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An application of the reflection principle to the transient analysis of the M / M /1 queue
Author(s) -
Towsley Don
Publication year - 1987
Publication title -
naval research logistics (nrl)
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 68
eISSN - 1520-6750
pISSN - 0894-069X
DOI - 10.1002/1520-6750(198706)34:3<451::aid-nav3220340310>3.0.co;2-8
Subject(s) - reflection (computer programming) , transient (computer programming) , interval (graph theory) , queueing theory , burke's theorem , queue , reflection principle (wiener process) , mathematics , queueing system , mathematical analysis , combinatorics , discrete mathematics , computer science , queue management system , statistics , fork–join queue , operating system , knowledge management , diffusion process , innovation diffusion , geometric brownian motion , programming language
Abstract This paper applies the well‐known reflection principle for random walks to the analysis of the transient M/M/ 1 queueing system. A closed‐form solution is obtained for the probability that exactly i arrivals and j departures occur over an interval of length t in an M/M/ 1 queueing system that contains n users at the beginning of the interval. The derivation of this probability is based on the calculation of the number of paths between two points in a two‐dimensional −y coordinate system that lie above the x axis and touch the x axis exactly r times. This calculation is readily performed through the application of the reflection principle.