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                     American International Journal of Multidisciplinary Scientific Research; Vol. 5, No. 3; 2019 
                                       ISSN 2638-1249   E-ISSN 2638-1273 

Published by Centre for Research on Islamic Banking & Finance and Business, USA 
 

     9 
 

 

Single-Server Queue System of Shuttle Bus Performance: Federal University of 
Technology Akure as Case Study 

 
 

Sidiq Okwudili Ben 
Department of Geography 

College of Education Ikere, Ekiti, Nigeria 
Email: sidiqoben@yahoo.com 

 
 

 
Abstract 
This study has examined the performance of University transport bus shuttle based on utilization using a Single-server queue 
system which occur if arrival and service rate is Poisson distributed (single queue) (M/M/1) queue. In the methodology, Single-
server queue system was modelled based on Poisson Process with the introduction of Laplace Transform. It is concluded that 
the performance of University transport bus shuttle is 96.6 percent which indicates a very good performance such that the 
supply of shuttle bus in FUTA is capable of meeting the demand. 
 

Keywords: Single-Server Queue System; Shuttle Buses; FUTA  
 

1. Introduction 
The issues arising from transportation has continually subjected to various debates in the urban societies. Globally, several 
attempts have been made to tackle the challenges, although the situation is not getting much better (Aderamo, 2012; Adanikin, 
Olutaiwo, and Obafemi, 2017). Managing transport infrastructures is crucial to facilitate accessible, affordable, reliable, safe, and 
efficient that movement of people and goods, which can be achieved by continous assessment of transport performance 
indicators.  Among the various modes of transport, the road transport is highly predominant which offers door-to-door service 
as in the case of the Federal University of Technology Akure (FUTA). Road transport infrastructure in the University is 
commonly plied by shuttle buses for the movement of the students, staff, members and non-members of the Institution to and 
from the University on a daily basis. 

Among the noticeable transportation problems in the University are traffic congestion; longer commuting; public 
transport inadequacy; difficulties for tricycles to have access to routes being plied by shuttle buses, challenge of freight 
distribution from one end of the University to another end, and other challenges which all impacts the performance of the 
University transport shuttle. This study concentrates on the performance of University transport bus shuttle with the aim of 
examining the bus shuttle efficiency based on utilization. 

Adeniran and Kanyio (2019) have laid a foundation of model on single-server queue system which this study will 
absolutely rely on. A similar study was conducted by Adanikin, Olutaiwo, and Obafemi (2017) on the performance study of 
University of Ado Ekiti (UNAD) transit shuttle buses. They adopted traffic volume, speed, density and revenue as main 
parameters of performance of transport shuttles, and find that the morning peak period (8.00am to 9.00am) has 234 
vehicles/hr, evening peak period (2.00pm to 3.00pm) has 284 vehicles/hr, while the off-peak period (11.00am to 12.00pm) 
has 156 vehicles/hr. Also, The average stopping time was 6.55 minutes, average interval between arrivals of motorists was 16.40 
seconds, the average queue length was 14.23 people, and the average waiting time at the bus-stop 4.17 minutes. These values 
were obtained using the queuing theory and shows much commuters time is lost on transit queues. This study focuses on the 
performance of bus terminal in FUTA, and does not factor in other parameters such as peak period, traffic volume, traffic speed, 
density, and others. 

 
2. Methodology 
2.1 Queuing System  
The concept of queue was first used for the analysis of telephone call traffic in 1913 (Copper, 1981; Gross and Harris, 1985; 
Bastani, 2009). In a system that deals with the rate of arrival and service rate, waiting time is inevitable and it is always 



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influenced by queue length. It is therefore crucial to minimize the waiting time to the lowest level in the bus terminal (Jain, 
Mohanty and Bohm, 2007). This is referred to as queuing system (Adeniran and Kanyio, 2019).  
The basic application of queue is shown in Figure 1, also the basic quantities are: 

i. Number of customers in queue L (for length); 
ii. Time spent in queue W for (wait) 

 

Figure1: Basic application of queue  

Source: Adeniran and Kanyio (2019) 

Examples of queue system are: 

1. Single-server queue system: This is also referred to as single queue, single server. It is simple if arrivals and services are 
Poisson distributed (M/M/1) queue. It has limited number of spots and not difficult. Figure 2 depicts single-server 
queue system. 

