This document analyzes data collected from a grocery store to determine the optimal number of checkout counters to open during different times of day. It uses Markov chain and queuing theory models. The Markov chain analysis finds transition probabilities between morning and evening customer arrival rates. It determines the expected arrival rate is highest in the evenings for busy days. The queuing model calculates performance measures like expected wait times for different numbers of open counters. It finds 3 counters minimizes wait times while maintaining high counter utilization. The analysis provides recommendations to the store manager for optimizing resource allocation during peak hours.

1. Course no: IE 5309-001
Stochastic Process
Semester: Spring 2016
Deciding the number of counters to open
in a Store using Stochastic Processes
Submitted to:
Dr. Bill Corley
Professor, Department of IMSE
The University of Texas at Arlington
Submitted by:
Group # B
Anas Fareed Mohammed
Md Mamunur Rahman
Ukesh Chawal
Date of submission: May 04, 2016

2. Table of Contents
1. Introduction...............................................................................................................................................1
2. Markov Chain model ................................................................................................................................2
2.1. Methodology......................................................................................................................................2
2.2. Data requirements..............................................................................................................................2
2.3. Results................................................................................................................................................2
2.3.1. Unit Step Transition Matrix........................................................................................................2
2.3.2. Long run probabilities.................................................................................................................4
3. Queuing Theory Model.............................................................................................................................4
3.1. Methodology......................................................................................................................................4
3.2. Expected customer arrival rate...........................................................................................................5
3.3. Results................................................................................................................................................6
4. Conclusion ..............................................................................................................................................11
APPENDIX A - DATA SET FOR SLOW DAYS......................................................................................12
APPENDIX B - DATA SET FOR BUSY DAYS ......................................................................................17
REFERENCES ...........................................................................................................................................21

3. Page | 1
1. Introduction
Happy customers are the main capital of a chain store. While buying grocery or other daily necessities from
a chain store, queuing is a very common phenomenon. To keep the customers happy doing shopping, it is
one of the important things to complete the checkout procedures without keeping them waiting for a longer
period of time. When a store opens it doesn’t have many customers in the opening few hours, the real rush
starts later in the day. The morning employees are busy in arranging the stacks and doing other cleaning
and managerial stuff. In the later part of the day is where the rush starts and counters get busy. A good
manager is the one who is prepared for the challenges in advance. In order to help the manager to prepare
in advance of what is going to come in the rush hours, we are analyzing the store for a year and applying
Markov Chain and Queuing Theory concepts to avoid busy lines and high customer waiting time during
the evening time or the rush hour.
The data has been collected from a chain store (Indo-Pak) by asking the manager of the store. The average
current waiting time in queue is two minutes which he assumes is the problem and he wants to reduce it to
one minute. Also he wants to predict how many employees he needs, to make his customer happy. This
will help him to know how to schedule his employees so he can have maximum number of employees
available during the peak hours.
Our group will study the queue nature of a store (Indo-Pak) and apply Queuing theory and Markov Chain
concept to answer the following questions-
▪ What is expected customer arrival rate in the rush hour on a particular day?
▪ How many check-out counters are required to be opened to minimize the operating cost of the
store and to keep the customers happy?
▪ What is the expected no of customers waiting in the queue for check-out?
▪ What is the expected waiting time of the customers in the queue?
▪ How will the business fare in the future?

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2. Markov Chain model
2.1. Methodology
Since Markov property is a memory less property, we can know as much information of the distribution
of any X(t) with only the latest information. In our case we try to get the distribution of the customer
arrival rate in the later part of the day when it is going to get busy i.e. in the evening time with the
information we have from the customer arrival rate of the morning in that particular day.
We also have two groups of study and prediction. One is for the days when the business going to be
slow on the days like Monday, Tuesday, Wednesday and Thursday. These days are called “slow days”.
The second group is when the business is going to be busy on the days like Friday, Saturday and Sunday.
These are called “busy days”.
For each category a Unit step transition matrix is calculated to know the probabilities of customer
arrival rates in the evening time, long run probabilities are calculated to know how the business is
going to fare in the future,
2.2. Data requirements
The shop after opening takes some time to settle and normal. It also takes some time or the customers
to know about the shop. Keeping this in mind, after the shop was run for a buffer period of one year,
the average customer arrival rate in the morning and in the evening was collected for all the days in the
second year. It was then segregated into two groups- “slow days” and “busy days”.
The data for the slow days in in appendix 1 and for busy days in appendix 2.
The average customer arrival rate was again segregated into 4 categories-
● high -H- 76-100 customers per hour
● average -A- 51-75 customers per hour
● low-L- 26-50 customers per hour
● very low-VL- 0-25 customers per hour
2.3. Results
2.3.1. Unit Step Transition Matrix
Unit step transition matrix was calculated to know the probabilities of having high, average, low or
very low customer arrival rate in the evening time from the arrival rate of the morning time of
respective group from the data collected. This was done by getting the number of days the customer
arrival rate changed from VL to L, VL to A, H to A….from the data we have. The table formed by
doing this is:-
Slow days VL L A H sum
VL 6 17 9 14 46
L 5 26 23 17 71
A 6 19 36 16 77
H 0 4 4 6 14
208

