Modeling Queuing Phenomenon at Petrol Pumps
-A Case Study of Bharat Petroleum
Rakesh Kumar
Lecturer
School of Mathematics
Shri Mata Vaishno Devi University, Sub Post Office, Katra-182320 (J&K)-INDIA
Abstract
The queuing problems are more frequent everywhere. A significant amount of time and
resources are wasted if we do not have a suitable mechanism to deal with the rising
congestion/queuing problems. This study is a part of a consultancy project which is
undertaken at Bharat Petroleum. The problems of queues at various refueling points have
been studied and the key reasons which result into long queues have been identified. The
main objective is to study the queuing phenomena of vehicles and minimize the queues at
various refueling points. The relevant data have been collected and the queuing analyses
of different queues have been performed using TORA (Techniques of Operations
Research Applications) software. Finally, some useful suggestions have been made.
Introduction
The problems of queues/ waiting lines are very common in our everyday life. Queues are
usually seen at bus stop, ticket booths, petrol pump, bank counter, traffic lights and so on.
Queuing theory deals with the mathematical description of behavior of queues. Queuing
theory can be applied to a variety of operational situations where it is not possible to
predict accurately the arrival rate of customers and service rate of service facilities. In
particular, it can be used to determine the level of service (either the service rate or the
number of service facilities).

Email: [email protected], [email protected]
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The present study is a part of a consultancy project which is undertaken at Bharat
Petroleum petrol pump in Greater Noida, India. Bharat Petroleum petrol pump is one and
only petrol pump in Greater Noida near the heart of the city adjacent to Ansal Plaza. It is
the only petrol pump between Greater Noida and Noida 24 km expressway. Due to one
and only fuel outlet in the city long queues can easily be seen in the service area.
Company deals in supply of petrol, diesel, Xtra Premium, turbojet and lubricants from
Bharat petroleum. Company has 8 refueling pumps, 4 for two wheelers and 4 for four
wheelers and heavy vehicles. The petrol filling station has employed 12 service
executives, 8 collection executives, 1 accountant and 1 supervisor working under the
ownership of a Retired Army officer.
The petrol pump was established in 15 January 2005 and was the second petrol pump
after one in the Surajpur industrial area. The petrol pump is located at Pari Chowk the
entry point and the most important square of Greator Noida. The completion and opening
of Ansal Plaza one and only multiplex, increased business of the petrol pump
significantly. Nearness to the Knowledge Parks also added to business because it added
thousands of customers in its consumer pool.
The problems of queues at various refueling points have been studied and the key reasons
which result into long queues have been identified. The suitable queuing models have
been developed for different queues by studying the arrival and service patterns of
customers. The models have been solved using TORA software.
Objectives of the study
The objectives of the study are:
(i)
To understand and identify the queuing problems at petrol pumps.
(ii)
To perform quantitative decision making using the appropriate operations
research models.
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(iii)
To develop suitable queuing models for minimizing the vehicle queues at various
refueling points.
(iv)
To decide up on the opening of one more petrol pump.
Methodology
The methodology to solve the problem involves the following:
1. Primary data collection and direction observations
2. Secondary Data.
1. Primary data collection and direction observations:
Several visits have been made to the petrol pump. The working staff was interviewed
regarding the type of problems they were facing in queuing handling. The primary data
concerning the arrival pattern of vehicles and their service pattern have been collected
using the format given below:
PRIMARY DATA
BHARAT PETROLEUM
QUEUE NO:
SR.
NO.
VEHICLE
NO.
PRODUCT NAME:PETROL/SPEED
TIME IN
HOURS
MINUTES
TIME START SERVING
SEC
HOURS
MINUTES
SEC
TIME OUT
HOURS
MINUTES
SEC
1/λn
Where 1/λn = Interarrival time , 1/μn = service time
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1/μn
2. Secondary Data
The secondary data pertaining to the number of service staff and their salary, number of
supervisors and manger with their salary, cost of operating the machines, profit per unit
and other overheads have also been collected.
Formulation of Queuing Model
At each refueling point there is only one queue and one service executive. The vehicles
join a particular queue one by one and they are served one by one on first-come, first
served (FCFS) basis. There is no limit on vehicles joining a particular queue. The primary
data collected shows that there was complete randomness in arrival as well as service
patterns. An M/M/1 queuing model has been proposed for each queue, which is based on
the following assumptions:
(i) The vehicles arrive at a particular queue one by one and follow Poisson distribution
with parameter λ, where λ is the mean arrival rate.
