Idle and busy periods in stable M/M/k queues

Abstract

This paper finds the first and second moments of the number of arrivals in a stable M/M/k queue during an idle period, i.e. in a period when at least one server is idle. These and similar results are used along with renewal theory asymptotics to find the first and second moments of the proportion of arrivals occurring while there are at least ξ customers waiting for service. The asymptotics of the second moment for large ξ are established.