Introduction to Simulation:
Simulation is one of the most powerful tools available to
decisionmakers responsible for the design and operation of complex processes
and systems. It makes possible to study, analysis and evaluation of a situation
that would not be otherwise possible. In an increasingly competitive world,
simulation has become an essential problemsolving methodology for engineers,
designers, and managers.
Simulation can be defined as the process of designing a model
of a real system and conducting an experiment with these models of system and, or
evaluating various strategies for the operation of the system. Thus, it is
important that the model we design in such a way, model behaviour mimics the behaviour
of a real system.
Simulation allows us to study the situation even though we are
unable to experiment directly with the real system, either the system
doesn’t exist or it is too difficult or expensive to directly
manipulate it.
We consider the simulation to include both constructions of the model and experimental use of the model for studying problems. Thus, we can think of
simulation as an experiment and applied methodology which seeks to:
i. Describes
the behaviour of the system.
ii. Use of
model to predict future behaviour i.e. the effect that will be produced by
changes in the system or in its method of operation.
When To Use Simulation?
Following are some of the purposes for which simulation may
use:
a. Simulation is very useful for experiments with the
internal interactions of a complex system, or of a subsystem within a complex
system.
b. Simulation can be employed to experiment with new
designs and policies, before implementing them.
c. Simulation can be used to verify the results
obtained by analytical methods and to reinforce the analytical techniques.
d. Simulation is very useful in determining the
influence of changes in input variable on the output of the system.
e. Simulation helps in suggestion modification in the system under investigation for its optimal performance.
Advantages:
Simulation has several advantages. First of all basic
concept of simulation is easy to understand and hence often easier to justify
to management or customer than some of the analytical methods. Also, a
simulation model may be more. Because its behaviour has been compared to that of a real system or because it requires fewer simplifying assumptions and hence
captures more of true characteristics of the system understudied.
Other Advantages Include:
Other Advantages Include:
a. We can taste new designs, layouts, etc. without
assigning resources to their implementation.
b. It can be used to explore new staffing policy,
operating procedure, design rules, organizational structure, information flows,
etc. with out disturbing the ongoing operation.
c. It allows us to identify a bottleneck (jams) in
information, matters, products, flows and test option for increasing the flow
rates.
d. It allows us to test a hypothesis about how or why a certain phenomenon occurs in the system.
e. It allows us to control time. Thus we can operate
the system several month or years of experience in a matter of seconds allowing
us to quickly look at a long time or we can slow down the phenomenon for study.
f. It allows us to gain insights into how a model the system actually works and understanding of which variables are most important
up to a performance.
g. Its great strength is its ability to let us
experiment with new and unfamiliar situations.
Disadvantages:
Even though simulation has many strengths and advantages, it is
not without drawbacks. And some are:
a. Simulation is an art that requires specialized trainers
and therefore, skill labels of practice vary widely. The utility of the study
depends upon the quality of the model and the skill of the models.
b. Gathering highly reliable input data can be time
consuming and the resulting data is sometimes highly comprised of insufficient
data or poor management decisions.
c. Simulation models are input and output models i.e.
they yield the portable output of the system for a given input. These are, therefore, run rather than solved. They do not yield optional rather they serve
as a tool for analysis of the behaviour of a system under conditions specified by
the experiments.
The Technique of Simulation – Monte Carlo Method:
Monte Carlo simulation is a computerized mathematical technique to generate random sample data, based on some known distribution for
numerical experiments. This method is applied to risk quantitative analysis and
