An Analytical Approximation for the Throughput of a Closed Fork/Join Network with Multi-Station Input Subnetworks

Fork/join stations are used for modeling synchronization between entities, and fork/join queueing networks are natural models for a variety of communication and manufacturing systems: Parallel computer networks, fabrication/assembly systems, supply chains and material control strategies for manufacturing systems. Exact solutions of general fork/join networks can only be obtained by using numerical methods to analyze the underlying Markov chains. However, this method is computationally feasible only for networks with small population size and number of stations. In this paper, we present a simple approximation method to estimate the throughput of a closed queueing network that features a single fork/join station receiving inputs from multi-station subnetworks. Our technique uses aggregation to estimate the arrival process from input subnetworks. Given the estimated arrival process, we then derive closed form approximate expression for the network throughput by analyzing a simplified Markov chain. A numerical study shows that the proposed approximation is fairly effective, particularly for large network sizes.