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.

In this note, we formulate a ‘one-sided’ version of Wormald’s differential equation method. In the standard ‘two-sided’ method, one is given a family of random variables that evolve over time and which satisfy some conditions, including a tight estimate of the expected change in each variable over one-time step. These estimates for the expected one-step changes suggest that the variables ought to be close to the solution of a certain system of differential equations, and the standard method concludes that this is indeed the case. We give a result for the case where instead of a tight estimate for each variable’s expected one-step change, we have only an upper bound. Our proof is very simple and is flexible enough that if we instead assume tight estimates on the variables, then we recover the conclusion of the standard differential equation method.