Reduction Methods for Tandem Queuing Systems

A tandem stage queuing system has n stages A₁, A₂, …, A_n in series, where stage A_i consists of m_i parallel channels each having the same constant service time s_i . In addition, one of the stages may be a single-channel stage A_j having variable service times that are always ≧s_i/m_i for i ≠ j. Customers arrive at the first stage and proceed through the stages, in order, on a first-come, first-served, unlimited queuing basis. For any sequence of customer arrival times, the time spent in the system by each customer is independent of the order of the stages. It follows that certain steady-state and time- and customer-dependent problems for the system and for its stages, involving waiting time, queue length, and busy period, can for any interarrival distribution be reduced to corresponding problems for a system of fewer stages, possibly a single stage.