Equatorial Journal of Computational and Theoretical Science, 2 (1): 1-5
Journal Homepage: www.erjournals.com
ISSN: 0184-7937
QUEUING THEORY APPLICATION AT TICKET WINDOWS IN
RAILWAY STATIONS (A STUDY OF THE LAGOS TERMINUS,
IDDO, LAGOS STATE, NIGERIA)
Ituen-Umanah, Williams Udoh
Department of Business Administration (M.Sc.Operations Research), University of Lagos,
willise2n@yahoo.com; +2347035690969
ABSTRACT
Queuing is an inevitable situation experienced in everyday life. In any service system, a queue forms whenever
current demand exceeds the current capacity to serve. In this paper, an analysis of the queuing situation at the
ticket windows in the Nigerian Railway Corporation Lagos Terminus was carried out. To achieve this, primary
data was obtained by observation and an appropriate queuing model was used to analyze the system and
calculated its performance measures. Results showed that increasing the number of servers lead to a decrease in
the average waiting time of passengers in queue and a decrease in the average waiting time of passenger in the
system. Therefore by applying Queuing theory, the waiting time of passengers can be determined and improved
upon. The study therefore recommends that management in railway stations can use results from queuing
analysis in making decisions that increase the efficiency of operations at their ticket windows.
Keywords: Arrival rate, Service rate, Traffic intensity, Performance measures, Waiting times
Citation: Ituen-Umanah, W. U. (2017). Queuing Theory Application at Ticket Windows in Railway Stations
(A Study Of The Lagos Terminus, Iddo, Lagos State, Nigeria). Equatorial Journal of Computational and
Theoretical Science, 2 (1): 1- 5.
INTRODUCTION
Railway transportation is a major form of landbased transportation that convey large number of
people and goods, using wheeled vehicles (trains
and trams) that run on a track known as a railroad
or railway. Railway transportation offers much
potential because of its advantages like relative
safety, dependability, affordability and the
capability of transforming an economy through
mass movement of people, goods and services
(Igwe, Oyelola, Ajiboshin & Raheem, 2013). In
Nigeria, railway transportation has been in
existence since 1898, however in 1955, an Act
establishing the Nigerian Railway Corporation
(NRC) – giving it the right to construct and operate
railway services in Nigeria – was passed into law
(Akwara, Udaw & Ezirim, 2014). NRC also own
and operate railway stations across the country with
the Lagos Terminus as its busiest station. A railway
station basically consists of at least one-track side
platform and a station building. Inside the station
building, various operations are carried out. The
Ticketing operation is one of the most important
operations carried out in a railway station, as tickets
purchased by passengers enable access to services
desired, while payments received from passengers
form huge portions of revenue accrued from
services offered by the station. In big stations like
the Lagos Terminus, ticketing officers attend to
passengers through barricaded windows known as
“ticket windows”. In some railway stations, ticket
windows are not sufficient and where enough
windows are physically available, all may not be
operational, resulting in queues. There are also
observed cases where passengers are not seen at
ticket windows for long periods, creating a situation
where servers are idle for such periods. A lack of
orderliness, owing to ticketing officers attending
first to passengers they know, at the expense of
other passengers that arrived earlier and cases of
negative passenger behaviour (like collusion or
jockeying), increase the waiting line dilemma at
ticket windows. In addition, the inability of
management in stations to determine the waiting
times of passengers at their ticket windows and if
these waiting times are adequate or to be improved
Equatorial Journal of Computational and Theoretical Science, 2 (1): 1-5
Page 1
Ituen-Umanah (2017)
on, add to the difficulty they face in solving
queuing problems affecting their ticketing
operations.
OBJECTIVES OF THE STUDY
The main objective of this paper is to analyze the
queuing situation at ticket windows in railway
stations, using the Lagos Terminus in Iddo, Lagos
State, Nigeria as a reference point. Specifically, the
study is set to:
1 observe the queuing system at the ticket
windows.
2 identify problems arising from the observed
queuing system.
3 establish an appropriate model for the
queuing system with the aim of
understanding and describing aspects of the
problems identified.
4 attempt quantitative analysis of the
performance measures of the model
established.
5 determine the waiting times of passengers at
the ticket windows and if these waiting
times are affected by alterations in the
queuing system.
LITERATURE REVIEW
The inconveniences associated with waiting in
queue to access service (or receive action) are
experienced by people, things and signals.
