Equilibrium and efficient clustering of arrival times to a queue

We consider a game of decentralized timing of jobs to a single server (machine) with a penalty for deviation from some due date and no delay costs. The jobs sizes are homogeneous and deterministic. Each job belongs to a single decision maker, a customer, who aims to arrive at a time that minimizes his deviation penalty. If multiple customers arrive at the same time then their order is determined by a uniform random draw. If the cost function has a weighted absolute deviation form then any Nash equilibrium is pure and symmetric, that is, all customers arrive together. Furthermore, we show that there exist multiple, in fact a continuum, of equilibrium arrival times, and provide necessary and sufficient conditions for the socially optimal arrival time to be an equilibrium. The base model is solved explicitly, but the prevalence of a pure symmetric equilibrium is shown to be robust to several relaxations of the assumptions: inclusion of small waiting costs, stochastic job sizes, random sized population, heterogeneous due dates and non-linear deviation penalties.