Seigha Gumus
Department of Research, Collaboration and Consultancy
South-South Office, National Centre for Technology
Management Niger Delta University, Nigeria
E-mail: seighagumus@gmail.com
Gordon Monday Bubou
Department of Research, Collaboration and Consultancy
South-South Office, National Centre for Technology
Management Niger Delta University, Nigeria
E-mail: gbubou@gmail.com
Mobolaji Humphrey Oladeinde
Production Engineering Department Faculty of
Engineering
University of Benin, Nigeria
E-mail: moladeinde@uniben.edu.ng
Submission: 28/11/2016
Revision: 16/12/2016
Accept: 25/12/2016
ABSTRACT
The
study evaluated the queuing system in Blue Meadows restaurant with a view to
determining its operating characteristics and to improve customers’
satisfaction during waiting time using the lens of queuing theory. Data was
obtained from a fast food restaurant in the University of Benin. The data
collected was tested to show if it follows a Poisson and exponential
distribution of arrival and service rate using chi square goodness of fit. A
95% confidence interval level was used to show the range of customers that come
into the system within a time frame of one hour and the range of customers
served within that time frame. Using the M/M/s model, the arrival rate, service
rate, utilization rate, waiting time in the queue and the probability of
customers likely balking from the restaurant was derived. The arrival rate (λ)
at Blue Meadows restaurant was about 40 customers per hour, while the service
rate was about 22 customers per hour per server. The number of servers present
in the system was two. The average number of customers in the system in an hour
window was 40 customers with a utilization rate of 0.909. The paper concludes
with a discussion on the benefits of performing queuing analysis to a
restaurant.
1. INTRODUCTION
Queuing
theory also known as Random System Theory is the body of knowledge about
waiting lines and is now an entire discipline within the field of operations
research (NOSEK; WILSON, 2001; KAVITHA;
PALANIAMMAL, 2014; RAMAKRISHNA; MOHAMEDHUSIEN, 2015).
In
fact, queuing theory has become a valuable tool for operations managers
(RAMAKRISHNA; MOHAMEDHUSIEN, 2015). The authors maintain that waiting has
become part of everyday life. For example, queuing system has been employed in
our day to day commercial (as well as socio-political) lives (KAVITHA;
PALANIAMMAL, 2014).
Some
scholars maintain that we queue or wait in line to get served in commercial
outfits like checkout counters, banks, super markets, fast food restaurants
etc. (KAVITHA;
PALANIAMMAL, 2014), grocery stores, post offices, to waiting on hold for
an operator to pick up telephone calls, waiting at an amusement park to go on
the newest ride (MANDIA, 2009). Others according to the authors include:
waiting in lines at the movies, campus dining rooms, the registrar’s office for
class registration, at the Division of Motor Vehicles etc. We equally queue in
other socio-political settings like queuing to vote, waiting in line to be
attended to by a public servant in government offices, etc.
Waiting
causes not only inconvenience, but also frustrate people’s daily lives. Thus,
unmanaged queues are detrimental to the gainful operation of service systems
and results in a lot of other managerial problems (YAKUBU; NAJIM, 2011).
In
order to reduce the frustrations of customers, managers adopt certain measures
like multiple line/multiple checkout systems (RAMAKRISHNA; MOHAMEDHUSIEN,
2015). The duo further stated that, in recent years, many banks,
credit unions, as well as fast food providers have shifted to a queuing system
whereby customers wait for the next available cashier, as this removes the frustrations
of "getting in a slow line" since one slow transaction does not
affect the throughput of the remaining customers.
Queuing
theory according to Dharmawirya and Adi (2011) was particularly suitable to be
applied in a fast food or restaurant settings, since it has an associated queue
or waiting line where customers who cannot be served immediately have to queue
for service. Blue Meadows is a fast food
restaurant selling fast food cuisines with a minimal table service for its
customers.
