An Upper Bound Solution for Homogeneous Fork/Join Queuing Systems

In this paper, we use dynamic-bubblesort technology to analyze general first-in-first-out K-queue homogenous fork/join queuing (HFJ) systems for any K >; 2. Jobs arrive with a mean rate λ and a general arrival distribution. Upon arrival, a job forks into K tasks. Task k, k = 1, 2,...,K, is assigned to the kth queuing system, which is a first-in-first-out server with a general service distribution and an infinite capacity queue. A job leaves the HFJ system as soon as all its tasks complete their service. We mathematically prove an upper bound solution for the mean response time that we denote by T_K. The upper bound solution of general K-queue HFJ systems for any K ≥ 2 is very simple and practical-one only needs to simulate a small number of queues (e.g., 16 queues). The tightness is evaluated by comparing with the simulation of thousands of queues for three different HFJ cases. The maximum offset for our upper bounds over a//the simulations is less than 5 percent. The corresponding source codes (reusable) are offered on our website for others to use.