  We consider a discrete population of users with homogeneous service demand
who need to decide when to arrive to a system in which the service rate
deteriorates linearly with the number of users in the system. The users have
heterogeneous desired departure times from the system, and their goal is to
minimize a weighted sum of the travel time and square deviation from the
desired departure times. Users join the system sequentially, according to the
order of their desired departure times. We model this scenario as a
non-cooperative game in which each user selects his actual arrival time. We
present explicit equilibria solutions for a two-user example, namely the
subgame perfect and Nash equilibria and show that multiple equilibria may
exist. We further explain why a general solution for any number of users is
computationally challenging. The difficulty lies in the fact that the objective
functions are piecewise convex, i.e., non-smooth and non-convex. As a result,
the minimization of the costs relies on checking all arrival and departure
order permutations, which is exponentially large with respect to the population
size. Instead we propose an iterated best-response algorithm which can be
efficiently studied numerically. Finally, we compare the equilibrium arrival
profiles to a socially optimal solution and discuss the implications.