5. Page | 3
Busy days VL L A H sum
VL 1 14 7 16 38
L 4 21 15 23 63
A 1 9 9 20 39
H 0 5 5 8 18
158
After getting the distribution we calculate the probabilities for Unit Step Transition Matrix, as shown
below:-
Slow days VL L A H
VL 6/46 17/46 9/46 14/46
L 5/71 26/71 23/71 17/71
A 6/77 19/77 36/77 16/77
H 0/14 4/14 4/14 6/14
Busy days VL L A H
VL 1/38 14/38 7/38 16/38
L 4/63 21/63 15/63 23/63
A 1/39 9/39 9/39 20/39
H 0/18 5/18 5/18 8/18
The Unit Step Transition Matrix attainted is:-
Slow days VL L A H
VL 0.13 0.37 0.20 0.30
L 0.07 0.37 0.32 0.24
A 0.08 0.25 0.47 0.21
H 0.00 0.29 0.29 0.43
Busy days VL L A H
VL 0.03 0.37 0.18 0.42
L 0.06 0.33 0.24 0.37
A 0.03 0.23 0.23 0.51
H 0.00 0.28 0.28 0.44
The unit step transition matrix is not only just for the year we collected data, but it can be used in
the years to come to know what would be the customer arrival rate in the evening after we know the
customer arrival rate of the morning.
For example if we are in the year 3 on a Monday, the customer arrival rate in the morning was
average ‘A’ so by looking in the table we can say there is 47% chance that it is going to be average

6. Page | 4
in the evening, 25% chance to be low and 21% chance to be high. Therefore we calculate the
expected value of the customer arrival rate and decide number of counters to open.
2.3.2. Long run probabilities
Usually when the business runs for a good amount of years it becomes established and well known
and the customer base stabilizes. The long run probabilities were calculated to prove this fact and it
did.
VL L A H
Slow days 0.05 0.3 0.36 0.29
As we can see that there is 36% chance of having an average customer arrival rate in the slow days
and 29% chance for high and 30% chance for low. Which was better than most of the days during
our data collection year.
VL L A H
Busy days 0.025 0.28 0.25 0.44
In the busy days it gets more rewarding with 44% chance for a high customer arrival rate and 25%
chance for average and 28% for low customer arrival rate, which is way better than most of the days
during our data collection years.
3. Queuing Theory Model
3.1. Methodology
Queue is a line of people waiting for something (service) and Queuing Theory is a Mathematical study
of waiting lines, using models to show results, within arrival, service, and departure processes.
Elements of a Queuing Model
● source of customers - finite or infinite
● customers - interarrival time distribution
● queue - finite or infinite capacity
● queue discipline
● # servers

7. Page | 5
● service time distribution
● jockeying, balking, reneging
Steady-state Measures Of Performance
● Ls = expected number of customers in system
● Lq = expected number of customers in queue
● Ws = expected waiting time in system
● Wq = expected waiting time in queue
 = expected number of busy servers
Balking of Queue
Some customers decide not to join the queue due to their observation related to the long length of queue,
insufficient waiting space or improper care while customers are in queue.
3.2. Expected customer arrival rate
In this section, expected customer arrival rate at evening is calculated applying law of total expectation and
morning to evening one step state transition matrix.
Table: Expected customer arrival rate for slow days
One step transition matrix from
morning to evening shift (slow
days)
Customer arrival rate
Typ
e
Rang
e
VL L A H
Min value Max value
Very
low
VL
0-24
VL
0.130 0.370 0.196 0.304
41.85 66.15
Low L 25-49 L 0.070 0.366 0.324 0.239 43.31 67.55
Average A 50-74 A 0.078 0.247 0.468 0.208 45.13 69.34
High H >74 H 0.000 0.286 0.286 0.429 53.57 78.00

