Two different kinds of heavy-traffic limit theorems have been proved for s-server queues. The first kind involves a sequence of queueing systems having a fixed number of servers with an associated sequence of traffic intensities that converges to the critical value of one from below. The second kind, which is often not thought of as heavy traffic, involves a sequence of queueing systems in which the associated sequences of arrival rates and numbers of servers go to infinity while the service time distributions and the traffic intensities remain fixed, with the traffic intensities being less than the critical value of one. In each case the sequence of random variables depicting the steady-state number of customers waiting or being served diverges to infinity but converges to a nondegenerate limit after appropriate normalization. However, in an important respect neither procedure adequately represents a typical queueing system in practice because in the (heavy-traffic) limit an arriving customer is either almost certain to be delayed (first procedure) or almost certain not to be delayed (second procedure). Hence, we consider a sequence of (GI/M/S) systems in which the traffic intensities converge to one from below, the arrival rates and the numbers of servers go to infinity, but the steady-state probabilities that all servers are busy are held fixed. The limits in this case are hybrids of the limits in the other two cases. Numerical comparisons indicate that the resulting approximation is better than the earlier ones for many-server systems operating at typically encountered loads.
This paper describes the Queueing Network Analyzer (QNA), a software package developed at Bell Laboratories to calculate approximate congestion measures for a network of queues. The first version of QNA analyzes open networks of multiserver nodes with the first-come, first-served discipline and no capacity constraints. An important feature is that the external arrival processes need not be Poisson and the service-time distributions need not be exponential. Treating other kinds of variability is important. For example, with packet-switched communication networks we need to describe the congestion resulting from bursty traffic and the nearly constant service times of packets. The general approach in QNA is to approximately characterize the arrival processes by two or three parameters and then analyze the individual nodes separately. The first version of QNA uses two parameters to characterize the arrival processes and service times, one to describe the rate and the other to describe the variability. The nodes are then analyzed as standard GI/G/m queues partially characterized by the first two moments of the interarrivaltime and service-time distributions. Congestion measures for the network as a whole are obtained by assuming as an approximation that the nodes are stochastically independent given the approximate flow parameters.
The purpose of this document is to summarize main points of the paper, "Numerical Inversion of Laplace Transforms of Probability Distributions", and provide R code for the Euler method that is described in the paper. The code is used to invert the Laplace transform of some popular functions. Context Laplace transform is a useful mathematical tool that one must be familiar with, while doing applied work. It is widely used in Queueing models where probability distributions are characterized in terms of transforms.
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