Biased processor sharing in fork-join queues

We consider a fork-join system in which a fixed amount of computational resources has to be distributed among the K tasks forming the jobs. The queueing disciplines of the fork- and join- queues are First Come First Served. At each epoch, at most K tasks are in service while the others wait in the fork-queues. We propose an algorithm with a very simple implementation that allocates the computational resources in a way that aims at minimizing the join-queue lengths, and hence at reducing the expected job service time. We study its performance in saturation and under exponential service time and provide a methodology to derive the relevant performance indices. Explicit closed-form expressions for the expected response time and join-queue length are given for the cases of jobs consisting of two, three and four tasks