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Sample calculation for VL scenario
Maximum customer arrival rate:
𝐸(𝑋) = ∑
𝑛
𝑖=1
𝐸(𝐴𝑖) 𝑃(𝐴𝑖)
= 24*0.130 + 49*0.370 + 74*0.196 + 100*0.304
= 66.15
Table: Expected customer arrival rate for busy days
One step transition matrix from morning to evening
shift (busy days)
Customer arrival rate
VL L A H Min value Max value
VL 0.026 0.368 0.184 0.421 50.00 74.42
L 0.063 0.333 0.238 0.365 47.62 71.98
A 0.026 0.231 0.231 0.513 55.77 80.28
H 0.000 0.278 0.278 0.444 54.17 78.61
3.3. Results
The numbers of counters to be opened in the supermarket will mainly depend upon the strategy the manager
is following. After having a brief discussion with the current manager of Indopak the following critical
values are determined for further calculations.
● Maximum number of people in the queue = 5 customers
● Maximum waiting time in the queue = 4.5 minutes
● Minimum utilization of the counters = 60%
If the manager is optimist then s/he will consider maximum expected customer arrival rate calculated in the
previous section. If the manager is pessimist then s/he will consider the minimum expected customer arrival
rate.
Number of counters versus queue performance
Figure: Number of customers vs number of counters (busy days- optimist high arrival rate)
29.4
26.8
7.4
3.3 2.8
28.4
24.8
4.8
0.7 0.2
0.00
5.00
10.00
15.00
20.00
25.00
30.00
35.00
1 2 3 4 5
#ofcustomers
# of counters
Ls Lq

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According to the above plot, if three counters are kept open then the number of customers waiting in the
system as well as in the queue become reasonable.
Figure: Waiting time vs number of counters (busy days- optimist high arrival rate)
According to the above plot, if three counters are kept open then the waiting time (in minutes) in the system
as well as in the queue become reasonable.
Figure: Counter utilization vs number of counters (busy days- optimist high arrival rate)
According to the above plot, if three counters are kept open then the counter utilization becomes 87% which
is pretty good.
22.4
20.4
5.7
2.5 2.1
21.7
18.9
3.7
0.5 0.1
0.00
5.00
10.00
15.00
20.00
25.00
1 2 3 4 5
Waitingtime(minutes)
# of counters
Ws Wq

10. Page | 8
Table: Results summary for slow days
VL L A H
Type
Optim
ist
Pessim
ist
Optim
ist
Pessimi
st
Optim
ist
Pessim
ist
Optim
ist
Pessim
ist
# of servers 3 2 3 2 3 2 3 3
Ls 3.71 2.72 3.96 3.01 4.31 3.46 7.11 2.30
Lq 1.51 1.32 1.71 1.57 2.00 1.95 4.52 0.51
Ws 3.37 3.89 3.51 4.17 3.73 4.60 5.47 2.57
Wq 1.37 1.89 1.51 2.17 1.73 2.60 3.48 0.57
C_bar 2.20 1.39 2.25 1.44 2.31 1.50 2.59 1.79
Server utilization 0.73 0.70 0.75 0.72 0.77 0.75 0.86 0.60
lamda_lost (per hour) 0.00 0.00 0.00 0.00 0.01 0.00 0.17 0.00
lamda_lost (per shift) 0.02 0.00 0.03 0.01 0.07 0.02 1.34 0.00
For a generic explanation, the first column of the table is explained below and the rest of the table follows
the same pattern. If three counters are open
 The expected number of customers in the system is 3.71.
 The expected number of customers in the queue is 1.51.
 The expected waiting time in the system is 3.37 minutes
 The expected waiting time in the queue is 1.37 minutes
 On average 2.2 out of 3 servers will be busy at any point of time
 73 percent of the time the servers will be busy serving the customers
 Per hour almost no customer will be lost due to bulking
 Per shift about 0.02 customers will be lost due to bulking
Table: Results summary for busy days
VL L A H
Type
Optim
ist
Pessim
ist
Optim
ist
Pessimi
st
Optim
ist
Pessim
ist
Optim
ist
Pessim
ist
# of servers 3 2 3 2 4 3 3 3
Ls 5.68 5.34 4.95 4.26 3.45 2.48 7.41 2.35
Lq 3.20 3.67 2.55 2.68 0.77 0.62 4.79 0.54
Ws 4.58 6.40 4.12 5.37 2.58 2.67 5.65 2.60
Wq 2.58 4.41 2.12 3.37 0.58 0.67 3.66 0.60
C_bar 2.48 1.67 2.40 1.59 2.68 1.86 2.61 1.81
Server utilization 0.83 0.83 0.80 0.79 0.67 0.62 0.87 0.60
lamda_lost (per hour) 0.05 0.04 0.02 0.01 0.00 0.00 0.20 0.00
lamda_lost (per shift) 0.42 0.31 0.18 0.09 0.00 0.00 1.62 0.00
The explanation for this table follows the same pattern as above.

