Analysis of Power Management for Energy and Delay Trade-off in a WLAN

Transcription

1 Analysis of Power Management for Energy and Delay Trade-off in a WLAN Mahasweta Sarkar and Rene.L Cruz Department of Electrical and Computer Engineering University of California at San Diego La Jolla, California msarkar,cruz@ece.ucsd.edu Abstract Energy efficient operation is of paramount importance for battery-powered wireless nodes. In an effort to conserve energy, standard protocols for WLANs [1] have the provision for wireless nodes to sleep periodically. In this paper we consider the problem of optimizing the timing and duration of sleep states for a wireless node fed by a stream of packets with the objective of minimizing power consumption with respect to a QoS constraint. The QoS parameter we have focused on is average delay. We assume a memoryless Bernoulli model for the arriving packets as well as fixed energy cost for waking up and going to sleep Using a dynamic programming formulation, coupled with a duality argument, we solved the optimization problem numerically. The solutions from the numerical calculations strongly suggest that the optimal policy (that which minimizes average power consumption subject to an average delay constraint) is such that the wireless node should transit to the sleep state only when there are no data packets remaining for it to serve. Using a branching process analysis, we were able to derive closed form expressions for the optimal sleep duration, as well as the associated minimal rate of power consumption. Keywords: power management, WLAN, sleep-wakeup policy. I. INTRODUCTION There are numerous existing and upcoming wireless applications (e.g based WLANS, sensor networks, ad hoc networks, RFIDs, paging systems) that require a number of mostly-dormant wireless nodes to be communicating with each other or with a central base station, intermittently. Since these nodes are battery powered, they are severely energy constrained. Hence in order to save energy and thereby prolong the system lifetime, it makes sense for these nodes to transit to a sleep state when they are not communicating with their peers. Studies have shown [] that being in an idle state (instead of being in the sleep state, during the non communication phases) entails energy consumption that is comparable to being in a receiving or transmitting state. This further emphasizes the benefits of operating in the sleep state with regards to power savings. We define a sleeping node as a wireless node that has switched off its receiving, transmitting and channel sensing circuitry. Thus, while in the sleep state the wireless node has no packet processing capabilities. However since the major power hungry circuitries [] are turned off during the sleep state, the wireless node consumes very little power while in this state. If a wireless node remains in the sleep state for too long it can result in delay penalties for the packets that accumulate for it at the central node (Access Point or similar). On the other hand, it can save substantial power for itself if it remains in the sleep state for longer durations. Thus an optimal policy may be desired which enables the wireless node to save maximum power by sleeping for the longest duration while not violating the system s packet delay constraint. In this paper we are generally interested in the problem of optimizing the timing and duration of sleep states of a wireless node with the objective of minimizing power with respect to a QoS constraint. The QoS parameter we have focused on is average delay. Going by our definition of a sleeping node - the main challenge of the sleep mechanism lies in the wireless nodes incapability to wake up as soon as a packet arrives for it during its sleep state. Various protocols have been suggested to by-pass this problem [3],[4],[5]. Most of these protocols utilize a separate wakeup radio which essentially fulfils the role of an out-ofband paging channel. This radio periodically listens for wakeup beacons and could be designed to be less power hungry than the main data radio. In this paper, we take a different view point in addressing the above problem. We try to calculate the optimal sleep duration as a function of average packet delay tolerance and packet arrival rate. The wireless node is then allowed to sleep for the pre-determined optimal number of slots. We believe, that our work fits in very well, with the IEEE WLAN standard [1] as well. The power saving mode (PSM) in the Infrastructure model of an system has the provision for the wireless nodes to notify the AP, at the time of association, of their sleep duration. The AP then allocates enough resources to support the sleeping nodes during their sleep states. The nodes wake up after their stipulated sleep duration to hear the beacon frame, transmitted by the AP. The beacon has information regarding packets buffered for the sleeping nodes at the AP. If the wireless node realizes (from the beacon message) that the AP has packets buffered for it, it stays awake for the entire duration of the beacon transmission and requests the AP to deliver its packets until its buffer is emptied. However, the standard does not specify any algorithm or scheme that the wireless nodes should use to declare their sleep durations. Our work is an attempt to provide insight into determining a sleep duration which minimizes average power consumption subject to an average delay constraint. To help gain a better understanding of the general problem, we considered a simple model where there is a single transmitter

