Discrete-Time Competing Queues with Geometric Service Requirements: Stability, Parameter Estimation and Adaptive Control

In this paper, several discrete-time queues compete at the beginning of each time slot for the attention of a single server with infinite capacity. The service requirements are geometric with queue dependent parameters, independent from customer to customer and independent from the arrivals. The stability of such systems is characterized under the action of an arbitrary, not necessarily Markovian, service allocation policy. Maximum likelihood estimates are obtained in explicit form for the service rates and their strong consistency is established. The results are then applied to the problem of adaptively controlling such systems when the service rates are unknown, under the long-run average performance criterion with instantaneous cost linear in the queue sizes.