The M/M/∞Service System with Ranked Servers in Heavy Traffic

✍ Scribed by G. F. Newell (auth.)

• Publisher: Springer-Verlag Berlin Heidelberg
• Year: 1984
• Tongue: English
• Leaves: 141
• Series: Lecture Notes in Economics and Mathematical Systems 231
• Edition: 1
• Category: Library

✦ Synopsis

We are concerned here with a service facility consisting of a large (- finite) number of servers in parallel. The service times for all servers are identical, but there is a preferential ordering of the servers. Each newly arriving customer enters the lowest ranked available server and remains there until his service is completed. It is assumed that customers arrive according to a Poisson process of rate A , that all servers have exponentially distributed service times with rate ~ and that a = A/~ is large compared with 1. Generally, we are concerned with the stochastic properties of the random function N(s ,t) describing the number of busy servers among the first s ordered servers at time t. Most of the analysis is motivated by special applications of this model to telephone traffic. If one has a brunk line with s primary channels, but a large number (00) of secondary (overflow) channels, each newly arriving customer is assigned to one of the primary channels if any are free; otherwise, he is assigned to a secondary channel. The primary and secondary channels themselves could have a preferential ordering. For some purposes, it is convenient to imagine that they did even if an ordering is irrelevant.

✦ Table of Contents

Front Matter....Pages N2-XI
Introduction....Pages 1-6
Limit properties for a » 1....Pages 6-12
Descriptive properties of the evolution....Pages 12-19
The overflow distribution....Pages 19-67
Joint distributions....Pages 67-89
A diffusion equation....Pages 89-96
Transient properties....Pages 96-109
Equilibrium properties of the diffusion equation....Pages 109-115
Equivalent random method....Pages 115-122
Back Matter....Pages 123-129

✦ Subjects

Operations Research/Decision Theory; R & D/Technology Policy; Economic Theory