Performance analysis of parallel processing systems

A centralized parallel processing system with job splitting is considered. In such a system, jobs wait in a central queue, which is accessible by all the processors, and are split into independent tasks that can be executed on separate processors. This parallel processing system is modeled as a bulk arrival M X /M/c queueing system where customers and bulks correspond to tasks and jobs, respectively. Such a system has been studied in [1, 3] and an expression for the mean response time of a random customer is obtained. However, since we are interested in the time that a job spends in the system, including synchronization delay, we must evaluate the bulk response time rather than simply the customer response time. The job response time is the sum of the job waiting time and the job service time. By analyzing the bulk queueing system we obtain an expression for the mean job waiting time. The mean job service time is given by a set of recurrence equations. To compare this system with other parallel processing systems, the following four models are considered: Distributed/Splitting (D/S), Distributed/No Splitting (D/NS), Centralized/Splitting (C/S), and Centralized/No Splitting (C/NS). In each of these systems there are c processors, jobs are assumed to consist of set of tasks that are independent and have exponentially distributed service requirements, and arrivals of jobs are assumed to come from a Poisson point source. The systems differ in the way jobs queue for the processors and in the way jobs are scheduled on the processors. The queueing of jobs for processors is distributed if each processor has its own queue, and is centralized if there is a common queue for all the processors. The scheduling of jobs on the processors is no splitting if the entire set of tasks composing that job are scheduled to run sequentially on the same processor once the job is scheduled. On the other hand, the scheduling is splitting if the tasks of a job are scheduled so that they can be run independently and potentially in parallel on different processors. In the splitting case a job is completed only when all of its tasks have finished execution. In our study we compare the mean response time of jobs in each of the systems for differing values of the number of processors, number of tasks per job, server utilization, and certain overheads associated with splitting up a job. The M X /M/c system studied in the first part of the paper corresponds to the C/S system. In this system