Stable and superstable customer policies in queues with balking and priority options

Abstract

We discuss decision-making in queueing systems where individual customers have one or both of two options: (1) option to join or balk, (2) option to choose priority through a payment made on arrival. Each customer's objective is to maximize the expected net gain, defined as 0 if he balks, and as R - b - hE( W) if he joins, where R is his reward, b is his payment, h is his unit time waiting loss, and E( W) is his expected waiting time. The models considered are: (A) a GI/ M/ s/ N queue with either FIFO or LIFO rule, and a balking option, (B) an M/ M/ s/ N queue with a priority option, (C) an M/ M/ s/ N queue with balking and priority options. Rational customers will limit their choice to a certain subset of actions, derived by considering the actions of other customers. When the subset consists of just one action we call it optimal. The analysis focuses on policies, that is, decision rules selecting an action as a function of arrival state (customers in system), R and h. Stable (self-sustaining) and superstable (optimal) policies are of special interest. A stable policy may not exist and it is not necessarily unique. In model A a superstable policy exists except, possibly, for N = ∞ with LIFO rule. For model B we give necessary and sufficient conditions that a policy is stable and induces either FIFO or LIFO service order. Similar results are derived in special cases of model C.