Multi-Server Queues with Intermediate Buffer and Delayed Information on Service Completions

Abstract

A controller with an unlimited buffer receives messages to be dispatched to c servers downstream in the network. However, the queue sizes at the individual servers are not known exactly, since information on each service completion reaches the controller only after some random delay. The controller can dispatch messages as soon as they arrive, in a cyclic manner, to the c servers. Alternatively, s/he can wait until full information is gained and dispatch a waiting message to a server only when s/he is sure that the server is free. Another strategy is to maintain a limited intermediate buffer in front of the servers, and forward messages to this buffer when information on service completion reaches the controller. If a server completes a job and the intermediate buffer is non-empty, it starts serving a job from this buffer with no delay. Such situations are common in many real life processes (such as passport control procedures, or at large waiting rooms in public offices) where customers wait, in front of c servers, to be served. A customer walks (=delay) to the next idle server when s/he sees her/his “waiting number” flashing on the screen. We analyze this model when the underlying process is the M/M/c queue and the information delay is exponential. We use both: i) probability generating functions of the multi-dimensional state space to calculate the boundary probabilities, and ii) matrix geometric approach to derive the stability condition of the system. We show that the intermediate buffer scheme reduces queue sizes and waiting times. Numerical examples are presented.