An analysis of the equilibrium strategies for route-choosing customers in a two-station queueing system

We investigate a two-station queueing system where strategic customers must be sequentially serviced at both stations. We prove that an established property - that the distribution of the total time spent in a two-station system is independent of the chosen route when services times are exponentially distributed - is not a general one by providing a counterexample with deterministic service times. In doing so, we also prove a concomitant property - that any routing strategy is an equilibrium - is peculiar to a system with a particular assumption of an exponential service time distribution. Using simulations, we show that - depending on the distribution of service times - there can be (1) cases with three equilibria, (2) cases with one pure strategy equilibrium, and (3) cases with one mixed strategy equilibrium.