decisionmaking problems. This method is used by the professionals of various
profiles such as finance, project management, energy, manufacturing,
engineering, research & development, insurance, oil & gas,
transportation, etc.
This method was first used by scientists working on the atom
bomb in 1940. This method can be used in those situations where we need to make
an estimate and uncertain decisions such as weather forecast predictions.
Important Characteristics:
Following are the three important characteristics of
MonteCarlo method.
a. Its output must generate random samples.
b. Its
input distribution must be known.
c. Its
result must be known while performing an experiment.
Advantages:
a. Easy to
implement.
b. Provides statistical sampling for numerical experiments using the computer.
c. Provides an approximate solution to mathematical problems.
d. Can be used for both stochastic and deterministic problems.
b. Provides statistical sampling for numerical experiments using the computer.
c. Provides an approximate solution to mathematical problems.
d. Can be used for both stochastic and deterministic problems.
Disadvantages:
a. Time
consuming as there is a need to generate a large number of sampling to get the
desired output.
b. The results of this method are only the approximation of true values, not the exact.
b. The results of this method are only the approximation of true values, not the exact.
Flow Diagram:
The following illustration shows a generalized
flowchart of Monte Carlo simulation.
Problem Depicting Monte Carlo Method:
This method is applied to solve both deterministic as well as
stochastic problems. There are many deterministic problems also which are solved
by using random numbers and interactive procedure of calculations. In such a case, we convert the deterministic model into a stochastic model, and the results
obtained are not exact values, but only estimates.
For example, we shall consider the problem
taking integral of a single variable over a range which corresponds to finding
the area under graph representing the function f(x). Let us suppose that f(x)
is positive and has a and b as bounds above by c. Then as
shown in the figure, the function f(x) will be contained within the rectangle of
sides c and (ba). Now, we can pick up points at random within the rectangle
and determine whether they lie beneath the curve or not. The points selected
are assumed to be obtained from a uniformly distributed random number generator.
Two successive samplings are made to get X and Y coordinates so that X is in
the range a to b and Y is in the range 0 to c. The fraction of points that fall or below the curve will be approximately the ratio
of the area under the curve to the area of the rectangle. If N points are
drawn and n of them fall under curve then;
Here,
Curve = f(x),
n=randomly selected points laying
inside the curve,
N=total numbers of points
selected,
Area of rectangle=c*(ba)
Now we have;
Which is the mathematical statement of the Monte Carlo method?
The accuracy increase as N increases. After enough
points have taken, the value of the integral (i.e. the area under the curve
represented by the function f(x)) is obtained by n/N * c * (b  a).
The computational technique is shown in the figure. At each trial, the value of x is selected at random between ‘a’ and ‘b’, say X0. Similarly, the second
random number is selected between 0 and C to give y. If y <= f(x0) then
point is inside the curve and count ‘n’ otherwise point will not lie in the
curve and the next point will be picked.
The application of the Monte Carlo Method for evaluation of Pi (Ï€) is converting a deterministic model into a stochastic model.
Some examples that use random sampling in problemsolving are as follows:
a. To find
the area of irregular surface figure
b. Numerical
Integration of singlevariable function
c. A
Gambling Game.
d. Random
Walk Problem
Determine the value of Pi (Ï€) using Monte Carlo Method:
We use random number generation method to determine the sample
points that lie inside or outside the curve. Let (x_{0}, y_{0})
be an initial guess for the sample point than from a linear congruential method of
random number generation:
X_{i+1 }= (a_{xi} + c) mod m
Y_{i+1 }= (a_{yi} +c) mod m
Where a & c are constants, m is the upper limit of generated
random number. If y ≤ y_{i} then
increment n.
Example:
We have circle equation = x^{2} + y^{2} = 1 or
y = √(1  x²) Now, generate the random numbers x and y within the interval 0
and 1.
For x: x_{0} = 27, a = 17, c = 0, m = 100
For y: y_{0} = 47, a = 17, c = 0, m = 100
X’

