Application of Exponential Gamma Distribution in Modeling Queuing Data

International Journal of Trend in Scientific Research and Development (IJTSRD)
Volume 4 Issue 2, February 2020
@ IJTSRD | Unique Paper ID – IJTSRD30097
Application
Distribution
Ayeni Taiwo Michael, Ogunwale Olukunle Daniel, Adewusi Oluwasesan Adeoye
Department of Statistics, Ekiti
ABSTRACT
There are many events in daily life where a queue is formed. Queuing
theory is the study of waiting lines and it is very crucial in analyzing the
procedure of queuing in daily life of human being. Queuing theory applies
not only in day to day life but also in sequence of computer
networks, medical field, banking sectors etc. Researchers have applied
many statistical distributions in analyzing a queuing data. In this study, we
apply a new distribution named Exponential
a data on waiting time of bank customers before service is been rendered.
We compared the adequacy and performance of the results with other
existing statistical distributions. The result shows that the Exponential
Gamma distribution is adequate and also performed better than
existing distributions.
KEYWORDS: Exponential-Gamma distribution, Queuing theory, AIC, BIC,
Networks
INRODUCTION
There are many events in daily life where a queue is
formed. Queuing theory is the study of waiting lines and it
is very crucial in analyzing the procedure of queuing in
daily life of human being. Queuing theory applies not only
in day to day activities but also in sequence of computer
programming, networks, medical field, banking sectors
etc. A queuing theory called a random service theory is
one of the issues in mathematics so that the existing
techniques in the queuing theory have substantial
importance in solving mathematical problems and
analyzing different systems [1].
Statistical distributions are very crucial in describing and
predicting real life occurrence. Although, many
distributions have been developed and there are always
rooms for developing distributions which are more
flexible and capable of handling real world application.
Lately, studies have shown that some real life data cannot
be analyzed adequately by existing distributions. At times,
it may be discovered to follow distributions of so
combined form of two or more random variables with
known probability distributions. In light of their
adequacies, variety in usage and performance, statistical
distributions have received a numerous attentions from
various researchers such as; [2],[3],[4] and [5]. Therefore
this study aimed to examine the adequacy and
performance of the new Exponential-Gamma distribution
to other exiting probability distributions using the data on
February 2020 Available Online: www.ijtsrd.com e
30097 | Volume – 4 | Issue – 2 | January-February 2020
Application of Exponential-Gamma
Distribution in Modeling Queuing Data
Department of Statistics, Ekiti State University, Ado-Ekiti, Nigeria
daily life where a queue is formed. Queuing
theory is the study of waiting lines and it is very crucial in analyzing the
procedure of queuing in daily life of human being. Queuing theory applies
not only in day to day life but also in sequence of computer programming,
networks, medical field, banking sectors etc. Researchers have applied
many statistical distributions in analyzing a queuing data. In this study, we
apply a new distribution named Exponential-Gamma distribution in fitting
e of bank customers before service is been rendered.
We compared the adequacy and performance of the results with other
