BIBLIOGRAPHY ACOSTA, RODLYNN MAY M. AND NORMAN,...
BIBLIOGRAPHY
ACOSTA, RODLYNN MAY M. AND NORMAN, CLEVENSON C. APRIL
2008. Analysis on the Number of Students Serviced at the Cashier’s Counter. Benguet
State University, La Trinidad, Benguet.
Adviser: Marycel H. Toyhacao
ABSTRACT
This study was conducted to determine the arrival distribution of students and
service time rendered by the cashier at the counter of Benguet State University
Administration and the queue discipline of the students waiting at the cashier counter.
Poisson and exponential distribution were employed in this study. It has been
found that the average number arrival at the cashier counter is 76 students per hour and
the average number of students that the cashier counter can serviced per hour is 26. This
discrepancy of student’s arrival and number of students serviced at the cashier’s counter
resulted to long queues.
The model was identified to fit the enrollment operation with three cashier
counters in the administration. The measuring capacity utilization (ρ) shows that 97.44%
of time the counters are busy. The average time a student spent at the system is 0.5142
hours, which means that a customer will receive complete service for approximately
30.85 minutes including the time the students waiting in the line. The average time in the
queue is 0.475738 hours which is approximately 28.54 minutes, which is the time spent
waiting in the line. The remaining time, which is W-Wq, is the time spends in the
counter, 0.038462 hours or 2.30772 minutes.
To improve the operating characteristics of the queueing of students serviced at
the cashier’s counter; some suggestions may be offered to the management to change the
arrival rates in a number of ways, such as providing discounts or other incentives for the
early enrollees. Adding one to the existing number of cashier counter will help reduce
number of students waiting in long queues. Results shows that if we add another counter,
the average number of students in the system was reduced from 39 to 14 students per
hour.; the average time students spends in the system is reduced from 30.8 minutes to
10.75 minutes and the probability that an arriving student has to wait for service was
reduced from 95% to 39%. More counter, the better the service.
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TABLE OF CONTENTS
Pages
Bibliography...……………………………………………………………………………..i
Abstract………………………………………………………………………………....…i
Table of Contents...………………………………………………………....…………... iii
INTRODUCTION
Background of the Study………………………………………………………..….1
Objectives of the Study………………………………….….…………………...…..2
Importance of the Study…………………………………..……………………...…3
Scope and Delimitation of the Study…………………..………………………...…3
REVIEW OF LITERATURE
Studies of Queueing…………………………………………...…………………….4
THEORETICAL FRAMEWORK
Arrival
Distribution…………………………………………………………....…..8
Service
Distribution…………………………………………………………....….8
Multiserver Multiqueue Model……………………………………………..……..9
Definition of Terms……………………………………………………………....10
METHODOLOGY
Locale and Time of the Study………………………………………………...……12
Data Used…………………………………...…………………………………...…12
Data Analysis………………………………………………………………...…….12
iii
RESULTS AND DISCUSSION
Arrival Distribution……………………………………………………………...…13
Service Time Distribution……………………………………………………...…..15
Operating Characteristics of Students
Serviced at the Cashier Counter……………………………………………….......17
SUMMARY, CONCLUSION, and RECOMMENDATION
Summary………………………………………………………………………...…19
Conclusion……………………………………………………………………...….19
Recommendation……………………………………………………………...…...20
LITERATURE CITED……………………………………………………………...…..21
iv
1
INTRODUCTION
Background of the Study
Waiting in lines is part of everyday life. Whether it is waiting in line at a grocery
store to buy items or checking out at the cash registers, or waiting at an amusement park
to go on the newest ride, a lot of time waiting is spent. People, wait in lines at the movies,
campus dining rooms, the Registrar’s Office for class registration, and many more.
Most people hate waiting. But reduction of the waiting time usually requires extra
investments. To decide whether or not to invest, it is important to know the effect of the
investment on the waiting time. So models and techniques are needed to analyze such
situations according to Mohr (1983).
Medhi (1991) in his Stochastic Models in Queueing System, states that attention
is paid to methods for the analysis and applications of queueing models. It is particularly
useful for the design of these systems in terms of layout, capacities and control.
Cox and Smith (1973) on “Queues” write that queueing model is used to
approximate a real queueing situation or system, It is characterized by arrival process of
customers, behavior of customers,service times and discipline. Arrival process of
customers usually assumes that the interarrival times are independent and have a common
distribution. In many practical situations customers arrive according to a Poisson stream
(i.e., exponential interarrival times). Customers may arrive one by one, or in batches.
Second is the behavior of customers. They may be patient and willing to wait for a long
time or customers maybe impatient and leave after a while. Third is the service time
which usually assumes that the service times are independent and identically distributed,
Analysis on the Number of Students Serviced at the Cashier’s Counter
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and that they are independent of the interarrival times. It can also occur that service times
are dependent of the queue length. For example, the processing rates of the machines in a
production system can be increased once the number of jobs waiting to be processed
becomes too large. Fourth is the service discipline. Customers can be served one by one
or in batches. There are many possibilities for the order in which they enter service.” First
come, first served”, in the order of arrival wherein the first one who arrived is the one
who will be entertained first like in the paying of fees especially in the enrollment of the
students.
Upon entering school, students must first enroll. Considering the enrollment
procedure of the College of Arts and Sciences students, students must present their class
card to the teacher who is assigned to assess the students to be recorded at the checklist.
After presenting the grades, the students will be given an enrollment form to be filled up
before assessment. The students will then go to the Dean’s office to be assessed for their
fees and then proceed to the Administration building for paying. There, students wait for
their turn to be entertained. However, there are only three (3) counters who will render
the sevice in the Administration. It is expected that a line will be formed. This is a system
where students arrive, join a waiting line, wait their turn, serve by a multiple server and
depart. It is described as the queueing system.
Objectives of the Study
This study aimed the following:
1. To determine the arrival distribution of students at the cashiers counter of BSU
Administration;
Analysis on the Number of Students Serviced at the Cashier’s Counter
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2. To determine the service time rendered by the cashier to the students at the
counter;
3. To determine the queueing model and operating characteristics of the student
enrollees at the cashier’s counter; and
4. To determine the economic analysis of the operating characteristics of queueing
of students enrollees at BSU Admistration.
Importance of the Study
The result of the study provides information to the management on the proper
management of queue and on what level of services will they offer. The arrival rate
specifies the average number of students arriving at the cashier’s counter per hour which
could help the management decide whether or not to make changes or choose alternatives
to improve the situation that involves waiting lines.
