Sojourn time in a single-server queue with threshold service rate control (Q2789363)

The paper under review considers a single-server queueing system, where customers arrive according to a Poisson stream with rate \(\lambda\) and receive service in order of arrival. The service requirements are exponentially distributed with mean 1. The rate of the server can be either \(\mu_0\) or \(\mu_1\) and this service rate can be adapted at random inspection times that occur according to a Poisson stream with rate \(\gamma\). For convenience, it is assumed that \(\mu_1>\mu_0\) even if under the stability condition this assumption may be removed. When the number of customers in the system is above the threshold \(K\), the service rate is upgraded to the high rate \(\mu_1\), otherwise it is downgraded to the low rate \(\mu_0\). The aim of the paper is to determine the stationary distribution of the sojourn time. This is a challenging problem, since due to adaptable service rate, the sojourn time does not only depend on the state seen at arrival, but it also depends on future arrivals. The paper under review determines the Laplace transform of the stationary sojourn time and describes a procedure to compute all moments. The special case of continuous inspection is analyzed first, where the service rate immediately changes once the threshold is crossed. Then, the analysis is extended to random inspection times. This extension requires the development of a new methodological tool, namely, matrix generating functions. The power of this tool is that it can also be used to analyze generalizations to phase-type services and inspection times.