On transient queue-size distribution in a finite-buffer model with threshold waking and early setup policy

A finite-buffer queueing system with threshold waking and early setup policy is investigated. The arrival stream is governed by a Poisson process while service times are assumed to be generally distributed. The natural FIFO processing discipline is used. Every time when the system empties, a type-specific energy saving policy is initialized that is a mixture of the classical N-type policy and an early setup mechanism. Namely, if the level of accumulated messages reaches M≤N, a generally distributed setup time is started, during which the service station achieves full readiness for processing. If, at the completion epoch of the setup time, the state of the system (the number of accumulated messages) equals at least N, then the service begins immediately. Otherwise, the service station waits (being ready for processing) for the Nth arrival. The representation for the Laplace transform of the transient queue-size distribution is obtained using the analytical approach based on the idea of embedded Markov chain, the formula of total probability, linear algebra and renewal theory. A numerical example and simulational study are attached.