Y’

√(1  x²)

In/Out

0.59

0.99

0.962

In

0.03

0.83

0.99

In

0.51

0.11

0.86

In

0.67

0.87

0.74

Out

Now,
Points inside the curve (n) = 3
Points inside the rectangle (N) = 4
Value of pi = (n/N)*4
= 3
We have to generate a random number for x and y. For x (random
number be in range 2 to 5) & for y (random number be in range 8 to 125).
Here, the area of the rectangle under the given condition = (5  2)
* (125  8) = 351, also we know, I = n/N * area of rectangle
Now, we can select the random points inside the curve (using the
random number generation method).
For x: x0 = 23, a = 17, c = 0, m = 50
Fir y: y0 = 61, a = 59, c = 0, m = 125
X

X’
= X*0.1

Y

X’^{3}

In/Out

23

2.3

59

12.167

Out

41

4.1

99

68.921

Out

47

4.7

91

103.823

In

49

4.9

119

117.649

Out

33

3.3

21

35.937

In

11

1.1

114

1.331

Out

37

3.7

101

50.653

Out

We get,
Points inside the curve (n) = 2
Points inside the rectangle (N) = 6
I = n/N * area of rectangle
I = 2/6 * 351
I = 117
Comparison of Simulation and Analytical Method:
Once we have built a mathematical model, it must then be
examined to see how it can be used to answer the questions of interest in
the system is supposed to represent. If the model is simple enough, it may
be possible to work with its relationships and quantities to get an exact,
analytical solution.
In the d = v*t example, if we know the distance to be travelled
and the velocity, then we can work with the model to get t = d/v as the time
that will be required. This is a very simple, closedform solution obtainable
with just paper and pencil, but some analytical solutions can become
extraordinarily complex, requiring vast computing resources; inverting a large
nonsparse matrix is a wellknown example of a situation in which there is an analytical formula known in principle, but obtaining it numerically in a given instance is far from trivial.
If an analytical solution to a mathematical model is available
and is computationally efficient, it is usually desirable to study the model in
this way rather than via a simulation. However, many systems are highly
complex, so that valid mathematical models of them are themselves complex,
precluding any possibility of an analytical solution. In this case, the model
must be studied using simulation, i.e., numerically exercising the model
for the inputs in question to see how they affect the output measures of
performance.
While there may be an element of truth to pejorative old saws
such as “method of last resort” sometimes used to describe simulation, the fact
is that we are very quickly led to simulation in many situations, due to the sheer complexity of the systems of interest and of the models necessary to
validly represent them.
Given, then, that we have a mathematical model to be studied
using simulation (henceforth referred to as a simulation model), we must
then look for particular tools to execute this model (i.e. actual simulation).
Difference Between Simulation and Analytic:
Basis

Simulation

Analytic

Input
Parameterization

Measured or Invented

Measured or invented
(with certain limitations)

