Queuing theory and its applications

Queuing Theory and its
Applications
A PROJECT REPORT ON THE TOPIC OF ADVANCE OPERATION RESEARCH
Prepared by
DEBASIS MOHANTY (UEBA19001)
05 Aug 2020
Advance Operation Research
Term IV, EMBA – BA

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Contents
Queueing Theory: An Introduction ...........................................................................................................2
Basic Structure of a Queuing System ........................................................................................................2
An Elementary Queuing Process ...............................................................................................................3
Kendall’s Notations....................................................................................................................................3
Terminology, Notations and Formulas: .....................................................................................................3
Queue Formulas for M/M/1 (Performance Measures):..............................................................................4
Queue Formulas for M/M/s (Performance Measures):..............................................................................5
Problem Description....................................................................................................................................5
PROBLEM STATEMENT I......................................................................................................................5
PROBLEM STATEMENT II ....................................................................................................................5
Modelling approach.....................................................................................................................................6
Priority-Discipline Queues.........................................................................................................................6
Non-Preemptive Priorities model...............................................................................................................6
Preemptive Priorities model.......................................................................................................................7
Problem Analysis .........................................................................................................................................7
Analysis......................................................................................................................................................7
Conclusion .................................................................................................................................................8
Cost Benefit Analysis.................................................................................................................................8
Some other analysis ...................................................................................................................................8
Exhibits.........................................................................................................................................................8
Bibliography...............................................................................................................................................13

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Queueing Theory: An Introduction
Queueing theory is the mathematical study of waiting in lines, or queues. Queueing theory, along with
simulation, are the most widely used operations-research and management-science techniques. Its main
objective is to build a model to predict queue lengths and waiting times to make effective business decisions
related to resources’ management and allocation to provide a given service.
In this project, I will discuss about Queueing Theory and some of its practical applications. We all have
experienced the annoyance of having to wait in a queue. We wait in line at supermarkets to check out, we
wait in line in banks and post offices and we wait in line at fast food restaurants. But we as customers do
not like waiting. And the managers of these establishments also don’t like their customers to wait as it may
cost them business.
So, the first question that arises is that “Why is there waiting?”
To which the answer is that there is more demand for service than there is an available facility for that
service.
And “Why is this so?”
For which there could be a number of reasons such as, shortage of available servers, limitation of space,
economic limitations etc.
These limitations can be removed with the expenditure of capital. And to know how much service should
then be made available, one needs to know:
1. How many people will form the queue?
2. How long must a customer wait?
Queuing theory attempts to answer these questions through detailed mathematical analysis.
GOAL: The ultimate goal is to achieve an economic balance between cost of service and the cost
associated with the waiting for that service.
Basic Structure of a Queuing System
Please see Exhibit 1.
Input Source (Calling population): The size is the total number of customers that might require service
from time to time, i.e., the total number of distinct potential customers. This population from which arrivals
come is referred to as the calling population.
Arrival process: It describes how the customers arrive to the system, and the distribution of the customers’
arrival.
Service mechanism: It is articulated by the number of servers, and whether each server has its own queue
or there is one queue feeding all servers, and the distribution of customer’s service times
Queue discipline: It refers to the rule that a server uses to choose the next customer from the queue when
the server completes the service of the current customer (e.g. FIFO: first-in, first-out; LIFO: last-in, first-
out; priority-based; random selection).
Service Mechanism: It refers at a given facility, the customer enters one of the parallel service channels
and is completely serviced by that server. A queueing model must specify the arrangement of the facilities