Considering the railway transportation subsector in
Nigeria, while it is generally believed that
passenger patronage has dropped owing to high
operational cost and inadequate funding, I wish to
add that operational inefficiency as well as poor
service delivery also contribute to this drop in
patronage – a fact corroborated by Agunloye and
Oduwaye (2011), where amongst other things, their
study showed that 80% of railway transport patrons
in Lagos metropolis agreed that services offered are
ineffective and inadequate. Part of these poor
services is experienced at the ticket windows in
railway stations, especially in terms of their queuing
situations. This problem is not limited to Nigeria
alone as Shanmugasundaram and Umarani (2015)
pointed out that getting tickets in an Indian railway
station is difficult and the government battles to
meet this ever increasing demand of almost a billion
people. However, they affirmed that the application
of queuing theory in tackling the problem has been
feasible and effective. Yang, Li and Zhao (2014)
conducted a study on “Passenger flow simulation in
urban subway station based on Anylogic process
and software” at the Hangzhou station in China and
concluded that reducing passenger queuing time
involves optimizing the queuing system by opening
ticket windows in the station dynamically and at a
reasonable cost during operation. Xu, Gao and Ou
(2007) researched on “Service performance analysis
and improvement for a ticket queue with balking
customers”, focusing on queuing systems managed
by ticket technology and showing that different
ticket queues have significantly different balking
probabilities when customer patience is low and
system traffic is high. They proposed improvement
in ticket queue technology, viewing it as a new area
for researchers to explore, owing to its wide spread
adoption in government agencies and commercial
organizations. Moving a step further, Ghosal,
Chaturvedi, Taywade and Jaisankar (2015) in their
work on “Android application for ticket booking
and ticket checking in suburban railways” proposed
the use of online ticketing applications, mobile
phones and other devices as a remedy to stress and
tension endured by passengers standing in long
queues to book tickets. The enthusiasm of
stakeholders in applying queuing theory to improve
their ticketing operations (especially using
technology and innovation) is encouraging and
should be used channeled into making good
decisions for the benefit of the railway transport
sub-sector in the country.
METHODOLOGY
The research design used in this study is the case
study design. Primary data was obtained from direct
observation of the queuing system at the ticket
windows in the Lagos Terminus. Observation
recording sheets, a stop watch and writing materials
were the necessary tools to be used in data
collection. The study population is made up of all
passengers that come to the Lagos Terminus to
purchase tickets and all servers that make the
ticketing operation possible. The sample for the
study is conveniently drawn from all passengers
and servers involved in the ticketing operation from
11a.m to 4p.m, Monday to Friday, for a period of
ten days.
Model specification:
Queuing theory uses models to represent various
types of queuing systems encountered in real life.
Basically, queuing models are logical descriptions
of the behaviour of queuing systems. The
Equatorial Journal of Computational and Theoretical Science, 2 (1): 1-5
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Ituen-Umanah (2017)
appropriate queuing model for this study was
established as:
M/M/c/∞/FCFS
The foregoing describes a queuing system with
Poisson probability distributed arrival from an
infinite calling population, exponentially distributed
service time, multiple but identical servers in
parallel, infinite system capacity and a first-come
first-served queue discipline.
For this model, the following general assumptions
were made:
1
Arrival of passengers followed a Poisson
process.
2
Passenger arrival was independent and
the arrival rate did not change overtime.
3
A single waiting line was formed and
each arrival waited to be served
regardless of the length of the queue.
4
Service times were exponentially
distributed and the mean service rate
was constant for each server.
5
The system had multiple but identical
servers in parallel.
6
No passenger left the queue without
being served.
7
Infinite capacity of the system.
8
First-come first-served queue discipline
Let
=
=
and
c
ρ
=
=
number of servers
Traffic intensity ( =
),
The performance measures used in this study for the
M/M/c/ /FCFS queuing model, are given as
follows:
(i)
Probability that the system is idle (Po)
P0 = [ ∑
+
(ii)
The average number of passengers in
queue (Lq)
Lq = [
(
(
) ] P0
(iii)
The average number of passengers in the
system (Ls)
Ls = Lq +
(iv)
The average waiting time for a passenger
in queue (Wq)
Wq =
(v)
The average waiting time for a passenger
in the system (Ws)
Ws
=
Wq +
The numbers of servers (c), as well as the values of
the mean arrival rate ( ), mean service rate (µ) and
the traffic intensity (ρ), are necessary in
determining the performance measures for the
model.
arrival rate
service rate
RESULTS AND FINDINGS
Presentation of analyzed data
Table 1: Summary and presentation of analyzed primary data for passengers that arrived and were served.
