Chapter03: System Simulation
3-1:: The Technique of
Simulation::
Question: Describe the
technique of simulation?
# Define simulation.
Answer:The term simulation describes any
procedure of establishing a dynamic mathematical model and deriving a solution
numerically. Simulation methods produce solutions in steps. Each step gives the
solution for one set of conditions and the calculation must be repeated to
expand the range of solution. System simulation solves problems by the
observation of the performance, over time, of a dynamic mathematical model of
the system.
Simulation/ Technique of simulation: “Simulation” or “System simulation” is use to describe any
procedure or technique of establishing a dynamic mathematical model of a system
and deriving a solution numerically. Simulation technique produce solution in
steps, each step gives the solution for one set of conditions and the
calculation must be repeated to expand the range of solution.
Simulation process solves the equations of the model, step by
step, with increasing values of time. As a result, the current values at any
step of the computation represent the state of the system being modeled at that
point in time.
q Question: What do you understand by system
simulation?
Answer:
System simulation:System simulation is considered to be a numerical computational
technique used conjunction with dynamic mathematical models
Simulation
is use to describe any procedure or technique of establishing a dynamic
mathematical model of a system and deriving a solution numerically. Simulation
technique produce solution in steps, each step gives the solution for one set
of conditions and the calculation must be repeated to expand the range of
solution.
Simulation process solves the equations of the model, step by
step, with increasing values of time. As a result, the current values at any
step of the computation represent the state of the system being modeled at that
point in time.
3-2:: The Monte Carlo
Method::
Question: Describe
Monte-Carlo method as a particular numerical computational technique. (’01)
Answer:
A particular numerical computation method, called the Monte
Carlo method, consists of experimental sampling with random numbers. We can
define Monte Carlo simulation to be a scheme employing random numbers, that is
U(0,1) random variates, which is used for solving certain stochastic or
deterministic problems where the passage of time plays no sustentative role.
Thus, Monte Carlo simulations are generally static rather dynamic. For example,
the integral of a single variable over a given range corresponds to finding the
area under the graph representing the function. Suppose the function, f (x) is positive and has lower and
upper bounds a andb, respectively. Suppose, also, the
function is bounded above by the value c. As shown in figure, the graph of the
function is then contained within a rectangle with sides of
lengthc,
and b – a..
Figure: The Monte Carlo Method.
If we pick points at random within the rectangle, and determine
whether they lie beneath the curve or not, it is apparent that, providing the
distribution of selected points is uniformly spread over the rectangle, the
function of points falling on or below the curve should be approximately the
ratio of the area under the curve to the area of the rectangle. If N points are used and n of them fall under the curve, then,
approximately,
 |
The accuracy improves as the number N increases. When it is decided that sufficient points have been
taken, the value of the integral is estimated by multiplying n/N by the area of
the rectangle, c(b – a).
The
computational technique is illustrated in figure. For each point, a value of x is selected at random between a and b,
say X0. A second random
selection is made between 0 and c to
give Y. If 
the point is accepted in the count n,
otherwise it is rejected and the next point is picked.
the point is accepted in the count n,
otherwise it is rejected and the next point is picked.
3-3:: Comparison of
Simulation and Analytical Methods::
Question: Give
comparison of simulation technique method and analytical technique method.
Answer: Comparison of simulation technique method
and analytical technique method are given below:
|
Simulation Technique Method
|
Analytical Technique Method
|
|
q Simulation gives specific solutions
rather than general solutions. Each execution of simulation tells only
whether a particular set of conditions did or did not meet the goal.
|
q Where, analytical solution gives general
solution.
|
|
q Many simulation runs may be needed to
find a maximum.
|
q Mathematical solution is preferable,
when the solution being sought is maximizing condition.
|
|
q Various types of complex problems can be
solved through simulation.
|
q The range of problems that can be solved
mathematically is limited. Mathematical techniques require that the model be
expressed in some particular format.
|
|
q There need a little abstraction for no
abstraction to apply simulation methods.
|
q Sometimes, the degree of abstraction
required to apply analytical method is too severe. It reduces the degree of
accuracy.
|
|
q The ideal way of using simulation is an
execution of mathematical solutions that might have been obtained at the cost
of too much simplification.
|
q Sometimes, it needs too much
simplification to form a model for analytical solution.
|
|
q Simulation easily removes many
limitations on a system, such as physical stop, finite time delays, nonlinear
forces, etc.
|
q These limitations make solvable
mathematical model in solution.
|
|
q Simulation will provide a quicker or
more convenient way of deriving results.
|
q Many analytical results occur in the
form of complex series or integrals that require extension evolution.
|
3-4:: Experimental
Nature of Simulation::
Question: Describe
experimental nature of simulation.
