Single Class Queueing Networks with Discrete and Fluid Customers on the Time Interval R

We discuss a model for general single class queueing networks which allows discrete and fluid customers and lives on the time interval R. The input for the model are the cumulative service time developments, the cumulative external arrivals and the cumulative routing decisions of the queues. A path space fixed point equation characterizes the corresponding behavior of the network. Monotonicity properties imply the existence of a largest and a smallest solution. Despite the possible non-uniqueness of solutions the sets of solutions have several nice properties. The set valued solution map is partially upper semicontinuous with respect to a quasi-linearly discounted uniform metric on the input paths space. In addition to this main result, we investigate convergence of approximate solutions, measurability, monotonicity and stationarity. We give typical examples for situations where solutions are non-unique and unique.