# Continuity theorems for the $M/M/1/n$ queueing system

*Stochastic Processes Seminar Series*

*by Professor Vyacheslav M. Abramov*

*Institution:*School of Mathematical Sciences, Monash University

*Date: Thu 7th July 2005*

*Time: 1:15 AM*

*Location: Room 213, Richard Berry Building*

*Abstract*: In this report continuity theorems are discussed for the number of

losses during a busy period of the $M/M/1/n$ queue, when the

service time probability distribution, slightly different in

certain sense from the exponential distribution, is approximated

by that exponential distribution. Continuity theorems are obtained

in the form of one or two-side stochastic inequalities. In this

discussion we show how the bounds of these inequalities are

changed if one or other assumption, associated with specific

properties of the service time distribution (precisely described

in the report), is done. Specifically, some parametric families of

service time distributions are discussed, and uniform estimations

(given for all possible values of the parameter) and local

estimations (where the parameter is fixed and takes only the given

value) are established. The analysis of the report is based on the

level crossing approach and some characterization properties of

exponential distribution.

*For More Information:* Aihua Xia at xia@ms.unimelb.edu.au