Power-Efficient Rate Scheduling in Wireless Links Using Computational Geometric Algorithms

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1 Power-Efficient Rate Scheduling in Wireless Links Using Computational Geometric Algorithms Mingjie Lin ISL, Department of Electrical Engineering Stanford Universiy, Stanford, CA Yashar Ganjali CSL, Department of Electrical Engineering Stanford University, Stanford, CA ABSTRACT Energy efficiency has become increasingly critical in designing and operating wireless networks, especially for mobile ad hoc networks consisting of portable mobile wireless computing/communication devices powered by limited battery capacity. Since the energy required to transmit a given amount of data is a convex and monotonically increasing function of the transmission rate [5, 12], theoretically one can improve energy efficiency by transmitting data at lower rates. Unfortunately, low data rates result in longer transmission duration and larger communication delay at receiving end, which is usually undesirable. How to optimally schedule transmission process to both minimize the total power consumption and observe all time constraints (available times and transmission deadlines) is a challenging and interesting problem. In this paper, we propose a technique to solve the above rate scheduling problem by transforming it into finding the shortest path between two vertices of a two dimensional polygon, which yields an elegant analytical solution and easy-to-prove optimality. To the best of our knowledge, this is the first solution to the rate scheduling problem in its general form. 1. INTRODUCTION In the past few years, there has been a lot of research on wireless transmission power control. Most of the pioneering research in the area of energy-constrained communication can be coarsely classified into three groups. The first category has focused on transmission schemes to minimize the transmission energy per bit. For example, Verdu discusses some optimal strategies that minimize the energy per bit required for reliable transmission in the wide-band regime [14]. El Gamal et al. propose an optimal scheduling algorithm to minimize transmission energy by maximizing the transmission time for buffered packets [6]. In the second category, research is mainly focused on mitigating the effect of interference that one user can cause to others in order Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. IWCMC 06, July 3 6, 2006, Vancouver, British Columbia, Canada. Copyright 2006 ACM /06/ $5.00. to conserve wireless transmission energy. Instead of directly performing energy conservation, the main approach ranges from obtaining distributed power control algorithms ([7, 15]) to determining the information theoretic capacity achievable under interference limitations [1, 9]. A recent study by Bambos and Kandukuri [2] and also Chockalingam and Zorzi [3] considers minimizing the power subject to a specified amount of information being successfully transmitted and compares the energy efficiency, defined as the ratio of total amount data delivered and total energy consumed, of several medium access protocols. In this paper, we expand on the work in [6, 12, 5] and consider a more generalized problem that captures more characteristics of many real-time applications including wireless mobile ad hoc networks. We attempt to compute an optimal schedule to minimize total transmission energy of transmitting real-time data items, while satisfying all hard time constraints. In order to simplify the presentation, we use wireless real-time video transmission as an example. It should be clear that the results we obtain can easily apply to other problem settings. The following summarizes our main contributions and main differences between our results and prior ones. 1. We consider the rate scheduling problem in its most general setting and propose a simple geometric approach to solve the problem, whereas all previous studies are some special cases of the rate scheduling problem and based on purely algebraic approaches. Our geometric approach is very intuitive and more importantly, can naturally lead to two efficient online algorithms that are more practical in real applications. 2. The running time of the proposed geometric algorithm is O(N) even for the generalized form, where N is the number of packets to be transmitted. This is a significant improvement over the previously known result (which works only for a special case of the rate scheduling problem) with a running time of O(N 2 ). 3. Our problem formulation has broad expressing power for practical scheduling problems in several areas such as real-time multimedia wireless transmission [6, 12], CPU energy conservation, warehouse order processing management, etc. We believe that the proposed geometric method can be applied to other similar optimization problems, and has more profound implications. The remainder of the paper is organized as the follows. In