existing statistical distributions. The result shows that the Exponential-
Gamma distribution is adequate and also performed better than the
Gamma distribution, Queuing theory, AIC, BIC,
How to cite this paper
Michael | Ogunwale Olukunle Daniel |
Adewusi Oluwasesan Adeoye
"Application of Exponential
Distribution in Modeling Queuing Data"
Published in
International
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456
6470, Volume
Issue-2, February
2020, pp.839
www.ijtsrd.com/papers/ijtsrd30097.pdf
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
(http://creativecommons.org/
by/4.0)
daily life of human being. Queuing theory applies not only
t also in sequence of computer
programming, networks, medical field, banking sectors
etc. A queuing theory called a random service theory is
one of the issues in mathematics so that the existing
techniques in the queuing theory have substantial
in solving mathematical problems and
Statistical distributions are very crucial in describing and
predicting real life occurrence. Although, many
distributions have been developed and there are always
distributions which are more
flexible and capable of handling real world application.
Lately, studies have shown that some real life data cannot
be analyzed adequately by existing distributions. At times,
it may be discovered to follow distributions of some
combined form of two or more random variables with
known probability distributions. In light of their
adequacies, variety in usage and performance, statistical
distributions have received a numerous attentions from
4] and [5]. Therefore
this study aimed to examine the adequacy and
Gamma distribution
to other exiting probability distributions using the data on
waiting time of bank customers before service is being
rendered, using the model selection criteria like the
Akaike information criterion (AIC), Bayesian information
criterion (BIC) and the log likelihood function (
METHODS
The Exponential-Gamma distribution was developed by
[6] and its pdf is defined as
1 1 2
( ; , ) , , , 0
( )
x e
f x x
α α λ
λ
α λ λ α
α
+ − −
= >
Γ
With the mean and variance;
1
2α
α
µ +
=
and ( )V x
α α λα
=
The cumulative distribution function is defined as
( )F x
λγ α
=
The survival function for the distribution defined by
( ) 1 ( )S x F x= − was obtained as;
e-ISSN: 2456 – 6470
February 2020 Page 839
Gamma
Data
Ekiti, Nigeria
How to cite this paper: Ayeni Taiwo
Michael | Ogunwale Olukunle Daniel |
Adewusi Oluwasesan Adeoye
"Application of Exponential-Gamma
Distribution in Modeling Queuing Data"
Published in
tional
Journal of Trend in
Scientific Research
and Development
(ijtsrd), ISSN: 2456-
6470, Volume-4 |
2, February
2020, pp.839-842, URL:
www.ijtsrd.com/papers/ijtsrd30097.pdf
Copyright © 2019 by author(s) and
International Journal of Trend in
Scientific Research and Development
Journal. This is an Open Access article
distributed under
the terms of the
Creative Commons
Attribution License (CC BY 4.0)
http://creativecommons.org/licenses/
waiting time of bank customers before service is being
he model selection criteria like the
Akaike information criterion (AIC), Bayesian information
criterion (BIC) and the log likelihood function (Ɩ).
Gamma distribution was developed by
1 1 2
( ; , ) , , , 0
( )
x
x e
f x x
α α λ
α λ λ α
+ − −
= > (1)
With the mean and variance;
1+
(2)
( )
( )
( )2 1
2 2
2
α α
α
α α λα
λ +
− +
= (3)
The cumulative distribution function is defined as
( , )
2 ( )
x
α
λγ α
α
=
Γ
(4)
The survival function for the distribution defined by
was obtained as;
IJTSRD30097

