manuscript aratan
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Abstract - The Cebu South Bus Terminal
(CSBT) has continuously developed over the years.
Recently,
The terminal had put up a conducive waiting and
ticketing area for the customers’ satisfaction. But
there is more to customer satisfaction. There are
several factors to be considered in meeting each
demand a customer wants. If the terminal has
succeeded in attracting customers, another factor that
needs consideration is the customers queuing time.
Waiting lines abound in all sorts of service systems
and they are non-value-added occurrences. For
customers, having to wait for service can range from
being acceptable, to being annoying, to being a
matter of life and death.
This study involves the analysis of the existing
ticketing system of Alcoy via Argao Ceres Bus Liner
of CSBT with which propositions are based. Assessed
in this research are the arrival of customers and thenumber of customers being served per unit time.
In the process of collecting factual data, the
inquiry made use of actual observation, interview and
tally method. Furthermore, the analysis made use of
goodness-of-fit test, hypothesis testing, queuing
theory and simulation.
With the foregoing findings, the company can
address the length of queue in the system through
adding the number of servers to be able to minimize
total system cost per unit time.
Keywords: queuing time, bus terminal,
ticketing system, waiting line, simulation
I. INTRODUCTION
Queuing theory is the study of queue or waiting lines.
Some of the analysis that can be derived using queuing
theory include the expected waiting time in the queue, the
average time in the system, the expected queue length, the
expected number of customers served at one time, the
probability of balking customers, as well as the
probability of the system to be in certain states, such as
empty or full.
Waiting lines are a common sight in bus terminals
especially during weekends and holidays. Hence, queuing
theory is suitable to be applied in a bus terminal setting
since it has an associated queue or waiting line where
customers who cannot be served immediately have to
queue (wait) for service. There are 11 servers working at
the same time. Based on personal interview and
observation, the Alcoy via Argao Ceres Bus Liner server
has the longest waiting line. This study uses queuing
theory to evaluate the said waiting line in CSBT at Cebu
City, Cebu. In addition, this study seeks to illustrate the
usefulness of applying queuing theory in a real case
situation, specifically, in finding the ideal number of
servers that would have the minimum total system cost at
maximum system capacity.
A. Objectives
1. To make a cost analysis of the system.2. To determine the ideal number of servers at
maximum system capacity.
B. Scope and Limitations of the Study
The study’s main subject is the queuing analysis
of the Alcoy via Argao Ceres Bus Liner ticketing
area. The focus of the study is not on redesigning the
process but rather on the ideal number of servers that
would minimize the total cost of the system. This
study focused only on the dense influx of passengers
in the ticketing area.
C. Statement of Assumptions
1. Salary of ticket officer per day is Php180.00.
2. The ticket officers work 8 hours a day.
3. Cost of customer waiting per hour is
Php20.00. This is for calculation purposes.
METHODOLOGY
Based on personal interview and actual observation,
the Alcoy via Argao Ceres Bus Liner server has the
Queuing Analysis of Ticketing System of Alcoy via Argao
Ceres Bus Liner at Cebu So=uth Bus Terminal
*Joahnna Jane C. Aratan, Jessabelle A. Caminero, Jaynamae N. Campo, Jessa
Rona C. Cepada, Estella Ching S. Dacutan, Ma. Gracelyn A. Demoral
Department of Industrial Engineering, Cebu Institute of Technology-University Cebu City,
Philippines, 6000
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longest waiting line. After determining the longest
waiting line, tally method was used in getting the number
of arrivals and the customers being served per unit time.
II. RESULTS
A. Influx Rate of Passengers
The following are the influx rate of passengers for
each day per peak season. These indicate the number of
customers per minute who arrives and joins the queue.
Figure 1. Influx Rate of Passengers from Monday to
Friday (From 6:05 PM to 7:00 PM)
B. Calculations
GOODNESS-OF-FIT TEST OF ARRIVAL RATE OF
PASSENGERS
Chi-Square Goodness-of-Fit test is commonly used to
test association of variables in two-way tables, where the
assumed model of independence is evaluated against the
observed data. In general, the chi-square test statistic is
of the form
The Poisson distribution is used to model the number of events occurring within a given time interval.
The formula for the Poisson probability mass
function is
Lambda
(λ) is the shape parameter which indicates the average
number of events in the given time interval.
The following is the plot of the Poisson probability
density function for four values of λ.