2 Section 2 we set up the framework for the minimum energy real-time video transmission scheduling problem. Then, in Section 3 we introduce a new geometric approach to model and solve the transmission process. We prove that the generalized optimal rate scheduling problem can be transformed into the problem of finding shortest Euclidean distance between two s inside a polygon. For the perpose of comparison, we also use the proposed method to solve other variations of the rate scheduling problem in Section??. 2. RATE SCHEDULING PROBLEM We consider the down-link channel in a wireless network involving a single transmitter and a single receiver. We assume that N packets arrive to the transmitter node at times t 1, t 2,..., t N, such that 0 = t 1 t 2 t N T. We denote the length of packet i by L i. After receiving packet i (and buffering it if needed) the transmitter starts sending the packet at time s i for a duration of τ i until the finish time f i = s i + τ i. For each packet i we introduce a departure deadline time d i. This is a bound on the finish time of the packet transmission and is used to model such constraints as real-time traffic requirements, and finite buffer sizes. Therefore, a set of pairs (s f i) ( i [1, N]) fully describes a transmission schedule, which we denote by Π. We assume four time constraints for the transmission scheduling: 1. The transmitter node needs to receive a packet before sending it, i.e. i [1, N], s i t i. This reflects the causality time constraint. 2. Each packet must be sent out before its departure deadline, i.e. i [1, N], f i d i. 3. Transmission time has to be physically realistic, i.e. f i > s i. 4. We assume that packets are transmitted in the order they are received (FIFO service model), with at most one packet being sent at any given time. In Appendix A we show that this is the only interesting case to be considered. Definition 1. A transmission rate schedule Π is feasible if and only if the following two conditions are satisfied: 1. i [1, N], 0 t i s i f i d i T ; 2. 0 s 1 < f 1 s 2 < f 2 s N < f N T. The power needed to transmit a unit amount of data is a function of the transmission rate r, and is denoted by w(r). Thus, for a particular sequence of packets, different feasible transmission schedules yield different amounts of total transmission energy. Among all feasible transmission schedules, our objective is to find the optimal one, i.e. the schedule which minimizes the total transmission energy given a channel power function w(r) and time constraints for a particular set of packets. Formally, we can describe the problem as follows. Given: (1) a sequence of packets with size L arrival time t and departure deadline d where i [1, N]; and (2) a wireless channel with power function w(r), find a feasible schedule Π = {(s f i) i [1, N]} that minimizes the total transmission energy. Here we have assumed that the transmission rate is fixed during [s f i]. Later, we will show that this is the only case we need to consider since varying the transmission rate only increases energy needed to transmit the packet. The power function of a wireless channel, w(r), represents how much energy needs to be spent in order to transmit a unit amount of data at a certain rate r. We assume that w(r) satisfies three conditions: w(r) > 0, w(r) is monotonically increasing in r, and w(r) is strictly convex in r. These conditions have been justified by considering several typical coding schemes [5]. A more recent study by Cu et al. [4] presents some experimental data and confirms these assumptions. Given a power function of a wireless channel, the energy required to transmit a packet of size L at data transmission rate r is given by the energy function L w(r). In other words, if the packet transmission starts at time s and finishes at time f, and the transmission rate remains fixed in [f, s], the energy needed to transmit the packet would be L w( L ). f s 3. OPTIMAL OFF-LINE SCHEDULING AL- GORITHM In this section, we present a novel geometric approach to solve the rate scheduling problem. We first introduce a new type of diagram, called RT diagram to represent the arrival, departure, and the lengths of different packets. We prove that the optimal rate scheduling problem is equivalent to finding the shortest Euclidean path between two s inside a two-dimensional polygon in this new setting. This is a classic problem in computational geometry and several fast algorithms exist for solving it. We note that the rate scheduling problem is an off-line problem in the sense that all the packet arrival times and the packet departure deadlines are assumed to be known ahead of time. Let us consider a given sequence of packets where packet i has a size of L starting time t and departure deadline d i (1 i N). We construct a two dimensional diagram to represent the arrival sequence of packets and the corresponding departure deadline sequence, as follows. The x-axis represents the time. The y-axis represents the accumulative amount of data. We have two curves in this graph: first, the accumulative amount of data which has been received up to the present 1 ; and second, the accumulative amount of data that must have been sent out up to the present time (Figure 1). More precisely, we construct the following set of vertices in a two dimensional Euclidean space: i [1, N], (x 2i 1, y 2i 1) = (t i 1 k=1 L k) and (x 2 y 2i) = (t i k=1 L k); i [1, N], (x 2i 1, y 2i 1) = (d i 1 k=1 L k) and (x 2 y 2i) = (d i k=1 L k); We connect the vertices in the two sequences above, i.e. (x 1, y 1),... (x 2N, y 2N ) and also (x 1, y 1),..., (x 2N, y 2N ). We also connect the heads and tails of the too sequences, i.e. the (x 1, y 1) is connected to (x 1, y 1) and the (x 2N, y 2N ) is connected to (x 2N, y 2N ). The resulting shape is a polygon, which we denote by P. We call this graph the 1 Here, we assume that packets are received instantaneously. This assumption can be easily relaxed as we will see in Section 4.2.

