Robust Box Bounds: Throughput Guarantees for Closed Multiclass Queueing Networks with Minimal Stochastic Assumptions

To use queueing theory to analyze real systems such as computer communications networks, we make assumptions that are strictly speaking untrue. The problem is especially severe for multiclass networks with FIFO service and greatly diiering service times. This paper provides an exact analysis for bounds, for cases with greatly relaxed assumptions. Service times can have general (IFR) distributions, diierent by class even at FIFO nodes. Routing can be arbitrary, including dependencies along the route, provided the number of visits to each node is a random variable. Only the mean service time and mean visit rates at nodes need be speciied. A new lower throughput bound is found which gives a minimum guaranteed throughput for each class; together with the familiar multiclass asymptotic upper bounds they give a convex feasible region in a multidimensional throughput space. A detailed analysis is given for systems with FIFO and innnite-server nodes, and the extension to processor-sharing nodes is described. Because the feasible region may be a complicated shape which is diicult to visualize, the results can be re-interpreted as a set of bounds on the separate throughputs. This is equivalent to a circumscribed rectangular region called the \Robust Box Bounds". Computation of these bounds is carried out by a novel technique based on interval arithmetic as implemented in BNR Prolog language. These bounds can be effectively used for the performance analysis of computer networks as well as other types of computing systems.