We consider a GI/GI/1 queue with impatient customers in heavy traffic. We use the solution of an approximating singular diffusion control problem to construct an admission control policy for the queue. The approximating control problem does not admit a so-called pathwise solution. Hence, the resulting admission control policy depends on second-moment data. We prove asymptotic optimality of the constructed policy using weak-convergence methods. Key words: impatience; reneging; deadlines; asymptotic optimality; admission control; singular control; reflected Ornstein-Uhlenbeck process; diffusion approximation; heavy traffic MSC2000 subject classification: Primary: 90B22, 93E20; secondary: 60K25, 68M20 OR/MS subject classification: Primary: queues, balking and reneging, diffusion models, limit theorems, optimal control; secondary: probability, stochastic model applications History: Received September 13, 2004; revised August 28, 2005, June 2, 2006, and March 22, 2007 1. Introduction. Models of systems that involve customers queueing for service can be made more realistic by incorporating the impatience of these customers. A natural way to incorporate impatience is to assume that each customer independently abandons the queue, or reneges, if his service has not begun within a certain amount of time, which could be a random variable drawn from a given distribution. Many papers have studied such queueing systems with reneging customers from a performance evaluation standpoint; see, for example, Afanas Several control problems are natural in queueing systems with impatient customers. In this paper we consider one such control problem. Suppose the manager of a GI/GI/1 queue with impatient customers receives a payment of $p > 0 when a customer enters the queue. However, the system manager must refund any reneging customer $r. A reasonable assumption is that r > p; that is, the customer receives additional compensation for not having received service. We assume that the customers are not strategic: They do not renege simply to make money. Rather, their time to reneging is drawn from an exponential distribution. The only control available to the manager is the ability to deny an arriving customer admission into the queue. For a FIFO queue, the likelihood that an admitted customer reneges before receiving service is increasing in the number of queued customers. Therefore, it may be in the manager's interest to exercise admission control and deny entry to customers when the queue is large, incurring an effective opportunity cost of $p per customer, rather than paying out $r when customer reneging is likely. We assume that the manager has access to a bank account with an interest rate $ (with continuous compounding). When is large, the system manager has no reason to turn away customers, because future payouts are heavily discounted. Also, if the customer reneging rate is tiny, so that the chance of any one customer reneging is very small, the system manager will allow entry to all customers. In either case, our problem is uninteresting. Hence, we assume p/r < 1 + / −1 so that neither of the aforementioned situations occurs. (Although the part of the parameter space that yields a nontrivial problem is not initially obvious, our analysis shows that p/r < 1 + / −1 is a sufficient condition for a nontrivial problem, and suggests the condition is necessary as well.) The manager's goal is to minimize the infinite-horizon expected discounted cost of refunds to reneging customers and exercising admission control. The control problem as stated above is essentially intractable for general arrival and service processes. Therefore, we lower our aspirations somewhat and look for admission control policies that perform well in some limiting parametric regime. The regime we consider is the so-called balanced heavy-traffic regime, where we
Ward and Kumar REF look at the diffusion control formulation for admission control in a GI/GI/1 with impatient customers.
Asymptotically Optimal Admission Control of a Queue with Impatient Customers
Math. Oper. Res.
Math. Oper. Res.
Mathematics Computer Science