3 Rate- diagram or RT diagram for short. We also call the s (x 1, y 1), and (x 2N, y 2N ) the starting and ending s of the RT diagram respectively. The polygon P defined here can be seen as a simple twodimensional transformation of the conventional one-dimensional time digram. Figure 1 illustrates a simple example of how to construct an RT diagram. Since packet sizes are represented in y-axis, data transmission rate can be explicitly represented by the slope of a line segment. t 1 t 2 d 1 t 3 d 2 d 3 t 4 Figure 2: A feasible schedule can be represented as a path inside RT-diagram. We only need to consider piece-wise linear paths as candidates of the optimal schedule. In this graph, the dashed line cannot represent the optimal schedule. t 1 d 1 t 2 t 3 d 2 d 3 t 4 Arrival times Departure Deadlines Figure 1: A Simple Example of Rate- Diagram. Let us consider a transmission schedule Π, and plot the accumulative amount of data transmitted as a function of time on the RT diagram. It is easy to verify that if Π is a feasible transmission schedule, then the curve corresponding to Π lies inside the polygon P, and connects the starting of the RT diagram to its ending. Conversely, any continuous simple curve C connecting the starting and ending s of the RT diagram that lies inside P is a feasible schedule if and only if at any (x, y) along C, the tangent of C is non-negative, i.e., data rate at time x is non-negative. Next, we prove that among all feasible schedules, the one corresponding to the shortest path between the starting and ending s of the RT diagram is the optimal one with minimum total transmission energy. This is a very interesting result, and makes the solution to the transmission rate scheduling problem very intuitive, and easy to obtain. We prove the main result in two steps. First, we show that among all feasible schedules that can be represented inside P, only those that are piece-wise linear should be considered as possible candidates of an optimal scheduling solution. Then, we show that among the schedules which can be represented by a piece-wise linear curve inside P, the one with the shortest length consumes the least amount of energy. Since there is a unique curve representing any feasible schedule, from now on, we will use the curve representing a schedule, and the schedule itself interchangeably. Lemma 1. We partition the time-line [0, T ] into 2N 1 segments and denoted the i-th interval by ξ i (1 i 2N 1). Any two consecutive intervals are separated by an arrival or departure (Figure 2). If Π is an optimal transmission schedule, then during any interval ξ the curve corresponding to Π is a straight line connecting its two ends in ξ i (1 i 2N 1). Proof. During each interval ξ there is no packet arrival and departure. Let (a b i) and (a i+1, b i+1) denote two ends of ξ i. To simplify notation, let l be b i+1 b i and τ be a i+1 a i. If the system sends at a fixed rate during this time interval, the transmission rate would be l/τ. We denote the actual transmission rate at time t [0, τ] by R(t) (using a shift in time). We need to show that the energy consumed by transmitting packets at a fixed rate is less than the energy consumed by any other schedule, i.e., τ 0 w(r(t)) R(t)dt l w(l/τ), (1) where, (1) R(t) 0, for any t [0, τ]; (2) w(r) is the channel transmission function which is positive, monotonically increasing, and convex; and (3) τ R(t)dt = l. 0 We note that w(r) r is a convex function of r. This is because w(r) is a non-negative, convex and increasing function of r. Now, we consider t as a random variable drawn according to the uniform distribution U(0, τ), and we let Y denote a random variable defined by Y = R(t). Due to basic probabilistic analysis, we know E[Y ] = τ R(t) 1 dt = l. 0 τ τ and E[w(Y ) Y ] = E[w(R(t)) R(t)] = τ w(r(t)) R(t) 0 1 dt = 1 τ w(r(t)) R(t)dt. Moreover, w(e[y ]) E[Y ] = τ τ 0 w( l ) l. Using Jensen s inequality, we have E[w(Y ) Y ] τ τ w(e[y ]) E[Y ], which completes the proof. We are ready to state the main theorem of this section. Theorem 1. Among all feasible schedules for an instance of the scheduling problem, the one corresponding to the shortest path that connects the starting and ending s of the RT diagram, has the minimum energy consumption. The proof of this theorem comes in Appendix B. Theorem 1 shows that the optimal