Strategic timing of arrivals to a finite queue multi-server loss system

We provide Game-theoretic analysis of the arrival process to a multi-server system with a limited queue buffer, which admits customers only during a finite time interval. A customer who arrives at a full system is blocked and does not receive service. Customers can choose their arrival times with the goal of minimizing their probability of being blocked. We characterise the unique symmetric Nash equilibrium arrival distribution and present a method for computing it. This distribution is comprised of an atom at time zero, an interval with no arrivals (a gap), and a continuous distribution until the closing time. We further present a fluid approximation for the equilibrium behaviour when the population is large, where the fluid solution also admits an atom at zero, no gap, and a uniform distribution throughout the arrival interval. In doing so, we provide an approximation model for the equilibrium behaviour that does not require a numerical solution for a set of differential equations, as is required in the discrete case. For the corresponding problem of social optimization, we provide explicit analysis of some special cases and numerical analysis of the general model. An upper bound is established for the price of anarchy (PoA). The PoA is shown to be not monotone with respect to population size.