Modeling the M(t)/M/s(t) Queue with an Exhaustive Discipline

The M/M/s queueing model assumes arrival rates, service rates, and number of servers to be constant in time. It is straightforward to extend the model to allow a time-varying arrival rate and a time-varying service rate, and the resulting models can be solved numerically using standard solvers for ordinary differential equations. The extension to a time-varying number of servers requires consideration of the behavior of a server that is providing service when scheduled to leave. Past work has (implicitly or explicitly) assumed a pre-emptive discipline, where the customer rejoins the queue. We demonstrate how to model an exhaustive discipline that is often more realistic, where servers complete their current service before leaving. We discuss how to compute the virtual waiting time distribution for this model and outline its extension to incorporate customer abandonments.