Application of Single-Server Queue System in Performance Analysis of Shuttle Bus Operation: A Case Study of Federal University of Technology Akure

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, Singleserver queue system was modelled based on Poisson Process with the introduction of Laplace Transform. Also, PASTA was introduced in queuing systems with Poisson arrivals. It is concluded that the performance of University transport bus shuttle is 93 percent which indicates a very good performance such that the supply of shuttle bus in FUTA is capable of meeting the demand. This study can be improved upon by examining the peak and off-peak period of traffic in the two major corridors (North gate and South gate) of FUTA, the economic cost of operating bus shuttle services can also be examined.

shuttle bus in FUTA is capable of meeting the demand. This present study was conducted barely two weeks after the study of Sidiq was conducted, and will replicate the methodology of Sidiq as culled from Adeniran and Kanyio (2019).

Concept of Queuing
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 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.  (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.  (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; 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 The Poisson Process (PP) is important in queue theory due to its outstanding properties. According to Adan and Resing (2015), queuing system is achieved as "let N(t) be the number of arrivals in [0, t] for a PP with rate λ, i.e. the time between successive arrivals is exponentially distributed with parameter λ and independent of the past. Then N(t) has a Poisson distribution with parameter λt. This is culled from Adeniran and Kanyio (2019).
The mean, and coefficient of variation of N(t) are Mean: E(N(t)) = λt; Coefficient of Variation: c 2 N(t) = 1 λt ………………… Equation 2 By the memoryless property of Poisson distribution, it can be verified that P(arrival in (t, t + ∆t]) = λ∆t + 0(∆t) ………………… Equation 3 Hence, when ∆t is small, P(arrival in (t, t + ∆t)) ≈ λ∆t ………………… Equation 4 In each small time interval of length ∆t the occurrence of an arrival is equally likely. In other words, Poisson arrivals occur completely randomly in time. The Poisson Process is an extremely useful process for modelling purposes in many practical applications. An important property of the Poisson Process is called "PASTA" (Poisson Arrivals See Time Averages).
PASTA is meant for queuing systems with Poisson arrivals, (M/./. systems), arriving vehicles find on average the same situation in the queuing system as an outside observer looking at the system at an arbitrary point in time. More precisely, the fraction of vehicles finding on arrival the system in some state A is exactly the same as the fraction of time the system is in state A.

Laplace Transform
The Laplace transform LX(s) of a nonnegative random variable X with distribution function f(x) is define as: 5 It can be noted that and L 1 X(0) = E((e −sX ) 1 )|s=0 Correspondingly, 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: Moreover, when Z with probability P equals X, with probability 1 − P equals Y, then 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 b. Suppose X is a random variable which follows an Erlang − r distribution with rate λ. Then X can be written as: where Xi are i.i.d. exponential with rate λ. Therefore 2.2.3 Basic queuing systems Kendall's notation shall be used to describe a queuing system as denoted by: A/B/m/K/n/D …………………. Equation 15 (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 follows 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), Average service rate (μ)
It is very difficult to solve these differential equations. However, when we focus on the limiting or equilibrium behaviour of this system, it is much easier. It was revealed by (Sztrik, 2016)  The rate matrix of the system is:

Conclusion and Recommendation
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. It is concluded that the performance of University transport bus shuttle is 93 percent which indicates a very good performance such that the supply of shuttle bus in FUTA is capable of meeting the demand. This result is very close to that of Sidiq (2019) which finds that the performance of University transport bus shuttle is 96.6 percent. This study can be improved upon by examining the peak and off-peak period of traffic in the two major corridors (North gate and South gate) of FUTA, the economic cost of operating bus shuttle services can also be examined.