(ii) There is only one server at each queue and the vehicles are served one by one on
FCFS basis. The service times are independently, identically and exponentially
distributed with mean rate μ.
(iii) The capacity of each queue is infinite, meaning thereby any number of vehicles can
join a particular queue.
Here, we have four M/M/1 parallel queues.
The mean arrival rate λ and the mean service rate μ of the four queues have been
calculated from the data as follows-
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Queue No.
λ
μ
1
2
0.023355322
0.022886094
0.025971451
0.025546022
3
4
0.012843312
0.011187608
0.015487253
0.017223105
Solution of the model
The Queuing model has been solved using the TORA software and the following results
have been obtained. Various measures of performance like p0 –the probability that there
is no customer in the queue, Ls- average number of customers in the system, Lq- average
number of customers in the queue, Ws- average waiting time of a customer in the system,
Wq- average waiting time of a customer in the queue, the server’s utilization and the
server’s idle time have been computed for all the queues.
QUEUEING OUTPUT ANALYSIS
Title: Bharat Petroleum, Greater Noida
Comparative Analysis
Table 1.
Scen
ario
(Queue)
C (No.
of
Servers)
Lambda
Mu
L'da
p0
Ls
Lq
Ws
(sec)
Wq
(sec)
Server
Utilizati
on (%)
Server
Idle
Time
(%)
1
1
0.0233
0.0259
0.0233
0.1007
8.9271
8.0279
382.233
343.730
0.8992
0.1007
2
1
0.0228
0.0255
0.0228
0.1041
8.6041
7.7082
375.953
336.808
0.8958
0.1041
3
1
0.0128
0.0154
0.0128
0.1707
4.8575
4.0282
378.214
313.645
0.8292
0.1707
4
1
0.0111
0.0172
0.01119
0.3504
1.8536
1.2040
165.6863
107.6248
0.6495
0.3504
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Results & Discussions
Through the study it has been found that the most congested period is from 9:00 AM to
11:00 AM in morning and 4:00 pm to 6:00 PM in the evening. This is the time when
most of the persons go to or come from their office and colleges, thus increasing the
inflow of vehicles at the petrol pump. Rest all the times, they have limited number of
vehicles which they can easily serve and the vehicles in the queue at that time were one
or two vehicles waiting for their service. This suggests that Management can go for some
part- time employees during the peak time periods and the cost of hiring such employees
will also be low.
The first two queues are for Petrol and the third and fourth queues are for Speed. From
table-1, we can see that the average queue lengths in queues 1 and 2 are higher than in
queues 3 and 4. Consequently, the average waiting times in queues 1 and 2 are higher
than that of queues3 and 4. The comparative analysis of all the four queues in table-1
provides a quantitative basis for analyzing the queuing phenomena at petrol pump.
The individual analysis of the four queues have been given in the annexure-I. In
annexure-I, the probabilities of ‘n’ (n upto 20) number of customers in the queue have
also been shown which help to deal with the uncertain queuing formations. The petrol
pump manager can better decide quantitatively on the number of service executives
required at a particular time period, the utilization of service facility, the idle periods and
the delays faced by the customers in different queues. Such analysis will definitely help
the manager efficiently run the facility. Earlier, the managing staff used to take decisions
qualitatively, which was resulting into wastage of time and resources.
The cost of opening another filling point is very high in comparison of the margin profit
generating from that extra filling point. And the rate of arrival of the customers is very
fluctuating so the idea of opening another filling point is not appropriate. Some more
filling stations are soon opening in Greater Noida mainly one from Indian Oil at sector
Delta, which can change the customers’ inflow so any further decision can only be taken
after these competitors are functional.
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Conclusion
The queuing problems under consideration have been studied quantitatively. Suitable
queuing models have been made and quantitative results have been obtained. The
software results allow the manager of petrol pump to compare the various measures of
performance of different queues. This analysis can help manager take decisions more
precisely as compared to the decisions based on intuition and judgment.
References
1. Taha, H. A., Operations Research, 7th Ed. (2005).
2. Trivedi, K.S., Probability and Statistics with Reliability, Queuing and Computer
Science Applications, 15th Ed. (2003).
3. Kumar, Rakesh and Khan, Nuzhat, Customer Service and Queues at Big Bazaar,
Gyanpratha (Accman Journal of Management Science), Vol. 1, No. 2 (July, 2009) 107113.
