We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

We consider an M/G/1 retrial queue, where the service time distribution has a finite exponential moment. We show that the tail of the queue size distribution is asymptotically given by a geometric function multiplied by a power function. The result is obtained by investigating analytic properties of probability generating functions for the queue size and the server state.

This study proposes an advanced method to factor in the contributions of individual group members engaged in an integrated group project using peer assessment procedures. Conway et al. proposed the Individual Weight Factor (IWF) method for peer assessment which has been extensively developed over the years. However, most methods associated with IWF use a simple average algorithm which still has limitations in effectively reflecting abnormal situations. Therefore, this study proposes an iterative method for measuring a fair IWF by considering each assessor's reliability. The proposed algorithm considers each assessor's reliability which is evaluated by an analysis of variance of the scores given by the assessor. This relative reliability factor is used to improve the fairness of the IWF and reduce the effect of unfair or unreliable assessors' scores. A comparison with previous methods demonstrates that the proposed method resulted in substantially more fair grades being given to individual students for group projects.

We study a fork-join processing network in which jobs arrive according to a Poisson process and each job splits into m tasks, which are simultaneously assigned to m nodes that operate like M/M/s queueing systems. When all of its tasks are finished, the job is completed. The main result is a closed-form formula for approximating the distribution of the network's response time (the time to complete a job) in equilibrium. We also present an analogous approximation for the distribution of the equilibrium queue length (the number of jobs in the system), when each node has one server. Kolmogorov-Smirnov statistical tests show that these formulae are good fits for the distributions obtained from simulations.

Abstract: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.