Model
Components

Virtually anything

Composed of limited
basic building blocks

Model
Outputs

Anything that can be
measured

Equilibrium measures

Effort
To Construct Model

Arbitrary

Modest

Computational
Cost

Typically large

Typically small

Underlying
Concepts

Probability or
Statistics

Algebra to stochastic
processes

Special
Properties

Credible

Insight, optimization

Experimental Nature of Simulation:
The simulation technique makes no specific attempt to isolate
(separate) the relationship between any particular variables; instead, it
observes how all variables of the model change with time. The relationship between the variables must be derived from these observations.
Simulation is, therefore, essentially an experimental problemsolving
technique. Many simulation runs have to be made to understand the relationships
involved in the system, so the use of simulation in a study must be planned as
a series of experiments.
Types of System Simulation:
A simulator is a device, computer program, or system that
performs a simulation. A simulation is a method for implementing a model over
time. There are three types of commonly uses simulations:
1. Live:
Simulation involving real people operating real systems.
a. Involve individuals or groups
b. May use actual equipment
c. Should provide a similar area of operations
d. Should be close to replicating the actual activity
2. Virtual:
Simulation involving real people operating simulated systems.
Virtual simulations inject HumanInTheLoop in a central role by
exercising:
a. Motor control skills (e.g. flying an aeroplane)
b. Decision skills (e.g. committing fire control resources
to action)
c. Communication skills (e.g. members of a C4I team)
3. Constructive:
Simulation involving simulated people operating simulated
systems. Real people can stimulate (make inputs) but are not involved in
determining outcomes. Constructive simulations offer the ability to:
a. Analyze concepts
b. Predict possible outcomes
c. Stress large organizations
d. Make measurements
e. Generate statistics
f. Perform analysis
Distributed Lag Model:
If the regression model includes not only the current but also
the lagged (past) values of the explanatory variables (the x's) it is called a
distributed lag model. If the model includes one or more lagged values of the
dependent variable among its explanatory variables, it is called an
autoregressive model. This is known as a dynamic model.
In other word, Distributed Lag Model is defined as a type of model
that have the property of changing only at fixed interval of time and based on
current values of variables on other current values of variables and values
that occurred in previous intervals.
In economic studies, some economic data are collected over
uniform time intervals such as a month or year. This model consists of linear
algebraic equations that represent continuous system but data are available at
fixed points in time.
For Example: Mathematical Model of National Economy:
Let,
C = Consumption
I = Investment
T = Taxes
G = Government Expenditures
Y = National Income
Then
C = 20 +
0.7 (Y – T)
I
= 2 + 0.1 Y
T
= 0.2 Y
Y = C + I
+ G
All the equation are expressed in billions of rupees. This is
static model and can be made dynamic by lagging all the variables as follows:
C = 20 + 0.7(Y_{1 }– T_{1})
I = 2 + 0.1Y_{1}
T = 0.2Y_{1}
Y = C_{1 }+ I_{1 }+ G_{1}
_{}
Any variable that can be expressed in the form of its current
value and one or more previous value is called a lagging variable. And hence this
model is given the name distributed lag model. The variable in a previous
interval is denoted by attaching –n suffix to the variable. Where n indicates
the n^{th} interval.
Advantages of Distributed Lag Model:
a. Simple to understand and can be computed by hand,
computers are extensively used to run them.
b. There is no need for special programming language to organize the simulation task.
b. There is no need for special programming language to organize the simulation task.
Cobweb Model:
Cobweb theory is the idea that price fluctuations can lead to
fluctuations in supply which cause a cycle of rising and falling prices.
In a simple cobweb model, we assume there is an agricultural market where supply can vary due to variable factors, such as the weather.
Assumptions of Cobweb theory:
a. In an agricultural market, farmers have to decide
how much to produce a year in advance, before they know what the market price
will be. (supply is price inelastic in shortterm)
b. A key determinant of supply will be the price from the previous year.
c. A low price will mean some farmers go out of business. Also, a low price will discourage farmers from growing that crop in the next year.
d. Demand for agricultural goods is usually price inelastic (a fall in price only causes a smaller % increase in demand).
b. A key determinant of supply will be the price from the previous year.
c. A low price will mean some farmers go out of business. Also, a low price will discourage farmers from growing that crop in the next year.
d. Demand for agricultural goods is usually price inelastic (a fall in price only causes a smaller % increase in demand).
1. If there is a very
good harvest, then supply will be greater than expected and this will cause a fall in price.
2. However, this fall in
price may cause some farmers to go out of business. Next year farmers may be put off by the low price and produce something else. The consequence is that if we have one year of low prices, next year farmers reduce the supply.
3. If supply is reduced,
then this will cause the price to rise.
4. If farmers see high
prices (and high profits), then next year they are inclined to increase supply because that product is more profitable.
In theory, the market could fluctuate between high price and low
price as suppliers responds to past prices.
Cobweb Theory and Price Divergence:
The price will diverge from the equilibrium when the supply curve is more elastic than the demand curve, (at the equilibrium point).
If the slope of the supply curve is less than the demand
curve, then the price changes could become magnified and the market more
unstable.
Cobweb Theory and Price Convergence:
At the equilibrium point, if the demand curve is more elastic than the supply curve, we get the price volatility falling, and the price will
converge on the equilibrium
Limitations of Cobweb theory:
1. Rational Expectations:
The model assumes farmers base next year’s supply purely on
the previous price and assume that next year’s price will be the same as last
year (adaptive expectations). However, that rarely applies in the real world.
Farmers are more likely to see it as a ‘good’ year or ‘bad year and learn from
price volatility.
