Power control in saturated fork-join queueing systems

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 First Come First Served queues, namely the service-queue and the join-queue. The former stores the tasks waiting to be served while the latter stores the served tasks waiting to be joined. Under heavy load conditions, the variance of the service times associated with the tasks tends to cause long join-queue lengths. In this work, we propose an algorithm to dynamically control the servers' speeds (e.g., via frequency scaling), that aims at reducing the power consumption of the servers whose join-queue lengths are longer than the others'. Under Markovian assumptions, we provide a model for the performance evaluation of the system in saturation that allows us to derive the expression for the steady-state distribution, the system's throughput and balance index. Finally, we derive the analytical expression for the marginal state probabilities of each server and provide upper and lower bounds for the expected power consumption.