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and the number of servers (parallel channels) at each one. Most elementary models assume one service
facility with either one server or a finite number of servers.
The time elapsed from the commencement of service to its completion for a customer at a service facility
is referred to as the service time (or holding time).
An Elementary Queuing Process
A basic business Queue model have two approaches:
• Analytical Model: The one which is built on differential equations which has arrivals and services.
• Simulation: Complex system with multiple stations (For e.g. CAT Exam, Air Traffic controller)
The basic goal as stated earlier is to minimize the service and waiting costs, to reduce the waiting time of
customers and optimize the service time because the resources are limited. Please see Exhibit 2.
As we have already suggested, queueing theory has been applied to many different types of waiting-line
situations. However, the most prevalent type of situation is a single waiting line (which may be empty at
times) forms in the front of a single service facility, within which are stationed one or more servers. Each
customer generated by an input source is serviced by one of the servers, perhaps after some waiting in the
queue (waiting line). We can see Exhibit 3 for a depiction.
Kendall’s Notations
The notations is given by M/M/s/K where, 1st two M stands for Markovian probability distributions,
follow by s which stands for number of servers. The K stands for Maximum Queue length or the Queue
capacity.
We have other queue models like M/D/s, M/G/s and M/𝐸 𝑘/s were D is Deterministic, G is General
(normal) and 𝐸 𝑘 is Erlang Distribution.
M = exponential distribution (Markovian)
D = degenerate distribution (constant times)
G = general distribution (any arbitrary distribution allowed)
For example, the M/M/s model, assumes that both interarrival times and service times have an
exponential distribution and that the number of servers is s (any positive integer). The M/G/1 model
discussed assumes that interarrival times have an exponential distribution, but it places no restriction on
what the distribution of service times must be, whereas the number of servers is restricted to be exactly 1.
Terminology, Notations and Formulas:
Unless otherwise noted, the following standard terminology and notation will be used:
State of system = number of customers in queueing system.
Queue length = number of customers waiting for service to begin. OR (state of system minus number of
customers being served.)
𝑵(𝒕) = number of customers in queueing system at time t (𝑡 ≥ 0).
𝑷 𝒏(𝒕) = probability of exactly n customers in queueing system at time t.
s = number of servers (parallel service channels) in queueing system.
𝛌 𝒏= mean arrival rate (expected number of arrivals per unit time) of new customers when n customers are
in system.

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𝝁 𝒏 = mean service rate for overall system (expected number of customers completing service per unit
time) when n customers are in system. Note: 𝜇 𝑛represents combined rate at which all busy servers (those
serving customers) achieve service completions.
𝝆 = overall system utilization. 𝜌 =
𝛌
𝑠𝜇
(utilization factor)
𝑳 𝒒 = expected number of customers in queue
𝑳 𝒔 = expected number of customers in the system
𝑾 𝒒 = expected time spent in the queue
𝑾 𝒔 = expected time spent in the system
Relationships between L,W, 𝑳 𝒒, and 𝑾 𝒒
Assume that λn is a constant λ for all n. It has been proved that in a steady-state queueing process,
L = λW (Because John D. C. Little provided the first rigorous proof, this equation sometimes is referred
to as Little’s formula.) Furthermore, the same proof also shows that 𝑳 𝒒 = λ𝑾 𝒒 . Under these
circumstances, 1/ λ and 1/μ are the expected interarrival time and the expected service time, respectively.
The arrival process is assumed to follow Poisson Distribution and the service rate is assumed to follow a
and Exponential distribution. Please see Exhibit 4.
If the λn are not equal, then λ can be replaced in these equations by λ̅, the average arrival rate over the
long run. (We shall show later how λ̅ can be determined for some basic cases.)
Now assume that the mean service time is a constant, 1/μ for all 𝑛 ≥ 1. It then follows that
𝑊𝑠 = 𝑊𝑞 + (1/μ)
These relationships are extremely important because they enable all four of the fundamental quantities
L,W, 𝐿 𝑞, and 𝑊𝑞 to be immediately determined as soon as one is found analytically. This situation is
fortunate because some of these quantities often are much easier to find than others when a queueing
model is solved from basic principles.
Queue Formulas for M/M/1 (Performance Measures):
Average Arrival rate : λ
Average Service rate : μ
Average number of customers in service: 𝜌 =
λ
𝜇
Probability of exactly “n” customers in the system: 𝑃𝑛 = 𝜌 𝑛(1 − 𝜌)
Probability of “k” or more customers in the system: 𝑃(𝑛 ≥ 𝑘) = 𝜌 𝑘
Average number of customers in the system: 𝐿 𝑠 = [λ/(μ - λ)]
Average number of customers in the queue: 𝐿 𝑞= 𝐿 𝑠 – (λ/μ) OR [𝜌λ/(μ - λ)]
Average time in the system: 𝑊𝑠 = [1/(μ - λ)]
Average time in the queue: 𝑊𝑞 = 𝑊𝑠 – (1/μ) OR [𝜌/(μ - λ)]

