Power Controlled Random Access

Transcription

1 1 Power Controlled Random Access Aditya Dua Department of Electrical Engineering Stanford University Stanford, CA Abstract The lack of an established infrastructure, and the vagaries of the wireless channel make the design of ad-hoc networks a challenging task. Cross-layer design for ad hoc networks has been widely advocated, because it helps overcome several sub-optimalities which are introduced by designing each layer in isolation. A cross-layer design encompassing all layers of the protocol stack is hard to achieve, and hence research often looks at pairwise optimizations. One such design methodology for joint link scheduling and power control, which optimizes over the physical and MAC layer is considered here, for a cellular network scenario. By using a heuristic relaxation, the NP-hard problem of joint power control and scheduling is transformed into one of power controlled random access, which can be cast as a convex optimization problem. A physical interpretation is provided for this relaxation. An outage analysis and a Markov chain based endto-end delay distribution analysis is presented for the proposed scheme, and the efficacy of the scheme is also studied through simulations. I. INTRODUCTION An ad-hoc wireless network is a collection of wireless nodes that self configure to form a network without the aid of any established infrastructure. The lack of an established infrastructure and the hostile nature of the wireless channel make the design of ad hoc networks a challenging task. The philosophy of cross-layer design for ad hoc networks has been strongly advocated in research [1], [2]. However, optimizing across all layers of the protocol stack is a hard task to accomplish, and research often focusses on pairwise optimizations. One such pairwise optimization, namely joint link scheduling and power control is investigated here (physical

2 2 and MAC layer), in the context of a cellular network. The formulation can easily be extended to an ad hoc network scenario. Transmit power control has been extensively studied, both for cellular networks and ad hoc networks [3], [4]. The objectives of power control are to help mitigate multiple access interference (MAI), and to help conserve battery power, which is a precious resource for wireless nodes. Several centralized and distributed power control schemes have been proposed in literature [3], [4], [5]. It has been shown that transmit power control can enhance both communication capacity and lifetime of ad hoc networks [6]. Multiple access schemes can be classified into two broad classes: random access (e.g. ALOHA) and link scheduling. The latter is of interest because MAI can be mitigated by regulating access to the shared wireless channel. Producing optimal schedules however is an NP-complete problem [7], and the focus has been heuristic algorithms [8]. The motivation behind joint link scheduling and power control is to eliminate some of the interference via design of good link schedules, which then reduces the burden on the power control algorithm. Only a few results are available on the problem of joint link scheduling and power control. In [9], this problem is solved in a centralized way via two alternating phases, that respectively search for an admissible set of users, and their transmit powers. A distributed solution to this problem is presented in [10], where the problem of optimally emptying the buffer is formulated in a dynamic programming framework. In [11], joint scheduling and power control problem (along with routing) has been studied for one-dimensional multihop networks. The paper is organized as follows: In Sec. II the system model is discussed and problem of joint link scheduling and power control is posed. In Sec. III, a heuristic relaxation is introduced, which converts the problem into one of power controlled random access (PCRA). This problem is in the form a geometric program, and can be easily cast into a convex optimization problem [12]. Some variants of the original problem are also presented. In Sec. IV, the probability of outage is derived for AWGN and Rayleigh faded channels, and a Markov chain based end-to-end delay distribution analysis is presented. Simulation results are presented in Sec. V, and the paper concludes in Sec. VI.

3 3 II. SYSTEM MODEL Let us consider an uplink CDMA scenario with N wireless nodes communicating with a base station, using direct-sequence spread-spectrum for multiple access. The base station uses a matched-filter type demodulator. The system is assumed to be time slotted, so that a new transmission can commence only at the beginning of a slot. A packet is assumed to be received without error if the signal-to-interference plus noise ratio is above a target SINR for the duration of the slot in which it is transmitted. Each node has a peak power constraint P 0. Let P i denote the transmit power of the i th node, such that 0 P i P 0. Let δ i be a indicator variable defined as 1 ; i th node active δ i = 0 ; else. (1) Let G i denote the gain from the i th node to the base station. The gain can incorporate effects such as path loss, shadowing, multipath fading, antenna gain, coding gain etc. Let σ 2 denote the additive white Gaussian noise (AWGN) power at the base station. If the i th node is active (δ i = 1), its SINR γ i is given by γ i = P i G i, (2) N P j G j ρ 2 jiδ j + σ 2 j=1 j i where ρ ji is the cross-correlation between the spreading sequences of the i th and j th nodes. Define N 1 vectors, P = [P 1,..., P N ] T and δ = [δ 1,..., δ N ] T, such that 0 P P 0 and δ {0, 1} N. The problem of joint power control and scheduling is to determine in each transmission slot the feasible set of nodes that can transmit, i.e., the vector δ, and their corresponding transmit powers, i.e., the vector P. This choice is subject to the SINR of all active nodes being above the target SINR. The NP-hard nature of the problem is evident, and can be attributed to the fact that the δ i s are binary valued.

