On the service time in a workload-barrier M/G/1 queue with accepted and blocked customers

Abstract

We consider an M/G/1 queue with a workload barrier at finite level K > 0. Arriving customers are accepted into the system only if their waiting time plus service time does not exceed K; otherwise they are blocked and cleared immediately. This ‘accept/block’ regulation causes the pre-arrival ‘prior’ pdf (probability density function) of service time for all arrivals, to transform into the pdf of the ‘posterior’ service time reflecting the effect of the barrier at level K on both accepted and blocked customers. We derive the pdf of the posterior service time, its expected value, the expected numbers of customers accepted or blocked in a busy cycle, and a specific value of K which is greater than the median of the prior pdf, where equal expected numbers of customers are accepted and blocked in a busy cycle. We propose an expression for distance, to measure the discrepancy between the posterior and prior pdfs. Examples are given. This service-time viewpoint is of theoretical interest, and has potential applications in various related stochastic models.