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Queue Formulas for M/M/s (Performance Measures):
Average server utilization: 𝜌 =
λ
𝑠𝜇
Average number of customers in queue: 𝐿 𝑞 =
((
𝛌
𝜇
)
𝑠
)λμ
(𝑠−1)! (𝑠𝜇−λ)
Average number of customers in system: 𝐿 = 𝐿 𝑞 +
λ
𝜇
Average number of customers spends waiting in the queue: 𝑊𝑞 =
𝐿 𝑞
λ
Average number of customers spends waiting in the system: 𝑊 = 𝑊𝑞 +
1
μ
Probability of zero customers: 𝑃0 =
1
∑ (
1
𝑘!
(
λ
𝜇
)
𝑘
𝑠−1
𝑘=0 +
1
𝑠!
(
λ
𝜇
)
𝑠
+
𝑠𝜇
𝑠𝜇−λ
Probability of n customers in system: 𝑃𝑛 =
(
λ
𝜇
)
𝑛
𝑛!
𝑃0 𝑓𝑜𝑟 𝑛 ≤ 𝑠
𝑃𝑛 =
(
λ
μ)
𝑛
𝑠! 𝑠(𝑛−𝑠)
𝑃0 𝑓𝑜𝑟 𝑛 > 𝑠
Problem Description
PROBLEM STATEMENT I
The County Hospital Example with the M/M/s Model. For the County Hospital emergency room problem,
the management engineer has concluded that the emergency cases arrive pretty much at random (a Poisson
input process), so that interarrival times have an exponential distribution. She also has concluded that the
time spent by a doctor treating the cases approximately follows an exponential distribution. Therefore, she
has chosen the M/M/s model for a preliminary study of this queueing system. By projecting the available
data for the early evening shift into next year, she estimates that patients will arrive at an average rate of 1
every 1/2 hour. A doctor requires an average of 20 minutes to treat each patient. What is the queue
performance at steady state? Do we require one more doctor?
To answer this the management engineer did his primary analysis and using the above queuing formula and
found out the performance measure. As per the server utilization is concern he/she was pretty much
convinced that the utilization percentage came down from 67% to 33% (Please see Exhibit 5), however
he/she wanted to further analyse the emergency situation to give an all round report to the management as
everyone might question him because it is obvious that whenever you increase resource then there
utilization gets divided.
PROBLEM STATEMENT II
The County Hospital Example with Priorities. For the County Hospital emergency room problem, the
management engineer has noticed that the patients are not treated on a first-come-first-served basis. Rather,
the admitting nurse seems to divide the patients into roughly three categories: (1) critical cases, where
prompt treatment is vital for survival; (2) serious cases, where early treatment is important to prevent further
deterioration; and (3) stable cases, where treatment can be delayed without adverse medical consequences.
Patients are then treated in this order of priority, where those in the same category are normally taken on a