The study also reminds students, to practice and develop the value of patience and
willingness to wait for their turn.
The result of the study serves as a basis for the other researchers who will be
interested to conduct queueing analysis.
Scope and Delimitation
The study was conducted at Benguet State University Administration Building,
La Trinidad, Benguet, during enrollment period for the Second semester school year,
2007-2008. The data gathering was executed on October 22-26, 2008. The main focus of
the study is the application of queueing analysis of the students serviced at the cashiers
counter regarding paying their enrolment fees.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
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REVIEW OF RELATED LITERATURE
Queue is everywhere. It is seen everywhere, paying for bills, line at a
supermarket counter, line at the banks machine, line of cars at a traffic light show.
Waiting line is undesirable for all parties times, considering the following studies
concerning queues.
Studies on Queueing
Mohr (1983) cited that Arrival Distribution for a waiting line involved
determining how many costumers arrive in a period of time. He determined the number
of customers in a 1-hour period, since the number of customer each hour is not
necessarily constant.
Queueing analysis is applied also on the parallell simulation of queueing networks
as studied by Sun, Z. et. al. (1991). Its limitations and potentials in performace evaluation
review on the queueing delay and cell loss for combined traffic sources in ATM networks
in United Kingdom as the main purpose. The development of broadband ISDN based on
the ATM introduces new network characteristics different to those traditional channel
based networks, typically queueing delay and cell loss when combined traffic sources
share the same bandwidth capacity. The paper reports the work on studying the new
network characteristics so that the bandwidth resource can be utilized efficiently with
restricted queueing delay and cell loss.
Another application of queueing was undertaken by Denisov and Sapozhnikov
(2006) of the Department of Mathematics/Boole Centre for Research in Informatics,
University College Cork, Cork, Ireland on the distribution of the number of customers in
the symmetric M/G/1 queue wherein they consider an M/G/1 queue with symmetric
Analysis on the Number of Students Serviced at the Cashier’s Counter
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service discipline. In this research they show that this conjecture is true if service
requirements have an Erlang distribution. They also show by a counter example,
involving the hyper exponential distribution, that the conjecture is generally not true.
In Madrid, Spain, Artalejo, Economou and Lopez-Herrero (1985) of the
Department of Statistics used queueing model on their research entitled “Analysis of a
Multiserver Queue with Setup Times” that deals with the analysis of an M/M/c queueing
system with setup times. This queueing model captures the major characteristics of
phenomena occurring in production when the system consists in a set of machines
monitored by a single operator, which carry out an extensive analysis of the system
including limiting distribution of the system state, waiting time analysis, busy period and
maximum queue length.
Beaubrun (2007) analyzes the traffic distribution and blocking probability in
future wireless networks through Poisson and Exponential distribution. Traditional
analysis of teletraffic in such networks assumes that call arrivals follow a Poisson
process, as each cell is being modeled as an M/G/c/c queueing system. Such a model has
not been explicitly addressed in the literature, the main contribution is to propose a
solution which enables to evaluate both traffic distribution and blocking probability
within each cell of the service area. Result analysis reveals that coefficient of variation of
call arrivals has more impact on the network performance than coefficient of variation of
channel holding time.
Wei (2003) conducted a study on grids. Grid computing has emerged as an
important new field, distinguished from conventional distributed computing by its focus
on large-scale resource sharing, innovative applications, and high-performance
Analysis on the Number of Students Serviced at the Cashier’s Counter
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orientation. A Grid integrates and coordinates resources and users that live within
different control domains with the goal of delivering non-trivial quality of service (QoS).
Consequently, task management and scheduling is of central importance for the Grid-
based systems. In this paper, the task dispatching and selecting on the distributed Grid
computing system by a multiserver multiqueue (MSMQ) model, and propose a modeling
and analysis technique based on Stochastic Petri Net (SPN) methods are used. An
approximate analysis technique is also proposed to reduce the complexity of the model.
Queues in the Philippines are common scenery and queueing discipline is not that
practiced. Thus, this study is conducted regarding long lines. However, the application of
queueing analysis here in the Philippines is not limited and the utilization is not yet fully
explored. It is due to lack of device or strategy in recording the data necessary for the
analysis and even the availability in some packages, is limited.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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THEORETICAL FRAMEWORK
Cox and Smith (1993) state that queueing theory deals with mathematical models
of various kinds of real queues, example is the situations where congestion occurs due to
randomness in arrival and service times and customers have to wait for service. Queues
occur when the current demand for service exceeds the capacity of service facilities, and
the main purposes of the theory is to provide means for designing/modifying/optimizing
service systems in such a way as to reduce the likelihood of queues, the customer’s
waiting times and so on. Analysis of queueing models may include determining the
distributions of the queue length, waiting times and the duration of the busy/idle periods
for servers.
Trivedi (1982) states that a queue or queueing system is a system which includes
a random input stream of requests which needs service and a mechanism which provides
that service. Typical examples of queues are telephones exchanges, customer’s queues at
checkout counters on business institutions in an airport, and a network of time sharing
computers.
Since cashiers counter of Benguet State University Administration Building
consist of a three queue and three service facility and assumes that arrivals are unlimited
and can be formally treated in a Poisson distribution. Service times are assumed to take
on the form of an Exponential distribution.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Arrival Distribution
For many waiting lines, the arrivals occurring in a given period of time appear to
have a random pattern-that is, while there may be a good estimate of the total number of
arrivals expected, each arrival is independent of other arrivals and cannot predict when
the arrival will occur. In this case, Poisson distribution will be used because it provides a
good description of the arrival pattern.
The Poisson probability distribution is defined as:
P(x) = λx e – λ
for x = 0, 1, 2, . . .
x!
Where, in waiting line applications,
x = number of arrivals in a specified period of time
λ= average of expected number of arrivals for the specified
period of time
e = 2.71828
Service Time Distribution
A service fine probability distribution is needed to describe how long it takes a
student to wait for his turn and to pay. Since students have different amount of fees,
cashier’s service times vary. Thus, exponential probability distribution will be used.
The exponential probability distribution is defined as:
f(x)
=
μe-μx
for x ≥ 0
where:
x = service time
µ = average or ecpected number of units that the service facility
Analysis on the Number of Students Serviced at the Cashier’s Counter
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can handle in a specific period of time
e = 2.71828
Multiserver-Multiqueue Model
A logical extension of the single-channel waiting line is the multiserver
multiqueue single channel waiting line. By multiple-channel waiting lines we mean that
two or more counter or service locations are present. It allows us to examine the situation
where the number of counters in service facility may assume any finite value
To do this, the following short-hand notation to solve the formula given is used.
