Application of Two Forms of the Erlang Distribution Law in Queueing Theory

Abstract

The purpose of this work is to compare two forms of the Erlang distribution law: ordinary and normalized. The ordinary Erlang distribution is a special case of a more general gamma distribution and its mathematical expectation depends on the order of the distribution k. For a normalized distribution, the mathematical expectation will not depend on the order of the distribution k; this is the meaning of the normalization operation. Consequently, these two forms of the Erlang distribution law will differ in their numerical characteristics. The paper considers the problem of how these two forms of the distribution law are applied in the theory of queuing and how they affect the main characteristic of queuing systems (QS) - the average delay of claims arriving for service in the system. The rest of the QS characteristics are derivatives of the average delay. For this, three different QSs are considered, including the Erlang distribution law. The method of spectral decomposition of the solution of the Lindley integral equation was used as a mathematical apparatus for studying the QS. For the practical application of the results obtained, the method of moments of the theory of probability was used. Basically, queueing systems G/G/1 with general distribution laws are in demand for modeling data transmission systems for various purposes, including computer and telecommunication networks. This is especially true since for G/G/1 systems there is no solution for the average latency in the final form for the general case. Therefore, such systems are investigated using examples with particular distribution laws.