Robust bounds and throughput guarantees for closed multiclass queueing networks

Abstract

To use queueing theory for analyzing real computing systems, we may make assumptions that are strictly speaking untrue. The problem is especially severe for multiclass systems with widely differing service times. This paper provides an exact analysis for bounds for systems with greatly relaxed assumptions. Service times can have arbitrary NBUE distributions, different by class even at FIFO nodes. Routing can be arbitrary, including dependencies along the route, provided the number of visits to a device per response cycle is random with a known expectation. Only the mean service time and mean visit rates at nodes need to be specified. 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 queueing network models of systems with infinite-server nodes as well as queueing nodes with various service disciplines: FIFO, processor sharing, and priority (preemptive as well as non-preemptive) scheduling. Because the feasible region may be a complicated shape which is difficult 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. A method for computing approximate system throughput from the box bounds is also proposed in the paper. Using Little's law and utilization law with the bounds on throughput bounds on response times and device utilizations are obtained. These analytic techniques can be effectively utilized for analyzing the performance of distributed systems as well as other types of computing systems.