The number of overlapping customers in Erlang-A queues: an asymptotic approach

In this paper, we investigate the number of customers that overlap or coincide with a virtual customer in an Erlang-A queue. Our analysis starts with the fluid and diffusion limit differential equations to obtain the mean and variance of the queue length. We then develop precise approximations for waiting times using fluid limits and the polygamma function. Building on this, we introduce a novel approximation scheme to calculate the mean and variance of the number of overlapping customers. This method facilitates the assessment of transient overlap risks in complex service systems, offering a useful tool for service providers to mitigate significant overlaps during pandemic seasons.