Stochastic Storage Processes pp 19-47 | Cite as

# The Queue G/G/1 and Associated Random Walk

Chapter

## Abstract

We consider the single-server queueing system where successive customers arrive at the epochs

*t*_{0}(= 0),*t*_{1},*t*_{2}, ..., and demand services times υ_{1}, υ_{2}, ... . The inter-arrival times are then given by*u*_{ n }=*t*_{ n }−*t*_{n−1}(*n*≥ 1). Let*X*_{ k }= υ_{ k }−*u*_{ k }(*k*≥ 1), and*S*_{0}≡ 0,*S*_{ n }=*X*1 +*X*_{2}+ ⋯ +*X*_{ n }(*n*≥ 1). We assume that the*X*_{ k }are mutually independent random variables with a common distribution; the basic process underlying this queueing model is the random walk {*S*_{ n }}. To see this, let*W*_{ n }be the waiting time of the*n*th customer and*I*_{ n }the idle period (if any) that just terminates upon the arrival of this customer. Then clearly for*n*≥ 0$$
{W_{n + 1}} = \max \left( {0,{X_{n + 1}} + {W_n}} \right),\;\;\;{I_{n + 1}} = - \min \left( {0,{X_{n + 1}} + {W_n}} \right).
$$

(1)

## Keywords

Random Walk Queue Length Busy Period Idle Period Queue Length Process
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer-Verlag New York Inc. 1980