4 4 III. PROBLEM FORMULATION To make the problem more tractable, we relax the binary valued constraint on the variables δ i. In particular, we replace the discrete variable δ i {0, 1} by a continuous variable i [0, 1]. A physical interpretation can be associated with this relaxation. The variable i can be thought of as the probability of the i th node being active in a transmission slot. Thus, δ i can be treated as a Bernoulli random variable with parameter i. We can now define the notion of expected SINR for active links, as the ratio of received signal power to the expected interference plus noise power. Thus, if the i th node is active, its expected SINR γ i is given by γ i = P i G i. (3) N P j G j ρ 2 ji j + σ 2 j=1 j i Thus, by a relaxation on the binary valued variables δ i, the problem of joint power control and scheduling is transformed into one of power controlled random access (PCRA), where the objective is to determine for each node the probability of transmission and the transmit power. The minimum acceptable SINR criterion for error free packet reception is now replaced by a minimum expected SINR criterion, i.e. γ i. We now aim to express the problem of power controlled random access as a convex optimization problem. Since the primary goal here is power control, the objective function of the convex optimization problem should involve some metric of the transmit power. We choose the l -norm as a metric, which is a measure of the maximum power spent by any node in a slot. It is easy to see that merely optimizing such an objective subject to SINR constraints would drive all transmission probabilities to 0, which is a trivial solution. Hence, we incorporate the transmission probabilities in the objective function by maximizing min i i, or alternatively minimizing max i 1/ i (l -norm). This constraint encourages each node to be active with a high probability, while the power constraint discourages a node from being active when the interference on the channel is high (other nodes are active with high probability). Clearly, the well known power v/s delay (or throughput) dilemma for multiple access networks is captured by this problem formulation [10].

5 5 We can now formulate a scalarized version of the multi-criterion optimization problem as follows minimize P + λ 1 subject to 0 P P 0 (4) 0 1 γ i, 1 i N, where λ is a scalarization parameter and = [ 1,..., N ] T. Varying λ will generate a family of solutions, which characterize the Pareto-optimal surface for the scalarized problem. Thus, λ quantifies the tradeoff between two competing objective functions. In particular, it captures the power v/s delay (or throughput) tradeoff for multiple access networks. By introducing dummy variables t P and t, the problem in (4) can be converted to the following form minimize t P + λt 1 P i t 1 P subject to 1, 1 i N t P P (5) where γ 1 i t 1 i = P 1 i G 1 i 1, 1 i N i 1, 1 i N γ 1 i 1, 1 i N, N j=1 j i P j G j ρ 2 ji j + σ 2 P 1 i G 1 i (6) is a posynomial, and the formulation in (5) is in the form of a Geometric program (GP) [12]. Positivity

6 6 constraints are implicit in a GP. We now define the (2N + 2) 1 vector x = [x 1,..., x 2N+2 ] T such that log(p i ) 1 i N x i = log( i ) N + 1 i 2N log(t P ) i = 2N + 1 log(t ) i = 2N + 2, (7) where log( ) denotes the natural logarithm. Combining (7) with (5), and taking logarithm of the objective function and constraints, we get the following convex optimization problem minimize log (exp(x 2N+1 ) + λexp( x 2N+2 )) subject to x i x 2N+1 0, 1 i N x 2N+1 ln(p 0 ) 0 x 2N+2 x i 0, N + 1 i 2N (8) x i 0, N + 1 i 2N N G j ρ 2 ji log exp(x j + x j+n ) + σ 2 x i 0, 1 i N, G i j=1 j i where exp( ) denotes the exponential function. The convexity of the problem follows from the convexity of the affine and log-sum-exp functions. Let us call the problem in (4) PCRA-1. Within the above framework, we can formulate three more variants of PCRA-1. The motivation behind studying these variants is to investigate the effect of reducing the degrees of freedom on the efficacy of PCRA-1. In the first variant, PCRA-2, we allow nodes to have different transmit power levels, but force them to have G i

