A note on the GI/GI/\(\infty\) system with identical service and interarrival-time distributions (Q1876054)

The paper deals with the steady state distribution -- in particular with the first two moments -- of the number of busy servers in a GI/GI/\(\infty\) system in which the service time distribution \(H\) is identical to the interarrival time distribution \(F\), i.e. \(H=F\). This particular system arises in the analysis of tandem service systems, where the first node is a finite server group fed by an infinite pool of customers, and the second node is an infinite server group and all service times are i.i.d. By specializing Takac's classical results for the binomial moments in general GI/GI/\(\infty\) systems, for the mentioned special case \(H=F\) three expressions for the variance \(V\) of the number of busy servers are given. In the special case of a mixture of a constant and an exponentially distributed time the resulting explicit expression shows that an approximation given by Rajaratnam and Takawira via simulation results is pretty close to the exact value. For the special case of a gamma distribution the resulting explicit formula for the variance verifies an expression conjectured by Rajaratnam and Takawira.