Constructing Matrix Exponential Distributions by Moments and Behavior around Zero

This paper deals with moment matching of matrix exponential (ME) distributions used to approximate general probability density functions (pdf). A simple and elegant approach to this problem is applying Padé approximation to the moment generating function of the ME distribution. This approach may, however, fail if the resulting ME function is not a proper probability density function; that is, it assumes negative values. As there is no known, numerically stable method to check the nonnegativity of general ME functions, the applicability of Padé approximation is limited to low-order ME distributions or special cases. In this paper, we show that the Padé approximation can be extended to capture the behavior of the original pdf around zero and this can help to avoid representations with negative values and to have a better approximation of the shape of the original pdf. We show that there exist cases when this extension leads to ME function whose nonnegativity can be verified, while the classical approach results in improper pdf. We apply the ME distributions resulting from the proposed approach in stochastic models and show that they can yield more accurate results

Improved Load Balancing in Large Scale Systems using Attained Service Time Reporting

Our interest lies in load balancing jobs in large scale systems consisting of multiple dispatchers and FCFS servers. In the absence of any information on job sizes, dispatchers typically use queue length information reported by the servers to assign incoming jobs. When job sizes are highly variable, using only queue length information is clearly suboptimal and performance can be improved if some indication can be provided to the dispatcher about the size of an ongoing job. In a FCFS server measuring the attained service time of the ongoing job is easy and servers can therefore report this attained service time together with the queue length when queried by a dispatcher. In this paper we propose and analyse a variety of load balancing policies that exploit both the queue length and attained service time to assign jobs, as well as policies for which only the attained service time of the job in service is used. We present a unified analysis for all these policies in a large scale system under the usual asymptotic independence assumptions. The accuracy of the proposed analysis is illustrated using simulation. We present extensive numerical experiments which clearly indicate that a significant improvement in waiting (and thus also in response) time may be achieved by using the attained service time information on top of the queue length of a server. Moreover, the policies which do not make use of the queue length still provide an improved waiting time for moderately loaded systems