Overflow analysis for finite waiting room systems

A system of multiple primary queues with finite waiting rooms, where blocked jobs are allowed to overflow onto a common secondary queue, also with finite waiting room, is considered. Poisson arrivals and exponentially distributed service times, possibly different ones, are assumed at each primary service facility. Service facilities consist of a single server or multiple servers. Based on a simple iterative expression for the Laplace-Stieltjes transform of the interoverflow time distribution, the peakedness of the overflow process from primary queues is studied as a function of the number of primary servers and buffers as well as the original primary load. Behaviors and relations are identified for extreme cases (low offered load or large waiting rooms) and illustrated with examples. An approximation consisting of a natural extension of Hayward's formula to finite waiting room systems is proposed to estimate the blocking seen by the overflow traffic offered to the secondary service facility. Numerical examples covering a wide range of systems show that the approximation is good over a wide range of loads for systems with a single primary queue, as well as at high load for systems with multiple primary queues.>