M/G/k with Exponential Setup

In this paper, we consider the M/G/k queueing system with setup times. This particular queueing model is common in manufacturing systems, where idle machines are turned off to save on operating costs, as well as in server farms, where idle servers are turned off to conserve power. While recent literature has analyzed the M/M/k system with exponential setup times, no closedform solutions were obtained. We provide the first analytical closed form expressions for the mean response time, limiting distribution of the system states, as well as the z-transform for the number of jobs in system for the M/M/k system with exponential setup times. In particular, we prove the following decomposition property: the mean response time of the M/M/k system with exponential setup times differs from the mean response time of an M/M/k system without setup times, by a constant factor, which is the mean of the exponential setup time. Using matrix analytic methods and simulations, we show that the above decomposition property may also hold for the M/G/k system with exponential setup times.