2. Price Divergence Is Unrealistic And Not Empirically Seen:
The idea that farmers only base supply on last year’s price
means, in theory, prices could increasingly diverge, but farmers would learn
from this and preempt changes in price.
3. It May Not Be Easy Or Desirable To Switch:
A potato grower may concentrate on potatoes because that is
his speciality. It is not easy to give up potatoes and take to aborigines.
4. Other Factors Affecting Price:
There are many other factors affecting price than a farmers decision
to supply. In global markets, supply fluctuations will be minimized by the role
of importing from abroad. Also, demand may vary. Also, supply can vary due to
weather factors.
5. Buffer Stock Schemes:
Governments or producers could band together to limit price
volatility by buying surplus.
Steps of Simulation Study:
1. Problem Formulation:
The study begins with defining the problem statement. It can
be developed either by the analyst or client. If the statement is provided by the client, then the analyst must take extreme care to ensure that the problem is
clearly understood. If a problem statement is prepared by the simulation
analyst, the client must understand and agree with the
formulation. Even with all of these precautions, the problem may need to be reformulated as the simulation study progresses.
2. Setting of Objectives and Overall Project Plan:
Another way to state this step is to "prepare a
proposal." The objectives indicate the questions to be answered by the simulation
study. Whether the simulation is appropriate or not is to be decided at this
stage. The overall project plan should include a statement of the alternative
systems and a method for evaluating the effectiveness of these alternatives. The
plan includes several personnel, number of days to complete the task, stages
in the investigation, output at each stage, cost of the study and billing
procedures if any.
3. Model Conceptualization:
Model is a simplification of reality. The realworld system
under investigation is abstracted by a conceptual model. It is recommended that
modelling begins with a simple model and grows until a model of appropriate
complexity has been achieved. For example, consider the model of a
manufacturing and material handling system. The basic model with the arrivals,
queues, and servers is constructed. Then, add the failures and shift schedules.
Next, add the materialhandling capabilities. Finally, add special
features. Constructing an excessive complex model will add to the cost of the
study and the time for its completion, without increasing the quality of the
output.
Maintaining client involvement will enhance the quality of the
resulting model and increase the client's confidence in its use.
4. Data Collection:
This step involves gathering the desired input data. The
data changes over the complexity of the model. Data collection takes a huge amount
of total time required to perform a simulation. It should be started at early
stages together with model building. The collection of data should be relevant
to the objectives of the study.
5. Model Translation:
The conceptual model constructed in Step 3 is coded into a
computer recognizable form, an operational model. The suitable simulation
language is used.
6. Verified:
Verification is concerning the operational model. Is it
performing properly? If the input parameters and logical structure of the model are
correctly represented in the computer, then verification is completed.
7. Validated:
Validation is the determination, that the model is an accurate
representation of the real system. This is done by calibration of the model; an
iterative process of comparing the model to the actual system behaviour. This
process is repeated until model accuracy is acceptable.
8. Experimental Design:
The alternatives to be simulated must be determined. For each the scenario that is to be simulated, decisions need to be made concerning the
length of the simulation run, the number of runs (also called replications),
and the length of the initialization period.
9. Production Runs And Analysis:
The production runs, and their subsequent analysis is used to
estimate measures of performance for the system design that is being
simulated.
10. More Runs:
After the completion of the analysis of runs, the simulation the analyst determines if additional runs are needed and any additional experiments
should follow.
11. Documentation And Reporting:
There are two types of Documentation: Program and Progress.
Program documentation is necessary for numerous reasons. If the program is
going to be used again by the same or different analysts, it may be necessary
to understand how the program operates. This will enable confidence in the
program so that the client can make decisions based on the analysis. Also, if
the model is to be modified, this can be greatly facilitated by adequate documentation.
Progress reports provide a chronology of work done and decisions made. It is
the written history of a simulation project. The result of all the analyses
should be reported clearly and concisely. This will enable the client to review
the final formulation.
12. Implementation:
If the client has been involved throughout the study period,
and the simulation analyst has followed all of the steps rigorously, then the likelihood of a successful implementation is increased.
Time Advancement Mechanism:
The simulation models we consider will be discrete, dynamic,
and stochastic. Discrete event simulation concerns the modelling of a system as
it evolves over time by a representation in which the state variables change
instantaneously at separate points in time.
These points in time are the one which an event occurs, where an event
is defined as an instantaneous occurrence that may change the state of a
system.
Because of the dynamic nature of discreteevent simulation
model, we must keep track of the current value of simulation time as the
simulation proceeds and we also need a mechanism to advance simulation time from
one variable to another. We call the variable in a simulation model that gives
the current value of simulation time the simulation clock or simulation time.
Simulation time means the integral clock time and not the time
a computer was taken to carry out the simulation. Two principal approaches for advancing the
simulation clock are:
a. Next Event Time Advance
b. Fixed
Increment Time Advance
Queuing Models and its Characteristics:
The Queuing theory provides predictions about waiting times,
the average number of waiting customers, the length of a busy period and so forth.
These predictions help us to anticipate situations and to take
appropriate measures to shorten the queues.
