dBm), respectively. ϴ includes 20 or- dered power levels which are linearly spaced between and including θmin and θmax. The SINR threshold Γ is set to 3 dB. δ is set to different values to evaluate the per- formances under different cell locations. Other parame- ters are shown in Table 1 [13]. For each set of parame- ters, the results are averaged over 10,000 repetitions. We compare the proposed algorithms with the scheme named NCPC in [5] and an ARQ scheme. For NCPC, we set the parameter, the extension radius of the center cell, to 1.15r, which is the same as the simulation in [5]. We refer the reader to [5] for details of NCPC. In the ARQ scheme, BS1 transmits at odd slots and BS2 transmits at even slots. Note that if all the users in a cell are complete, only the base station of the other cell will transmit at the remaining time slots. In each cell, the base station re- transmits each uncoded packet until all the users in its cell has received it. The transmit power is determined as follows. At a particular slot, the base station BSb first selects a target user Ui which requires the packet to be broadcast while has the longest distance from the base station. Then, use Equation (4) to calculate the expected energy consumption Ebi(θm) for each θm. Finally, select the power level with minimal Ebi(θm) as the transmit power in that slot. For small scale networks, we compare the perform- ances of DP, GT and FP with NCPC and ARQ. In Figure 3, we first evaluate the performances of different cell locations with N = 4 and K = 4 and δ varies from 0.3 to 1.9. When δ = 0.3, the two cells are almost totally over- lapped with each other. When δ = 1.9, the two cells are almost separated. We can find that both DP and GT al- ways perform better than NCPC and ARQ. In particular, Table 1. Simulation parameters[13]. Parameter Value Cell radius r 1 Km Bandwidth 20 MHz No. of resource blocks 100 BS antenna gain 11 dBi MS antenna gain 0 dBi Minimum coupling loss53 dB Path loss 128.1+37.6log10(dbi), dbi in Km Noise density -165 dBm/Hz Noise figure 9 dB L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 317 Figure 3. Energy consumption for N=4, K=4 and various δ. compared with NCPC when δ = 0.3, DP and GT can re- duce energy consumption by 37% and 34%, respectively. Compared with ARQ, when δ = 0.3, DP and GT can re- duce energy consumption by 26% and 22%, respectively. To investigate the effects of number of users, we run simulations with N = 4, δ = 1.0 and K varies from 2 to 4. The simulation results are shown in Figure 4. DP and GT also outweigh NCPC and ARQ in all the scenarios. Especially, compared with NCPC, both DP and GT can reduce energy consumption by about 50% when K = 2. Compared with ARQ, when K = 4, DP and GT can re- duce energy consumption by 17% and 16%, respectively. In both Figure 3 and Figure 4, the performance gap be- tween DP and GT is less than 6%, which means the GT performs very well and is close to the optimal scheme. In both Figure 3 and Figure 4, FP performs worse than other schemes. The reason is that, when the number of users is small, the “worst” user is not located near the cell boundary with high probabilities, where the worst user means it has the longest distance from base station. That means the transmit power of FP is too large and FP is energy inefficient. For large scale networks, we evaluate the performance of GT, FT, NCPC and ARQ with changing K and δ. Considering the computational complexity of DP, it is not taken into comparison. In Figure 5, the parameters are set to N = 60, K = 50 and δ varies from 0.3 to 1.9. Both GT and FP perform better than NCPC and ARQ, no matter how the cells are located. In particular, compared with NCPC, when δ = 0.3, GT and FP can reduce energy consumption by 19% and 15%, respectively. Compared with ARQ, when δ = 1.9, GT and FP can reduce energy consumption by 50% and 47%, respectively. We also evaluate the performances with N = 60, δ = 1 and K var- ies from 20 to 100. The simulation results are shown in Figure 6. Both GT and FP also require less transmit en- ergy. In particular, compared with NCPC, when K = 100, GT and FP can reduce energy consumption by 18% and 14%, respectively. Compared with ARQ, when K = 100, GT and FP can reduce energy consumption by 57% and 55%, respectively. In both Figure 5 and Figure 6, the performance of FP is comparable to that of GT. In Fig- ure 6, the gap is less than 5% when K ≥ 40. As the num- ber of users increases, the gap between GT and FP be- comes smaller and smaller. The reason is that, when number of users is large, the users with worst channels are located near the cell boundary. In that case, the transmit power selected by GT is close, or even equal, to that selected by FP. Therefore, their performances are very close. 6. Conclusions In this paper, we consider the problem of minimizing the expected transmit energy consumption for a two-cell broadcasting system. On one hand, we apply the innova- tive network coding to reduce the number of retransmis- sions. On the other hand, we use power control to control Figure 4. Energy consumption for N=4, δ=1 and various K. Figure 5. Energy consumption for N= 60, K= 50 and various δ. L. Y. HUANG ET AL. Copyright © 2013 SciRes. CN 318 Figure 6. Energy consumption for N=60, δ=1 and various K. the received SINR such that the transmit efficiency can be improved. Besides, we also consider the scheduling problem of the two base stations. We first propose an optimal algorithm called DP, in the sense of minimizing expected energy consumption, which uses dynamic pro- gramming to choose the optimal action for each state. To reduce complexity, we also propose two sub-optimal greedy algorithms, called GT and FP, respectively, for large scale networks. Simulation shows that GT has a very close performance to that of the optimal DP when the network is small. When the number of users is very large, both FP and GT outweigh NCPC and ARQ. Al- though the algorithms are designed for two-cell systems in this paper, they can be extended to multiple-cell sys- tems. 7. Acknowledgements The work described in this paper was partially supported by a grant from the University Grants Committee of the Hong Kong Special Administrative Region, China (Pro- ject No. AoE/E-02/08). Part of this work was done when the first author vis- ited Yonsei University in 2012 Fall. REFERENCES [1] S. Vadgama, “Trends in Green Wireless Access,” FU- JITSU Scientific & Technical Journal, Vol. 45, 2009, pp. 404-408. [2] S. Zeadally, S. U. Khan and N. Chilamkurti, “En- ergy-Efficient Networking: Past, Present, and Future,” The Journal of Supercomputing, Vol. 62, No. 3, 2012, pp. 1093-1118. HUdoi:10.1007/s11227-011-0632-2UH [3] Ericsson, “Sustainable Energy Use in Mobile Communi- cations,” White paper, Aug. 2007. [4] S. Y. R. Li, R. W. Yeung and N. 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