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We then went ahead and checked our output using TORA Software.
Cross checking the values using TORA software
Input for the slow days
Output for the slow days

12. Page | 10
For a generic explanation, the second scenario is explained below and the rest of the table follows the same
pattern which resemble the first column of the table above from excel. If three counters are open
 The expected number of customers in the system is 3.71.
 The expected number of customers in the queue is 1.50.
 The expected waiting time in the system is 0.056 hours which is 3.36 minutes
 The expected waiting time in the queue is 0.02280 hours which is 1.368 minutes
The value matches with that of excel output as expected.
Input for the busy days
Output for the busy days
The explanation for this output follows the same pattern as above.

13. Page | 11
4. Conclusion
This project facilitated our group to study the queue nature of a store (Indo-Pak) and apply Queuing theory
and Markov Chain concept. Using Markov Chain we predicted the customer arrival rate in the evening by
knowing the customer arrival rate of the morning time of any day. We also calculated that in the long run
we have average customer arrival rate (50-74 customer arrival rate) 36% of time and high customer arrival
rate(75-100) 29% of time for the slow days whereas in the busy days we have high customer arrival rate
for 44% of the time and average customer arrival rate for 25% of the time. Thus the business will fare good
in the future.
Using queuing theory we determined the expected customer arrival rate on a particular day, the expected
number of customers in the system, the expected number of customers in the queue, the expected waiting
time in the system, the expected waiting time in the queue, the expected number of counters to open to
maximize the customer satisfaction, the expected number of servers who will be busy at any point of time
serving customers, what percentage of the time the servers will be busy serving the customers, the expected
number of customers lost due to bulking per hour as well as per shift.

14. Page | 12
APPENDIX A - DATA SET FOR SLOW DAYS
Serial number morning category evening category
1 53 A 75 H
2 47 L 50 A
3 46 L 33 L
4 57 A 37 L
5 23 VL 22 VL
6 18 VL 54 A
7 28 L 92 H
8 31 L 96 H
9 56 A 47 L
10 63 A 58 A
11 61 A 53 A
12 70 A 22 VL
13 77 H 55 A
14 43 L 67 A
15 34 L 25 L
16 77 H 50 A
17 76 H 49 L
18 43 L 43 L
19 66 A 42 L
20 80 H 29 L
21 44 L 59 A
22 36 L 37 L
23 28 L 86 H
24 13 VL 83 H
25 32 L 75 H
26 66 A 91 H
27 32 L 25 L
28 67 A 45 L
29 24 VL 22 VL
30 69 A 51 A
31 34 L 56 A
32 60 A 61 A
33 16 VL 25 L
34 20 VL 21 VL
35 64 A 28 L
36 55 A 59 A
37 78 H 77 H
38 37 L 100 H
39 50 A 56 A
40 47 L 48 L
41 52 A 27 L