2 and a single receiver. The receiver is the wireless node whose mode we wish to control. The transmitter can give commands to the receiver regarding its sleep state, and forwards incoming streaming data to the receiver appropriately. We consider packet transmission from the transmitter (e.g a wired Access Point (AP)) to a wireless node over a static channel. To reduce its power consumption, the wireless node is capable of going to sleep. While in the sleep state, it is incapable of receiving packets from the AP. The AP has the provision to buffer packets for the wireless node while it sleeps. These packets are transmitted to the wireless node when it wakes up. Clearly, the packets incur delay whenever the wireless node goes to sleep. On one hand the wireless node can greatly reduce its power consumption by sleeping for long periods of time. On the other hand, the longer it sleeps the greater is the delay incurred by its packets which have to be buffered at the AP during these long sleep durations. In our problem, we have a constraint on the maximum average delay that the data packets can tolerate at the AP, namely D max. Given this constraint of D max, we seek optimal sleep durations for the wireless node that minimizes its average power consumption. We formulated this as an optimization problem and solved it numerically using dynamic programming. The solutions from the numerical calculations strongly suggest that the optimal policy (that which minimizes average power consumption subject to an average delay constraint) is such that the AP should only command the receiver to sleep when there is no data queued at the transmitter. The system thus behaves as a single server queue with vacations. We were able to derive closed form expressions for the optimal sleep duration, as well as the associated minimal rate of power consumption. The rest of the paper is organised as follows: Section II gives a detailed description of our system model. Section III gives a mathematical formulation of the problem. Section IV outlines the Dynamic Programming equations. Section V discusses the results obtained numerically from the DP. Section VI describes the closed form expressions derived for the optimal sleep duration and section VII concludes the paper. II. SYSTEM MODEL We now describe the model of the system under study in more detail. Time is divided into equal length intervals called slots, which are indexed by integers. Packets arrive randomly to the Access Point (AP), to be transmitted to the wireless node across a static channel. We assume that packet arrivals at the AP are memoryless and modeled by a Bernoulli process. Let p be the probability of a packet arriving in a slot. We also assume that a packet always arrives at the beginning of a slot. It can be served in the same slot it arrived or during a later slot. Thus the AP has the capability to queue packets for the wireless node. We define P n to be the power consumed by the wireless node in slot n, a n to be a random variable that represents the number of packets that arrive at the AP at the beginning of slot n, u n to be the number of packets transmitted to the wireless node in slot n and x n to be the number of packets queued at the AP at the end of slot n, also called the backlog of slot n. We require that u n (x n 1 + a n ). The backlog process x n, therefore, satisfies the recursion x n+1 x n u n + a n (1) The wireless node can be in one of the two following states - Sleep or Awake. There is a power cost associated with each state. Let P s denote the power required per slot while in Sleep state and P a denote the power required per slot while in Awake state. In addition, let P as denote the power required for switching from Awake to Sleep state and P sa be the power required for switching from Sleep to Awake state. We assume P a P s []. We characterize the system state in each slot by the tuple (x n, r n ), where where x n backlog at the end of slot n, and r n sleep state of the system during slot n r n 0; wireless node awake during slot n 1; wireless node asleep during slot n but will be awake in slot n + 1 k; wireless node asleep during slot n, n + 1,..., n + k 1, but will be awake during slot n + k III. PROBLEM STATEMENT We require that the average delay suffered by data packets at the AP be no more than D max. We let x E[x n ]. According to Little s result, assuming that the backlog process x n is stationary, the average delay is given by x/p. Thus, we require that, x pd max We approach the problem, by considering it as an optimal control problem over a finite time horizon [8] say over slots 0, 1,,..., N 1. In order to reflect the average delay constraint, we consider policies, such that x n ] pd max N () E[ Let A be the set of all such policies. We consider the following optimization problem: } min E[ P n ] subject to A (3) We use a similar approach as in [6] to solve the above equation. Let f(d max ) min A } E[ P n ] (4)

