The M/M/1 Fork-Join Queue with Variable Sub-Tasks

The fork-join queue models parallel resources where arriving jobs divide into various number of sub-tasks that are assigned to unique devices within the parallel resource. Each device in the parallel resource is modeled by M=M=1 queueing servers. A job completes execution and departs the parallel resource after all its sub-tasks complete execution. This paper analyzes N-server fork-join queues where arriving jobs divide into 1 K N sub-tasks that are assigned to unique servers of the fork-join queue. There is no known closed-form solution for N > 2 fork-join queues. The paper presents an O(log K) algorithm for computing the mean response time pessimistic and optimistic bounds and for computing the mean response time approximation of the fork-join queue. The error bounds for the response time bounds and approximation are presented.