schedule corresponds to the shortest feasible distance between the starting and ending of the RT diagram. If we consider the edges of the constraint polygon as walls, and any feasible schedule as a rope connecting the starting and ending s, we can find the optimal schedule by tightening the rope as much as possible (which yields the shortest length). Fortunately, this problem, i.e. finding the shortest Euclidean path between two s inside a polygon, is a well-studied problem in computational geometry and several algorithms have been proposed to solve the problem [8]. The algorithm proposed by Lee and Preparata [10] finds the shortest path between two s inside a simple polygon in linear time, once a triangulation is known. In 1986, Tarjan and Van Wyk

4 [13] have developed a linear-time algorithm for triangulating simple polygons. Therefore, the shortest path problem can be solved in two stages and with total computational complexity of O(N) where N is the total number of polygon edges. Therefore, the complexity of finding the optimal schedule is O(N) as well. To the best of our knowledge, this is the first solution to the rate scheduling problem in its general form. One reason might be the fact that solving this problem algebraically is rather difficult. However, as we have just seen, using RT diagrams makes the problem very easy to understand and solve. Figure 3 illustrate an example of the problem in this setting and the optimal scheduling solution. t 1 t 2 t 3 t 4 Figure 4: An instance of the optimal rate scheduling problem where packet have different arrival times, and the same departure deadline. d s2 t 1 t 2 d 1 t 3 f 2 d 2 d 3 t 4 t 1 t 2 d 1 t 3 d 2 d 3 t 4 Figure 3: The optimal scheduling problem where packets have different arrival times and different departure deadlines. We can find the starting (ending) time of transmission of each packet, by intersecting the rate curve, with the horizontal line corresponding to the start (end) of the packet length. 4. SPECIAL CASES AND EXTENSIONS OF THE OFF-LINE SCHEDULING PROBLEM In the proceeding, we present two examples of how powerful this technique is, by showing how it can be applied to the special cases studied before. We also show how we can extend our technique to solve problems in a more general setting. 4.1 Scheduling with the same departure deadline Scheduling N packets with various arrival times and the same departure deadline is a special case of the optimal rate scheduling problem. In this case for all i [1, N], we have d i = T. This problem has been studied by El Gamal et al. [6, 12]. They give an algebraic solution for the optimal schedule, which can be computed in O(N 2 ). Here is the formal description of the problem: Given 1. a sequence of packets with size L starting time t where i [1, N]; 2. a common departure deadline D for all packets; and 3. a wireless channel with power function w(r). Find A feasible schedule Π = {(s f i) i [1, N]} that minimizes the total transmission energy N i=1 Li w( L i f i s i ). Figure 4 illustrates the RT diagram for an instance of the rate scheduling problem with the same departure deadline, along with the corresponding optimal schedule. We can see that finding optimal solution in this case is equivalent to Figure 5: Extension to the off-line scheduling problem when arrival data rate and departure data rate are finite. finding a convex hull of a given set of s. This example shows how using the RT diagrams and the main theorem of this paper can simplify solving these kind of scheduling problems. We also note that using RT diagrams reduces the computational complexity of finding the optimal schedule from O(N 2 ) to O(N). 4.2 Extension to the off-line scheduling problem Throughout the present paper, and in previous work on the rate scheduling problem ([6, 12]), it has always been assumed that packets become available and depart instantaneously (i.e. with infinite rate). This, certainly, is not true in practice. A realistic model would be to assume packets are sent/received with a certain finite rate; which means their departure/arrival takes some time. This behavior is captured in Figure 5 by drawing sloped lines for arrivals and departures. Interestingly, after we construct the corresponding RT diagram, we can easily verify that all the conclusions drawn before are still valid. Therefore, the optimal scheduling problem can still be solved by locating the shortest path inside the polygon P which connects the starting and ending s of the RT diagram. For this general type of problem, it is imaginable that the close form solution is very difficult to obtain through a purely algebraic method. We also note that the computational complexity of this algorithm is still O(N). 5. CONCLUSION We studied an generalized version of the optimal rate

