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

When to arrive at a queue with earliness, tardiness and waiting costs