An Analytical Throughput Approximation for Closed Fork/Join Networks

Queueing networks featuring fork/join stations are natural models for a variety of computer and manufacturing systems. Unfortunately, an exact solution for a Markovian fork/join network can only be obtained by analyzing the underlying Markov chain using numerical methods, and these methods are computationally feasible only for networks with small population sizes and numbers of service stations. In this paper we present a new, simple, and accurate analytical approximation method to estimate the throughput (and other performance metrics) of a closed queueing network that features a single fork/join station receiving inputs from general subnetworks. An extensive numerical study illustrates the high accuracy of our proposed technique, especially for networks with large populations and numbers of stations. It also shows that the accuracy of our approximation method improves with increasing population size, deteriorating network balance, and increasing number of stations when the added stations weaken the network balance. Furthermore, our method has significant computational advantages compared to simulation and existing approximation techniques, the latter of which are in general less accurate than ours and in many cases even fail to provide a solution in our numerical study. We also bound analytically the relative error of our method for a broad class of networks, which provides theoretical support for some of our numerical observations.