International Journal of Trend in Scientific Research and Development (IJTSRD) @ www.ijtsrd.com eISSN: 2456-6470
@ IJTSRD | Unique Paper ID – IJTSRD30097 | Volume – 4 | Issue – 2 | January-February 2020 Page 840
( , )
( ) 1
2 ( )
x
S x α
λγ α
α
= −
Γ
(5)
While the corresponding hazard function defined by
( )
( )
( )
f x
h x
S x
= was obtained as;
1 1 2
2
( )
2 ( ) ( , )
x
x e
h x
x
α α λ α
α
λ
α λγ α
+ − −
=
Γ −
(6)
The cumulative hazard function for distribution defined by
0
( ) ( ( )) log(1 ( )) ( )
x
H x W F x F x h x dx= = − − ≡ ∫ and
was obtained as;
( , )
( )
2 ( ) ( , )
x
H x
xα
λγ α
α λγ α
=
Γ −
(7)
A. Maximum Likelihood Estimator
Let 1 2X ,X ,...,Xn be a random sample of size n from
Exponential-Gamma distribution. Then the likelihood
function is given by;
1
1
11
L( , ; ) exp 2
( )
n
n n
i i
ii
x x x
α
αλ
α λ λ
α
+
−
==
   
= −   Γ   
∑∏ (8)
by taking logarithm of (9), we find the log likelihood
function as;
1 1
log( ) log log log ( ) ( 1) log 2
n n
i i
i i
L n n n x xα λ λ α α λ
= =
= + − Γ + − −∑ ∑(9)
Therefore, the MLE which maximizes (9) must satisfy the
following normal equations;
'
1
log ( )
log log
( )
n
i
i
L n
n x
α
λ
α α =
∂ Γ
= − +
∂ Γ
∑ (10)
1
log
2
n
i
i
L n n
x
α
λ λ λ =
∂
= + −
∂
∑ (11)
The solution of the non-linear system of equations is
obtained by differentiating (9) with respect to ( , )α λ
gives the maximum likelihood estimates of the model
parameters. The estimates of the parameters can be
obtained by solving (10) and (11) numerically as it cannot
be done analytically. The numerical solution can also be
obtained directly by using python software using the data
sets.
In this study, we applied the Akaike Information Criterion
(AIC), Bayesian Information Criterion (BIC) and the log
likelihood function (Ɩ) to compare the new developed
Exponential-Gamma distribution with the existing
probability distributions such as the Exponential and the
Gamma distributions.
The AIC is defined by;
ˆ2 2 ( )AIC k In= − l (12)
where ݇ is the number of the estimated parameter in the
model
ˆl is the maximized value of the model.
The BIC is defined by;
ˆ( ) 2 ( )In n k In− l (13)
where ݇ is the number of the estimated parameter in the
model
n is the number of observations
ˆl is the maximized value of the model.
The approach in (12) and (13) above is used when
comparing the performance of different distributions to
determine the best fit model. To select the appropriate
model by considering the number parameters and
maximum likelihood function; the AIC, BIC and likelihood
function are examined; consequently an acceptable model
has smaller AIC and BIC value while the log likelihood
value is expected to be greater. The Python software was
used for the comparison of the performance of the
Exponential-Gamma, Exponential and Gamma
distributions.
RESULTS
The analysis of the data set was carried out by using
python This data has been previously used by [7], [8] and
[9]. It represents the waiting time (measured in min) of
100 bank customers before service is being rendered. The
data is as follows:
Table1
0.8, 0.8, 1.3, 1.5, 1.8, 1.9, 1.9, 2.1, 2.6, 2.7, 2.9, 3.1, 3.2,3.3,
3.5, 3.6, 4.0, 4.1, 4.2, 4.2, 4.3, 4.3, 4.4, 4.4, 4.6, 4.7,4.7, 4.8,
4.9, 4.9, 5.0, 5.3, 5.5, 5.7, 5.7, 6.1, 6.2, 6.2, 6.2,6.3, 6.7, 6.9,
7.1, 7.1, 7.1, 7.1, 7.4, 7.6, 7.7, 8.0, 8.2, 8.6,8.6, 8.6, 8.8, 8.8,
8.9, 8.9, 9.5, 9.6, 9.7, 9.8, 10.7, 10.9, 11.0,11.0, 11.1, 11.2,
11.2, 11.5, 11.9, 12.4, 12.5, 12.9, 13.0, 13.1, 13.3, 13.6,13.7,
13.9, 14.1, 15.4, 15.4, 17.3, 17.3, 18.1, 18.2, 18.4, 18.9, 19.0,
19.9,20.6, 21.3, 21.4, 21.9, 23.0, 27.0, 31.6, 33.1, 38.5.
Table2: Summary of the data
Parameters Values
n 100
Min 0.800
Max 38.500
Mean 9.877
Variance 52.37411
Skewness 2.540292
Kurtosis 1.472765
The results from the table 2 above indicated that the
distribution of the data is skewed to the right with
skewness 2.540292. This shows that Exponential-Gamma
has the ability to fit a right skewed data. Also, it was
observed that the kurtosis is 1.472765 which is lesser than
3. This implies that the distribution has shorter and lighter
tails with a light peakedness when compared to that of the
Normal distribution.

