Asymptotic Optimal Control of a data transmission queue in Heavy traffic with imperfect channel

Abstract: This study investigates the asymptotic optimal control of a data transmission queue operating under heavy traffic conditions with an imperfect channel. This model considers a single-server N-policy queue where packets arrive according to a Poisson process and transmissions are subject to channel failures, leading to retransmissions that rejoin the queue. The server remains inactive until the queue reaches a threshold N, after which it continues service until the system empties. Steady-state balance equations are developed to obtain explicit probability distributions for OFF and ON states, and heavy traffic scaling is used to derive asymptotic expressions for queue length, server utilization, and cost. The analysis establishes a reduced cost function capturing the trade-off between holding and activation costs and shows that the optimal activation threshold follows a classical square-root heavy traffic law. Numerical illustrations and simulations compare exact steady-state optimization with heavy traffic approximations, highlighting the conservatism and near optimal performance of asymptotic policies as system utilization approaches saturation.

Keywords: Data transmission queue, N-policy, imperfect channel, retransmission, heavy traffic, asymptotic optimal control