Where:
λ = average arrival rate of the students
μ = average service rate per counter
s = number of available counter
1. the percentage of time the counter are busy or the probability that the
counters
are
busy
λ
ρ =
sμ for sμ>λ
2. the probability that no students in the ounter
1
Po =
n
s
λ
λ
∑ μ
μ
+
!
n
!
s 1
( − (ρ / s))
3. the average time a student spent in the queue waiting for the service
Analysis on the Number of Students Serviced at the Cashier’s Counter
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s
ρ Po
!
s 1
( − (ρ / s))
Wq =
sμ 1
( − (ρ / s))
4. the average time a student spends in the system (waiting time + service time)
1
W = Wq +
μ
5. the average number of students in the queue waiting for service
Lq = λWq
6. the average number of students in the service system
L = λW
Definition of Terms
The following are defined for the purpose of the study:
Arrival defines the way customers enter the system. Mostly the arrivals are
random with random intervals between two adjacent arrivals. Typically the arrival is
described by a random distribution of intervals also called Arrival Pattern.
Arrival Rate is the average number of students arriving per time period denoted
by lambda.
Cashiers counter is a window where cashier’s stay and does its work.
Exponential distributions are a class of continuous probability distribution. They
are often used to model the time between independent events that happen at a constant
average rate.
Poisson distribution is a discrete probability distribution that expresses the
probability of a number of events occurring in a fixed period of time if these events
occur with a known average rate, and are independent of the time since the last event.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Queue is also known as the waiting line, that is, the place where a certain number
of customers spend most of their time waiting for service in the queueing system.
Queueing discipline represents the way the queue is organized.
Queueing system considered to be the waiting lines or the queues as well as the
number of available service facility which contains one or more server.
Service Time, as applied to queuing systems, is a period of time
describing how long it takes a student to be serviced.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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METHODOLOGY
Locale and Time of the Study
The study was conducted at Benguet State University, La Trinidad, Benguet
during the second semester, school year 2007-2008.
Data Gathering Tool
Data were gathered through individual observations among the Benguet State
University students who paid their fees during enrollment period at the cashiers counter
(Administration Building), on the 22nd until 26th day of October 2007. The data gathered
were the arrival time, waiting time and service time.
Data Analysis
The summarized data were utilized in the application of Poisson probability
distribution to explain the arrival pattern of the Benguet State University students
serviced at the Cashier’s counter. Exponential probability distribution was also used to
provide a good description of the service time distribution. The Multiserver-Multiqueue-
Single-Channel Waiting line Model was utilized to determine the queue data
characteristics.
The data gathered were encoded,summarized and were analyzed using Microsoft
Excel. Average number of students arriving per hour (λ) and the average number of
students that can be served per hour (μ) were computed using the Microsoft Excel.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
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RESULTS AND DISCUSSION
Arrival Distribution
Table 1.1 presents the summary for the arrivals of students per hour interval, the
mean of the arrivals, the total number of arrivals and the overall mean.
Time interval 1:00-2:00 had the highest students arrival of 869 with the highest
mean of 174 students arrivals per hour. This time interval had the highest arrival because
students were done processing their enrollment and now ready to pay, where, time
interval 4:01-5:00 had the lowest number of arrivals of 62 and lowest mean of 12. This is
because the time interval that was considered was 30 minutes wherein cashiers are
leaving the counters and the closing time for the administration.
The students arrivals were not scheduled in paying and occurred in an
unpredictable manner, a random pattern appears to exist. Mohr (1983) determined that
indeed, arrival is not constant in a period of time. Thus, to determine the student arrivals,
the Poisson distribution provides a good description of the arrival pattern which is
xe λ
λ −
P( X = x) =
defined by as
!
x
for x=40, 42, 44,…,60
where λ= 76 and e=2.71828.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Table 1.1 Summary of arrivals and arrival means
HOUR INTERVAL
# OF STUDENTS
MEAN ARRIVAL/HOUR
8;00-9:00 97
19
9:01-10:00 440
88
10:01-11:00 467
93
11:01-12:00 232
46
1;00-2;00 869
174
2;01-3:00 558
112
3:01-4:00 310
62
4:01-5:00 62
12
overall mean
76
Table 1.2 presents the Poisson students arrival distribution of Administration
cashier counter ,where we would expect 40 students arriving in an hour 1.75% of the
time, exactly 45 arrival in an hour 4.11 % of the time, we would expect 50 students
arriving in an hour 5.58% of the time, and so on.
The study by Beaubrun (2007) also uses the same distribtution we used in this
study where in, he uses Poisson in assuming the call arrivals which its main contribution
is to propose a solution which enables to evaluate both traffic distribution and blocking
probability within each cell of the service area.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Table 1.2 Arrival distribution
X=NUMBER OF ARRIVALS
P(X≤X)
40 0.017464586
42 0.026379436
43 0.036264753
44 0.04556745
45 0.052535877
46 0.055773802
47 0.054701229
48 0.049712752
49 0.041981451
50 0.033028964
51 0.024267891
52 0.054701229
53 0.052637032
54 0.049712752
55 0.04609728
56 0.041981451
57 0.037562351
58 0.033028964
59 0.02855046
60 0.024267891
Service Time Distribution
Table 2.1 presents the summary for the number of students serviced by the cashier
per one hour interval, the mean of the students serviced by the cashier, the total number
of students who were serviced and the overall mean of the students being served by the
cashier counter per day.
Time interval 1:00-2:00 had the highest mean students that were served by the
cashier counter of 40 which was equal to the highest students arrival 869. This implies
that the numbers of arrivals per hour was 76 and the number of students that was being
served by the cashier counter per hour was 26. Thus there is the discrepancy between the
arrivals of students per hour and the students serviced by the cashier counter per hour is
Analysis on the Number of Students Serviced at the Cashier’s Counter
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the reason for queueing. Time interval 8:00-9:00 had the lowest number of students who
were served for a reason that students are still processing their enrollment procedures.