Blue
Meadows restaurant is situated in the University of Benin, behind the
Postgraduate School. This fast food restaurant operates in a manner that
customers can take away their orders immediately after payment or sit down at
the premises to enjoy their meal. However, clients suffer unnecessary delays,
especially during peak periods. This shows a need of a numerical model for Blue
Meadows’ management to understand the situation better.
Thus,
the aim of the study was to assist Blue Meadows solve this problem by decreasing
customers’ waiting time by modeling a queuing theory to simulate the waiting
lines. It is intended to show that queuing theory satisfies the model when
tested with a real – case scenario.
The
remainder of the paper is structured as follows: the next section provides a
background into queuing theory, its associated terminology, its applications
and its relationship to customer satisfaction, as well as a review of related
works. Next, we discuss the Blue Meadows’ Model. This is followed by the presentation
of analysis and results. The results are then discussed, and of course we
conclude with a summary.
2. THEORETICAL BACKGROUND
The
queuing theory is known as Random System Theory which has the solutions for
statistical interference and problem of behavior and optimization in queuing
system (KAVITHA;
PALANIAMMAL, 2014). It is the
formal study of waiting lines which has now become an area of scientific
inquiry, sub-discipline within operations research (COPE et al., 2011).
The
origin of queuing theory dates back over a century. Indeed, Chowdhurry (2013)
confirms that the study of waiting lines was one of the oldest and most widely
employed quantitative analysis techniques. However, Johannsen’s “Waiting Times
and Number of Calls” (an article published in 1907 and reprinted in Post Office
Electrical Engineers Journal, London, October, 1910) seems to be the first
paper on the subject.
It
had its early research work in the early 1900s by a Danish engineer named A.K
Erlang of the Copenhagen Telephone Company (COOPER, 1990; COPE et al. 2011;
CHOWDHURRY, 2013; RAMAKRISHNA; MOHAMEDHUSIEN, 2015). Erlang is claimed to have
derived several important formulas for teletraffic engineering that today bore
his name.
According
to Ramakrishna and Mohamedhusien (2015), it was only after World War II that
works on waiting line models were extended to other kinds of problems. The
authors maintain that, today a wide variety of seemingly diverse problem
situations are recognized as being described by the general waiting line model.
Indeed, queuing theory has many applications
in human endeavors, some of which include: telephony; manufacturing;
inventories; dams; supermarkets; computer and information communication systems
and networks; call centers; hospitals, banking, etc. (SZTRIK, 2010). Nosek and Wilson
(2001) confirm that queuing theory has been used extensively by the service
industries.
Undoubtedly,
there are numerous factors that affect a customer’s perception of the waiting
experience, some of which include: physical, psychological and emotional. If
there were to be no queue at all, it would create the impression that the value
of the attraction is to some extent diminished. However, one may observe that
attractions with short queues tend to attract less public. So, in principle, it
is important not to aim at eliminating queues, but instead concentrate on
giving people an option to join the queue, or skip part of the queue and spend
the time somewhere else.
The
dynamics of queues have been analyzed by using steady-state mathematics.
Essentially, it is purely a mathematical approach that is employed in the
waiting line analysis (KAVITHA; PALANIAMMAL, 2014). While various models
constitute several queuing systems (KAVITHA; PALANIAMMAL, 2014), such queuing
processes are described by using the Kendall-Lee (1953) notation which uses
mnemonic characters that specify the queuing system:
A/B/C/D/E/F
–
A: Specifies the nature of the arrival process.
–
B: Specifies the nature of the service times.
–
C: Specifies the number of parallel servers
–
D: Specifies the queue discipline.
–
E: Specifies the maximum number of entities in the
system.
–
F: Specifies the size of the population from which
entities are drawn.
2.1.
Characteristics
of a queuing process
The
queuing theory considers mainly six general characteristics of any queuing
processes:
i.