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first-come-first-served basis. A doctor will interrupt treatment of a patient if a new case in a higher-priority
category arrives. Approximately 10 percent of the patients fall into the first category, 30 percent into the
second, and 60 percent into the third. Because the more serious cases will be sent to the hospital for further
care after receiving emergency treatment, the average treatment time by a doctor in the emergency room
actually does not differ greatly among these categories. The management engineer has decided to use a
priority-discipline queueing model as a reasonable representation of this queueing system, where the three
categories of patients constitute the three priority classes in the model. Because treatment is interrupted by
the arrival of a higher-priority case, the preemptive priorities model is the appropriate one. Given the
previously available data (𝜇=3 and 𝜆=2), the preceding percentages yield λ1=0.2, λ2=0.6 𝑎𝑛𝑑 λ3=1.2.
Calculate the expected waiting times in the queue (so excluding treatment time) for the respective priority
classes when there is one (𝑠=1) or two (𝑠=2) doctors on duty. Also find out the corresponding results for
the preemptive and non preemptive priorities model to show the effect of preempting.
Modelling approach
Priority-Discipline Queues
In priority-discipline queueing models, the queue discipline is based on two priority system.
• Static : Doesn’t change throughout the execution
• Dynamic: Changes after some time
We have two models based on priority system. They are:
• Non-Preemptive Priorities model
• Preemptive Priorities model
With non-preemptive priorities, a customer being served cannot be ejected back into the queue
(preempted) if a higher-priority customer enters the queueing system. Therefore, once a server has begun
serving a customer, the service must be completed without interruption. With preemptive priorities, the
lowest-priority customer being served is preempted (ejected back into the queue) whenever a higher-
priority customer enters the queueing system. A server is thereby freed to begin serving the new arrival
immediately.
When a server does succeed in finishing a service for a higher priority customer, so a preempted customer
normally will get back into service again and will eventually finish its treatment after waiting again in the
queue. Both the queue model will follow queue discipline of FIFO according to its priority.
Non-Preemptive Priorities model
To calculate the waiting time let 𝑊𝑘 be the steady state expected waiting time in the system (including
service time) for a member of priority class k. Then
𝑊𝑘 =
1
ABk−1Bk
+
1
𝜇
for k = 1,2, . . . , N
Where 𝐴 = 𝑠!
𝑠𝜇 −λ
𝜌 𝑠
∑
𝜌 𝑗
𝑗!
𝑠−1
𝑗=0 + 𝑠𝜇
𝐵0 = 1,
𝐵 𝑘 = 1 −
∑ λ 𝑖
𝑘
𝑖=1
𝑠𝜇
Using the above equations, we can create a excel model or any functions which can solve the purpose.
However, the underlying algorithm in shown in Exhibit 6.

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Preemptive Priorities model
Similarly, for preemptive model waiting time let 𝑊𝑘 be the steady state expected waiting time in the
system and it is expressed as per below formula:
𝑊𝑘 =
1/𝜇
𝐵 𝑘−1 𝐵 𝑘
Using the above equations, we can create a excel model, however, the underlying algorithm in shown in
Exhibit 7.
Problem Analysis
Analysis
Steady-state results from the priority-discipline models for the County Hospital problem can be solved by
using the excel templates available in the CD resources provided along with our book.
Using the sheet, we can compute all the four fundamental quantities L,W, 𝐿 𝑞, 𝑊𝑞 and the resource
utilization. The details are present in the Exhibit 8 and Exhibit 9 tables.
However, for the preemptive queue model specific the probabilities of arrivals are provided which is not
considered when run in the excel model. As the excel formulas treat the arrivals as a Poisson distribution.
So, manually subtracting the expected service time 1/𝜇 from each waiting time of the queue calculating
the cumulative arrival rate using the model used in the previous problem will do the needful. The
calculation is done below for each priority class separately.
Priority Class 1 (10% probability):
Arrival rate (𝜆) 0.2
Service rate ( 𝜇) 3
Number of servers (s) 2
Average time in the system (𝑊1) 0.33370
Waiting time = 𝑊1-
1
𝜇
= 0.33370 - 0.33333 = 0.00037
Arrival rate (𝜆) 0.8
𝑊1−2 =
1
4
𝑊1 +
3
4
𝑊2 so 𝑊2 =
4
3
[0.33937 −
1
4
(0.33370)] = 0.34126
1
𝜇
= 0.34126 - 0.33333 = 0.00793
Arrival rate (𝜆) 2
𝑊1−3 = 0.1𝑊1 + 0.3𝑊2 + 0.6𝑊3 so, 𝑊3 =
1
0.6
[0.375 − 0.1(0.33370) − 0.3(0.34126)] = 0.39875