7 7 a common probability of transmission. Thus, PCRA-2 can be formulated as minimize P + λ subject to 0 P P 0 (9) 0 1 γ i, 1 i N, where is a scalar, and in the definition of γ i in (3) we set i = i. Another variant we consider is PCRA-3, where the nodes are allowed to transmit with different probabilities, but are forced to transmit at a common power level. PCRA-3 can be formulated as minimize P + λ 1 subject to 0 P P 0 (10) 0 1 γ i, 1 i N, where P is a scalar, and in the definition of γ i in (3) we set P i = P i. In yet another variant, PCRA-4, all nodes are assigned a common power level and transmission probability. PCRA-4 can be formulated as minimize P + λ subject to 0 P P 0 (11) 0 1 γ i, 1 i N, where P and are scalars, and in the definition of γ i in (3) we set P i = P, i = i. PCRA-2,3 and 4 can be easily cast into geometric programs, and hence into convex optimization problems by following a procedure similar to that for PCRA-1.

8 8 IV. OUTAGE AND DELAY ANALYSIS The special structure of the solution obtained for PCRA-1 (through simulations) makes it amenable to outage and delay analysis. We assume that all pairs of spreading codes have identical cross-correlations, i.e., ρ ij = ρ i, j, i j. Based on simulation results in Sec. V, we make three key observations. The first observation is that PCRA-1 yields equal received power at the base station for all nodes, i.e., P i G i = c (a constant). The second observation is that all nodes are assigned equal probabilities of transmission, i.e., i = i. The third observation is that at the optimal point, all nodes have an expected SINR equal to the target SINR, i.e., γ i = i. Now, based on the three observations and from our definition of expected SINR (3) we have = c (N 1)cρ 2 + σ 2. (12) A. Probability of Outage A link is said to be in outage if the received SINR on the link falls below the target SINR level. We first consider the probability of outage for an AWGN channel, i.e., when the gains G i are constant. We denote by P out (i k) the probability of outage on the i th link, conditioned on the presence of k actively interfering nodes. Since all nodes have equal received power at the base station, and transmit with the same probability, we expect the probability of outage to be identical on all links. If k interfering links are active, the received SINR for the i th link, γ i k is and P out (i k) = 1 P (γ i k γ i k = c kcρ 2 + σ 2, (13) ). Using (12), this simplifies to P out (i k) = 1 P (k < (N 1) ). Under heavy traffic conditions, i.e., when every node has a packet to transmit in each slot, k has a binomial distribution. Thus, the unconditioned probability of outage is P out = N 1 k= (N 1) ( ) N 1 k (1 ) N 1 k. (14) k We now consider the probability of outage in a Rayleigh faded channel. The received power on each link has an exponential distribution, with mean value P i G i (= c). We replace P i G i by E i, where E i is

9 9 exponentially distributed with mean c. The expected SINR on the i th link, γ (fade) i is γ (fade) i = ρ 2 E i. (15) N E j + σ 2 Now, the probability of outage conditioned on the presence of k actively interfering nodes is given by P out (i k) = 1 P ( γ (fade) i j=1 j i ), which can be expressed as P out (i k) = 1 P E = E j E i σ2 ρ 2 γ j S 0 ρ 2, (16) j i where S denotes the set of active interfering nodes, and S = k. All E j s (j S) are i.i.d. exponential random variables with mean c, and the probability density of E ( ) i is given by f(y) = ρ2 ρ exp 2 y u( y), ρ 2 c c where u(y) is the unit step function. Thus, E is a sum of (k + 1) independent random variables. The Laplace transform of the probability density of E, denoted by LE(s), is given by the product of the Laplace transforms of the individual probability densities. Thus, we have ( ) ( ) k 1 1 LE(s) = 1 + cs 1 c. (17) ρ 2 s Using (17), for y 0, we get the probability density of E as ( ) k ( ) 1 ρ 2 ρ 2 f E (y) = 1 + ρ 2 c exp c y, y 0. (18) From (16) and (18) we get the conditional probability of outage as ( ) k ( ) 1 P out (i k) = 1 exp σ2. (19) 1 + ρ 2 c Since k has a binomial distribution, we can uncondition to obtain the probability of outage P out as B. Delay Analysis ( ) ( ) N 1 P out = 1 exp σ2 1 ρ2. (20) c 1 + ρ 2 We can now model the buffer at each node as a Markov chain, using the number of backlogged packets as the state of the Markov chain. We assume that each node generates a packet in each slot with a probability p.