Answer: The simulation technique makes no
specific attempt to isolate the relationships between any particular variables.
Instead it observes the way in which all the variables of the model change with
time. Relationships between the variables must be derived from these
observations.
Simulation is, therefore, essentially an experimental
problem-solving technique. Many simulation runs have to be made to understand
the relationships involved in the systems, so the use of simulation in a study
must be planned as a series of experiments.
3-5:: Types of System
Simulation::
Question: Describe
different types of system simulation.
Answer: The different types of system simulation
are given below:
- Continuous
simulation: Continuous simulation technique
generally applied on continuous model. Continuous simulation concerns the
modeling over time of a system by a representation in which the state
variables change continuously with respect to time. Typically, continuous
simulation models involve differential equations that give relationships
for the rates of change of the state variables with time. If the
differential equations are particularly simple, they can be solved
analytically to give the values of the state variables for all values of
time as a function of the values of the state variables at time 0. For
most continuous modes analytic solutions are not possible.
Numerical-analysis techniques, e.g., Runge-Kutta integration are used to
integrate the differential equations numerically, given specific values
for the state variables at time 0.
- Discrete
simulation/Discrete-Event Simulation:
Discrete simulation technique generally applied on discrete model.
Discrete-event simulation concerns the modeling of a system as it evolves
over time by representation in which the state variables change
instantaneously at separate point in time. In more mathematical terms, the
system can change at only a countable number of points in time. These points in tome are the once at
which an event occurs, where an event is defined as an instantaneous
occurrence that may change the state of the system. Although
discrete-event system simulation could conceptually be done by hand
calculations, the amount of data that must be stored and manipulated for
most real world systems dictates that discrete-event simulation be done on
a digital computer.
Science some systems are neither completely discrete nor
completely continuous, the need may arise to construct a model with aspects of
both discrete-event and continuous simulation, resulting in a combined
discrete-continuous simulation. There are three fundamental types of
interactions that can occur between discretely changing and continuously
changing state variables:
q A discrete event may cause a discrete
change in the value of continuos state variable.
q A discrete event may cause the
relationship governing a continuos state variable to change at a particular
time.
q A continuous state variable achieving a
threshold value may cause a discrete event to occur or to be scheduled.
- Monte Carlo simulation( a special
type of simulation):
A particular numerical computation method, called the Monte
Carlo method, consists of experimental sampling with random numbers. We can define
Monte Carlo simulation to be a scheme employing random numbers, that is U(0,1)
random variates, which is used for solving certain stochastic or deterministic
problems where the passage of time plays no sustentative role. Thus, Monte
Carlo simulations are generally static rather dynamic.
[N.B.:- both
analytical and numerical technique may applied for continuous and discrete
model.]
3-6:: Numerical
Computational Technique for Continuous Models::
Question: Describe
numerical computational technique with the example of continuous models.
Answer: To illustrate the general numerical
technique of simulation based on a continuous model, we can consider the
following example.
A builder observer that the rate at which he can sell houses
depends directly upon the number of families who do not yet have a house. As
the number of people without houses diminishes, the rate at which he sells
houses drops. Let H be the potential
number of households, and y be the
number of families with houses. The situation is represented in figure-1; the
horizontal line at H is the total
potential market for houses.
SHAPE
Figure: Sale of house and air
conditioners.
The curve for y
indicates how the number of houses sold increases with time. The slope of the
curve (i.e., the rate at which y increases) decreases as H – y gets less. This reflects the slowdown of sales as the market
becomes saturated. Mathematically, the trade can be expressed by the equation
at
[N.B.: Follow book
(page: 44-46) to complete the answer)
…………………………………….
