Improving Service by Informing Customers About Anticipated Delays

This paper investigates the effect upon performance in a service system, such as a telephone call center, of giving waiting customers state information. In particular, the paper studies two M/M/s/r queueing models with balking and reneging. For simplicity, it is assumed that each customer is willing to wait a fixed time before beginning service. However, customers differ, so the delay tolerances for successive customers are random. In particular, it is assumed that the delay tolerance of each customer is zero with probability \beta , and is exponentially distributed with mean \alpha -1 conditional on the delay tolerance being positive. Let N be the number of customers found by an arrival. In Model 1, no state information is provided, so that if N \ge s, the customer balks with probability \beta ; if the customer enters the system, he reneges after an exponentially distributed time with mean \alpha -1 if he has not begun service by that time. In Model 2, if N = s + k \ge s, then the customer is told the system state k and the remaining service times of all customers in the system, so that he balks with probability \beta + (1 - \beta )(1 - q k ), where q k = P(T > S k ), T is exponentially distributed with mean \alpha -1 , S k is the sum of k + 1 independent exponential random variables each with mean (s\mu ) -1 , and \mu -1 is the mean service time. In Model 2, all reneging is replaced by balking. The number of customers in the system for Model 1 is shown to be larger than that for Model 2 in the likelihood-ratio stochastic ordering. Thus, customers are more likely to be blocked in Model 1 and are more likely to be served without waiting in Model 2. Algorithms are also developed for computing important performance measures in these, and more general, birth-and-death models.