(n, k) fork-join queues are prevalent in popular distributed systems, erasure coding based cloud storages, and modern network protocols like multipath routing, estimating the sojourn time of such queues is thus critical for the performance measurement and resource plan of computer clusters. However, the estimating keeps to be a well-known open challenge for years, and only rough bounds for a limited range of load factors have been given. This paper developed a closed-form linear transformation technique for jointly-identical random variables: An order statistic can be represented by a linear combination of maxima. This brand-new technique is then used to transform the sojourn time of non-purging (n, k) fork-join queues into a linear combination of the sojourn times of basic (k, k), (k + 1, k + 1), ..., (n, n) fork-join queues. Consequently, existing approximations for basic fork-join queues can be bridged to the approximations for non-purging (n, k) fork-join queues. The uncovered approximations are then used to improve the upper bounds for purging (n, k) fork-join queues. Simulation experiments show that this linear transformation approach is practiced well for moderate n and relatively large k.
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 e 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