Figure 2. Plot of the Poisson Probability Density
Function for Four Value of λ
Waiting lines are a direct result of arrival and service
variability. They occur because random, highly variable
arrival and service patterns cause systems to be
temporarily overloaded. In many instances, the
variabilities can be described by theoretical distributions.
In fact, the most commonly used models assume that the
customer arrival rate can be described by a Poisson
distribution and that the service time can be described bya negative exponential distribution. Figure 8 illustrate
these distributions.
0
0.1
0.2
0.3
0 1 2 3 4 5 6 7 8 9 10 11 12
Customersper timeunit
R e l a t i v e f r e q u e n c y
Figure 3. Poisson Distribution (rate)
The following are the summary of results:
Day
Ave. no.
of
arrivals
per min
χ c2 χ t
2Interpretatio
n
Mon 1.727 7.237 9.488Poisson
Distributed
Tue 1.636 7.928 9.488Poisson
Distributed
Wed 3.364 12.384 15.507Poisson
DistributedThu 1.873 8.008 12.592
Poisson
Distributed
Fri 2.091 10.219 15.507Poisson
Distributed
Ave 2.138 9.1552 12.516Poisson
Distributed
Table 1. Summary of Arrivals’ Results Using
Goodness-of-Fit Test (From Monday to
Friday)
The following are sample calculations:
x f o f ox P(x) f e (f o - f e)2/
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f e
0 13 00.177
89.7773 1.062257
1 17 170.307
1
16.888
00.000743
2 8 16
0.265
2
14.585
1 2.973142
3 8 240.152
78.3975 0.018814
4 7 280.065
93.6262 3.139006
5 2 100.031
41.7270 0.043155
Σf o=
55
Σf ox
=
95
ΣP(x)
=
1.000
0
Σf e =
55.001
1
Xc2 =
7.2371
λ = 1.727
Table 2. Monday Arrivals’ Results Using Goodness-of-
Fit Test
*x is the number of customer arrival
*f o is the observed frequency of x
*λ is the customer arrival rate
* P(x) is the probability of x
*f e is the expected frequency of x
Six-Step Hypothesis Testing:
Step 1. State the null and alternative hypotheses:
Ho: The arrival of customers is Poisson Distributed.
Ha: The arrival of customers is not PoissonDistributed.
Step 2. Choose the level of significance: α , the
probability of making a Type I Error if H0 is true.
α = 0.05
Step 3. Determine the critical values for the level of significance α ,
d f = k-p-1 = 6-1-1 = 4
Xt2 = 9.488
Step 4. Decision Rule
|χ c2|≥|χ t
2|: Reject Ho
|χ c2|<|χ t
2|: Do not Reject Ho
Ste p 5. Calculate the test statistic
Xc2 = 7.2371
Step 6. Conclusion
Since χ c2<χ t
2: Do not Reject Ho
Therefore, the arrival of customers is Poisson Distributed.
λ = Σf ox/Σf oλ = 95/55
λ = 1.727
Figure 4. Graph of Poisson Distribution (Rate)
from Monday to Friday
There are numerous queuing models from which
analysts can choose. Naturally, much of the success of the
analysis will depend on choosing an appropriate model.
Model choice is affected by the characteristics of the
system under investigation. The main characteristics are
population source, number of servers (channels), arrival
and service patterns and queue discipline (order of
service). The approach to use in analyzing a queuing problem depends on whether the potential number of
customers is limited. There are two possibilities: infinite-
source and finite-source populations. In the case of this
study, the possibility is infinite-source.
The following is a list of symbols used for infinite-
source models.
