Optimum Bribing for Queue Position

In this paper we consider that relative position in queue is determined according to the size of a customer’s bribe (which is paid before the customer sees the queue length). Such a policy allows the customer himself to affect his own queue position, rather than the classical approach of assuming that a customer is preassigned to some (possibly continuous) priority class. For the case of Poisson arrivals, arbitrary service time distribution, and arbitrary distribution of customer bribe, we obtain the average waiting time for customers as a function of their bribe. We consider both preemptive and nonpreemptive disciplines. Examples are presented for various bribing distributions, which demonstrate that many well-known priority queuing systems are special cases of this bribing situation. Furthermore, a cost function is defined after we introduce the notion of an impatience factor (which converts seconds of wait into dollars). Conditions for optimum bribing are then determined, where the optimization refers to minimizing the average cost subject to a mean bribe constraint. An example for exponential service and exponential bribing is carried out and the results are plotted.