10 10 Since all links have the same outage probability, the Markov chains associated with each node are statistically identical. If a node is in state n, in the subsequent transmission slot it can either go to state (n 1) or (n + 1), or remain in state n. If n = 0, the node can go to state 1, or remain in state 0. An upward transition (n to n + 1) can occur if a new packet is generated and either no transmission is attempted or an attempted transmission fails. A downward transition (n to n 1) occurs if no new packet is generated and an attempted transmission is successful. No transition (n to n) occurs if no new packet is generated and no transmission is attempted or an attempted transmission fails, or if a packet is generated and an attempted transmission is successful. Let q denote the probability of a failed transmission. Under heavy traffic conditions, we have q = P out (20). Let µ ij denote the probability of transition from state i to state j in a given slot. Based on the foregoing discussion, we have p(1 ) + p q j = i + 1, i 0 µ ij = (1 p)(q + 1 ) + p(1 q) j = i, i > 0 (1 p)(1 q) j = i 1, i > 0 (1 p) + p (1 q) j = i = 0. (21) The state transition matrix for such a system would be semi-infinite and have a tri-diagonal form. By solving the balance equations, the steady state distribution π = [π 1, π 2,...] can be obtained, which in turn can be used to characterize the delay distribution. In steady state, a newly generated packet sees k backlogged packets ahead of it in the queue with probability π k. The number of transmission slots required for successful transmission has a geometric distribution with parameter (1 q). Given a new packet that is generated when the system is in state k, its total delay (queueing and transmission) is a sum of (k + 1) i.i.d geometric random variables, which is a negative binomial random variable. Thus, the conditional delay distribution P d (n k) is given by ( ) n 1 P d (n k) = [1 (1 q)] n k 1 [ (1 q)] k+1, n k + 1. (22) k

11 11 From the steady state distribution, the delay distribution P d (n), i.e., the probability that a packet experiences an end-to-end delay of n slots, can be obtained as P d (n) = k P d (n k)π k, n k + 1. (23) V. SIMULATION RESULTS The simulation scenario comprises of N = 8 wireless communicating with a base station, using spreading codes with cross-correlation ρ = 1/15 for multiple access. The peak power constraint is set at P 0 = 1. The gains G i are kept constant for the simulation runs. It is observed through simulations that PCRA-1 yields equal probabilities of transmission for all nodes, i.e., it degenerates to PCRA-2. Thus, the outage and delay analysis presented in Sec. IV for PCRA-1 is applicable to PCRA-2. Also, the performance of PCRA-3 and PCRA-4 is seen to be very similar. PCRA-3 yields equal probabilities of transmission for all nodes, except for the one with lowest gain. We focus on the performance of PCRA-2 and PCRA-4 here. Fig. 1 depicts the Pareto-optimal curve for PCRA-2 (obtained by varying λ), for different target SINR levels. Different points on the curve depict a tradeoff between the transmit power and probability of transmission (power v/s delay tradeoff). Fig. 2 depicts the Pareto-optimal curve for PCRA-4. Fig. 3 depicts the mean buffer occupancy versus offered load (or probability of packet generation p), for different λ (or equivalently ), for an SINR target of = 12. It is interesting to note that the maximum offered load the system can cope with does not increase monotonically with the probability of transmission. High probability of transmission implies stronger contention for shared wireless channel, which hurts system throughput. Fig. 4 depicts the performance of PCRA-4. It can be seen that PCRA-4 is stable over a larger range of offered loads, compared to PCRA-2. For the chosen scenario, PCRA-2 and PCRA-4 are stable for p 0.5 and p 0.7 respectively. However, this comes at the cost of higher transmit power. For example, with = 0.5 and p = 0.25, PCRA-4 requires 2 db more power than PCRA-2, for a target SINR = 12. As another example, with = 0.8 and p = 0.3, PCRA-4 requires 3.3 db more power than PCRA-2. However, PCRA-4 has a more parsimonious

