Analysis of the fork-join queue

A fork-join queue is considered as a typical model of parallel processing systems with arrival and departure synchronizations. An approach for obtaining the transient and the steady-state solutions of the fork-join queue in terms of the virtual waiting times, which can be used to obtain the response time and the delay between the fork and the join instants, is presented. With the restriction of the service time being the Erlangian distribution, the formulations result in a form of a series, the worth of which is in actually computing numerical results. The joint-state-occupancy probabilities can be easily deduced from the results. The approximate solutions in the steady state are presented for the cases where the interarrival time distributions are exponential and hyperexponential, and the service-time distributions are exponential and two-state Erlangian. The parallelism achievable with synchronization constraints is examined from the approximations