On value functions for FCFS queues with batch arrivals and general cost structures Abstract   We develop a unified framework for analyzing and optimizing costs for systems of FCFS queues with batch arrivals, setup delays and a general nonlinear cost structure that includes costs associated with energy used, setup times and Quality of Service (QoS) measures. We focus on the MXG1 and GeoXG1 queues with i.i.d. service times, but our results hold also for arbitrary i.i.d. batch structures where service times within a batch may depend on the batch size and have different, possibly dependent, distributions. We use the notion of value functions from the theory of Markov decision processes (MDPs), along with exponential cost functions, to develop the notion of, and simple expressions for, value generating functions. These can be used to find efficient energy control and job dispatching policies.

A ProcessorSharing model for the Performance of Virtualized Network Functions The parallel execution of requests in a Cloud Computing platform, as for Virtualized Network Functions, is modeled by an MXM1 ProcessorSharing (PS) system, where each request is seen as a batch of unit jobs. The performance of such paralleled system can then be measured by the quantiles of the batch sojourn time distribution. In this paper, we address the evaluation of this distribution for the MXM1PS queue with batch arrivals and geometrically distributed batch size. General results on the residual busy period (after a tagged batch arrival time) and the number of unit jobs served during this residual busy period are first derived. This enables us to provide an approximation for the distribution tail of the batch sojourn time whose accuracy is confirmed by simulation.