Table 2.1 Summary of average students serviced per hour
HOUR
COUNTER COUNTER COUNTER
MEAN
INTERVAL
I
II
III
SERVICED/HOUR
8;00-9:00 6
5
18
2
9:01-10:00 45
150
101
20
10:01-11:00 187
146
168
33
11:01-12:00 151
184
112
30
1;00-2;00 209
164
202
40
2;01-3:00 221
154
186
37
3:01-4:00 155
152
162
31
4:01-5:00 3
70
85
11
overall mean
26
The exponential probability distribution was used to describe the service time,
since students demanded a different service that depends on what amount they are
supposed to pay, thus the Cashiers counter service times varied. The exponential
distribution is defined by ƒ(x) = μe-μt for x ≥ 0 where μ=26 and e=2.71828. Table 2.2
presents the exponential distribution for the service time thus, 98.94% of the students are
expected to be serviced in 6 minutes or less (t=0.1), 99.97% in 12 minutes or less (t=0.2),
so on and 100% of the students to be serviced in 36 minutes (t=.6).
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Table 2.2 Service time distribution ofassumed values of service time (in hours)
X=SERVICE TIME (IN HOURS)
P(SERVICE TIME ≤X)
0.1 0.983427279
0.2 0.999725345
0.3 0.999995448
0.4 0.999999925
0.5 0.999999999
0.6 1
0.7 1
0.8 1
0.9 1
1.0 1
Operating Characteristics of Students
Serviced at the Cashier’s Counter
Using the Poisson arrivals and exponential service times identical servers and a
first come first serve discipline, operating characteristics for the multiserver-multiqueue-
single-channel waiting line model was presented in Table 3.
Table 3 presents the teller utilization rate which is the percentage of the time a
counter is busy. The total service rate must be greater than the arrival ratet that is kμ > λ.
If sμ ≤ the average number of customer spends in the system (L) and the average time a
student spends in the system (W) both become infinitely large. In table 3, the utilization
rate is clear that the formula is applicable since λ/kμ < 1 thus the probability that the
servers were busy is 97% . the probability that the cashier is idle is .5%. This implies that
the server in free for .3 minutes upon the start of the operation thus most of the time the
cashier is busy. On the average, there are 39 students in the system. The average time a
customer spends in the system is .51 hour or 30 minutes. The average time a customer
spends in the queue before being served is .47 hour or 28 minutes. The average number
Analysis on the Number of Students Serviced at the Cashier’s Counter
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of students in the queue is 36 students and the probability that an arriving customer has to
wait for service is .95.
Suppose another counter is added with the same arrival rate and service rate and a
first in-first-out discipline. Based on table 3 the probability that the servers are busy is
reduced by 24% from 97% to 73%. The probability that the server is idle is increased
from .5% to 3.4% or from .3 minutes to 2.09 minutes.
Table 3. Economic analysis on the operating characteristics of the queueing of students
serviced at the cashiers counter
SERVERS
(s)
ρ=λ/sμ P(0) L
W Lq Wq Pw
3 0.974358
0.00586
39 0.5142 36 0.475738
0.95125
4 0.730769
0.03489
14
0.17925
11
0.14078
0.3942
where:
L= average number of units in the system
W= the average time a unit spends in the system (waiting time + service)
Lq= the average number of units in the ueue waiting for service
Wq= the average time a unit spends in the queue waiting for service
Pw= the probability that an arriving students has to wait for service
The average number of student in the system is reduced by 25 from 39 to 14
students. The average time a student spends in the system is reduced by 20 minutes from
.5142 hour to .17925 hour or from 30 minutes to 10 minutes. The average number of
students in the queue is reduced by 25 from 36 to 11 students. The probability that an
arriving customer will wait is reduced from 95% to 39%.
The assumption is clear that four counters will greatly improve operating
characteristics of the system.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
Poisson and exponential distribution were employed in this study. The average
number of students who arrive is 76 students per hour and the average students that the
cashier counter can service per hour were 26, that results to long lines of students.
The model was identified to analyze the enrollment operation for there are three
cashier counters in the administration. The measuring capacity utilization (ρ) shows
th97.44% of time the counters are busy. The average time a student spent at the system is
.5142 hours, which means that a customer will receive complete service for
approximately 30.85 minutes including the time the students waiting in the line. The
average time in the queue is 0.475738 hours which is approximately 28.54 minutes, that
is the time spent waiting in the line. The remaining time, which is W-Wq is the time
spent in the counter, 0.038462 hours or 2.30772 minutes.
Conclusion
It is the variability in arrival and service distribution that causes waiting lines.
Waiting line model allows to estimate performance by predicting average system
utilization, average number of students in the service system, average number of students
in the waiting line, average time a student spends in the system and the average time
students wait in line.
The result on the operating characteristics indicates that students wait an average
of .5142 hours or 31 minutes. Which appears to be undesirable for students to wait for
their turn for almost half an hour. And shows that long waits suggest a lack of concern by
the management or, can be linked to a perception of poor service management. But
Analysis on the Number of Students Serviced at the Cashier’s Counter
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sometimes, it’s the students fault why they were served that long because they arent
following the schedule given by the management for them to enroll.
Recommendation
The operating characteristics of the queueing of students serviced at the cashier’s
counter suggest the management to change the arrival rates in a number of ways, such as
providing discounts for the early enrollees. Adding one counter will help reduce a
number of students waiting in long queues. Results show that if another counter is added,
the average number of students in the system is reduced from 39 to 14 students per hour;
the average time students spends in the system is reduced from 30.8 minutes to 10.75
minutes and the probability that an arriving student has to wait for service is reduced
from 95% to 39%. The more the counter, the better the service is.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
21
LITERATURE CITED
BAROVKOV,A.A. Asymptotic Methods in Queueing Theory. P.63.
BEAUBRUN, R. 2007. International Journal of Wireless Information Networks,
Volume
14.
BELL, MARTIN L. 1992. Mathematical Methods in Queueing Theory. P.5.
CHANG, W. 1980 Single Server Queueing Processess in Computing Systems, IBM
Systems Journal, Pp.42-50.
COX, D. R. AND SMITH W. L. 1993 Queues (New York; Wiley) Pp.6-12.
DENISOV, DENIS AND SAPOZHNIKOV, A. 2006. Proofs of Queueing Formula. P.39.
J.R. ARTALEJO, A. ECONOMOU AND M.M. LOPEZ-HERRERO. 1985. An
Introduction To The Research Queueing Package. In Proc. 1985 Winter
Simulation
Conf., Pp. 257-262.
K.S. TRIVEDI. 1982 Probability and Statistics with reliability, queueing and
computer science application. PreticeHall, Englewood Cliffs.Pp. 134-149.
KRINIK, A., RUBINO, G., MARCUS, D., SWIFT R., KASFY, H.,LAM, H.
2005.
Dual Process to Solve Single Server Sytem; In Journal of Statistical
Planning and
Inference.P.135.
M.A. TURNQUIST AND J.M. SUSSMAN. 1977. Toward guidelines for Designing
Experiments in Queueing Simulation. In Simulation, Pp. 137,144.