5 scheduling problem 2 and developed a novel geometric algorithm which is not only intuitive to understand but also achieves O(N) computational complexity. Our result is a significant improvement over previous results in generality and solution complexity. Additionally, we proved that if transmission of data items can be reordered, the optimal scheduling problem becomes NP-hard and therefore indicated for the first time the fundamental performance limit of this kind of optimal scheduling problem. Finally, our problem formulation captures the balance trade-off between processing rate and processing cost in several categories of practical scheduling problems, e.g., real-time multimedia wireless transmission, CPU energy conservation, and warehouse order processing management, etc. We believe that our proposed scheduling algorithms can be applied to other similar optimization problems, and has more profound implications. 6. ACKNOWLEDGEMENT We would like to thank Mohsen Bayat Abtin Keshavarzian, and Vahbod Pourahmad for their useful comments and discussions. 7. REFERENCES [1] N. Bambos. Toward power-sensitive network architectures in wireless communications. IEEE Personal Commun., 5:5059, June [2] N. Bambos and S. Kandukuri. Power control multiple access (pcma). Wireless Networks, [3] A. Chockalingam and M. Zorzi. Energy efficiency of media access pro-tocols for mobile data networks. IEEE Trans. Commun., 46: , Nov [4] S. Cu, A. J. Goldsmith, and A. Bahai. Energy-constrained modulation optimization for both uncoded and coded systems. Submitted to IEEE Trans. on Wireless Communications. [5] A. El Gamal, C. Nair, B. Prabhakar, E. Uysal-Biyikoglu, and S. Zahedi. Energy-efficient scheduling of packet transmissions over wireless networks. In Proceedings of the IEEE Infocom, volume 3, pages , June [6] A. El Gamal, C. Nair, B. Prabhakar, E. Uysal-Biyikoglu, and S. Zahedi. Energy-efficient scheduling of packet transmissions over wireless networks. In Proc. IEEE Infocom, volume 3, pages , June [7] A. Goldsmith. Capacity and dynamic resource allocation in broadcast fading channel. In 33rd Annu. Allerton Conf. Communication, Control and Computing, page [8] L. J. Guibas and J. Hershberger. Optimal shortest path queries in a simple polygon. In Proceedings of the third annual symposium on Computational geometry, pages 50 63, [9] S. Hanly and D. Tse. Power control and capacity of spread-spectrum wireless networks. Automat., 35(12): , Due to page limit, we focused on the off-line version and omitted all discussion of the on-line rate scheduling problem, interested reader can refer to [11] for more details on this. [10] D. T. Lee and F. Preparata. Euclidean shortest paths in the presence of rectilinear barriers. Networks, 14: , [11] M. Lin and Y. Ganjali. Power-efficient rate scheduling in wireless links using computational geometric algorithms. In mingjie/lin06.pdf, [12] B. Prabhakar, E. Uysal-Biyikoglu, and A. El Gamal. Energy-efficient packet transmission over a wireless link. IEEE/ACM Transactions on Networking, 10(4): , August [13] R. E. Tarjan and C. J. V. Wyk. A linear-time algorithm for triangulating simple polygons. In Proceedings of the eighteenth annual ACM symposium on Theory of computing, pages , [14] S. Verdu. Spectral efficiency in the wideband regime. IEEE Trans. Information Theory, 48: , June [15] P. Viswanath, V. Anantharam, and D. Tse. Optimal sequences, power control and capacity of synchronous cdma systems with linear mmse multiuser receivers. IEEE Trans. Inform. Theory, 45: , Sept APPENDIX A. PACKET TRANSMISSION ORDER Theorem 2. When reordering is allowed, optimal rate scheduling problem is NP-hard in the strong sense. We first devise a special case of this general scheduling problem, and show that resulted scheduling problem can be reduced to a well-known NP-hard problem 2-bin packing; hence, the above