For functionals of multitype closed queueing networks, a conditional job-observer property is shown which provides more insight into the classical job-observer property. Applications and examples are given, including the classical job-observer property for the number of customers in a network, a representation of cycle time distributions and a basic formula for sojourn times.

A cyclic service system is composed of K channels (queues) and a single cyclically roving server who typically takes a positive amount of time to switch between channels. Research has previously focused on evaluating and computing performance measures (notably, waiting times) of fixed template routing schemes under three main service disciplines, the exhaustive, gated and limited service regimes.

In this paper, probabilistic results are derived that allow control strategies and optimal policies to be considered for the first time. By concentrating on a new objective function, we are able to derive rules of index form amenable for direct implementation to dynamically control the system at suitably defined decision epochs. These rules utilize current system information, are of an adaptive nature, and are shown to emanate from a general physical principle.

In a closed cycle of exponential queues where the first and the last nodes are multiserver queues while the other nodes are single-server queues, the cycle-time distribution has a simple product form. The same result holds for passage-time distributions on overtake-free paths in Gordon–Newell networks. In brief, we prove Burke's theorem on passage times in closed networks.