15. Page | 13
42 36 L 40 L
43 14 VL 22 VL
44 78 H 82 H
45 63 A 42 L
46 43 L 39 L
47 36 L 49 L
48 50 A 20 VL
49 16 VL 68 A
50 52 A 42 L
51 60 A 52 A
52 67 A 89 H
53 12 VL 53 A
54 18 VL 91 H
55 14 VL 55 A
56 43 L 54 A
57 46 L 69 A
58 53 A 22 VL
59 68 A 74 A
60 70 A 65 A
61 12 VL 89 H
62 14 VL 32 L
63 57 A 45 L
64 47 L 98 H
65 56 A 51 A
66 15 VL 90 H
67 55 A 27 L
68 37 L 54 A
69 66 A 78 H
70 23 VL 48 L
71 29 L 56 A
72 71 A 62 A
73 11 VL 84 H
74 28 L 82 H
75 13 VL 21 VL
76 80 H 82 H
77 20 VL 42 L
78 53 A 74 A
79 65 A 93 H
80 46 L 36 L
81 26 L 53 A
82 47 L 51 A
83 47 L 73 A
84 18 VL 88 H

16. Page | 14
85 45 L 76 H
86 71 A 77 H
87 26 L 66 A
88 63 A 25 L
89 19 VL 81 H
90 36 L 44 L
91 18 VL 56 A
92 30 L 77 H
93 44 L 46 L
94 62 A 74 A
95 46 L 36 L
96 27 L 22 VL
97 28 L 81 H
98 32 L 91 H
99 68 A 45 L
100 27 L 32 L
101 39 L 83 H
102 38 L 49 L
103 52 A 72 A
104 38 L 23 VL
105 28 L 45 L
106 23 VL 83 H
107 10 VL 69 A
108 62 A 78 H
109 58 A 72 A
110 10 VL 41 L
111 22 VL 38 L
112 50 A 91 H
113 48 L 56 A
114 32 L 66 A
115 26 L 70 A
116 26 L 67 A
117 69 A 77 H
118 52 A 61 A
119 74 A 33 L
120 50 A 64 A
121 70 A 74 A
122 24 VL 48 L
123 19 VL 82 H
124 29 L 22 VL
125 63 A 24 VL
126 55 A 49 L
127 77 H 50 A

17. Page | 15
128 70 A 92 H
129 18 VL 33 L
130 13 VL 48 L
131 59 A 95 H
132 66 A 67 A
133 36 L 46 L
134 56 A 72 A
135 24 VL 37 L
136 21 VL 84 H
137 26 L 40 L
138 54 A 69 A
139 50 A 62 A
140 78 H 32 L
141 49 L 23 VL
142 47 L 39 L
143 60 A 96 H
144 62 A 49 L
145 20 VL 34 L
146 26 L 61 A
147 63 A 43 L
148 23 VL 76 H
149 48 L 88 H
150 19 VL 52 A
151 72 A 88 H
152 13 VL 91 H
153 63 A 100 H
154 74 A 71 A
155 37 L 97 H
156 40 L 44 L
157 42 L 97 H
158 33 L 83 H
159 72 A 58 A
160 16 VL 36 L
161 24 VL 40 L
162 51 A 35 L
163 75 H 28 L
164 64 A 74 A
165 46 L 31 L
166 37 L 39 L
167 60 A 54 A
168 21 VL 33 L
169 13 VL 82 H
170 61 A 23 VL

18. Page | 16
171 31 L 22 VL
172 39 L 49 L
173 25 L 48 L
174 76 H 85 H
175 56 A 51 A
176 27 L 28 L
177 61 A 74 A
178 62 A 23 VL
179 50 A 46 L
180 62 A 50 A
181 24 VL 46 L
182 14 VL 62 A
183 25 L 73 A
184 69 A 52 A
185 66 A 50 A
186 80 H 47 L
187 50 A 66 A
188 69 A 90 H
189 24 VL 29 L
190 22 VL 96 H
191 45 L 43 L
192 41 L 93 H
193 63 A 43 L
194 47 L 67 A
195 75 H 48 L
196 23 VL 23 VL
197 48 L 51 A
198 75 H 53 A
199 24 VL 52 A
200 39 L 50 A
201 41 L 62 A
202 34 L 58 A
203 20 VL 49 L
204 67 A 66 A
205 66 A 75 H
206 65 A 72 A
207 69 A 65 A
208 50 A 65 A

