Travel and Sojourn Times in Stochastic Networks

This study concerns travel times in a stochastic network in which units move among the nodes that process the units. The network may be closed or open, there may be several types of units and the processing at each node may depend on the numbers of units at the other nodes. This is a Jackson network when the nodes operate independently. The travel time on a "route" in this network is the time it takes for an arbitrary unit to traverse one of a series of nodes that constitute the route, when the network is in equilibrium. An example is the time for a unit to move from one set of nodes to another. We present an expression for the expectation of a general travel time. We also characterize the distribution of the travel time, and the sojourn times at the nodes, on an overtake-free path. This includes the known results on the product-form distribution of sojourn times at the nodes on overtake-free paths in Jackson networks.