  We consider a queueing facility where customers decide when to arrive. All
customers have the same desired arrival time (w.l.o.g.\ time zero). There is
one server, and the service times are independent and exponentially
distributed. The total number of customers that demand service is random, and
follows the Poisson distribution. Each customer wishes to minimize the sum of
three costs: earliness, tardiness and waiting. We assume that all three costs
are linear with time and are defined as follows. Earliness is the time between
arrival and time zero, if there is any. Tardiness is simply the time of
entering service, if it is after time zero. Waiting time is the time from
arrival until entering service. We focus on customers' rational behaviour,
assuming that each customer wants to minimize his total cost, and in
particular, we seek a symmetric Nash equilibrium strategy. We show that such a
strategy is mixed, unless trivialities occur. We construct a set of equations
that its solution provides the symmetric Nash equilibrium. The solution is a
continuous distribution on the real line. We also compare the socially optimal
solution (that is, the one that minimizes total cost across all customers) to
the overall cost resulting from the Nash equilibrium.
