Energy-efficient Scheduling of Delay Constrained Traffic over Fading Channels

A delay-constrained scheduling problem for point-to-point communication is considered: a packet of $B$ bits must be transmitted by a hard deadline of $T$ slots over a time-varying channel. The transmitter/scheduler must determine how many bits to transmit, or equivalently how much energy to transmit with, during each time slot based on the current channel quality and the number of unserved bits, with the objective of minimizing expected total energy. In order to focus on the fundamental scheduling problem, it is assumed that no other packets are scheduled during this time period and no outage is allowed. Assuming transmission at capacity of the underlying Gaussian noise channel, a closed-form expression for the optimal scheduling policy is obtained for the case T=2 via dynamic programming; for $T>2$, the optimal policy can only be numerically determined. Thus, the focus of the work is on derivation of simple, near-optimal policies based on intuition from the T=2 solution and the structure of the general problem. The proposed bit-allocation policies consist of a linear combination of a delay-associated term and an opportunistic (channel-aware) term. In addition, a variation of the problem in which the entire packet must be transmitted in a single slot is studied, and a channel-threshold policy is shown to be optimal.

💡 Research Summary

The paper addresses the problem of transmitting a single packet of B bits over a fading wireless channel within a hard deadline of T time slots, with the goal of minimizing the expected total energy consumption. The transmitter has causal knowledge of the channel: at each slot t the current channel gain gₜ is known, while future gains are unknown. Transmission follows the Shannon capacity of an additive white Gaussian noise (AWGN) channel, i.e., the number of bits transmitted in a slot is bₜ = log₂(1 + gₜEₜ), which can be inverted to give the required energy for a given number of bits: Eₜ(bₜ,gₜ) = (2^{bₜ} − 1)/gₜ.

The scheduling problem is formulated as a finite‑horizon dynamic programming (DP) problem. The state variable is the remaining bits βₜ, the action is the number of bits bₜ to send in the current slot, and the stage cost is the energy Eₜ(bₜ,gₜ). The DP recursion is

Jₜ(βₜ,gₜ) = min_{0 ≤ bₜ ≤ βₜ} { (2^{bₜ} − 1)/gₜ + \bar J_{t‑1}(βₜ − bₜ) },

where \bar J_{t‑1}(·) is the expected cost‑to‑go function (expectation taken over the unknown future channel gains). The terminal condition at the last slot (t = 1) is J₁(β₁,g₁) = (2^{β₁} − 1)/g₁, because all remaining bits must be transmitted regardless of channel quality.

Closed‑form solution for T = 2
When only two slots remain, the optimal number of bits to send in the first slot (t = 2) can be derived analytically. The objective becomes

(2^{b₂} − 1)/g₂ + E_{g₁}