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 α of the service capacity to station 1 and 1-α 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 α. 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 α = 0 and α = 1. We extend these results to k ≥ 3 stations, sample paths that allow for server breakdown and repair, and to a tandem arrangement of single-server tandem queues.