3 We use duality to solve the above equation, since direct evaluation of f(d max ) appears difficult. Defining β as a dual variable, where β 0, we define the dual function h(β) min E[ P n + β x n ] } β 0 (5) Assuming, we can calculate h(β), f(d max ) can be lower bounded as follows: f(d max ) min A min A min } E[ P n ] ( )} E[ P n ] + β E[ x n ] pd max N ( )} E[ P n ] + β E[ x n ] pd max N h(β) β(pd max N) so that, f(d max ) sup h(β) β(pdmax N) } (6) β 0 For sufficiently large N, it is apparent that h(β) is convex in β, and therefore in this case (6) is an equality. IV. DYNAMIC PROGRAMMING FORMULATION We calculate h(β), using the approach of dynamic programming [8]. We define, J N (x, r) min E[ P n + β x n x 0 x, r 0 r] (7) Then J N (x, r) satisfies the following recursions: For simplicity we assume here that p P robability(a n 1) E[a n ] J N+1 (x, 0) P a + βx + minp[j N (x, 0)] + (1 p)[j N ((x 1) +, 0)], P as + min k>0 p[j N (x + 1, k)] + (1 p)[j N (x, k)]}} where (x 1) + max(x 1, 0) J N+1 (x, 1) P s + P sa + βx J N+1 (x, k) P s + βx + p[j N (x, 0)] + (1 p)[j N ((x 1) +, 0)] + p[j N (x + 1, k 1)] + (1 p)[j N (x, k 1)] k > 1 The initial condition of the recursion is: J 0 (x, k) 0, x, k We augment the dynamic programming recursions to calculate: E[ P n x 0 x, r 0 r] and (8) x n x 0 x, r 0 r] (9) E[ for the corresponding optimal policy, for each value of β. V. RESULTS We performed a series of numerical calculations to evaluate the minimum average energy required per slot that satisfied the maximum allowed average delay (D max ) constraint. We developed a computer program, to implement the dynamic programming algorithm [8] for computing h(β) over a range of values of β, given a Bernoulli arrival distribution with a packet arrival probability of p in every slot, a fixed service rate of one packet per slot and fixed power costs of P a, P s, P as, P sa. We assumed P a P s []. While only finite horizon, finite state solutions can ever be exactly computed, we apply the value iteration algorithm descibed in [7], to compute good approximations to the optimal policy and corresponding average cost for the infinite horizon case. Let, M (x, r) be the optimal next sleep state when (x, r) is the current state of the system.our computer program yeilded the following result. As long as we had, P a P s, M (x, r) was always found to be of the form : M (x, r) 0; (i.e stay Awake) when x > 0 and r 0 k; (i.e go to Sleep for k slots) where k > 0 and x 0 and r 0 r 1; (no action)when r > 0 and x 0 (10) The value of k (the sleep duration) is dependant on D max and p. We tried to use induction to prove that the optimal policy, as suggested strongly by the numerical results, is always of this form, which is what intuition suggests. However, we were unable to do so thus far. We present some representative results related to specific scenarios in a graphical representation. We assume p.01, P a 1, P as.0001,p sa.01,p s.001 to obtain the results displayed in Figures 1 and. We see that the higher the average delay the system can tolerate, the lower is the power consumed per slot.this lower average power consumption per slot is reflected in longer sleep durations. VI. DERIVATION OF THE CLOSED-FORM EXPRESSION Next, we present an analysis of policies of the form in (10). We shall study the above policy with a branching process analysis to obtain the average power consumption and the average delay as a function of the sleep duration k.

4 Optimal Sleep Duration(in slots) Average Power Cost/slot Average Delay(in slots) Fig. 1. Delay vs.sleep Average Delay (in slots) Fig.. Delay vs. Power Let us assume that the receiver goes to sleep for k slots when there are 0 packets in the buffer. Packets for the sleeping node accumulate at the AP during these k slots. When the receiver wakes up after k slots, it serves these first batch of packets. A second batch of packets may arrive while packets from the first batch are being served and so forth until the backlog is cleared and the buffer occupancy goes down to 0 again. The receiver then goes to sleep again for k slots. If there are no packet arrivals during the sleep duration (k slots), the receiver goes back to sleep again for yet another k slots. Since the arrival rate p < service rate 1, the system is stable and hence reaches this 0 backlog state infinitely often. We define one Epoch as the time period between the onset of two consecutive Sleep cycles (of duration k slots). Thus each epoch consist of several sub-epochs. The first sub-epoch comprises of the k slots during which the receiver is in the sleep state. The second sub-epoch comprises of a random number of slots required to serve the batch of packets that arrives during the k sleep slots. In general, the nth sub-epoch comprises of the service time of the batch of packets that arrive during the (n-1)th sub-epoch. Obviously, the reciever is asleep during the first subepoch and awake during the remaining sub-epochs. An epoch terminates when the reciever has no more packets to serve and is ready to go to sleep again. Let L be a random variable that represents the duration of an epoch. Note that L k + A, where A is a random variable representing how long the node is awake during an epoch and k is the duration of the receiver s sleep state. Let Z n denote the length (i.e the number of slots) of the nth sub-epoch. Therefore, Epoch Length L Z 0 + Z n k + A n1 As mentioned previously, time is divided into equal unit length intervals called slots. Packets follow a Bernoulli arrival pattern, with probability p of a packet arrival in every slot. For simplicity we assume that when the receiver is awake, one packet can be served in each slot. Note that, Z n Average Power expended per slot Average Power expended in an epoch Average Epoch length P as + P s k + P sa + P a E[A] Average Backlog per slot Average Backlog over an epoch Average Epoch length E[Cumulative Backlog over an epoch] E[L] k + E[A] (11) Note that the minimum length of an epoch k + 1.

