We use a multidimensional diffusion process to approximate the dynamics of a queue served by many parallel servers. Waiting customers in this queue may abandon the system without service. To analyze the diffusion model, we develop a numerical algorithm for computing its stationary distribution. A crucial part of the algorithm is choosing an appropriate reference density. Using a conjecture on the tail behavior of the limit queue length process, we propose a systematic approach to constructing a reference density. With the proposed reference density, the algorithm is shown to converge quickly in numerical experiments. These experiments demonstrate that the diffusion model is a satisfactory approximation for many-server queues, sometimes for queues with as few as twenty servers.
J. G. Dai. Shuangchi He. "Many-server queues with customer abandonment: Numerical analysis of their diffusion model." Stoch. Syst. 3 (1) 96 - 146, 2013. https://doi.org/10.1214/11-SSY029