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Annexure-I
TORA Optimization System, Windows®-version 1.00
QUEUEING OUTPUT ANALYSIS
Title: Bharat Petrolium G. Noida
Scenario 1-- (M/M/1):(GD/infinity/infinity)
Lambda = 0.02336, Mu = 0.02597
Lambda eff = 0.02336, Rho/c = 0.89927
Ls = 8.92718, Lq = 8.02792
Ws = 382.23377, Wq = 343.73003
n
Probability, pn
Cumulative, Pn
0
0.10073
0.10073
1
0.09059
0.19132
2
0.08146
0.27278
3
0.07326
0.34604
4
0.06588
0.41191
5
0.05924
0.47115
6
0.05327
0.52443
7
0.04791
0.57233
8
0.04308
0.61541
9
0.03874
0.65415
10
0.03484
0.68899
11
0.03133
0.72032
12
0.02817
0.74849
13
0.02534
0.77383
14
0.02278
0.79661
15
0.02049
0.81710
16
0.01842
0.83552
17
0.01657
0.85209
18
0.01490
0.86699
19
0.01340
0.88039
20
0.01205
0.89244
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TORA Optimization System, Windows®-version 1.00
QUEUEING OUTPUT ANALYSIS
Title: Bharat Petrolium G. Noida
Scenario 2-- (M/M/1):(GD/infinity/infinity)
Lambda = 0.02289, Mu = 0.02555
Lambda eff = 0.02289, Rho/c = 0.89588
Ls = 8.60412, Lq = 7.70824
Ws = 375.95398, Wq = 336.80891
n
Probability, pn
Cumulative, Pn
0
0.10412
0.10412
1
0.09328
0.19740
2
0.08357
0.28097
3
0.07487
0.35584
4
0.06707
0.42291
5
0.06009
0.48300
6
0.05383
0.53683
7
0.04823
0.58505
8
0.04320
0.62826
9
0.03871
0.66697
10
0.03468
0.70164
11
0.03107
0.73271
12
0.02783
0.76054
13
0.02493
0.78547
14
0.02234
0.80781
15
0.02001
0.82782
16
0.01793
0.84575
17
0.01606
0.86181
18
0.01439
0.87620
19
0.01289
0.88909
20
0.01155
0.90064
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TORA Optimization System, Windows®-version 1.00
QUEUEING OUTPUT ANALYSIS
Title: Bharat Petrolium G. Noida
Scenario 3-- (M/M/1):(GD/infinity/infinity)
Lambda = 0.01284, Mu = 0.01549
Lambda eff = 0.01284, Rho/c = 0.82928
Ls = 4.85753, Lq = 4.02825
Ws = 378.21483, Wq = 313.64579
n
Probability, pn
Cumulative, Pn
0
0.17072
0.17072
1
0.14158
0.31230
2
0.11741
0.42970
3
0.09736
0.52706
4
0.08074
0.60780
5
0.06696
0.67476
6
0.05553
0.73028
7
0.04605
0.77633
8
0.03819
0.81452
9
0.03167
0.84618
10
0.02626
0.87244
11
0.02178
0.89422
12
0.01806
0.91228
13
0.01498
0.92725
14
0.01242
0.93967
15
0.01030
0.94997
16
0.00854
0.95851
17
0.00708
0.96560
18
0.00587
0.97147
19
0.00487
0.97634
20
0.00404
0.98038
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TORA Optimization System, Windows®-version 1.00
QUEUEING OUTPUT ANALYSIS
Title: Bharat Petrolium G. Noida
Scenario 4-- (M/M/1):(GD/infinity/infinity)
Lambda = 0.01119, Mu = 0.01722
Lambda eff = 0.01119, Rho/c = 0.64957
Ls = 1.85363, Lq = 1.20406
Ws = 165.68636, Wq = 107.62480
n
Probability, pn
Cumulative, Pn
0
0.35043
0.35043
1
0.22763
0.57806
2
0.14786
0.72592
3
0.09605
0.82197
4
0.06239
0.88435
5
0.04053
0.92488
6
0.02632
0.95120
7
0.01710
0.96830
8
0.01111
0.97941
9
0.00721
0.98663
10
0.00469
0.99131
11
0.00304
0.99436
12
0.00198
0.99633
13
0.00128
0.99762
14
0.00083
0.99845
15
0.00054
0.99900
16
0.00035
0.99935
17
0.00023
0.99958
18
0.00015
0.99972
19
0.00010
0.99982
20
0.00006
0.99988
11
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