 

Figure 2: Single-server queue system 

Source: Adeniran and Kanyio (2019) 

2. Multi-server queue system: This is comprises of single queue, many servers (M/M/c) queue. The c is referred to as 
Poisson servers. Figure 3 depicts multiple-server queue system. 

 

Figure 3: Multiple-server queue system 

Source: Adeniran and Kanyio (2019) 

In single-server queue system, arrival and service processes are Poisson such that 

a. Customers arrive at an average rate of λ per unit time; 



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b. Customers are serviced at an average rate of µ per unit time; 

c. Interarrival and inter-service time are exponential and independent; 

d. Hypothesis of Poisson arrivals is reasonable; and 

e. Hypothesis of exponential service times are not so reasonable (Adeniran and Kanyio, 2019) 

In order to explain how the queuing system works, there is need to first introduce the Poisson Process (PP). It has exceptional 
properties and is a very important process in queuing theory. To simplify the model, we often assume customer arrivals follow a 
PP. The Laplace Transform (LT) is also a very powerful tool that was adopted in the analysis (Trani, 2011). Apart from PP 
and LT, there is focus on the queue model itself (Adeniran and Kanyio, 2019). 

2.2 Modelling of Single Queue System 
2.2.1 Laplace Transform 
The Laplace transform LX(s) of a nonnegative random variable X with distribution function f(x) is define as: 

LX(s) = E(e−sX) = ∫ e−sX∞

x=0
 f(x)dx  ………………… Equation 1 

It can be noted that  

LX(0) = E(e−X .0) = E(1) = 1  ………………… Equation 2 

and 

L1
X(0) = E((e−sX)1)|s=0 

 = E(−Xe−sX)|s=0 

 = −E(X) ………………… Equation 3 

Correspondingly, 

L (k)
x

 (0) = (−1)kE(Xk) ………………… Equation 4 

There are many useful properties of Laplace Transform. These properties can make calculations easier when dealing with 
probability. For instance, let X, Y, Z be three random variables with  

Z = X +Y and X, Y are independent.  

Then the Laplace Transform of Z can be found as: 

LZ(s) = LX(s) · LY (s) ………………… Equation 5 

Moreover, when Z with probability P equals X, with probability 1 − P equals Y, then  

LZ(s) = PLX(s) + (1 − P)LY (s) ………………… Equation 6 

Laplace Transforms of some useful distributions can now be introduced. 

a. Suppose X is a random variable which follows an exponential distribution with rate λ. The Laplace Transform of X is  

LX(s) = 
(λ)

λ + s
  ………………… Equation 7 

b. Suppose X is a random variable which follows an Erlang − r distribution with rate λ. Then X can be written as: 

X = X1 + X2 + · · · + Xr  ………………… Equation 8 

where Xi are i.i.d. exponential with rate λ. Therefore, we have 

LX(s) = LX1(s) · LX2(s)… LXr(s) 

 = (
λ

λ + s
)

n

  ………………… Equation 9 

c. Suppose X is a constant real number c, then 

LX(s) = E(e−sX) 



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 = E(e−sc) 

 = e−sc   ………………… Equation 10 (culled from Adeniran and Kanyio, 2019) 
 

2.2.2 Basic queuing systems 
Kendall’s notation shall be used to describe a queuing system as denoted by:  
A/B/m/K/n/D   …………………. Equation 11 (Adan and Resing, 2016) 
 
Where 
 A: distribution of the interarrival times 
 B: distribution of the service times 
 m: number of servers 
 K: capacity of the system, the maximum number of passengers in the system including  the one being serviced 
  n: population size of sources of passengers 
 D: service discipline 
 
G shall be used to denote general distribution, M used for exponential distribution (M stands for Memoryless), D be used for 
deterministic times (Sztrik, 2016).  
 
A/B/m is also used to describe a queuing system, where:  
 A stands for distribution of interarrival times,  
 B stands for distribution of service times and  
 m stands for number of servers.  
 
