Research Reports from the Department of Operations

Authors

N.U. Prabhu

Document Type

Report

Publication Date

8-1-1967

Abstract

In the present survey, we follow the treatment given by Feller (1966). In Sections 2 and 3, we develop the basic results concerning the ladder processes arising from a general one-dimensional random walk. In Section 4, we apply these results to the waiting time in a single-server queue, with details worked out for the two cases where (i) the arrivals are Poisson, and (ii) service times are exponentially distributed. In Section 5, these two systems are again considered, and the concepts of Sections 2 and 3 are used to investigate the number of customers present in each system at certain epochs of time. The discussion of these simple cases reveals the essential features of the theory; a detailed account of the general single-server queue using ladder processes is given by Prabhu (1965b). It is the author's contention that the limiting properties of the queue (for example, the so-called heavy traffic behavior, inequalities for the mean waiting time, and other characteristics) arise as natural consequences of the results presented here, but we shall not discuss them in this paper.

Keywords

Operations research, Random walks (Mathematics), Markov processes, Poisson processes, Queuing theory, Distribution (Probability theory)

Publication Title

Technical Memorandums from the Department of Operations, School of Management, Case Western Reserve University

Issue

Technical memorandum no. 88

Rights

This work is in the public domain and may be freely downloaded for personal or academic use

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