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1
𝜇
= 0.39875 - 0.33333 = 0.06542
Please Exhibit 10 for the updated results for the preemptive priorities case model.
Conclusion
When 𝑠=1, the 𝑊𝑘− 1/𝜇 values for the preemptive priorities case indicate that providing just a single doctor
would cause critical cases to wait about 1 1/2 minutes (0.024 hour) on the average, serious cases to wait
more than 9 minutes, and stable cases to wait more than 1 hour. (Contrast these results with the average
wait of 𝑊𝑞 = 2/3 hour for all patients that was obtained in table for the first model under the first-come-
first-served queue discipline.) However, these values represent statistical expectations, so some patients
have to wait considerably longer than the average for their priority class. This wait would not be tolerable
for the critical and serious cases, where a few minutes can be vital. By contrast, the 𝑠=2 results in above
table (preemptive priorities case) indicate that adding a second doctor would virtually eliminate waiting for
all but the stable cases. Therefore, the management engineer recommended that there be two doctors on
duty in the emergency room during the early evening hours next year. The board of directors for County
Hospital adopted this recommendation and simultaneously raised the charge for using the emergency room!
Cost Benefit Analysis
The management engineer also calculated total cost to strengthen its above conclusion. So, we could see
that when we have two doctor it does costs less considering the waiting cost to be $100 per hour. Please
see Exhibit 11 for the details.
Some other analysis
A simulation using R simmer package is also conducted for better understanding of the above problem.
The details of the simulation result in provided in the Exhibit 12 and Exhibit 13.
The kink in the usage curve shows that the time consumed by the server in emptying the customers for the
sake of priority ones.
Exhibits
Exhibit 1

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Exhibit 2
Total Cost = Waiting Cost + Service Cost
𝐶 𝑤 ∗ 𝐿 + 𝐶𝑠 ∗ 𝑠
𝐶 𝑤 = 𝑊𝑎𝑖𝑡𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
𝐶𝑠 = 𝐶𝑜𝑠𝑡 𝑝𝑒𝑟 𝑠𝑒𝑟𝑣𝑒𝑟 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒
L = 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑢𝑠𝑡𝑜𝑚𝑒𝑟𝑠 𝑖𝑛 𝑠𝑦𝑠𝑡𝑒𝑚
s = 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑒𝑟𝑣𝑒𝑟𝑠
Exhibit 3
A single server queue model (M/M/1)
Exhibit 4
Exhibit 5
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9
Poisson Distribution
λ=1 λ=3 λ=5
0
0.1
0.2
0.3
0.4
1 2 3 4 5 6 7 8 9 10
Exponential Distribution
𝜇 = 1 𝜇 = 1.5 𝜇 = 2
Given an average arrival rate λ, the Poisson
distribution gives the probability a certain number of
customer x arrive in a given time. P(x) =
𝑒-λλ 𝑥
𝑥!
Given an average service time 𝜇, the Exponential
distribution gives the probability of service
provided by a server at time t. P(x) =𝑒−𝜇𝑡

10 | P a g e
Queue Performance/Operating Characteristics at
steady state
s = 1 s = 2 measure
Average server utilization (𝜌) 67% 33% busy
Average number of customers in the queue (Lq) 1.3333 0.0833 in queue
Average number of customers in the system (L) 2.0000 0.7500 in system
Average waiting time in the queue (Wq) 0.6667 0.0417 hour
Average time in the system (W) 1.0000 0.3750 hour
Probability (% of time) system is empty (Po) 0.3333 0.5000 empty
Exhibit 6
Non-Preemptive Priority Queue Algorithm
Patients Priority Arrival Time Burst Time System Time
(Completion Time)
Turnaround Time
(ST - AT)
Waiting Time
(TT - BT)
P1 3 0 8 8 8 0
P2 4 1 2 17 16 14
P3 4 3 4 21 18 14
P4 5 4 1 22 18 17
P5 2 5 6 14 9 3
P6 6 6 5 27 21 16
P7 1 10 1 15 5 4
Total 124 95 68
Queue P1 P5 P7 P2 P3 P4 P6
0 8 14 15 17 21 22 27
Exhibit 7
Preemptive Priority Queue Algorithm
Patients Priority Arrival Time Burst Time New BT System Time
(Completion
Time)
Turnaround
Time (ST - AT)
Waiting
Time (TT -
BT)
P1 3 0 8 3 15 15 7
P2 4 1 2 17 16 14
P3 4 3 4 21 18 14
P4 5 4 1 22 18 17
P5 2 5 6 1 12 7 1
P6 6 6 5 27 21 16
P7 1 10 1 11 1 0
Total 125 96 69
Queue P1 P5 P7 P5 P1 P2 P3 P4 P6
0 5 10 11 12 15 17 21 22 27