Arrival pattern of customers: inter-arrival times most
commonly fall into one of the following distribution patterns: A Poisson
distribution, a Deterministic distribution, or a General distribution. However,
inter-arrival times are most often assumed to be independent and memoryless,
which is the attributes of a Poisson distribution.
ii.
Service pattern: the service time distribution can be
constant, exponential, hyper exponential, hypo-exponential or general. The
service time is independent of the inter-arrival time
iii.
Number of servers: the queuing calculations change
depends on whether there is a single server or multiple servers for the queue.
A single server queue has one server for the queue. This is the situation
normally found in a grocery store where there is a line for each cashier. A
multiple server queue corresponds to the situation in a bank in which a single
line waits for the first of several tellers to become available.
iv.
Queue Lengths: the queue in a system can be modeled as
having infinite or finite queue length.
v.
System capacity: the maximum number of customers in a
system can be from 1 up to infinity. This includes the customers waiting in the
queue.
vi.
Queuing discipline: there are several possibilities in terms of the sequence of
customers to be served.
·
FCFS: First Come, First Served. This is the
most commonly used discipline applied in the real-world situations, such as
check-in counters at the airport.
·
LCFS: Last Come, First Served. This
illustrates a reverse order service given to customer versus their arrival.
·
SIRO: Service in Random Order.
·
PD: Priority Discipline. Under this discipline,
customers will be classified into categories of different priorities.
According
to Nosek and Wilson (2001), queuing management has been applied very
successfully in several service-oriented industries. For instance, many
researchers have previously used queuing theory to model the fast food
operation, and many service industries to reduce cycle time in a busy system
such as hospitals and restaurants as well as to increase throughput and
efficiency (see for example, SOMANI; DANIELS; JERMSTAD, 1982; PIERCE (II);
ROGERS; SHARP, 1990; ANDREWS; KHARWAL, 1991; PARSONS, 1993; JONES; DENT, 1994;
PROCTOR, 1994; ROSENFELD, 1997; BRANN; KULICK, 2002; CURIN; VOSKO; CHAN;
TSIMHONI, 2005; TYAGI; SAROA.; SINGH,
2014; OLADEJO; AGASHUA; TAMBER, 2015).
Nevertheless,
the type of queuing system a business uses is an important factor in
determining how efficient the business is run (ZHANG; NG; TAY, 2000). However,
there are several other determining factors for a restaurant to be considered a
good or a bad one; taste, cleanliness, the restaurant layout and settings are
some of the most important factors.
These
factors when managed carefully will be able to attract plenty of customers.
Besides attributes such as location, ambience and quality of food (AUTY, 1992),
other important factors to be considered is when a restaurant has succeeded in
attracting customers is the price and the customers queuing time (DHARMAWIRYA;
ADI, 2011; LI; LEE, 1994).
3. BLUE MEADOWS’ QUEUING MODEL
The
method employed in the data collection was by observation. The data collected
was the arrival time, inter-arrival time, waiting time and number of customers
in the queue at Blue Meadows restaurant for a period of five days (Monday –
Friday) with a time frame of one-hour window intervals from 09Hrs – 15Hrs
daily. Based on observation, it is concluded that the model that best
illustrates the operation of Blue Meadows is M/M/2.
This
means that the arrival and service time are exponentially distributed (Poisson
Process). The restaurant system consists of only two servers. However, the data
obtained has been tested to show that it fits both Poisson and exponential
distribution. However, distribution is
used for testing the goodness of fit for the set of data collected. The actual
frequencies are compared to the frequencies that theoretically would be
expected to occur if the data follows the Poisson distribution.
Assume
a random variable T represents either inter-arrival or service times.
The
random variable is said to have an exponential distributed with parameter µ, if
its probability density function is:
(1)
The
cumulative probabilities are:
(2)
The
expected value and variance of T are, respectively,
(3)
The
confidence intervals for average service rate and average arrival rate can be
estimated. Assuming service time and arrival time are identically independent
with N(0,1) then the 95% confidence interval for arrival can be:
[(mean arrival
time + 1.96*SE (mean arrival time)-1, (mean arrival time – 1.96*SE
(mean arrival time))-1] (4)
Where
SE(mean arrival time) = SD(mean arrival time) / n
Similarly,
95% confidence interval for service rate can be:
[(mean service time + 1.96*SE (mean
service time)]-1,
The
following queuing parameters are used to describe the model.
