On Markovian traffic with applications to TES processes
type: Item
instance of: scholarly article
title: On Markovian traffic with applications to TES processes
zbMATH Open document ID: 0812.60038
DOI: 10.1155/S1048953394000316
author: David L. Jagerman
author: Benjamin Melamed
published in: Journal of Applied Mathematics and Stochastic Analysis
publication date: 1995-03-16T00:00:00Z
full work available at URL: https://eudml.org/doc/47147
review text: Summary: Markov processes are an important ingredient in a variety of stochastic applications. Notable instances include queueing systems and traffic processes offered to them. This paper is concerned with Markovian traffic, i.e., traffic processes whose inter-arrival times (separating the time points of discrete arrivals) form a real-valued Markov chain. As such this paper aims to extend the classical results of renewal traffic, where interarrival times are assumed to be independent, identically distributed. Following traditional renewal theory, three functions are addressed: the probability of the number of arrivals in a given interval, the corresponding mean number, and the probability of the times of future arrivals. The paper derives integral equations for these functions in the transform domain. These are then specialized to a subclass, \(\text{TES}^ +\), of a versatile class of random sequences, called TES (transform-expand- sample), consisting of marginally uniform autoregressive schemes with modulo-1 reduction, followed by various transformations. TES models are designed to simultaneously capture both first-order and second-order statistics of empirical records, and consequently can produce high- fidelity models. Two theoretical solutions for \(\text{TES}^ +\) traffic functions are derived: an operator-based solution and a matric solution, both in the transform domain. A special case, permitting the conversion of the integral equations to differential equations, is illustrated and solved. Finally, the results are applied to obtain instructive closed- form representations for two measures of traffic burstiness: peakedness and index of dispersion, elucidating the relationship between them.
Mathematics Subject Classification ID: 60G10
Mathematics Subject Classification ID: 60K20
Mathematics Subject Classification ID: 60G07
Mathematics Subject Classification ID: 60G55
Mathematics Subject Classification ID: 60J27
Mathematics Subject Classification ID: 65C05
Mathematics Subject Classification ID: 90B15
Mathematics Subject Classification ID: 90B22
zbMATH DE Number: 722520
zbMATH Keywords: peakedness functional
zbMATH Keywords: peakedness function
zbMATH Keywords: index of dispersion for intervals
zbMATH Keywords: queueing systems
zbMATH Keywords: traffic processes
zbMATH Keywords: second-order statistics of empirical records
MaRDI profile type: Publication
OpenAlex ID: W2059404963
description: scientific article; zbMATH DE number 722520