……………………………………
……………………………………
3-7:: Numerical
Computation Technique for Discrete Models::
Question: Explain with
an example the Numerical Computation Technique for Discrete Model. (’01)
Answer:
To illustrate the general computational technique
of simulation with discrete models, consider the following example. A clerk
begins his day’s work with a pile of documents to be processed. The time taken
to process them varies. He works through the pile, beginning each document as
soon as he finishes the provides one, except that he takes a five minute break
if, at the time he finishes a document, it is an hour or more since he began
work or since he last has a brake. We assume the times to process the documents
are given. We will also keep a count of how many documents are left. The count
will be initially set to the number of documents at the beginning of the day,
and, in this simple model, we will assume that no documents arrive during the
day. The count will be decremented for each completed job, and work will stop
if the count goes to zero.
|
Document Number
I
|
Start Time
tb
|
Work Time tw
|
Cumulative Time
tc
|
Break Flag
F
|
Number of jobs
N
|
|
1
|
0
|
45
|
45
|
0
|
57
|
|
2
|
45
|
16
|
61
|
1
|
56
|
|
3
|
66
|
5
|
71
|
0
|
55
|
|
4
|
71
|
29
|
100
|
0
|
54
|
|
5
|
100
|
33
|
133
|
1
|
53
|
|
6
|
138
|
25
|
163
|
0
|
52
|
|
7
|
163
|
21
|
184
|
0
|
51
|
Table: Simulation of Document
Processing.
The computations involved can be organized as shown in table.
The ith row corresponds to the ith document. The first column numbers the
documents. There are then four columns giving various times, measured in
minutes from time zero. The second column gives the time the clerk begins to
work on a document, denoted by tb(i). The third column gives the
time required to work on the document, denoted by tw(i), and the
fourth column gives the time each document is finished. The fifth column
contains the cumulative time since work started or since the last break,
measured at the time each job is completed. This is denoted by tc(i).
There is a sixth column which contains a flag, denoted by F, that takes the
value 1 if the clerk should take a break after the ith document and the value 0
if he should not. The clerk works until there are no more documents, or the
time he finishes a document goes beyond some time limit.
The computation proceeds row by row, and from left to right.
The first row shows that starts on the first document at time zero. The
processing time is 45 minutes, so the job is finished at 45, with a cumulative
time (in this case, since the start of work) of 45. This is not long enough for
a break, so the flag is set to 0. The count, which was initialized to 57 jobs,
is dropped to 56.
Because the flag is zero and the count is not zero, the second
document is begun at 45. It needs 16 minutes for processing, which leads to a
cumulative time of 61, so the flag is set to 1 to indicate that a break should
be taken. Because of the five minute break, the third document starts at 66.
The computational-continues in this manner until either N, the count of
documents to be processed, drops to zero, or the finish time, tf,
reaches some limit representing the end of the work period.
3-8:: Distributed Lag
Models::
Models that have the properties of changing only at fixed
intervals of time, and of basing current values of the variables on other
current values and values that occurred in previous intervals, are called distributed lag models. They are used
extensively in econometric studies where the uniform steps correspond to a time
interval, such as a month or a year, over which some economic data are
collected. As a rule, these models consist of linear, algebraic equations. They
represent a continuous system, but one in which the data is only available at
fixed points in time.
As an example, consider the following simple mathematical model
of the national economy of a country. Let,
C be consumption,
I be investment,
T be taxes,
G be government
expenditure,
Y be national income.
C = 20 + 0.7(Y – T)
I
= 2 + 0.1Y
T
= 0.2Y
Y
= C + I + G
This is
a static mathematical model, but it can be made dynamic by picking a fixed time
interval, say one year, and expressing the current values of the variables in
terms of values at pervious intervals. Any variable that appears in the form of
its current value and one or more previous intervals said to be lagged
variable. Its value in a previous interval is denoted by attaching the suffix-n
to the variable, where n indicate the interval. The static model is made
dynamic by lagging all the variables, as follows:
Lagged variable: Any variable that appears in the form of
its current value and one or more previous intervals said to be lagged
variable. Its value in a previous interval is denoted by attaching the suffix-n
to the variable, where n indicate the interval.
3-10:: Progress of a Simulation Study::
SHAPE
Figure: The process of simulating.
Question: What do you
understand by system simulation? Why simulation is necessary? (’99)
Answer:
System simulation:System simulation is considered to be a
numerical computational technique used conjunction with dynamic mathematical
models.
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