A further attractive feature of the theory is quite an astonishing range of its applications. Some of the more prominent of these are
telephone conversation, machine repair, toll booths, taxi stands, inventory
control, the loading, and unloading of ships scheduling patients in the hospital
clinics, production flow and applications in the computer field concerning
program scheduling, etc.
A queuing system may be described as one having a service
facility, at which units of some kind (called customers) arrive for service and
whenever there are more units in the system than the service facility can
handle simultaneously, a queue or waiting line develops. The waiting units take
their turn for service according to a preassigned rule and after service, they
leave the system. Thus, the input to the system consists of the customers
demanding service and the output is the serviced customers.
Characteristics of the Queuing System:
1. The Arrival Pattern:
The arrival pattern describes how a
customer may become a part of the queuing system. The arrival time for any
customer is unpredictable. Therefore, the arrival time and the number of
customers arriving at any specified time intervals are usually random variables.
A Poisson distribution of arrivals corresponds to arrivals at random. In Poisson
distribution, successive customers arrive after intervals which independently
are and exponentially distributed. The Poisson distribution is important, as it
is a suitable mathematical model of many practical queuing systems as described
by the parameter “the average arrival rate”.
2. The Service Mechanism:
The service mechanism is a
description of the resources required for service. If there are an infinite number of
servers, then there will be no queue. If the number of servers is finite, then
the customers are served according to a specific order. The time taken to serve
a particular customer is called the service time. The service time is a
statistical variable and can be studied either as the number of services
completed in a given time or the completion period of service.
3. The Queue Discipline:
The most common queue discipline is
the “First Come First Served” (FCFS) or “Firstin, Firstout” (FIFO).
Situations like waiting for a haircut, ticketbooking counters follow FCFS
discipline. Other disciplines include “Last In First Out” (LIFO) where the last
customer is serviced first, “Service In Random Order” (SIRO) in which the
customers are serviced randomly irrespective of their arrivals. “Priority
service” is when the customers are grouped in priority classes based on
urgency. “Preemptive Priority” is the highest priority given to the customer
who enters into the service, immediately, even if a customer with lower
priority is in service. “Nonpreemptive priority” is where the customer goes
ahead in the queue but will be served only after the completion of the current
service.
The Number of Customers allowed in the System:
In certain cases, a service system
is unable to accommodate more than the required number of customers at a time.
No further customers are allowed to enter until space becomes available to
accommodate new customers. Such types of situations are referred to as finite
(or limited) source queue. Examples of finite source queues are cinema halls,
restaurants, etc.
On the other hand, if a service system can accommodate any number of customers at a time, then it is
referred to as an infinite (or unlimited) source queue. For example, in a sales
department, where the customer orders are received; there is no restriction on
the number of orders that can come in so that a queue of any size can form.
The Number of Service Channels:
The more the number of service
channels in the service facility, the greater the overall service rate of the
facility. The combination of arrival rate and service rate is critical for
determining the number of service channels. When there are several service
channels available for service, then the arrangement of service depends upon
the design of the system's service mechanism.
Parallel channels mean, several
channels providing identical service facilities so that several customers may
be served simultaneously. Series channel means a customer goes through successive
ordered channels before service is completed. A queuing system is called
a oneserver model, i.e., when the system has only one server,
and a multiserver model i.e., when the system has several
parallel channels, each with one server.
a. Arrangement Of Service Facilities In Series:
b. Arrangement Of Service Facilities In Parallel:
c. Arrangement of Mixed Service facilities:
Measuring the Performance Of The System:
To measure the performance of the system, we estimate the
following three qualities:
1. Estimate
the expected average delay in a d(n) queue of (n)
customers. The actual average delay for n customers depends on the interarrival and service time. From a single sum of simulation with customers delays
D1, D2… Dn, the estimate of d(n) is
given by:
Note that a customer could have a delay of zero in case of an
arrival finding the system empty or idle. If many delays were 0 then this could
represent the system providing very good service.
This estimate of d(n)
gives information about the system performance form the customer point of view.
2. Estimate the expected average number of customers
in the queue (but not being served), denoted by q(n). It is different from the average delay in the queue because it
takes over continuous time rather than over the customer (being discrete).
Let Q(t) = number of customers in queue at time (t), (t≥0).
T(n) = time required for ‘n’ delays in queue.
Pj = expected proportion of time that Q(t) = 1 (value of Pi
will be between 01)
Then, the average number of customer queue;
Single Server Queuing System:
Consider a single server queuing system, where the inter
arrival time A1, A2… are independent and identically distributed (IID) random
variables. A customer who arrives and finds the server idle enters the
service immediately and the service time S1, S2… of the successive customers
are IID random variables that are independent of interarrival times.
a. A customer who arrives and finds the server busy,
joins the end of the single queue.
b. After completing service for a customer1, the server chooses the next customer from the queue (if any) in a FIFO manner.
c. The simulation will begin in the empty and idle
state i.e. no customers are present and the service is idle.
d. At time 0, the system will begin waiting for the arrival
of the 1st customer will occur, after the 1^{st} interarrival time A1
rather than at time 0.
e. The simulation will continue until N numbers of
customers have completed their delays in a queue.
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