theorem 2 is proved. Suppose we have 2k + 1 packets denoted by {p i 1 i (2k + 1)} to schedule. The arrival time of packet i is t its departure deadline is d i and the length of the packet is L i. Let us assume that our packets are such that L k+1 L i for any 1 i 2k + 1 and i k + 1; for any 1 i 2k + 1 and i k + 1, s i equals 0, and t i equals T ; and s k+1 equals T/3 and t k+1 equals 2T/3. Since the length of the packet p k+1 is much larger than any of the other 2k jobs, we can easily show that in the optimal schedule we should allocate all its allowed time window for its transmission. Now, the problem becomes to partition all other 2k jobs into two time windows of W 1 = [0, T/3] and W 2 = [2T/3, T ]. To gain the least transmission energy, we must balance the packets transmitted during W 1 and W 2. In other words, we need to minimize the difference between the total length of the packets transmitted during W 1 and W 2. This is exactly the 2-bin packing problem, which we know is NP-complete. Therefore, the original problem is NP-hard in the strong sense. B. PROOF OF THE MAIN THEOREM Proof. Based on Lemma 1, we only need to consider piece-wise linear schedules. Let Π M denote a feasible schedule in RT diagram, where M is the number of segments, Π M = {π i i {1, M}}, and each π i represents the i-th line

6 segment in Π M. Also, let L(Π M ) and E(Π M ) denote total length and total transmission energy of schedule Π M in the RT diagram. We use mathematical induction on the number of line segments in Π M to prove the theorem. (1) Let us consider a case where M = 1. Lemma 1 implies the theorem in this case since the shortest distance between the starting and ending s of the RT diagram (two vertices of polygon P, which is simply a rectangle) is the straight line connecting them. (2) We assume for any M < k, Theorem 1 is true, i.e., for any other piecewise-linear schedule Π M, having L(Π M ) L(Π M ) implies E(Π ) E(Π M ). (3) Now, we need to show that the theorem is also true for M = k. We prove this by contradiction. Let Π denote the schedule with the shortest length in the RT diagram. We need to prove that E(Π ) is the minimum among all E(Π). Suppose this is NOT true, i.e., there exists a schedule Π with longer length than Π in RT diagram but with the minimum total transmission energy. We show this is not possible. We consider two cases: Case (2): Π and Π do not intersect (other than at the two ends of the polygon P ). This situation is depicted in Figure 7. Without loss of generality, we assume schedule Π is above Π in the RT diagram. Let π 1, π 2,, π M denote all consecutive line segments in schedule Π. We claim that slope(π 1) slope(π 2) slope(π M ). (4) π ι π ι+1 (a) Figure 8: Proof of the claim in Step 2. π ι (b) π ι+1 Π 2 Π 2 Π Π 1 L Π 1 t L Π Figure 6: Case 1 of the poof of Theorem 1. Case (1): Π and Π have at least one intersection, as depicted in Figure 6. Since L(Π 1 + Π 2) > L(Π 1 + Π 2), we either have L(Π 1) > L(Π 1), or L(Π 2) > Π 2). (2). Without loss of generality let us assume inequality 2 holds. Due to the inductive hypothesis (step 2 of our proof), we have E(Π 1) > E(Π 1). Therefore, E(Π 1 + Π 2) > L(Π 1 + Π 2). (3) which shows that Π can not be optimal, a contradiction. L t Π Π t Figure 9: The last line segments of Π and Π can not join. This is true because otherwise we have at least one pair of consecutive line segments π i and π i+1, where slope(π i) slope(π i+1) as shown in Figure8(a). Since Π is below Π, we always can lower x for a very small amount, which reduces L(π i) + L(π i+1) and therefore, E(Π i) + E(Π i+1) is reduced (based on Lemma 1 and induction Step 2). This is clearly not possible, therefore the claim is true. The exactly same argument can be applied to Π, which proves the following. slope(π 1) slope(π 2) slope(π M ) (5) Taking a closer look at equation 4 and 5 and the RT diagram we can see that at least one of Π and Π is not a feasible schedule since the Π and Π cannot join at the end (Figure 9). A similar argument can be applied to the case when schedule Π is above Π in RT diagram and still leads to a contradiction. In summary, due to the contradiction, there can not be a feasible schedule Π that satisfies both of conditions: L(Π M ) > L(Π M ) and E(Π M ) E(Π M ). Therefore, the proof is completed. Figure 7: Case 2 of the poof of Theorem 1.