Delay Constrained Scheduling over Fading Channels: Optimal Policies for Monomial Energy-Cost Functions

arXiv:0809.5009v1 [cs.IT] 29 Sep 2008 Delay Constrained Scheduling over Fading Channels: Optimal Policies for Monomial Energy-Cost Functions Juyul Lee and Nihar Jindal Department of Electrical and Computer Engineering University of Minnesota E-mail: {juyul, nihar}@umn.edu Abstract— A point-to-point discrete-time scheduling problem of transmitting B information bits within T hard delay deadline slots is considered assuming that the underlying energy-bit cost function is a convex monomial. The scheduling objective is to minimize the expected energy expenditure while satisfying the deadline constraint based on information about the unserved bits, channel state/statistics, and the remaining time slots to the deadline. At each time slot, the scheduling decision is made without knowledge of future channel state, and thus there is a tension between serving many bits when the current channel is good versus leaving too many bits for the deadline. Under the assumption that no other packet is scheduled concurrently and no outage is allowed, we derive the optimal scheduling policy. Furthermore, we also investigate the dual problem of maximizing the number of transmitted bits over T time slots when subject to an energy constraint. I. INTRODUCTION An opportunistic scheduling policy that adapts to the time- varying behavior of a wireless channel can achieve energy- efficient communication on the average in a long-term perspec- tive. However, this opportunistic approach may not be appro- priate for short-term deadline constrained traffic. This paper considers scheduling a packet over a finite time horizon while efficiently adapting to wireless (fading) channel variations and taking care of the deadline constraint. Our primal problem setting is the minimization of energy expenditure subject to a hard deadline constraint (i.e., a packet of B bits must be scheduled within finite T discrete-time slots) assuming that the scheduler has causal knowledge of the channel state information (CSI). Causal CSI means that the scheduler knows the past and current CSI perfectly, but does not know future CSI. The scheduler is then required to make a decision at each time slot given the number of unserved bits, the number of slots left before the deadline, and causal CSI, in order to minimize the total energy expenditure. At each time slot, the scheduler deals with the tension between serving more bits when the channel is good and leaving too many bits to the end. Likewise, we consider the dual (scheduling over a finite time-horizon) problem of maximizing the transmitted bits subject to a finite energy constraint. We also briefly discuss scheduling problems when the CSI is available non-causally. We assume that no other packet is scheduled simultaneously The work of J. Lee is supported by a Motorola Partnership in Research Grant. and the hard delay deadline must be met (i.e., no outage is allowed). These finite-time horizon scheduling problems can be applicable to regularly arriving packets with hard delay deadlines, e.g., VoIP and video streaming. Delay constrained scheduling over fading channel has been studied for various traffic models and delay constraints. Uysal- Biyikoglu and El Gamel [1] considered scheduling random packet arrivals over a fading channel and thus adapt (transmit power/rate) to both the channel state and queue state, and generally try to minimize average delay. Many references can be found in [1]. Most cases do not admit analytical closed- form solution for causal (or online) scheduling. Instead, they proposed causal algorithms with heuristic modifications from non-causal (offline) policies. References [2]–[4] take a slightly different perspective: single packet scheduling (no queue) with a hard delay deadline rather than an average delay constraint. The subject of this paper is the single-packet scheduling problem of [2] specialized to the case where the required energy E to transmit b bits under channel state g is governed by a convex monomial function, i.e., E = bn/g, where n denotes the monomial order. The biggest advantage of using this monomial cost function is that it yields closed-form so- lutions in various scenarios, unlike the Shannon-cost function setting described in [4]. As a result, it provides intuition on the interplay between the monomial order, delay deadline, and the channel states so that it ultimately suggests general ideas for a more general energy-cost function. Although the monomial cost does not hold for operating at capacity in an AWGN channel, according to Zafer and Modiano [5] and their reference [6], there is a practical modulation scheme that exhibits an energy-bit relation that can be well approximated by a monomial. Actually, Zafer and Modinano [5] considered the same problem but for a continuous-time Markov process channel in continuous-time scheduling, i.e., the scheduler can transmit at any time instant rather than discrete slotted time. Although they provided a solution in the form of a set o

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