University of Calicut

18 January 2019

Infinite Divisibility of Some Steady State Queue Characteristics

In this paper we consider various continuous/discrete time queueing models with general service time distribution. With an aim to identify queueing models we examine the structural properties like infinite divisibility of the stationary distribution of various characteristics of different queueing models and report some new results. Our investigation is further extended to bulk arrival models and multiple vacation queueing models, revealing the structural aspects of their steady state distributions.

Many of the well known distributions belong to the class of infinitely divisible distribu-tions. The concept of infinite divisibility and related concepts like self decomposability, stability etc. are useful in probability and statistics; these could be of potential impor-tance in both theoretical and practical problems. They basically concern certain structural aspects of distributions. In many situations, the verification of these prop-erties could be easier than that of certain other properties of the distribution. If such properties characterize or identify certain models fully or partially, one could utilize them with advantage in a model building problem or in the analysis of the model. The problem of identifiability occurs in all fields where stochastic modeling is widely used. In many practical situations, the main objective of the investigator is find the physical structure of the probability distribution of an observable random variable. Identification problems arise when the variable can be explained using several models. Manoharan et al. (2003) bring together the revelent materials on identifiability problems in diverse fields of queueing theory such as network, telecommunications, etc. In identification of queueing models the knowledge about the structural aspects of distributions frequently encountered in the theory of queues is very important. The structural properties such as unimodality, infinite divisibility, log-convexity, stability,etc. could provide us an insight about the behavior of the distributions.

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