The concert/cafeteria queueing problem: a game of arrivals

We introduce the concert or the cafeteria queueing problem: Fixed but a large number of users arrive into a queue which provides service starting at time 0. Users may arrive before 0. They incur a queued waiting cost α · W, where W is the time to wait in the queue until service, and service time cost β · (t + W), where t is the arrival time and t + W is the total time until service. Each user picks a mixed strategy for arrival to minimize E[αW + β(t + W)]. We analyze the system in an asymptotic regime and develop fluid limit for the resultant queueing system. The limiting system may be modeled as a non-atomic game for which we determine an equilibrium arrival strategy. In particular, we note that the equilibrium arrival strategy is to arrive uniformly between some τ0 < 0 and τ1 selected so that the queue is never empty. We note that larger the β/α, larger the queue. Furthermore, we note that the 'price of symmetric anarchy' of this system equals 2. In addition to modeling queue formation at large concerts or cafeterias in certain settings, the model may be relevant more generally, for instance, in explaining queue formation in DMV offices at opening time, and at retail stores at opening time during peak shopping season.