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Analysis of Fork/Join and Related Queueing Systems

Fork/join (F/J) requests arise in contexts such as parallel computing, query processing in parallel databases, and parallel disk access in RAID. F/J requests spawn K tasks that are sent to K parallel servers, and the completion of all K tasks marks the completion of an F/J request. The exact formula for the mean response time of K = 2-way F/J requests derived under Markovian assumptions (RF/J2) served as the starting point for an approximate expression for RF/JK for 2 < K ≤ 32. When servers process independent requests in addition to F/J requests, the mean response time of F/J requests is better approximated by RmaxK, which is the maximum of the response times of tasks constituting F/J requests. RmaxK is easier to compute and serves as an upper bound to RF/JK. We discuss techniques to compute RmaxK and generally the maximum of K random variables denoting the processing times of the tasks of a parallel computation XmaxK. Graph models of computations such as Petri nets—a more general form of parallelism than F/J requests—are also discussed in this work. Jobs with precedence constraints may require multiple resources, which are represented by a queueing network model. We also discuss various queueing systems related to F/J queueing systems and outline their analysis.