Hence M/M/1 denotes a system with Poisson arrivals, exponentially distributed service times and a single server.  
M/G/m denotes an m- server system with Poisson arrivals and generally distributed service times, and so on. 
In this section, the basic queuing models (M/M/1 system), which is a system with Poisson arrivals, exponentially distributed 
service times and a single server. The following part is retrieved from Queuing Systems (Adan and Resing, 2016). 

Firstly, it is assumed that inter-arrivals follow an exponential distribution with rate λ, and service time follows the exponential 
distribution with rate µ. Further, in the single service model, to avoid queue length instability, it is assume that: 
According to Adanikin, Olutaiwo and Obafemi (2017),  
 

Utilization (R) = 
Average Arrival Rate (λ)

Average service rate (μ)
 < 1 …………………….. Equation 12   

 
Here R is the fraction of time the server is working (called the utility factor) limiting probability pk in the M/M/1 system.  
The expected queue length L is given by 
 

E(L)  = ∑ i. pi

∞

i=0

 

= ∑ i. Ri 

∞

i=0

(1 − R) 

= R(1 − R) ∑ i. Ri 

∞

i=0

 

= R(1 − R) (∑〖i. Ri)i

∞

i=1

〗 

 = R(1-R) (
1

1−R
)i 

 = 
R

1−R
..............................Equation 13 (Adeniran and Kanyio, 2019) 

 
 

 



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3. Results and Discussions 
3.1. Traffic Survey 
3.1.1 Stopping time of shuttle bus 
Stopping time refers to the total time duration the shuttle bus spends at the bus stop. The stopping time is made up of: 

a) The boarding stop time “A”: This is also made up of the time taken to close the door = 15 seconds; time taken by the 
driver to check the traffic before take-off = 9 seconds; and time taken to park the bus and open the bus for commuters 
= 7 seconds. 

 A = (15+9+7) = 31 seconds 
b) The average boarding time per passenger = “B1” = 9 seconds 
c) Number of passengers boarding = n1 = 18 passengers.  

Mathematically, stopping time T = A + B1* n1  
Stopping Time = (31 + (9*18)) = (22+162) = 184 seconds 
Stopping Time = 3.07 minutes 
 
3.2 Waiting Time  
This is the length of time spent by the passengers at the bus stop before boarding a bus. It is also referred to as Delay. The 
queuing theory is employed in this study. 

a) Average arrival rate (λ) = 204 passengers/hour 

λ =
204

3600
  = 0.057 

b) Average service rate (μ) = 213 passengers/hour 

μ =
213

3600
 = 0.059 

c) Average interval between arrival =
1

λ
 

=
1

0.057
 = 17.54 Seconds 

d) Average interval between service rate =
1

μ
   

=
1

0.059
 = 16.95 Seconds 

e) Average queue length =
𝜆2

𝜇(𝜇−𝜆)
 

=
0.0572

0.059(0.059−0.057)
 = =

0.0033

0.00012
= 25.5 Passengers 

f) Average waiting time in the queue =
λ 

𝜇(𝜇−𝜆)
 

=
0.057

0.059(0.059−0.057)
 = =

0.057

0.00012
= 475 Seconds = 7.92 Minutes 

g) Average time spent in the system (bus stop) =
1

μ−λ 
 

=
1

0.059 − 0.057
= 500 Seconds = 8.33 Minutes 

h) Efficiency of shuttle bus operation based on bus stop utilization (R) = 
𝜆

𝜇
 

R =  
0.057

0.059
  = 0.966 

 
It is important to note that the Utilization factor is less than 1(R< 1), hence the performance of University transport 

bus shuttle is 96.6 percent which indicates a very good performance such that the supply of shuttle bus in FUTA is capable of 
meeting the demand.  
 
4. Conclusion and Recommendation 
This study has carefully explored the quantitative performance of University transport bus shuttle based on utilization using a 
Single-server queue system which occur if arrival and service rate is Poisson distributed (single queue) (M/M/1) queue. It is 
concluded that the performance of University transport bus shuttle is 96.6 percent which indicates a very good performance 
such that the supply of shuttle bus in FUTA is capable of meeting the demand. 
 
 
 



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