Programming and Computation: System Dynamics

Chapter 05: System Dynamics

5-1: Historical Background

{Exponential growth: The growth rate is directly proportional to current level.}

Question: Define System Dynamics. (’97)/Describe the principle of system dynamics study.

(or)Define System Dynamics study? What is the principle concern of the System Dynamics study? (’01)

Answer:The principal concern of a System Dynamics study is to understand the forces operating in a system in order to determine their influence on the stability or growth of the system. The output of the study will suggest some reorganization, or change in policy, that can solve an existing problem or guide developments away from potentially dangerous directions.

5-2:: Exponential Growth Models::

Question: Discuss exponential growth models and its application in simulation. (’97,’99)

Answer: Growth implies a rate of change. Mathematical models describing growth involve differential equations. We can consider the growth of a capital fund that is earning compound interest. If the growth rate coefficient is k (i.e., 100k% interest rate), then the rate at which the fund grows is k times the current size of the fund. Expressed mathematically, where x is the current size of the fund,

Where

at t=0

This is a first-order differential equation whose solution is the exponential function. The solution, in terms of the mathematical constant e, which has the approximate value 2.72, is

Following figure plots for various values of k and an initial value of 1. It can be seen that the fund grows indefinitely, whatever (positive) value of k is used, and it grows faster with greater values of k. Looking at the curve for k = 0.2, and picking the point x = 2, the corresponding slope at the point A has a certain value. Later, at the point B, where x has become twice its value at A, the slope has become twice as great as at A. Since the slope is measured as the first-order differential coefficient, this fact is simply the reflection of the law defining exponential growth: the growth rate is directly proportional to the current level.

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Question: How can we say the following data represents a exponential growth rate with the help of semi-logarithmic graph paper?(see table 5-1,page 86)

Answer: Another way of describing the exponential function is to say that the logarithm of the variable increases linearly with time. To test whether any particular set of data represents exponential growth, the logarithms of the data should be plotted against time. If the data then appear to fall on a straight line, the growth is exponential, and the slope of the straight line will be greater for larger growth rate coefficients.

We can plot given tabular data on semi-logarithmic graph paper where the horizontal lines are placed at logarithmic intervals. Plotting data on such paper is equivalent to taking the logarithm of the data and then plotting on normal linear graph paper. Following figure shows the gross national product for several countries plotted against year on semi-logarithm paper. The points fall reasonably well on straight lines, indicating exponential growth rates.

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Question: How can we calculate the growth rate coefficient from straight line on semi-logarithmic graph paper?

The growth tare coefficient can be estimated by picking two points of the straight line that best fits the data, and taking the (natural) logarithm of the ratio of the value. If the points are

and

, the result is

form which it is possible to derive k. In terms of the more familiar logarithms to the base 10, the corresponding result is

From table, for U.K--, the values for 1965 and 1971 are 35.3 and 55.6 respectively.

Log55.6/35.3=0.434(1971-1965)K

Log1.575=0.434´6´k

K=0.1973/.434´6=0.075768@7.5%

So the growth rate co-efficient for U.K., is 7.5%

Year	UK	USA	Thailand	Japan
1963	30.2	596	68.1	245
1965	35.3	692	84.3	321
1966	37.7	759	101.4	369
1967	39.8	804	108.3	437
1968	42.9	863	116.7	518
1969	45.7	929	128.6	605
1970	49.9	983	135.9	712
1971	55.6	1060	145.3	794
1972	61.1	1159	160.2	907

Question: What is time constant? Describe with respect to exponential growth model.

Answer:

Time constant: Sometimes the exponential growth rate coefficient k is expressed is the form of

so that

The solution for the exponential growth model then takes the form

The constant T is said to be a time constant since it provides a measure of how rapidly the variable x grows. For example, when t equals T, the variable is exactly e times its initial value

. If T is small, say 2, x reaches this level after two time units. If T is large, say 20, x only reaches that level after 20 time units.

The inverse relationship between k, the growth rate coefficient, and T, the time constant, means that a large coefficient is associated with a small time constant and, therefore, a more rapid rate of increasing. For example, a growth rate of 0.05, or 5% will double the size of a population in a little under 14 years, while a growth rate of 10% will do the same in just under 7 years. The value 14, 7 are the time constants of the two cases.