Utilization
factor, (5)
The
probability that the system shall be idle,
(6)
The
expected number of customers waiting in the queue,
(7)
The
expected number of customers in the system,
(8)
The
expected waiting time of customers in the queue,
(9)
The
expected waiting time a customer spends in the system,
(10)
4. ANALYSIS AND RESULTS
To
calculate for a 95% confidence interval for both service time and arrival time,
the following were obtained. The standard deviation and mean arrival time is
obtained from the data using spread sheet.
Table 1: Confidence
intervals for Arrival Rate and Service Rate
Days |
SD |
Arrival time |
Arrival Rate |
Service Time |
Service Rate |
||||
Monday |
1.38 |
0.98 |
1.56 |
38.46 |
61.22 |
2.33 |
3.86 |
15.54 |
24.69 |
Tuesday |
1.29 |
1.14 |
1.82 |
32.96 |
52.47 |
2.19 |
3.48 |
17.21 |
27.34 |
Wednesday |
0.92 |
1.20 |
1.65 |
36.14 |
50.00 |
2.22 |
3.07 |
19.51 |
26.97 |
Thursday |
1.15 |
1.25 |
1.93 |
31.14 |
47.86 |
2.11 |
3.24 |
18.47 |
28.41 |
Friday |
1.02 |
1.34 |
1.96 |
30.55 |
44.90 |
2.14 |
3.15 |
19.02 |
28.00 |
The
confidence intervals show the range of number of customers that arrive within
an hour time frame for each day and the range of number of customers served.
4.1.
Test
for Poisson distribution
This
test is statistically tested to show the pattern in which the customers arrive
at the system. The test was carried out for both peak and off-peak periods.
Peak periods were between 10:00hrs – 11:00hrs and 14:00hrs – 15:00hrs.
The
following data from Monday to Friday has been compiled to obtain Table 2 below:
Table 2: Relative Frequency
and probabilities for peak period
Arrivals |
Frequency |
|
Relative frequency |
|
0 |
0 |
0 |
0 |
0.0004 |
1 |
20 |
20 |
0.081 |
0.0459 |
2 |
21 |
42 |
0.085 |
0.0668 |
3 |
21 |
63 |
0.085 |
0.092 |
4 |
24 |
96 |
0.096 |
0.094 |
5 |
24 |
120 |
0.096 |
0.102 |
6 |
25 |
150 |
0.101 |
0.156 |
7 |
25 |
175 |
0.101 |
0.132 |
8 |
26 |
208 |
0.105 |
0.127 |
9 |
31 |
279 |
0.125 |
0.113 |
10 |
31 |
310 |
0.125 |
0.110 |
∑ = 248 |
∑ = 1463 |
in 1hour
Figure 1: Compares of Relative Frequency to Probabilities for peak period
Table 3: Relative Frequency
and probabilities for off peak period
Arrivals |
frequency |
|
Relative frequency |
|
0 |
0 |
0 |
0 |
0.000005 |
1 |
11 |
11 |
0.030 |
0.060 |
2 |
11 |
22 |
0.030 |
0.040 |
3 |
13 |
39 |
0.036 |
0.015 |
4 |
13 |
52 |
0.036 |
0.045 |
5 |
14 |
70 |
0.039 |
0.0211 |
6 |
15 |
90 |
0.041 |
0.0326 |
7 |
15 |
105 |
0.041 |
0.0396 |
8 |
15 |
120 |
0.041 |
0.0505 |
9 |
16 |
144 |
0.044 |
0.0423 |
10 |
16 |
160 |
0.044 |
0.0406 |
11 |
17 |
187 |
0.047 |
0.0449 |
12 |
17 |
204 |
0.047 |
0.0459 |
13 |
19 |
247 |
0.052 |
0.0490 |
14 |
19 |
266 |
0.052 |
0.0538 |
15 |
19 |
285 |
0.052 |
0.0565 |
16 |
19 |
304 |
0.052 |
0.0585 |
17 |
24 |
408 |
0.066 |
0.0621 |
18 |
24 |
432 |
0.066 |
0.070 |
19 |
25 |
475 |
0.069 |
0.074 |
20 |
41 |
820 |
0.113 |
0.122 |
∑ = 363 |
∑ = 4441 |
Figure 2: Compares of Relative Frequency to Probabilities
4.2.
