Optimal admission pricing policies for M/Ek/1 queues

This paper extends the Low‐Lippman M/M/1 model to the case of Gamma service times. Specifically, we have a queue in which arrivals are Poisson, service time is Gamma‐distributed, and the arrival rate to the system is subject to setting an admission fee p. The arrival rate λ(p) is non‐increasing in p. We prove that the optimal admission fee p* is a non‐decreasing function of the customer work load on the server. The proof is for an infinite capacity queue and holds for the infinite horizon continuous time Markov decision process. In the special case of exponential service time, we extend the Low‐Lippman model to include a state‐dependent service rate and service cost structure (for finite or infinite time horizon and queue capacity). Relatively recent dynamic programming techniques are employed throughout the paper. Due to the large class of functions represented by the Gamma family, the extension is of interest and utility.