Equilibrium and socially optimal arrivals to a single server loss system

We study a single server model with no queue and exponential services times, in which service is only provided during a certain time interval. A number of customers wish to obtain this service and can choose their arrival time. A customer that finds a busy server leaves without being served. We model this scenario as a non-cooperative game in which the customers wish to maximize their probability of obtaining service. We characterize the Nash equilibrium and the price of anarchy, which is defined as the ratio between the optimal and equilibrium social utility. In particular, the equilibrium arrival distribution has an atom at zero, a period with no arrival and is continuous on some interval until the closing time. We further generalize our analysis to take into account uncertainty regarding the population size, i.e. a game with a random number of customers. In the special case where the population size follows a Poisson distribution, we show that the continuous part of the distribution is uniform, which is not the case in general. Finally, we show that the price of anarchy is not monotone with respect to the population size; but rather uni-modal with values close to one for small and large populations.