Applied Mathematical Sciences

Applied Mathematical Sciences, Vol. 2, 2008, no. 24, 1161 – 1167

On a Truncated Erlangian Queueing System with

State – Dependent Service Rate, Balking and Reneging

M. S. El – Paoumy

Department of Statistics, Faculty of Commerce, Dkhlia, Egypt
Al-Azhar University, Girl’s Branch
[email protected]

Abstract

The aim of this paper is to derive the analytical solution of the truncated Erlangian
service queue with state-dependent rate, balking and reneging (M/ER/I/N (
??, )) . We
obtain
s nP,, the probabilities that there are “n” units in the system and the unit in the
service occupces stage “s” (s = 1, 2,…, r ) .We treat this queue for general values of r ,
k and N.

Keywords: truncated Erlangian service queue, balking, reneging.

1 INTRODUCTION

This paper considers the queueing system M/Er/1/N with state – dependent service
rate, balking and reneging concepts .The Erlang distrbution, denoted by Er is a special
case of the gamma distribution, is named after A.K. Erlang who pioneered queueing
system theory for its application to congestion in telphone networks .The nontruncated
queue: M/Er/1 was solved by Morse 3 at r =2 and white et al. 4 Who obtained the
solution in the form of a generating function and the probabilities could be obtained by
a power series expansion. Al Seedy 1 gave an analytical solution of the queue:

1162 M. S. El – Paoumy

M/Er/1/N with balking only. This work had been followed by Kotb 2 who studied the
analytical solution of the state – dependent Erlangian queue: M/Er/1/N with balking by
using a very useful lemma. In this paper we treat the analytical solution of the queue:
M/Er/1/N
()??, for finite capacity considering by using a recurrence relations .We
obtain
s nP,, the probabilities that there are “n” units in the system and the unit in serive
occupies stage “s” (
r s? ? 1 in terms of 0P .
The probability of an empty system
0P is also obtained .The discipline considered is
first in first out (FIFO).

2 THE PROBLEM ANALYSIS

Consider the single – channel service time Erlangian queue having r – service stages
each with rate
n?, with the state – dependent and reneging in the form:
?
? ?
? ?
? ? + ? + ? =?? ??o t e t ft
This means that the units are served with two different rates
1?r or 2?r depending
on the number of units in the system whether
k n??1 or N n k??+1 respectively.
Also, consider an exponential interarrival pattern with rate
.n?Assume ()??1 be
the probability that a unit balks (does not enter the queue).
where :
=?p(a unit joins the queue), ; 1 , 1 0N n??