# Exponential Breakdown

Exponential Model on Growth of Internet Networks

1989- 1996

Internet Network: A global network connecting millions of computers. More than 100 countries are linked into exchanges of data, news and opinions. The Internet is revolutionizing and enhancing the way we as humans communicate, both locally and around the globe. Simply put, the Internet is a network of linked computers allowing participants to share information on those computers.

Source: Hobbes' Internet Timeline v8.2 (http://www.zakon.org/robert/internet/timeline/)

Reporting Period Covered: This report summarizes and describes data reported to Zakon Company from 1989-1996

The following data is related to growth of Internet Networks during 1989-1996

Time Networks

1 Jul-89 650

2 Oct-89 837

3 Oct-90 2063

4 Jan-91 2338

5 Jul-91 3086

6 Oct-91 3556

7 Jan-92 4526

8 Apr-92 5291

9 Jul-92 6569

10 Oct-92 7505

11 Jan-93 8258

12 Apr-93 9722

13 Jul-93 13767

14 Oct-93 16533

15 Jan-94 20539

16 Jul-94 25210

17 Oct-94 37022

18 Jan-95 39410

19 Jul-95 61538

20 Jan-96 93671

21 Jul-96 134365

The exponential regression of the data can be established by a scatter gram

Computer Networks is getting a wide range of applications in day to day life. Almost all business activities is somehow related to computer network. The present study tries to estimate the rate of change in the number of networks based on some secondary data.

Equation: A regression line of the form can be suggested to model this data. Here Y represents the number of the internet networks in the month X since 1989.

Thus the exponential model can be written as equation:

Y = 448.380 (1.063)

Prediction : The model Y = 448.380 ( 1.063) can be used to predict the future

value. The above model can be used to predict the future number of internet networks for

example, in the year 2008 (x = 19 ), the number of internet network will be estimated at

502609327.40 ( see below).

Networks = 448.380 ( 1.063 )

= 448.380 ( 1120945.02 )

= 502609327.40

Even though the model explained the variability in networks is 97.86 %., the model assumes that the internet networks will increase exponentially. This model may not be suitable for long term prediction as the growth pattern may drastically change due to advancement in network technology

The Base:

In the model Y = ab the parameter b or base b is known as the compound growth rate.

The base b = 1.063 which means that the number of internet networks growth is

increasing by 6.3 % per year.

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#### Solution Preview

For Y = ab^X, if we take ln on both sides, we get ln Y = ln a + X ln b

This model ...

#### Solution Summary

The occurrence of exponential break-down is investigated.

Poisson and exponential distributions

Please see attached file for full problem description.

The maintenance department of a factory claims that the number of breakdowns of a particular machine follows a Poisson distribution with a mean of two breakdowns every 500hours. Let x denote the time (in hours) between successive breakdowns.

a. Find lambda and mu(x)

b. Write the formula for the exponential probability curve of x.

c. Sketch the probability curve.

d. Assuming that the maintenance department's claim is true, find the probability that the time between successive breakdowns is at most five hours.

e. Assuming that the maintenance department's claim is true, find the probability that the time between successive breakdowns is between 100 hours and 300 hours

f. Suppose that the machine breaks down five hours after its most recent breakdown. Based on the answer to part d, do you believe the maintenance departments claim?

Thank you for your help.

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