A Note on the Comparison between Bernoulli and Limited Policies in Vacation Models

Abstract

A vacation model is a single-server queueing system where the server occasionally takes a vacation, i.e., becomes unavailable to the customers for a period of time. Under a limited service policy with parameter m, m = 1,2,..., the server can serve a maximum of m consecutive customers before it has to take a vacation. Under a Bernoulli service policy with parameter p, 0 p 1, at every service completion, the server serves the next customer (if available) with probability p or starts a vacation with probability 1 - p. Under both policies, the server takes a vacation if no customer is present. In this note, the waiting time under a Bernoulli policy with parameter p is shown to dominate that under a limited policy with parameter m when the parameters are related by p = (m - 1)/m and the vacation lengths are identical and deterministic. The comparison is in the convex increasing ordering and holds both in the transient and steady- state regimes.