Analysis of fork/join queueing networks with retrials

In this paper, we consider a fork/join queueing network with retrials. Jobs arrive at the network according to a Poisson process with rate Л. Each node operates like a /M/1/B queueing system under a FCFS discipline. When each node has less then B+1 tasks, an arriving job is split into M tasks which are simultaneously assigned to the M nodes. The tasks are serviced independently, and their service times at node i have an exponential distribution with rate ц,-, i = 1, ..., M. When all of its M tasks are finished, the job is completed and exits the network. In other case, the job goes to the retrial orbit to retry for service after a random time. The retrial policy is assumed to be independent of the number of jobs in the orbit, i.e., a constant retrial policy. We represent the state of the network over time by the stochastic process x(t) = (r(t), rn(t), nM(t)), where r(t) denotes the number of jobs in orbit at time t, n(t) denotes the number of tasks in node i at time t. This is an (M + 1)-dimensional continuous time Markov chain on the state space X, X = {(r,щ,.., n_M) : r > 0,0 < n_i < B + 1,i = 1,..., m}. Applying a matrix-geometric approach, we obtain the stationary distribution of the number of jobs in the network under exponential assumptions. Using the distribution, we determine performance measures. Finally, some numerical examples and a section of conclusions commenting the main research contributions of this paper are presented. The results can be used for the performance analysis of multiprocessor systems and other modern distributed systems.

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