12 12 description and a lower computational complexity than PCRA-2. The plots in Fig. 3 and 4 were obtained by simulating PCRA-2 and PCRA-4 respectively over 20,000 transmission slots. VI. CONCLUSIONS The problem of joint power control and scheduling for the uplink of a DS-CDMA based cellular wireless network is considered in this paper. Based on a heuristic relaxation, the NP-hard problem is cast into a geometric program, which in turn can be formulated as a convex optimization problem. The problem formulation clearly captures the power v/s delay tradeoff associated with wireless multiple access networks. The formulation can easily be extended to an ad-hoc network scenario. Owing to the availability of computationally inexpensive algorithms for solving convex optimization problems in literature, the proposed schemes are amenable to real-time implementation. Distributed implementations of the proposed schemes are subject to further research. REFERENCES [1] A. Goldsmith, S.B. Wicker, Design challenges for energy-constrained ad-hoc wireless networks, IEEE Wireless Communications, vol. 9, no. 4, pp. 8-27, Aug [2] W. Stark, H. Wang, A. Worthen, S. Lafortune, D. Teneketzis, Low-energy wireless communication network design, IEEE Wireless Communications, vol. 9, no. 4, pp , Aug [3] G. Foschini, Z. Miljanic, A simple distributed autonomous power control algorithm and its convergence, IEEE Trans. Veh. Technol., vol. 40, pp , [4] N. Bambos, Toward power-sensitive network architectures in wireless communications: concepts, issues, and design aspects, IEEE Personal Communications, vol. 5, pp , Jun [5] J.P. Monks, V. Bhargavan, W.W. Hwu, A power controlled multiple access protocol for wireless packet networks, Proc. IEEE INFOCOM 2001, pp , Apr [6] J.P. Monks, J.P. Ebert, A. Wolisz, W.W. Hwu, A study of the energy saving and capacity improvement potential of power control in multi-hop wireless networks, Proc. IEEE Conf. LCN 2001, Nov [7] S. Ramanathan, E.L. Lloyd, Scheduling algorithms for multi-hop radio networks, IEEE ACM Trans. Networking, vol. 1, no. 2, Apr [8] E.L. Llyod, S. Ramanathan, Efficient distributed algorithms for channel assignment in multihop radio networks, Journal of High Speed Networks, 2: , [9] T. Elbatt, A. Ephremides, Joint scheduling and power control for wireless ad-hoc networks, Proc. IEEE INFOCOM 2002, pp , Jun

13 13 [10] N. Bambos, S. Kandukuri, Power-controlled multiple access schemes for next-generation wireless packet networks, IEEE Wireless Commun., vol. 9, no. 3, pp , Jun [11] B. Radunovic, J.Y. Le Boudec, Joint scheduling, power control and routing in symmetric, one-dimensional, multi-hop wireless networks, Technical Report IC/2002/84, EPFL, Lausanne, Switzerland, Dec [12] S. Boyd, L. Vandenberghe, Convex optimization and its engineering applications, Stanford University EE 364 course reader =12 =14 =16 = max(p i ) / Fig. 1. Pareto optimal curve for PCRA =12 =14 =16 = Transmit Power (P) / Fig. 2. Pareto optimal curve for PCRA-4

14 Mean buffer occupancy λ=0.05 ( =0.4532) λ=0.1 ( =0.5930) λ=0.2 ( =0.7753) λ=0.3 ( =0.8915) λ=0.4 ( =0.9791) = Offered load (p) Fig. 3. Mean buffer occupancy v/s offered load for PCRA λ=0.05 ( =0.3306) λ=0.1 ( =0.4302) λ=0.2 ( =0.5466) λ=0.5 ( =0.7192) λ=2.0 ( =0.8666) =12 Average buffer occupancy Offered load (p) Fig. 4. Mean buffer occupancy v/s offered load for PCRA-4