Computing performance bounds of fork-join parallel programs under a multiprocessing environment

We study a multiprocessing computer system which accepts parallel programs that have a fork-join computational paradigm. The multiprocessing computer system under study is modeled as K homogeneous servers, each with an infinite capacity queue. Parallel programs arrive at the multiprocessing system according to a series-parallel phase type interarrival process with mean arrival rate of h. Upon the program arrival, it forks into K-independent tasks and each task is assigned to an unique server. Each task´s service time has a k-stage Erlang distribution with mean service time of λ. A parallel program is completed upon the completion of its last task. This kind of queuing model has no known closed form solution in the general (K⩾2) case. In this paper, we show that by carefully modifying the arrival and service distributions at some imbedded points in time, we can obtain tight performance bounds. We also provide a computational efficient algorithm for obtaining upper and lower bounds on the expected response time. The methodology is flexible and allows one to trade-off the tightness of the bounds and computational cost