Independent Journal of Management &
Production
E-ISSN: 2236-269X
ijmp@ijmp.jor.br
Instituto Federal de Educação, Ciência e
Tecnologia de São Paulo
Brasil
Gumus, Seigha; Monday Bubou, Gordon; Humphrey Oladeinde, Mobolaji
APPLICATION OF QUEUING THEORY TO A FAST FOOD OUTFIT: A STUDY OF BLUE
MEADOWS RESTAURANT
Independent Journal of Management & Production, vol. 8, núm. 2, abril-junio, 2017, pp.
441-458
Instituto Federal de Educação, Ciência e Tecnologia de São Paulo
Avaré, Brasil
Available in: http://www.redalyc.org/articulo.oa?id=449551140011
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APPLICATION OF QUEUING THEORY TO A FAST FOOD
OUTFIT: A STUDY OF BLUE MEADOWS RESTAURANT
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.
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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.
Keywords: queuing theory; Poisson distribution; service rate; customer satisfaction;
fast food outfit; Blue Meadows 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
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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).
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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
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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.
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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).
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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)
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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)
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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
Monday
Tuesday
Wednesday
Thursday
Friday
SD
1.38
1.29
0.92
1.15
1.02
Arrival time
0.98 1.56
1.14 1.82
1.20 1.65
1.25 1.93
1.34 1.96
Arrival Rate
38.46 61.22
32.96 52.47
36.14 50.00
31.14 47.86
30.55 44.90
Service Time
2.33
3.86
2.19
3.48
2.22
3.07
2.11
3.24
2.14
3.15
Service Rate
15.54 24.69
17.21 27.34
19.51 26.97
18.47 28.41
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
0
1
2
3
4
5
6
7
8
9
10
0
20
21
21
24
24
25
25
26
31
31
∑ = 248
Relative frequency
0
20
42
63
96
120
150
175
208
279
310
∑ = 1463
0
0.081
0.085
0.085
0.096
0.096
0.101
0.101
0.105
0.125
0.125
0.0004
0.0459
0.0668
0.092
0.094
0.102
0.156
0.132
0.127
0.113
0.110
in 1hour
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Figure 1: Compares of Relative Frequency to Probabilities for peak period
Table 3: Relative Frequency and probabilities for off peak period
Arrivals
frequency
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0
11
11
13
13
14
15
15
15
16
16
17
17
19
19
19
19
24
24
25
41
∑ = 363
Relative frequency
0
11
22
39
52
70
90
105
120
144
160
187
204
247
266
285
304
408
432
475
820
∑ = 4441
0
0.030
0.030
0.036
0.036
0.039
0.041
0.041
0.041
0.044
0.044
0.047
0.047
0.052
0.052
0.052
0.052
0.066
0.066
0.069
0.113
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0.000005
0.060
0.040
0.015
0.045
0.0211
0.0326
0.0396
0.0505
0.0423
0.0406
0.0449
0.0459
0.0490
0.0538
0.0565
0.0585
0.0621
0.070
0.074
0.122
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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
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4.4.
Chi Square Goodness of Fit test for off-peak period
Observed frequency and theoretical frequency obtained are
This gives us a chi square
and
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
4.5.
is not rejected
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
Monday
Tuesday
Wednesday
Thursday
Friday
9 - 10
6.0769
6
6.818
5.273
5.538
10 -11
4.7
5
3.333
3.367
4.4
11 – 12
2.647
2.882
2.857
2.857
2.833
12 - 1
4.317
3.875
3.875
3.056
3.111
1–2
4.125
3
3
3.187
3.063
2–3
5.08
3.885
3.885
2.583
2.667
Table 5: Exponential distribution for peak period
Service rate (µ)
2.583
2.667
3.333
3.367
3.885
3.885
4.400
4.700
5.000
5.080
Expected value E(T)
0.38714
0.37495
0.30003
0.29700
0.25740
0.25740
0.22727
0.21276
0.20000
0.19685
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Figure 3: Probability Density Function for the exponential distribution. (Peak Period)
Table 6: Exponential distribution for off Peak Period
Service rate (µ)
2.647
2.833
2.857
2.857
2.882
3.000
3.000
3.056
3.063
3.111
3.187
3.875
3.875
4.125
4.317
5.273
5.538
6.000
6.077
6.818
Expected value E(T)
0.3777
0.3529
0.3500
0.3500
0.3469
0.3333
0.3333
0.3272
0.3264
0.3214
0.3137
0.2580
0.2580
0.2424
0.2316
0.1896
0.1805
0.1666
0.1645
0.1466
Figure 4: Probability Density Function for the exponential distribution. (Off Peak
Period)
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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
Monday
Tuesday
Wednesday
Thursday
Friday
Arrival Rate
47.0
40.6
42.0
37.6
36.0
Service Rate
19.0
21.13
22.64
22.38
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
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ISSN: 2236-269X
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DOI: 10.14807/ijmp.v8i2.576
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
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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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