19. Page | 17
APPENDIX B - DATA SET FOR BUSY DAYS
Serial number morning category evening category
1 10 VL 43 L
2 12 VL 92 H
3 47 L 56 A
4 36 L 79 H
5 46 L 55 A
6 37 L 23 VL
7 63 A 35 L
8 29 L 25 L
9 53 A 79 H
10 10 VL 79 H
11 30 L 28 L
12 42 L 31 L
13 51 A 84 H
14 18 VL 32 L
15 22 VL 37 L
16 24 VL 71 A
17 58 A 93 H
18 17 VL 77 H
19 44 L 99 H
20 12 VL 44 L
21 32 L 52 A
22 59 A 94 H
23 67 A 77 H
24 45 L 68 A
25 81 H 80 H
26 48 L 54 A
27 32 L 41 L
28 28 L 30 L
29 10 VL 70 A
30 18 VL 74 A
31 60 A 38 L
32 48 L 23 VL
33 32 L 61 A
34 26 L 36 L
35 68 A 99 H
36 80 H 86 H
37 40 L 40 L
38 29 L 49 L
39 30 L 75 H
40 23 VL 61 A
41 39 L 27 L

20. Page | 18
42 11 VL 98 H
43 23 VL 24 VL
44 65 A 84 H
45 26 L 81 H
46 32 L 52 A
47 30 L 98 H
48 59 A 38 L
49 26 L 35 L
50 40 L 71 A
51 31 L 28 L
52 82 H 82 H
53 21 VL 92 H
54 51 A 29 L
55 82 H 27 L
56 60 A 98 H
57 70 A 67 A
58 35 L 38 L
59 29 L 22 VL
60 33 L 100 H
61 12 VL 32 L
62 33 L 84 H
63 24 VL 81 H
64 77 H 43 L
65 78 H 54 A
66 10 VL 28 L
67 78 H 57 A
68 53 A 79 H
69 22 VL 91 H
70 46 L 34 L
71 36 L 40 L
72 48 L 61 A
73 17 VL 45 L
74 40 L 90 H
75 46 L 24 VL
76 26 L 26 L
77 80 H 88 H
78 34 L 48 L
79 75 H 86 H
80 20 VL 92 H
81 20 VL 93 H
82 33 L 59 A
83 51 A 85 H
84 45 L 34 L

21. Page | 19
85 48 L 57 A
86 52 A 93 H
87 49 L 85 H
88 48 L 98 H
89 23 VL 28 L
90 70 A 99 H
91 64 A 70 A
92 82 H 46 L
93 75 H 68 A
94 61 A 72 A
95 21 VL 96 H
96 12 VL 42 L
97 47 L 79 H
98 48 L 89 H
99 15 VL 78 H
100 62 A 49 L
101 77 H 90 H
102 30 L 89 H
103 50 A 53 A
104 33 L 90 H
105 75 H 100 H
106 25 L 82 H
107 20 VL 76 H
108 19 VL 46 L
109 14 VL 44 L
110 34 L 64 A
111 20 VL 93 H
112 15 VL 31 L
113 33 L 59 A
114 41 L 48 L
115 72 A 61 A
116 33 L 60 A
117 58 A 22 VL
118 44 L 79 H
119 46 L 66 A
120 65 A 25 L
121 31 L 76 H
122 42 L 34 L
123 66 A 46 L
124 15 VL 40 L
125 22 VL 56 A
126 19 VL 83 H
127 13 VL 65 A

22. Page | 20
128 49 L 90 H
129 78 H 73 A
130 35 L 32 L
131 76 H 48 L
132 46 L 28 L
133 75 H 29 L
134 14 VL 52 A
135 58 A 68 A
136 72 A 63 A
137 54 A 65 A
138 38 L 98 H
139 26 L 87 H
140 52 A 27 L
141 68 A 79 H
142 16 VL 78 H
143 62 A 98 H
144 39 L 86 H
145 13 VL 98 H
146 70 A 44 L
147 16 VL 25 L
148 50 A 55 A
149 50 A 87 H
150 53 A 90 H
151 75 H 68 A
152 50 A 98 H
153 73 A 85 H
154 63 A 85 H
155 59 A 88 H
156 33 L 94 H

23. Page | 21
REFERENCES
Sundarapandian, V. (2009). "7. Queueing Theory". Probability, Statistics and Queueing Theory. PHI
Learning.
home.snc.edu/eliotelfner/.../Team%204%20Queuing%20Analysis.ppt
Bose S.J., Chapter 1 - An Introduction to Queueing Systems, Kluwer/Plenum Publishers, 2002.