Table3: Estimates and performance of the distributions (bank customers)
Distribution Parameters log likelihood(Ɩ) AIC BIC
Exponential-Gamma
ˆα = 1.7
ˆλ = 0.4
-12.5 30.9 38.7
Exponential ˆα = 0.8 -320.6 645.2 650.3591
Gamma
ˆα =1.7
ˆλ = 0.5
-316.7 639.5 647.3
The estimates of the parameters, log-likelihood, Akaike information criterion (AIC) and Bayesian information criterion
(BIC) for the data on waiting time of 100 bank customers before service is being rendered is presented in Table 3 It was
observed that Exponential-Gamma provides a better fit as compared to Exponential and Gamma distributions since it has
highest value of log-likelihood (Ɩ) and the lowest value of Akaike information criterion (AIC) and Bayesian information
criterion (BIC). Hence, the Exponential-Gamma distribution performed better than other distribution compared.

CONCLUSION
daily life of human being. In this study we apply the new
Exponential-Gamma in modeling queuing data on bank
customers waiting time before been serviced. We
compared the adequacy and performance of the results of
new Exponential-Gamma distribution with the exiting
probability distributions. The new Exponential-Gamma
distribution is adequate and performed better than other
distribution compared. It also fits the data better than
other existing distributions. Therefore, for higher
precision in analyzing queuing data, the use of
Exponential-Gamma distribution is highly recommended
in various fields where analysis of queuing data is crucial.
References
[1] Dieter, F., S. Bart and B. Herwig. Performance
evaluation of CAI and RAI transmission modes in a
GI-G-1 queue. Computers & Operations Research, vol.
28, pp. 1299-1313, 2001.
[2] Zhou W., Analysis of a single-server retrial queue
with FCFS orbit and Bernoulli vacation. Applied
Mathematics and Computation, vol. 161: pp. 353-
364, 2005.
[3] Damodhar F Shastrakar, Sharad S Pokley 2018
Application Of binomial Distribution and Uniform
Distribution to Study the Finite Queue Length
Multiple Server Queuing Model International Journal
of Pure and Applied Mathematics vol. 120, pp 10189-
10205, 2018.
[4] O.S Adesina, G. Odularu, A. F Adedayo. Modeling
queuing system with inverse gamma distribution: a
spreadsheet simulation approach Anale. Seria
Informatică. vol. XVI, pp 55-60 2018.
[5] Menth, M., Henjes, R., Zepfel, C., & Tran-Gia, P. 2006.
Gamma-approximation for the waiting time
distribution function of the M/G/1 - ∞ queue. 2nd
Conference on Next Generation Internet Design and
Engineering, 2006. NGI ’06.
doi:10.1109/ngi.2006.1678232
[6] Ogunwale, O. D., Adewusi, O. A. and Ayeni, T. M.
Exponential-Gamma Distribution; international
journal of emerging technology and advanced
engineering; vol.9; pp 245-249, 2019.
[7] Ghitany, M. E., Atieh, B. and Nadarajah, S. Lindley
distribution and its application. Mathematics and
Computers in Simulation. vol.78,pp 493-506. 2008
[8] Oguntunde, P. E., Owoloko, E. A. and Balogun, O. S.
(2015), On A New Weighted Exponential
Distribution: Theory and Application; Asian Journal
of Applied Sciences vol. 9 pp 1-12, 2016
[9] Alqallaf, F., Ghitany, M. E. and Agostinelli, C. Weighted
exponential distribution: Different methods of
estimations. Applied Mathematics and Information
Sciences, vol. 9, pp 1167-1173, 2015

Application of Exponential Gamma Distribution in Modeling Queuing Data

More Related Content

What's hot (19)

Similar to Application of Exponential Gamma Distribution in Modeling Queuing Data (20)

More from ijtsrd (20)

Recently uploaded (20)

Application of Exponential Gamma Distribution in Modeling Queuing Data