Moment formulae for a class of mixed multi-job-type queueing networks

Queueing network models play an important role during each stage of a computer system's life cycle (from initial conception to system maturity), where in each stage broadly applicable performance analysis tools are needed. This paper presents new results which contribute to the foundations of a tool to support performance analysis and modeling activities. In dealing with some performance issues, it is important to be able to quantify distribution or moment information, because these quantities can influence system capacity and service and performance measures. It is also important that models include the effect of congestion adaptive I/O devices, in a stable and efficient manner, for this inclusion can significantly affect the outcome of studying certain performance issues. We address the problem of direct, recursive computation of moments of the queue size distributions at a class of service centers embedded in a mixed network of queues. The parameterized class includes state-dependent processing rates useful in modeling congestion adaptive I/O devices. We also present results for calculating moments of both the waiting time and virtual delay (work backlog) distributions at a class of service centers. In addition, we obtain a Little's Law type of relation between delay moments and queue size factorial moments. For a class of networks, an algorithm is given for the direct, recursive computation of the tail of the node delay distribution.