Chi
Square Goodness of Fit test for peak period
Using
Chi Square goodness of fit to test, the data for both peak and off peak
periods, it is shown that the observed frequency and theoretical frequency
obtained are and
This
gives us a chi square value of
With
a degree of freedom:
n
– 1 = 11 – 1
=
10
Where n = 11
and a 0.05 level of significance
from tables, the critical value of with 10 degree
of freedom is given as 18.31
4.3.
Hypothesis/Decision
rule:
Reject
Ho: if 18.31
otherwise do not reject Ho
18.31
Therefore,
Ho is not rejected
4.4.
Chi
Square Goodness of Fit test for off-peak period
Observed
frequency and theoretical frequency obtained are and
This
gives us a chi square value of
Degree
of freedom:
n
– 1 = 21 – 1
=
20
Using
0.05 level of significance, from tables, the critical value of with 20 degree
of freedom is 31.41
Reject : if 31.41
otherwise do not reject
31.41
Therefore
is not rejected
4.5.
Test
for exponential distribution
The
data obtained is also tested to show the pattern in which the service rate
follow. The test was carried out for both peak and off-peak periods.
Table 4: Daily count for
service rate
Days/time |
9 - 10 |
10 -11 |
11 – 12 |
12 - 1 |
1 – 2 |
2 – 3 |
Monday |
6.0769 |
4.7 |
2.647 |
4.317 |
4.125 |
5.08 |
Tuesday |
6 |
5 |
2.882 |
3.875 |
3 |
3.885 |
Wednesday |
6.818 |
3.333 |
2.857 |
3.875 |
3 |
3.885 |
Thursday |
5.273 |
3.367 |
2.857 |
3.056 |
3.187 |
2.583 |
Friday |
5.538 |
4.4 |
2.833 |
3.111 |
3.063 |
2.667 |
Table 5:
Exponential distribution for peak period
Service rate (µ) |
Expected value E(T) |
2.583 |
0.38714 |
2.667 |
0.37495 |
3.333 |
0.30003 |
3.367 |
0.29700 |
3.885 |
0.25740 |
3.885 |
0.25740 |
4.400 |
0.22727 |
4.700 |
0.21276 |
5.000 |
0.20000 |
5.080 |
0.19685 |
Figure 3: Probability Density Function for the exponential distribution.
