Customer joining strategies in Markovian queues with B-limited service rule and multiple vacations

This paper studies customer joining strategies in some single-server Markovian queues with batch limited service rule and multiple vacations. The server begins to take a vacation time as soon as a batch of F customers are served continuously. If the server finds that there are fewer than F customers present in the system at the completion instant of a vacation time, then he takes another until there are no less than F customers waiting after his returning. We consider both the fully observable case and the fully unobservable case, and get customer joining strategy in equilibrium in each case as well as their socially optimal joining strategy in the fully unobservable case. For each case, we find that there may be multiple equilibria but not all of them are stable, and stable equilibria do not always exist. For the fully observable queues, the stable equilibrium thresholds in a vacation period and in a service period are independent of F. For the fully unobservable queues, customers' equilibrium behavior is inconsistent with their socially optimal behavior, and there always exists an optimal F to maximize social welfare. So the system manager can achieve social optimization by controlling arrivals and the batch size F.

Automated summary

This paper investigates customer joining strategies in single-server Markovian queues with batch limited service rule and multiple vacations. The study considers both fully observable and fully unobservable cases and derives equilibrium and socially optimal joining strategies. The results show that there may be multiple equilibria in both cases, but stable equilibria are not always present. For fully observable queues, the stable equilibrium thresholds are independent of the batch size F. However, for fully unobservable queues, customers' equilibrium behavior diverges from their socially optimal behavior, and there exists an optimal F to maximize social welfare. Hence, the system manager can achieve social optimization by controlling arrivals and the batch size F.