  Consider a population of customers each of which needs to decide
independently when to arrive to a facility that provides a service during a
fixed period of time, say a day. This is a common scenario in many service
systems such as a bank, lunch at a cafeteria, music concert, flight check-in
and many others. High demand for service at a specific time leads to congestion
that comes at a cost, e.g., for waiting, earliness or tardiness. Queueing
Theory provides tools for the analysis of the waiting times and associated
costs. If customers have the option of deciding when to join the queue, they
will face a decision dilemma of when to arrive. The level of congestion one
suffers from depends on others behavior and not only that of the individual
under consideration. This fact leads customers to make strategic decisions
regarding their time of arrival. In addition, multiple decision makers that
affect each other's expected congestion, call for non-cooperative game
theoretic analysis of this strategic interaction. This common daily scenario
has prompted a research stream pioneered by the ?/M/1 model of Glazer and
Hassin (GH1983) that first characterized an arrival process to a queue as a
Nash equilibrium solution of a game. This survey provides an overview of the
main results and developments in the literature on queueing systems with
strategic timing of arrivals. Another issue is that of social optimality,
namely the strategy profile used by customers that optimizes their aggregate
utility. In particular, we review results concerning the price of anarchy
(PoA), which is the ratio between the socially optimal and the equilibrium
utilities.