MEDHI, J. 1991. Stochastic Models in Queueing Theory. Academic Press,
Boston.P.452.
MOHR, B. 1983. Controlled Queueing Systems. P.12.
MOOD, A. M.., GRAYBILL, F. A.., BOES, D. C. Introduction to the Theory of
Statistics-Third
Edition. P.94.
SUN, Z.; COSMAS, J.; SCHARF, E.M.; AND CUTHBERT, L.G. 1989. Parallel
Simulation of Queueing networks: Limiations and Potentials In Performance
Evaluation Review, Pp. 146,155.
TANNER, M.1995. Practical Queueing Analysis. New York McGrow Hill. P.3.
WEI, Y. 2003 Modeling and Performance Analysis of a Multiserver Multiqueue System
on the Grid P. 337.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
ACOSTA, RODLYNN MAY M. AND NORMAN, CLEVENSON C. APRIL
2008. Analysis on the Number of Students Serviced at the Cashier’s Counter. Benguet
State University, La Trinidad, Benguet.
Adviser: Marycel H. Toyhacao
ABSTRACT
This study was conducted to determine the arrival distribution of students and
service time rendered by the cashier at the counter of Benguet State University
Administration and the queue discipline of the students waiting at the cashier counter.
Poisson and exponential distribution were employed in this study. It has been
found that the average number arrival at the cashier counter is 76 students per hour and
the average number of students that the cashier counter can serviced per hour is 26. This
discrepancy of student’s arrival and number of students serviced at the cashier’s counter
resulted to long queues.
The model was identified to fit the enrollment operation with three cashier
counters in the administration. The measuring capacity utilization (ρ) shows that 97.44%
of time the counters are busy. The average time a student spent at the system is 0.5142
hours, which means that a customer will receive complete service for approximately
30.85 minutes including the time the students waiting in the line. The average time in the
queue is 0.475738 hours which is approximately 28.54 minutes, which is the time spent
waiting in the line. The remaining time, which is W-Wq, is the time spends in the
counter, 0.038462 hours or 2.30772 minutes.
To improve the operating characteristics of the queueing of students serviced at
the cashier’s counter; some suggestions may be offered to the management to change the
arrival rates in a number of ways, such as providing discounts or other incentives for the
early enrollees. Adding one to the existing number of cashier counter will help reduce
number of students waiting in long queues. Results shows that if we add another counter,
the average number of students in the system was reduced from 39 to 14 students per
hour.; the average time students spends in the system is reduced from 30.8 minutes to
10.75 minutes and the probability that an arriving student has to wait for service was
reduced from 95% to 39%. More counter, the better the service.
ii
TABLE OF CONTENTS
Pages
Bibliography...……………………………………………………………………………..i
Abstract………………………………………………………………………………....…i
Table of Contents...………………………………………………………....…………... iii
INTRODUCTION
Background of the Study………………………………………………………..….1
Objectives of the Study………………………………….….…………………...…..2
Importance of the Study…………………………………..……………………...…3
Scope and Delimitation of the Study…………………..………………………...…3
REVIEW OF LITERATURE
Studies of Queueing…………………………………………...…………………….4
THEORETICAL FRAMEWORK
Arrival
Distribution…………………………………………………………....…..8
Service
Distribution…………………………………………………………....….8
Multiserver Multiqueue Model……………………………………………..……..9
Definition of Terms……………………………………………………………....10
METHODOLOGY
Locale and Time of the Study………………………………………………...……12
Data Used…………………………………...…………………………………...…12
Data Analysis………………………………………………………………...…….12
iii
RESULTS AND DISCUSSION
Arrival Distribution……………………………………………………………...…13
Service Time Distribution……………………………………………………...…..15
Operating Characteristics of Students
Serviced at the Cashier Counter……………………………………………….......17
SUMMARY, CONCLUSION, and RECOMMENDATION
Summary………………………………………………………………………...…19
Conclusion……………………………………………………………………...….19
Recommendation……………………………………………………………...…...20
LITERATURE CITED……………………………………………………………...…..21
iv
1
INTRODUCTION
Background of the Study
Waiting in lines is part of everyday life. Whether it is waiting in line at a grocery
store to buy items or checking out at the cash registers, or waiting at an amusement park
to go on the newest ride, a lot of time waiting is spent. People, wait in lines at the movies,
campus dining rooms, the Registrar’s Office for class registration, and many more.
Most people hate waiting. But reduction of the waiting time usually requires extra
investments. To decide whether or not to invest, it is important to know the effect of the
investment on the waiting time. So models and techniques are needed to analyze such
situations according to Mohr (1983).
Medhi (1991) in his Stochastic Models in Queueing System, states that attention
is paid to methods for the analysis and applications of queueing models. It is particularly
useful for the design of these systems in terms of layout, capacities and control.
Cox and Smith (1973) on “Queues” write that queueing model is used to
approximate a real queueing situation or system, It is characterized by arrival process of
customers, behavior of customers,service times and discipline. Arrival process of
customers usually assumes that the interarrival times are independent and have a common
distribution. In many practical situations customers arrive according to a Poisson stream
(i.e., exponential interarrival times). Customers may arrive one by one, or in batches.
Second is the behavior of customers. They may be patient and willing to wait for a long
time or customers maybe impatient and leave after a while. Third is the service time
which usually assumes that the service times are independent and identically distributed,
Analysis on the Number of Students Serviced at the Cashier’s Counter
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2
and that they are independent of the interarrival times. It can also occur that service times
are dependent of the queue length. For example, the processing rates of the machines in a
production system can be increased once the number of jobs waiting to be processed
becomes too large. Fourth is the service discipline. Customers can be served one by one
or in batches. There are many possibilities for the order in which they enter service.” First
come, first served”, in the order of arrival wherein the first one who arrived is the one
who will be entertained first like in the paying of fees especially in the enrollment of the
students.
Upon entering school, students must first enroll. Considering the enrollment
procedure of the College of Arts and Sciences students, students must present their class
card to the teacher who is assigned to assess the students to be recorded at the checklist.
After presenting the grades, the students will be given an enrollment form to be filled up
before assessment. The students will then go to the Dean’s office to be assessed for their
fees and then proceed to the Administration building for paying. There, students wait for
their turn to be entertained. However, there are only three (3) counters who will render
the sevice in the Administration. It is expected that a line will be formed. This is a system
where students arrive, join a waiting line, wait their turn, serve by a multiple server and
depart. It is described as the queueing system.
