We analyze a sequence of single-server queueing systems with impatient customers in heavy traffic. Our state process is the offered waiting time, and the customer arrival process has a state dependent intensity. Service times and customer patient-times are independent; i.i.d. with general distributions subject to mild constraints. We establish the heavy traffic approximation for the scaled offered waiting time process and obtain a diffusion process as the heavy traffic limit. The drift coefficient of this limiting diffusion is influenced by the sequence of patience-time distributions in a non-linear fashion. We also establish an asymptotic relationship between the scaled version of offered waiting time and queue-length. As a consequence, we obtain the heavy traffic limit of the scaled queue-length. We introduce an infinite-horizon discounted cost functional whose running cost depends on the offered waiting time and server idle time processes. Under mild assumptions, we show that the expected value of this cost functional for the n-th system converges to that of the limiting diffusion process as n tends to infinity. © 2011 Elsevier B.V. All rights reserved.
Lee, C., & Weerasinghe, A. (2011). Convergence of a queueing system in heavy traffic with general patience-time distributions. Stochastic Processes and Their Applications, 121(11), 2507–2552. https://doi.org/10.1016/j.spa.2011.07.003