Using cumulant functions in queueing theory

A new procedure that obtains summary measures for the state distribution of state-dependent Markovian queueing systems and networks as they evolve over the transient period is developed. This procedure maintains computational tractability for multi-server systems and for large networks of a general topology. The procedure involves defining a partial differential equation that relates a moment generating function to the rates of possible changes in the state of the model in a small interval of time. The partial differential equation then yields a closed set of approximating differential equations by utilizing a truncated cumulant generating function. Numerically solving these differential equations describes low order cumulants that correspond directly to key summary measures of the state distribution. Numerical examples comparing the first and second cumulant of a common M/ M/1 queueing system and of queueing networks under various topologies to exact answers genererated via Kolmogorov equations demonstrate the accuracy of this procedure.