Dynamic Control of the Join-Queue Lengths in Saturated Fork-Join Stations

The analysis of fork-join queueing systems has played an important role for the performance evaluation of distributed systems where parallel computations associated with the same job are carried out and a job is considered served only when all the parallel tasks it consists of are served and then joined. The fork-join nodes that we consider consist of K >= 2 parallel servers each of which is equipped with two FCFS queues, namely the service-queue and the join-queue. The former stores the tasks waiting for being served while the latter stores the served tasks waiting for being joined. When the queueing station is saturated, i.e., the service-queues are never empty, we observe that the join-queue sizes tend to grow infinitely even if the expected service times at the servers are the same. In fact, this is due to the variance of the service time distribution. To tackle this problem, we propose a simple service-rate control mechanism, and show that under the exponential assumption on the service times, we can analytically study a set of relevant performance indices. We show that by selectively reducing the speed of some servers, significant energy saving can be achieved.