Sojourn times in G/M/1 fork-join networks

We consider a processing network in which jobs arrive at a fork-node according to a renewal process. Each job requires the completion of m tasks, which are instantaneously assigned by the fork-node to m task-processing nodes that operate like G/M/1 queueing stations. The job is completed when all of its m tasks are finished. The sojourn time (or response time) of a job in this G/M/1 fork-join network is the total time it takes to complete the m tasks. Our main result is a closed-form approximation of the sojourn-time distribution of a job that arrives in equilibrium. This is obtained by the use of bounds, properties of D/M/1 and M/M/1 fork-join networks, and exploratory simulations. Statistical tests show that our approximation distributions are good fits for the sojourn-time distributions obtained from simulations.