Objectives of the Study
This study aimed the following:
1. To determine the arrival distribution of students at the cashiers counter of BSU
Administration;
Analysis on the Number of Students Serviced at the Cashier’s Counter
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2. To determine the service time rendered by the cashier to the students at the
counter;
3. To determine the queueing model and operating characteristics of the student
enrollees at the cashier’s counter; and
4. To determine the economic analysis of the operating characteristics of queueing
of students enrollees at BSU Admistration.
Importance of the Study
The result of the study provides information to the management on the proper
management of queue and on what level of services will they offer. The arrival rate
specifies the average number of students arriving at the cashier’s counter per hour which
could help the management decide whether or not to make changes or choose alternatives
to improve the situation that involves waiting lines.
The study also reminds students, to practice and develop the value of patience and
willingness to wait for their turn.
The result of the study serves as a basis for the other researchers who will be
interested to conduct queueing analysis.
Scope and Delimitation
The study was conducted at Benguet State University Administration Building,
La Trinidad, Benguet, during enrollment period for the Second semester school year,
2007-2008. The data gathering was executed on October 22-26, 2008. The main focus of
the study is the application of queueing analysis of the students serviced at the cashiers
counter regarding paying their enrolment fees.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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REVIEW OF RELATED LITERATURE
Queue is everywhere. It is seen everywhere, paying for bills, line at a
supermarket counter, line at the banks machine, line of cars at a traffic light show.
Waiting line is undesirable for all parties times, considering the following studies
concerning queues.
Studies on Queueing
Mohr (1983) cited that Arrival Distribution for a waiting line involved
determining how many costumers arrive in a period of time. He determined the number
of customers in a 1-hour period, since the number of customer each hour is not
necessarily constant.
Queueing analysis is applied also on the parallell simulation of queueing networks
as studied by Sun, Z. et. al. (1991). Its limitations and potentials in performace evaluation
review on the queueing delay and cell loss for combined traffic sources in ATM networks
in United Kingdom as the main purpose. The development of broadband ISDN based on
the ATM introduces new network characteristics different to those traditional channel
based networks, typically queueing delay and cell loss when combined traffic sources
share the same bandwidth capacity. The paper reports the work on studying the new
network characteristics so that the bandwidth resource can be utilized efficiently with
restricted queueing delay and cell loss.
Another application of queueing was undertaken by Denisov and Sapozhnikov
(2006) of the Department of Mathematics/Boole Centre for Research in Informatics,
University College Cork, Cork, Ireland on the distribution of the number of customers in
the symmetric M/G/1 queue wherein they consider an M/G/1 queue with symmetric
Analysis on the Number of Students Serviced at the Cashier’s Counter
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5
service discipline. In this research they show that this conjecture is true if service
requirements have an Erlang distribution. They also show by a counter example,
involving the hyper exponential distribution, that the conjecture is generally not true.
In Madrid, Spain, Artalejo, Economou and Lopez-Herrero (1985) of the
Department of Statistics used queueing model on their research entitled “Analysis of a
Multiserver Queue with Setup Times” that deals with the analysis of an M/M/c queueing
system with setup times. This queueing model captures the major characteristics of
phenomena occurring in production when the system consists in a set of machines
monitored by a single operator, which carry out an extensive analysis of the system
including limiting distribution of the system state, waiting time analysis, busy period and
maximum queue length.
Beaubrun (2007) analyzes the traffic distribution and blocking probability in
future wireless networks through Poisson and Exponential distribution. Traditional
analysis of teletraffic in such networks assumes that call arrivals follow a Poisson
process, as each cell is being modeled as an M/G/c/c queueing system. Such a model has
not been explicitly addressed in the literature, the main contribution is to propose a
solution which enables to evaluate both traffic distribution and blocking probability
within each cell of the service area. Result analysis reveals that coefficient of variation of
call arrivals has more impact on the network performance than coefficient of variation of
channel holding time.
Wei (2003) conducted a study on grids. Grid computing has emerged as an
important new field, distinguished from conventional distributed computing by its focus
on large-scale resource sharing, innovative applications, and high-performance
Analysis on the Number of Students Serviced at the Cashier’s Counter
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6
orientation. A Grid integrates and coordinates resources and users that live within
different control domains with the goal of delivering non-trivial quality of service (QoS).
Consequently, task management and scheduling is of central importance for the Grid-
based systems. In this paper, the task dispatching and selecting on the distributed Grid
computing system by a multiserver multiqueue (MSMQ) model, and propose a modeling
and analysis technique based on Stochastic Petri Net (SPN) methods are used. An
approximate analysis technique is also proposed to reduce the complexity of the model.
Queues in the Philippines are common scenery and queueing discipline is not that
practiced. Thus, this study is conducted regarding long lines. However, the application of
queueing analysis here in the Philippines is not limited and the utilization is not yet fully
explored. It is due to lack of device or strategy in recording the data necessary for the
analysis and even the availability in some packages, is limited.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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7
THEORETICAL FRAMEWORK
Cox and Smith (1993) state that queueing theory deals with mathematical models
of various kinds of real queues, example is the situations where congestion occurs due to
randomness in arrival and service times and customers have to wait for service. Queues
occur when the current demand for service exceeds the capacity of service facilities, and
the main purposes of the theory is to provide means for designing/modifying/optimizing
service systems in such a way as to reduce the likelihood of queues, the customer’s
waiting times and so on. Analysis of queueing models may include determining the
distributions of the queue length, waiting times and the duration of the busy/idle periods
for servers.
Trivedi (1982) states that a queue or queueing system is a system which includes
a random input stream of requests which needs service and a mechanism which provides
that service. Typical examples of queues are telephones exchanges, customer’s queues at
checkout counters on business institutions in an airport, and a network of time sharing
computers.
Since cashiers counter of Benguet State University Administration Building
consist of a three queue and three service facility and assumes that arrivals are unlimited
and can be formally treated in a Poisson distribution. Service times are assumed to take
on the form of an Exponential distribution.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Arrival Distribution
For many waiting lines, the arrivals occurring in a given period of time appear to
have a random pattern-that is, while there may be a good estimate of the total number of
arrivals expected, each arrival is independent of other arrivals and cannot predict when
the arrival will occur. In this case, Poisson distribution will be used because it provides a
good description of the arrival pattern.
The Poisson probability distribution is defined as:
P(x) = λx e – λ
for x = 0, 1, 2, . . .
x!
