Work-Conserving Tandem Queues

Consider a tandem queue of two single-server stations with only one server for both stations, who may allocate a fraction alpha of the service capacity to station 1 and 1-alpha to station 2 when both are busy. A recent paper treats this model under classical Poisson, exponential assumptions. Using work conservation and FIFO, we show that on every sample path (no stochastic assumptions), the waiting time in system of every customer increases with alpha. For Poisson arrivals and an arbitrary joint distribution of service times of the same customer at each station, we find the average waiting time at each station for alpha=0 and alpha=1. We extend these results to k is greater than or equal to 3 stations, sample paths that allow for server breakdown and repair, and to a tandem arrangement of single-server tandem queues.