Article
Performance Analysis of
M/M
(a,b)
/1 Queuing Model
with Balking and
State-dependent Reneging
and Service
Kuntal Bakuli
1
Manisha Pal
1
Calcutta Statistical Association Bulletin
69(1) 76–86
© 2017 Calcutta Statistical Association, Kolkata
SAGE Publications
sagepub.in/home.nav
DOI: 10.1177/0008068317696551
http://csa.sagepub.com
Abstract
The article investigates a M/M
(a,b)
/1 queuing model with impatient customers. The size of a batch taken
up for service depends on the number of customers present in the waiting line. The server serves
a minimum of ‘‘a’’ and a maximum of ‘‘b’’ customers at a time. The service time is assumed to be
exponentially distributed with mean dependent on the batch size. Customers arrive to the system as
a Poisson process, and may leave on finding a long queue or may renege after waiting in the queue for
an exponentially distributed time. The model is analyzed to find different measures of effectiveness of
the system. A new measure of performance is also discussed. The approach adopted for analysis of the
model is based on embedded Markov chain.
Keywords
Bulk service, balking and reneging, Poisson arrival, embedded Markov chain, performance measures
AMS 2000 subject classification: 60J10
1. Introduction
Bulk service and bulk arrival are common phenomena in real life. Some common examples of systems
with bulk service are telecommunication, transportation, production, airline scheduling, and elevators.
Over the past few decades, much study has been carried out on bulk service queuing systems. Jaiswal
[1]
discussed a bulk service queue with variable capacity. Powell and Humblet
[2]
investigated general control
strategy for bulk service queues. Downton
[3]
studied the waiting time in bulk service model. Jayaraman,
Nadarajan, and Sitrarasu
[4]
analyzed a general bulk service system with arrival rate dependent on server
breakdown. Dshalalow
[5]
explored the D-policy for bulk queuing system. Jain and Singh
[6]
investigated
state-dependent bulk service queue with delayed vacations. However, very few authors considered bulk
1
Department of Statistics, University of Calcutta, India.
Corresponding author:
Kuntal Bakuli, Department of Statistics, University of Calcutta, Kolkata-700 019, India.
E-mail: kuntalbakuli17@gmail.com