  Suppose customers need to choose when to arrive to a congested queue with
some desired service at the end, provided by a single server that operates only
during a certain time interval. We study a model where the customers incur not
only congestion (waiting) costs but also penalties for their index of arrival.
Arriving before other customers is desirable when the value of service
decreases with every admitted customer. This may be the case for example when
arriving at a concert or a bus with unmarked seats or going to lunch in a busy
cafeteria. We provide game theoretic analysis of such queueing systems with a
given number of customers, specifically we characterize the arrival process
which constitutes a symmetric Nash equilibrium.