5-3:: Exponential Decay Model::

Question: Discuss exponential decay models and its application in simulation. (’97)

Answer:This model is closely, related to the exponential growth model. In this model variable decays from some initial value,

at a rate proportional to the current value. The model can, in fact, be interpreted as a negative growth model. The equation for the model is

at t=0

The solution is

The model is shown in the following figure for various values of k. Constant k is sometimes expressed in the form 1/T. The characteristic of the model is that the level, x , is divided by a constant factor for a given interval of time. In the interval of T time units, the level is divided by e. Since e is approximately 2.72, the level is reduced by a factor of 0.37. Each successive interval of T reduces the level by the same factor.

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Above figure illustrate the exponential delay. Here different delay model need different time units (delay time) to reach the same output level, x.

Example: An example of a decay model is an aging population, which is being diminished by deaths or breakdowns. Another example is the manner in which radioactive material decays.

[Side note:

Now

5-4:: Modified Exponential Growth Models::

Question: Illustrate modified exponential growth model with an example. (’01)

Answer:

X = The number of people who might buy the product,

x = The number of people who have brought the product,

X – x = The number of people who have not brought the product.

x = k(X – x)

x = 0 at t = 0

and the solution is

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Figure: Modified exponential curves.

5-5:: Logistic Curves::

Question: What is logistic curve? Explain logistic curve in terms of logistic function.(99)

Answer:

The logistic curve/S-shaped curve resembling the one shown in the following (fig.1).

Initially the curve turns upward, looking like the exponential growth model curve. Eventually the curve reaches a point of inflection, where it turns from an increasing to a decreasing slope. From there, the curve resembles a modified exponential curve. An example of this shape, in terns of the behavior of people buying a product, can be given. Initially the sales rate will tend to be proportional to the number sold, which is the condition governing the exponential growth curve. As more products are sold, the market begins to become saturated, the conditions of the modified exponential model take over.

The logistic function is, in effect, a combination of exponential and modified exponential functions that describes this process mathematically. The differential equation defining the logistic function is

Initially, when x is very much smaller than X, which is the market limit, the value of term X - x remains essentially constant at the value X. The equation for the logistic function, therefore, is approximately

which is the equation for the exponential growth model with a constant of kX. Much later, when the market is almost saturated, the value of x will be close to the value of X, so that it changes very little with time. The equation for the logistic function takes the approximate form,

which is the differential equation for modified exponential function with a constant of kX.

5-6:: Generalization of Growth Model (N.B. Please follow book, Page(91-100))

Residual market for residential telephone used in U.S. is dependent on the following three economic factors:

the number of households in the U.S., denoted by H_i;
Disposable personal income per capita, denoted by P_i and
Local service revenue per telephone, denoted by C_i.

Then the mathematical model takes the form

where

5-7: System Dynamic Diagrams:

Question: Discuss the basic structure of system dynamic model. (’99)

Answer:

The basic structure of a System Dynamics model is illustrated in the following figure(1). It consists of number of reservoirs, levels, interconnected by flow paths. The rates of flow are controlled by decision functions that depend upon

conditions in the system. The levels represent the accumulation of various entities of the system, such as inventories of goods, unfilled orders, numbers of

employees, etc. The current value of a level at any time represents the accumulated difference between the input and output flow for that level. Rates are defined to represent the instantaneous flow to from a level. Decision functions or, as they are also called, rate equations determined how rates depend upon the levels. Sometimes an auxiliary variable needs to be defined, usually by combining other variables mathematically by some relationship other than the integration implied by a rate equation and a level.

5-9: Multi Segment Models

q Question: Draw the system dynamics diagram for the following mathematical model of a computer club: (’01)

H: Number of total members;

y: Number of members present;

x: Number of members using UNIX;

Solution:

5-10:: Representation of Time Delays::

Representation of time delays for system dynamics:

When there is a need to represent a time delay, the simplest method is to introduce an appropriate time constant. Suppose there is a level x of outstanding orders, and it is known that it takes an average of D time units to fill an order. A System Dynamics representation would say that the outstanding order level is being depleted with a time constant of D time units, and would include the following decay equation:

In general, to represent a delay, it is necessary to identify the level that is controlled by the delay and apply the rule

, this is sometimes called an exponential delay because of the solution it produces.

Problem: Dhaka computer center operates their services for some period of time. Students come here to learn programming languages. Initially, student arrival rate was exponential form. But presently it decreases due to some economical factors. Students at first learn PASCAL programming language and after that some of them learn C language and then they leave the center.

Solution:

X®Number of interested students who want to join the center.

x ®Number of students who arrived the center.

y ®Number of students who learned PASCAL programming language.

y¢®Number of students who are interested in C programming language.

z ®Number of students who learned C programming language.