(Peak Period)
Table 6: Exponential
distribution for off Peak Period
Service rate (µ) |
Expected value E(T) |
2.647 |
0.3777 |
2.833 |
0.3529 |
2.857 |
0.3500 |
2.857 |
0.3500 |
2.882 |
0.3469 |
3.000 |
0.3333 |
3.000 |
0.3333 |
3.056 |
0.3272 |
3.063 |
0.3264 |
3.111 |
0.3214 |
3.187 |
0.3137 |
3.875 |
0.2580 |
3.875 |
0.2580 |
4.125 |
0.2424 |
4.317 |
0.2316 |
5.273 |
0.1896 |
5.538 |
0.1805 |
6.000 |
0.1666 |
6.077 |
0.1645 |
6.818 |
0.1466 |
Figure 4: Probability Density Function for the exponential distribution. (Off
Peak Period)
The
various mean for Monday to Friday of the data collected have been obtained
using formula:
Arrival rate =
Service rate =
Duration
of data collection = 180 minutes
Table 7: Arrival and Service
rate for Monday - Friday
Days |
Arrival Rate |
Service Rate |
Monday |
47.0 |
19.0 |
Tuesday |
40.6 |
21.13 |
Wednesday |
42.0 |
22.64 |
Thursday |
37.6 |
22.38 |
Friday |
36.0 |
22.64 |
Therefore,
Mean arrival rate = 40 Customers per
hour
Mean service rate = 22 Customers
per hour
The
average number of customers waiting in line = 17.9 Customers
Average
time spent waiting in line = 0.45 hours
Expected
waiting time for a customer in the system = 0.49 hours
Expected
number of customers in the system = 19.72 Customers
5. DISCUSSIONS
The
following results and test obtained are discussed in detail. The confidence
interval for both arrival and service time at 95% shows the range of number of
customers that come into the system and also the range of customers served on a
daily basis. It also shows that there are still some customers not being served
and are waiting for their turn in the queue to be served. This is however due
to the service provided by a server to a customer.
For
testing the data to show that it fits the Poisson distribution, Tables 2 &
3 shows values of the relative frequency to that of the probabilities for the
peak period. However, the plot in figure 4.2 shows a close relationship between
the probabilities and the relative frequencies. A goodness of fit test is also
calculated using chi square to show that the data obtained follows a Poisson distribution.
The theoretical frequency is compared to the actual frequency obtained. A 0.05
level of significance is used. The same procedure was also carried out for
off-peak period to show that the data (arrival rate) follows a Poisson
distribution.
The
plot in Figures 4.4 and 4.5 show the pattern in which the service rate follows.
From the plot, it shows that the random variable with parameter µ is
exponentially distributed. The random variable µ is plotted against the
expected value.
However,
from the calculations of the queuing parameters, it is shown that the
performance of the servers on average was sufficiently good. It can be seen
that the probability of the servers to be busy was 0.909 which was 90.9%. The
average number of customers waiting in a queue is Lq = 17.9 customers per 2-servers. The waiting time in a
queue per server is Wq =
0.45 hours which are normal time in a busy server. The plot in Figure 4.6 shows
the probability curve which takes the shape of the Probability Density
Function.
This
is a clear indication that the data obtained follows an exponential
distribution. Also, the utilization obtained was directly proportional to the
mean number of customers. This simply means that the mean number of customers
will increase as the utilization increases. The utilization rate at the
restaurant was highly above average at 0.909. This was the utilization rate
during breakfast and lunch time at week days. When the service rate was higher,
the utilization will be lower which reduces the probability of the customers
going away.
6. CONCLUSION
Providing
insight into the study of queue theory through the examination of the Blue
Meadows Waiting Line Model, our work presents a foundation for the development
of strategies that may enhance customer satisfaction in fast food restaurants
and other service industries. We evaluated the performance of single channels,
two servers in Blue-Meadows restaurant at the University of Benin. The
utilization rate at the restaurant was highly above average at 0.909.
This
gave the probability of the servers to be busy at 90.9%. The Model played a key
role in highlighting the operations effectiveness of the services rendered as
well as identifying need for improvements. In order to improve operations
within the waiting line, the service rate should be improved. This research can
help improve the quality service at Blue Meadows restaurant. The result of the
research work may serve as a reference to analyze the current system and
improve the next system.
However,
the restaurant can now estimate how many customers will wait in the queue and
how many will walk away each day. By anticipating the number of customers
coming and going in a day, the restaurant can set a target profit that should
be achieved daily depending on the purchases each customer makes. Some of the
limitations of the study included: the inaccuracy of results since some of the
data that were used were based on approximation. It is hoped that findings from
this study can contribute to the betterment of Blue-Meadows restaurant in terms
of its way of dealing with customers.
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