Symbol Represents
λ Customer arrival rate
μ Service rate per server
Lq The average number of customers
waiting for service
L s The average number of customers inthe system (waiting and/or being
served)
r The average number of customers
being served
ρ The system utilization
DAYλ
(cpm)
µ
(cpm)
Monday 1.727 1.450
Tuesday 1.636 2.150
Wednesda
y 3.364 3.160
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Thursday 1.873 1.800
Friday 2.091 2.850
Average2.138
2.282
Table 3. Summary of Results from ….Monday to
Friday
*cpm, customers per minute
No. of
servers1 2 3 4
(Lq) 13.91 0.41 0.14 0.07
(Ls) 14.85 0.88 0.45 0.31
(Wq) 6.51 0.19 0.07 0.03
(Ws) 6.94 0.41 0.21 0.14
(Pw) 0.94 0.47 0.31 0.23
Table 4. Effect of No. of Servers on the System
COST ANALYSIS
No. of
servers 1 2 3 4
Custome
r waiting
cost per
minute
4.94 0.29 0.15 0.10
Server
cost per
minute
0.38 0.75 1.13 1.50
Total
cost per
minute
5.32 1.04 1.28 1.60
Table 5. Total System Cost per minute
Sample computations:
Lq = λ 2 / [μ(μ-λ)]
= (2.138)2 / [2.282(2.282-2.138)]
= 13.91
Ls = λ / (μ-λ)
= 2.138 / (2.282-2.138)
= 14.85
Wq = λ / [μ(μ-λ)]
= 2.138 / [2.282(2.282-2.138)]
= 6.51 minutes/customer Ws = 1 / (μ-λ)
= 1 / (2.282-2.138)
= 6.94
Pw = 2.138 / 2.282
= 0.94
Each ticket officer is paid Php180 per day at
minimum. Therefore, the server cost per minute is
=(Php180/day)(1day/8hours)(1hour/60minutes)
= Php0.375/minute
Customer waiting cost per minute is L s x Php20/hour
=(14.85)(Php20/hour)(1hour/60minutes)
= Php4.94/minute
Total cost per minute is customer waiting cost per
minute plus server cost per minute. In calculating, we get,
= Php0.375/minute + Php4.94/minute
= Php5.32/minute
The following is the simulation layout of the existingsystem of Alcoy via Argao Ceres bus liner:
Figure 5. Simulation Layout of CSBT Alcoy via
Argao Ticketing Area
III. DISCUSSION
More customers queue at the Alcoy via Argao
ticketing area as 7:00 P.M. approaches. This is the time
where the Ceres bus liner has its last trip. This is also the
time where workers from different companies and
students flood in the terminal.
Since all the arrivals were Poisson Distributed,queuing theory was applicable. The effect of number of
servers on ticketing operation was computed and cost
analysis made possible. One of the requirements of any
practical system is that λ < μ, which means ρ < 1, failing
which an unstable system results. If the arrivals are faster
than the time in which they can be processed, the waiting
line and the waiting time will increase continuously, and
no steady state can be achieved.
Based on the results of the cost analysis, it was found
out that two (2) servers will minimize the total cost of the
system. Because the total cost will continue to increase
once the minimum has been reached.
Using simulation to aid in the analysis, it turned outthat the same number of servers is needed to lessen the
length of the queue.
IV. CONCLUSION
Based on the results, it was found out that the server of
Alcoy via Argao Ceres bus liner ticketing area is not
sufficient. Thus, there is a need to calculate for the ideal
number of servers for the system. By doing a cost and
simulation analysis of the system, the ideal number of
servers was determined. Based on the results, the bus liner
should have two (2) servers in order to minimize the total
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cost per minute of the system. By doing so, the company
could save an amount of Php5.32/minute –
Php1.04/minute = Php4.28/minute. Therefore, a total
annual savings of
=(Php4.28/minute)(60minutes/hour)(8hours/day)
(350days/year)
= Php719,040/year
V. RECOMMENDATION
Primary Recommendation
The researchers recommend that the Alcoy via
Argao Ceres bus liner ticketing area should have two
(2) servers in their ticketing area to minimize total
cost in the system.
Secondary Recommendation
Further research must be conducted in the CSBT
regarding the ticketing process, the layout of the
ticketing area, the capacity of the bus and other
operations in the bus terminal that needs to be
evaluated for further improvements.
REFERENCES
[1] http://ascelibrary.org/proceedings/resource/2/asc
ecp/387/41139/478_1?isAuthorized=no
[2] http://www.clancor.net/ticket-system.html
[3] http://www.daniweb.com/web-development/php/threads/227665
[4] http://www.lockmedia.com/solutions/bus_ticketi
ng_system_demo.asp
[5] http://www.stat.yale.edu/Courses/1997-
98/101/chigf.htm
[6] Stevenson, William J. Operations
Management. 9th Ed. Philippines: MacGraw-
Hill, 2007, ch. 14, pp. 720-722
[7] Levin, Rubin, Stinson and Gardner Quantitative
Approaches to Management, 6th Ed. Singapore:
McGraw-Hill Companies, Inc., 2001, ch. 18, pp.
813-830