INPUTS
DAY 1
DAY 2
DAY 3
DAY 4
DAY 5
DAY 6
DAY 7
DAY 8
DAY
9
DAY
10
Number of passengers that
arrived
Number of passengers
served
Total Time of observation
(hours)
C
Λ
µ
Ρ
P0
Lq
Ls
Wq (hours)
Ws (hours)
Sourceμ Authors’ compilation
442
396
401
423
415
433
409
381
397
401
421
381
382
395
407
406
390
374
384
392
5
5
5
5
5
5
5
5
5
5
2
88.4
84.2
0.5249
0.3116
0.3994
1.4993
0.0045
0.0164
2
79.2
76.2
0.5197
0.3160
0.3845
1.4239
0.0049
0.0180
2
80.2
76.4
0.5249
0.3116
0.3992
1.4489
0.0050
0.0181
2
84.6
79
0.5354
0.3026
0.4305
1.5014
0.0051
0.0178
2
83
81.4
0.5098
0.3247
0.3582
1.3779
0.0043
0.0166
2
86.6
81.2
0.5333
0.3044
0.4237
1.4902
0.0049
0.0172
2
81.8
78
0.5244
0.3120
0.3977
1.4464
0.0049
0.0177
2
76.2
74.8
0.5094
0.3250
0.3569
1.3756
0.0047
0.0181
2
79.4
76.8
0.5169
0.3185
0.3770
1.4109
0.0047
0.0177
2
80.2
78.4
0.5115
0.3232
0.3625
1.3855
0.0045
0.0173
Equatorial Journal of Computational and Theoretical Science, 2 (1): 1-5
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Ituen-Umanah (2017)
To have a holistic view of the queuing situation, the
entire 10days is considered and analyzed using the
same method. The results from this are shown
below:
Table 2: Results from analyzed data for the entire 10 days of observation as a whole.
INPUTS
RESULTS FOR THE ENTIRE 10
DAYS
Total number of passengers that arrived in 10 4098
days
3932
Total number of passengers served in 10 days
50
Total time of observation in 10 days (hours)
2
C
81.96
78.64
µ
0.5211
0.3148
P0
0.3885
Lq
1.4307
Ls
0.0047
Wq (hours)
0.0174
Ws (hours)
Sourceμ Authors’ compilation.
The queuing model used in this study is generally
referred to as the M/M/c queuing model, where c =
number of servers = 2. If we consider increasing the
servers from two to three and then four, we have the
following:
Table 3: Reaction of the waiting times to increase in server
INPUTS
c=2
c=3
0.5211
0.3474
ρ
0.0047
0.0007
Wq (hours)
0.0174
0.0134
Ws (hours)
c=4
0.2606
0.0001
0.0128
Source: Author’s compilation
As servers increased from two to three, the average
waiting time of a passenger in queue decreased
from 0.0047hours (16.92seconds) to 0.0007hours
(2.52seconds) which represents a total of 85.11%
decrease. Also the average waiting time of a
passenger in the system decreased from
0.0174hours (62.64seconds) to 0.0134hours
(48.24seconds), representing 22.99% decrease in
waiting. Furthermore, when the servers increased
from three to four, Wq decreased from 0.0007hours
(2.52seconds) to 0.0001hour (0.36seconds)
representing a total of 85.71% decrease in waiting,
while
Wq
decreased
from
0.0134hours
(48.24seconds) to 0.0128 hours (46.08seconds)
representing a 4.48 % decrease in waiting. This is
very important, as it shows that alterations can be
applied to improve the queuing system and reduce
waiting time of passengers.
CONCLUSION AND RECOMMENDATIONS
This paper demonstrates the importance of applying
queuing theory in solving waiting line problems at
ticket windows in railway stations. The results in
this study showed that the waiting times of
passengers can be measured and improved upon if
necessary. The results also showed that servers can
be assessed and judgments on their service rates are
less difficult to make when queuing theory is
applied. Based on the findings of this study, it is
concluded that proper analysis of the queuing
system at the ticket windows can aid the
management of railway stations in making better
decisions that can lead to a reduction in waiting
lines at ticket windows, improve efficiency of the
entire ticketing operation and ensure greater
passenger satisfaction of services offered. The
Nigerian Railway Corporation (NRC) can improve
access to tickets by selling them in public places
(other than their railway stations), through certified
Equatorial Journal of Computational and Theoretical Science, 2 (1): 1-5
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Ituen-Umanah (2017)
agents and on the internet through NRC approved
online platforms. This will encourage better
patronage and reduce waiting lines at ticket
windows. In addition, services of trained Operations
Research consultants are required to carry out
periodic analysis of the queuing situation in railway
stations and management in these stations should
implement recommendations
consultants.
made
by
the
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