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Exhibit 8
Queue Performance/Operating
Characteristics at a steady state Preemptive Priorities
s = 1 s = 2
Priority Classes (𝛌1 = 0.2, 𝛌2 = 0.6, 𝛌3 = 1.2) Class 1 Class 2 Class 3 Class 1 Class 2 Class 3
Average server utilization (𝜌) 67% 67% 67% 33% 33% 33%
Average number of customers in the queue (Lq) 0.00476 0.09221 1.23636 0.00230 0.03873 0.29231
Average number of customers in the system (L) 0.07143 0.29221 1.63636 0.06897 0.23873 0.69231
Average waiting time in the queue (Wq) 0.02381 0.15368 1.03030 0.01149 0.06454 0.24359
Average time in the system (W) 0.35714 0.48701 1.36364 0.34483 0.39788 0.57692
Exhibit 9
Characteristics at a steady state
Non-Preemptive Priorities
s = 1 s = 2
Priority Classes (𝛌1 = 0.2, 𝛌2 = 0.6, 𝛌3 = 1.2) Class 1 Class 2 Class 3 Class 1 Class 2 Class 3
Average server utilization (𝜌) 67% 67% 67% 33% 33% 33%
Average number of customers in the queue (Lq) 0.04762 0.19481 1.09091 0.00575 0.01989 0.05769
Average number of customers in the system (L) 0.11429 0.39481 1.49091 0.07241 0.21989 0.45769
Average waiting time in the queue (Wq) 0.23810 0.32468 0.90909 0.02874 0.03316 0.04808
Average time in the system (W) 0.57143 0.65801 1.24242 0.36207 0.36649 0.38141
Exhibit 10
Characteristics at a steady state
Preemptive Priorities
s = 2
Priority Classes (𝛌1 = 0.2, 𝛌2 = 0.6, 𝛌3 = 1.2) Class 1 Class 2 Class 3
Average server utilization (𝜌) 33% 33% 33%
Average number of customers in the queue (Lq) 0.00230 0.03873 0.29231
Average number of customers in the system (L) 0.06897 0.23873 0.69231
Average waiting time in the queue (Wq) 0.00037 0.00793 0.06542
Average time in the system (W) 0.34483 0.39788 0.57692
Exhibit 11
Input Data
Arrival rate () 2 hour
Service rate () 3 hour
Cost of Service 100 hour
Cost of Waiting 100 hour
Total Cost based on Queue
Servers Service Cost Wait Cost Total Cost
1 100.00 200.00 300.00
2 200.00 75.00 275.00
3 300.00 67.60 367.60
4 400.00 66.77 466.77
5 500.00 66.68 566.68

12 | P a g e
Exhibit 12
Preemptive QUEUE OUTPUT
Non - Preemptive QUEUE OUTPUT
Exhibit 13
Preemptive QUEUE RESOURCE USAGE
SlNo. name start_time end_time activity_time finished replication waiting_time
1 John0 2.00 26 24 TRUE 1 0.00
2 Patient0 0.00 38 24 TRUE 1 14.00
3 Jimmy0 33.00 57 24 TRUE 1 0.00
4 Maria0 43.00 69 24 TRUE 1 2.00
5 Patient1 0.18 81 24 TRUE 1 56.82
6 Patient2 8.64 93 24 TRUE 1 60.36
7 Patient3 21.13 105 24 TRUE 1 59.87
8 Patient4 28.09 117 24 TRUE 1 64.91
SlNo. name start_time end_time activity_time finished replication waiting_time
1 Patient0 0.00 24 24 TRUE 1 0.00
2 John0 2.00 36 24 TRUE 1 10.00
3 Patient1 0.18 48 24 TRUE 1 23.82
4 Jimmy0 33.00 60 24 TRUE 1 3.00
5 Maria0 43.00 72 24 TRUE 1 5.00
6 Patient2 8.64 84 24 TRUE 1 51.36
7 Patient3 21.13 96 24 TRUE 1 50.87
8 Patient4 28.09 108 24 TRUE 1 55.91

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Non - Preemptive QUEUE RESOURCE USAGE
Bibliography
1. Problem Statement taken from Introduction to Operation Research (10th
Edition) by Frederick S. Hillier,
Gerald J. Lieberman, Bodhibrata Nag, Preetam Basu.
2. The materials for analysis is taken from the CD resource provided along with the above book.
3. R studio to run simulation using simmer package.

Queuing theory and its applications

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