5 and We define the following terms: Y 0 B 0 B 1 k n1 x n cumulative backlog at the end of the first sub-epoch k n1 a n total number of packet arrivals at the end of first sub-epoch k+b 0 nk+1 a n total number of packets that arrived Thus, we have, during the service of the batch of B 0 packets Also, E[B 1 B 0 B 0 ] pb 0 (1) (13) E[B 0 ] pk (14) E[B 0] pk + (k 1)p k (15) Hence, E[B 1 ] pe[b 0 ] p(pk) p k Generalizing, we have, Similarly, E[B m ] pe[b m 1 ] p (m+1) k (16) E[B m ] pe[b m 1] + p E[B m 1 ] p E[B m 1 ] p m [kp(1 p)] + p E[B m 1 ] (17) After solving the above recursion, we get, E[B m ] pm [pk + p k(k 1)] + p m+1 k(1 p m ) (18) We now find a general expression for the expected cumulative backlog after each sub-epoch. Substituting values of E[B m 1 ] and E[B m 1] and simplifying, we get, Thus, E[Y m ] pm k(k 1)(p + 1) + p m+1 k m > 0 m1 E[Y m ] (p )k(k + 1) (1 p) Next, we find the average length of an epoch. Epoch length k + 1; ifb 0 0 Simplifying, we have, Thus, Therefore, k + B 0 + B ; otherwise E[L] k + 1 Probability(B 0 0) + E[B 0 ] + E[B 1 ] +... () (3) k 1 p + (1 p)k (4) E[A] k k (1 p) + (1 p)k k Average Backlog per slot E[Y 0] + E[Y 1 ] + E[Y ] +... Replacing with appropriate values, we get, Average Backlog per slot From Little s Formula, we get, Average maximum delay pk(k + 1) k + (1 p) k+1 (5) D max k(k + 1) k + (1 p) k+1 (6) Average Power expended per slot P as + P s k + P sa + P a E[A] (7) (8) E[Y m B m 1 ] ( B m 1 1) + p(b m 1)(B m 1 + 1) Therefore, (B m 1) (p + 1) + B m 1(p 1) E[Y m ] E[B m 1 ](p + 1) + E[B m 1](p 1) m > 0 (19) (0) (1) Hence; Average Power expended per slot (1 p)(p as + P s k + P sa ) + P a ((1 p) k+1 + pk) k + (1 p) k+1 It is of interest, to note from equation ( 6) that the sleep duration k is only a function of the average delay D max and packet arrival rate p. It is independant of the associated power costs like P a, P as, P sa and P s.

6 VII. CONCLUSION In this paper we have looked at the problem of optimizing the timing and duration of sleep states on wireless nodes, with the objective of minimizing power with respect to a QoS constraint, namely average packet delay. We considered a simple model where there is a single transmitter and a single receiver. We wish to control the state (sleep or awake) of the receiver. The transmitter can give commands to the receiver regarding its sleep state and forwards incoming streaming data to the receiver appropriately. We formulated this as an optimization problem and solved it numerically using dynamic programming. The solutions from the numerical calculations strongly suggest that the optimal policy (that which minimizes average power consumption subject to an average delay constraint) is such that the transmitter should only command the receiver to sleep when there is no data queued at the transmitter. We were able to derive closed form expressions for the optimal sleep duration as well as the associated minimal rate of power consumption. Future work will be focused on more elaborate models involving multiple users. REFERENCES [1] IEEE Computer Society, IEEE standard 80.11:wireless LAN medium access control (MAC) and physical layer (PHY) specifications. The institute of Electrical and Electronics Engineers, New York, NY, [] L.M. Feeney and M.Nilsson, Investigating the energy consumption of a wireless network interface in an ad hoc networking environment, Proc. of IEEE INFOCOM, April 001. [3] R. Zheng and R. Kravets, On demand power management for ad-hoc network In Proceedings of the nd Annual Joint Conference of the IEEE Computer and Communications Societies, May 003. [4] B.Chen, K.Jamieson, H.Balakrishnan, and R.Morris Span: An energyefficient coordination algorithm for topology maintenance in ad-hoc wireless networks, In Proc. of ACM/IEEE 7th Intl. Conference on Mobile Computing and Networking (MobiCom), July 001. [5] A.K. Salkintzis and C.Chamaz, An in-band power saving protocol for mobile data networks, IEEE Trans. on Communications, Vol.46, pp , September [6] B.E. Collins and R.L. Cruz, Transmission Policies for Time Varying Channels with Average Delay Constraints Proc.1999 Allerton Conference on Communications,Control and Comp., Monticello, IL, [7] Sennott, L.I.,Stochastic Dynamic Programming and the Control of Queueing Systems, New York: Wiley,1999 [8] Bertsekas, D.P Dynamic Programming and Stochastic Control, Vol.16, NJ:Prentice Hall, 1987