Where, in waiting line applications,
x = number of arrivals in a specified period of time
λ= average of expected number of arrivals for the specified
period of time
e = 2.71828
Service Time Distribution
A service fine probability distribution is needed to describe how long it takes a
student to wait for his turn and to pay. Since students have different amount of fees,
cashier’s service times vary. Thus, exponential probability distribution will be used.
The exponential probability distribution is defined as:
f(x)
=
μe-μx
for x ≥ 0
where:
x = service time
µ = average or ecpected number of units that the service facility
Analysis on the Number of Students Serviced at the Cashier’s Counter
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9
can handle in a specific period of time
e = 2.71828
Multiserver-Multiqueue Model
A logical extension of the single-channel waiting line is the multiserver
multiqueue single channel waiting line. By multiple-channel waiting lines we mean that
two or more counter or service locations are present. It allows us to examine the situation
where the number of counters in service facility may assume any finite value
To do this, the following short-hand notation to solve the formula given is used.
Where:
λ = average arrival rate of the students
μ = average service rate per counter
s = number of available counter
1. the percentage of time the counter are busy or the probability that the
counters
are
busy
λ
ρ =
sμ for sμ>λ
2. the probability that no students in the ounter
1
Po =
n
s
λ
λ
∑ μ
μ
+
!
n
!
s 1
( − (ρ / s))
3. the average time a student spent in the queue waiting for the service
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
10
s
ρ Po
!
s 1
( − (ρ / s))
Wq =
sμ 1
( − (ρ / s))
4. the average time a student spends in the system (waiting time + service time)
1
W = Wq +
μ
5. the average number of students in the queue waiting for service
Lq = λWq
6. the average number of students in the service system
L = λW
Definition of Terms
The following are defined for the purpose of the study:
Arrival defines the way customers enter the system. Mostly the arrivals are
random with random intervals between two adjacent arrivals. Typically the arrival is
described by a random distribution of intervals also called Arrival Pattern.
Arrival Rate is the average number of students arriving per time period denoted
by lambda.
Cashiers counter is a window where cashier’s stay and does its work.
Exponential distributions are a class of continuous probability distribution. They
are often used to model the time between independent events that happen at a constant
average rate.
Poisson distribution is a discrete probability distribution that expresses the
probability of a number of events occurring in a fixed period of time if these events
occur with a known average rate, and are independent of the time since the last event.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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11
Queue is also known as the waiting line, that is, the place where a certain number
of customers spend most of their time waiting for service in the queueing system.
Queueing discipline represents the way the queue is organized.
Queueing system considered to be the waiting lines or the queues as well as the
number of available service facility which contains one or more server.
Service Time, as applied to queuing systems, is a period of time
describing how long it takes a student to be serviced.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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METHODOLOGY
Locale and Time of the Study
The study was conducted at Benguet State University, La Trinidad, Benguet
during the second semester, school year 2007-2008.
Data Gathering Tool
Data were gathered through individual observations among the Benguet State
University students who paid their fees during enrollment period at the cashiers counter
(Administration Building), on the 22nd until 26th day of October 2007. The data gathered
were the arrival time, waiting time and service time.
Data Analysis
The summarized data were utilized in the application of Poisson probability
distribution to explain the arrival pattern of the Benguet State University students
serviced at the Cashier’s counter. Exponential probability distribution was also used to
provide a good description of the service time distribution. The Multiserver-Multiqueue-
Single-Channel Waiting line Model was utilized to determine the queue data
characteristics.
The data gathered were encoded,summarized and were analyzed using Microsoft
Excel. Average number of students arriving per hour (λ) and the average number of
students that can be served per hour (μ) were computed using the Microsoft Excel.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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RESULTS AND DISCUSSION
Arrival Distribution
Table 1.1 presents the summary for the arrivals of students per hour interval, the
mean of the arrivals, the total number of arrivals and the overall mean.
Time interval 1:00-2:00 had the highest students arrival of 869 with the highest
mean of 174 students arrivals per hour. This time interval had the highest arrival because
students were done processing their enrollment and now ready to pay, where, time
interval 4:01-5:00 had the lowest number of arrivals of 62 and lowest mean of 12. This is
because the time interval that was considered was 30 minutes wherein cashiers are
leaving the counters and the closing time for the administration.
The students arrivals were not scheduled in paying and occurred in an
unpredictable manner, a random pattern appears to exist. Mohr (1983) determined that
indeed, arrival is not constant in a period of time. Thus, to determine the student arrivals,
the Poisson distribution provides a good description of the arrival pattern which is
xe λ
λ −
P( X = x) =
defined by as
!
x
for x=40, 42, 44,…,60
where λ= 76 and e=2.71828.
Analysis on the Number of Students Serviced at the Cashier’s Counter
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Table 1.1 Summary of arrivals and arrival means
HOUR INTERVAL
# OF STUDENTS
MEAN ARRIVAL/HOUR
8;00-9:00 97
19
9:01-10:00 440
88
10:01-11:00 467
93
11:01-12:00 232
46
1;00-2;00 869
174
2;01-3:00 558
112
3:01-4:00 310
62
4:01-5:00 62
12
overall mean
76
Table 1.2 presents the Poisson students arrival distribution of Administration
cashier counter ,where we would expect 40 students arriving in an hour 1.75% of the
time, exactly 45 arrival in an hour 4.11 % of the time, we would expect 50 students
arriving in an hour 5.58% of the time, and so on.
The study by Beaubrun (2007) also uses the same distribtution we used in this
study where in, he uses Poisson in assuming the call arrivals which its main contribution
is to propose a solution which enables to evaluate both traffic distribution and blocking
probability within each cell of the service area.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
15
Table 1.2 Arrival distribution
X=NUMBER OF ARRIVALS
P(X≤X)
40 0.017464586
42 0.026379436
43 0.036264753
44 0.04556745
45 0.052535877
46 0.055773802
47 0.054701229
48 0.049712752
49 0.041981451
50 0.033028964
51 0.024267891
52 0.054701229
53 0.052637032
54 0.049712752
55 0.04609728
56 0.041981451
57 0.037562351
58 0.033028964
59 0.02855046
60 0.024267891
Service Time Distribution
Table 2.1 presents the summary for the number of students serviced by the cashier
per one hour interval, the mean of the students serviced by the cashier, the total number
of students who were serviced and the overall mean of the students being served by the
cashier counter per day.
Time interval 1:00-2:00 had the highest mean students that were served by the
cashier counter of 40 which was equal to the highest students arrival 869. This implies
that the numbers of arrivals per hour was 76 and the number of students that was being
served by the cashier counter per hour was 26. Thus there is the discrepancy between the
arrivals of students per hour and the students serviced by the cashier counter per hour is
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
16
the reason for queueing. Time interval 8:00-9:00 had the lowest number of students who
were served for a reason that students are still processing their enrollment procedures.
