Travel times in queueing networks and network sojourns

The paper first describes expected values of sojourn times for semi-stationary (or synchronous) networks. This includes sojourn times for units and sojourn times for the entire network. A typical sojourn time of a unit is the time it spends in a sector (set of nodes) while it travels through the network, and a typical network sojourn time is the busy period of a sector. The present results apply to a wide class of networks including Jackson networks with general service times, general Markov or regenerative networks, and networks with batch processing and concurrent movement of units. The results also shed more light on when Little’s law for general systems, holds for expectations as well as for limiting averages. Next, the paper describes the expectation of a unit’s traveltime on a general route in a ‘basic’ Markov network process (such as a Jackson process). Examples of travel times are the time it takes for a unit to travel from one sector to another, and the time between two successive entrances to a node by a unit. Finally, the paper characterizes the distributions of the sojourn times at nodes on certain overtake-free routes and the travel times on such routes for Markov network processes.