Table 2.1 Summary of average students serviced per hour
HOUR
COUNTER COUNTER COUNTER
MEAN
INTERVAL
I
II
III
SERVICED/HOUR
8;00-9:00 6
5
18
2
9:01-10:00 45
150
101
20
10:01-11:00 187
146
168
33
11:01-12:00 151
184
112
30
1;00-2;00 209
164
202
40
2;01-3:00 221
154
186
37
3:01-4:00 155
152
162
31
4:01-5:00 3
70
85
11
overall mean
26
The exponential probability distribution was used to describe the service time,
since students demanded a different service that depends on what amount they are
supposed to pay, thus the Cashiers counter service times varied. The exponential
distribution is defined by ƒ(x) = μe-μt for x ≥ 0 where μ=26 and e=2.71828. Table 2.2
presents the exponential distribution for the service time thus, 98.94% of the students are
expected to be serviced in 6 minutes or less (t=0.1), 99.97% in 12 minutes or less (t=0.2),
so on and 100% of the students to be serviced in 36 minutes (t=.6).
Analysis on the Number of Students Serviced at the Cashier’s Counter
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17
Table 2.2 Service time distribution ofassumed values of service time (in hours)
X=SERVICE TIME (IN HOURS)
P(SERVICE TIME ≤X)
0.1 0.983427279
0.2 0.999725345
0.3 0.999995448
0.4 0.999999925
0.5 0.999999999
0.6 1
0.7 1
0.8 1
0.9 1
1.0 1
Operating Characteristics of Students
Serviced at the Cashier’s Counter
Using the Poisson arrivals and exponential service times identical servers and a
first come first serve discipline, operating characteristics for the multiserver-multiqueue-
single-channel waiting line model was presented in Table 3.
Table 3 presents the teller utilization rate which is the percentage of the time a
counter is busy. The total service rate must be greater than the arrival ratet that is kμ > λ.
If sμ ≤ the average number of customer spends in the system (L) and the average time a
student spends in the system (W) both become infinitely large. In table 3, the utilization
rate is clear that the formula is applicable since λ/kμ < 1 thus the probability that the
servers were busy is 97% . the probability that the cashier is idle is .5%. This implies that
the server in free for .3 minutes upon the start of the operation thus most of the time the
cashier is busy. On the average, there are 39 students in the system. The average time a
customer spends in the system is .51 hour or 30 minutes. The average time a customer
spends in the queue before being served is .47 hour or 28 minutes. The average number
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
18
of students in the queue is 36 students and the probability that an arriving customer has to
wait for service is .95.
Suppose another counter is added with the same arrival rate and service rate and a
first in-first-out discipline. Based on table 3 the probability that the servers are busy is
reduced by 24% from 97% to 73%. The probability that the server is idle is increased
from .5% to 3.4% or from .3 minutes to 2.09 minutes.
Table 3. Economic analysis on the operating characteristics of the queueing of students
serviced at the cashiers counter
SERVERS
(s)
ρ=λ/sμ P(0) L
W Lq Wq Pw
3 0.974358
0.00586
39 0.5142 36 0.475738
0.95125
4 0.730769
0.03489
14
0.17925
11
0.14078
0.3942
where:
L= average number of units in the system
W= the average time a unit spends in the system (waiting time + service)
Lq= the average number of units in the ueue waiting for service
Wq= the average time a unit spends in the queue waiting for service
Pw= the probability that an arriving students has to wait for service
The average number of student in the system is reduced by 25 from 39 to 14
students. The average time a student spends in the system is reduced by 20 minutes from
.5142 hour to .17925 hour or from 30 minutes to 10 minutes. The average number of
students in the queue is reduced by 25 from 36 to 11 students. The probability that an
arriving customer will wait is reduced from 95% to 39%.
The assumption is clear that four counters will greatly improve operating
characteristics of the system.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
19
SUMMARY, CONCLUSION AND RECOMMENDATION
Summary
Poisson and exponential distribution were employed in this study. The average
number of students who arrive is 76 students per hour and the average students that the
cashier counter can service per hour were 26, that results to long lines of students.
The model was identified to analyze the enrollment operation for there are three
cashier counters in the administration. The measuring capacity utilization (ρ) shows
th97.44% of time the counters are busy. The average time a student spent at the system is
.5142 hours, which means that a customer will receive complete service for
approximately 30.85 minutes including the time the students waiting in the line. The
average time in the queue is 0.475738 hours which is approximately 28.54 minutes, that
is the time spent waiting in the line. The remaining time, which is W-Wq is the time
spent in the counter, 0.038462 hours or 2.30772 minutes.
Conclusion
It is the variability in arrival and service distribution that causes waiting lines.
Waiting line model allows to estimate performance by predicting average system
utilization, average number of students in the service system, average number of students
in the waiting line, average time a student spends in the system and the average time
students wait in line.
The result on the operating characteristics indicates that students wait an average
of .5142 hours or 31 minutes. Which appears to be undesirable for students to wait for
their turn for almost half an hour. And shows that long waits suggest a lack of concern by
the management or, can be linked to a perception of poor service management. But
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
20
sometimes, it’s the students fault why they were served that long because they arent
following the schedule given by the management for them to enroll.
Recommendation
The operating characteristics of the queueing of students serviced at the cashier’s
counter suggest the management to change the arrival rates in a number of ways, such as
providing discounts for the early enrollees. Adding one counter will help reduce a
number of students waiting in long queues. Results show that if another counter is added,
the average number of students in the system is reduced from 39 to 14 students per hour;
the average time students spends in the system is reduced from 30.8 minutes to 10.75
minutes and the probability that an arriving student has to wait for service is reduced
from 95% to 39%. The more the counter, the better the service is.
Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
21
LITERATURE CITED
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Analysis on the Number of Students Serviced at the Cashier’s Counter
/ Rodlynn May M. Acosta & Clevenson C. Norman. 2008
Document Outline
- Analysis on the Number of Students Serviced at the Cashier�s Counter
- BIBLIOGRAPHY
- ABSTRACT
- TABLE OF CONTENTS
- INTRODUCTION
- REVIEW OF RELATED LITERATURE
- THEORETICAL FRAMEWORK
- METHODOLOGY
- RESULTS AND DISCUSSION
- SUMMARY, CONCLUSION AND RECOMMENDATION
- LITERATURE CITED