Problem #7 NepalWireless.net - The following story is told by Dr. Mahabir Pun, t

Question

Problem #7 NepalWireless.net - The following story is told by Dr. Mahabir Pun, the Team Leader of Nepal Wireless Networking Project: The Beginning It started with a wish. In 1996. I wished to get Internet in my village for the first time after Himanchal High School got four used computers as presents from the students of a school in Australia. Internet and e-mail were quite new terms then. Turning Point I kept on asking people for ideas. I also wrote a very short e-mail to the BBC in 2001, asking if they knew anybody who could give me ideas (if there were any) to get a cheaper Internet connection to my remote village in Nepal. They took my interview and wrote the articles "Village in the Clouds Embraces Computers" and "Praise for 'Inspirational' Web Pioneer" about my school and the computers we had built in wooden boxes. That article changed everything: I got many responses with ideas from people all over the world. That was the first time I heard about Wi-fi (802.11b) wireless technology. The Network Now Starting in 2002, it has been already 14 years since Nepal Wireless is working in to bring broadband Internet in the rural areas. Our focus is to bring broadband Internet in the rural areas where no commercial Internet Service Providers would like to go. The ultimate goal is to maximize the benefit of the technology to the rural population. Nepal Wireless has built a network spanning 15 districts of Nepal. Altogether, 175 villages are now connected. The Investigation The Nepal Wireless network continually updates the technology while expanding connection to rural mountain villages. Of concern, is the network capacity at the Harmi Relay tower. On average, during peak times the Harmi Relay receives 55 megabytes of digital information per second with a standard deviation of 15 megabytes per second. The Harmi Relay can handle up to 85 megabytes of digital information per second before the system starts buffering. When the more than 97 megabytes are received in 1 second, the tower starts dropping data packets. Assume the data arrivals to the Harmi Relay are random and normally distributed. During peak times: a) What is the probability that the Harmi Relay will receive no more than 49 megabytes in one second? b) What is the probability that the Harmi Relay will receive between 22 and 67 megabytes in one second? c) In a one second period, what is the probability that the Harmi Relay will need to buffer information? d) In any one second, what is the probability that the Harmi Relay will start dropping packets?

Explanation / Answer

Solution:

Given that µ = 55 and = 15

a. The respective Z-score with X = 49 is

Z = (X - µ)/

Z = (49 – 55)/15

Z = -0.4

Using Z-tables, the probability is

P [Z -0.4] = 1 – P [Z > -0.4]

                    = 1 – 0.6554

                    = 0.3446

b. The respective Z-score with X = 22 is

Z = (X - µ)/

Z = (22 – 55)/15

Z = -2.2

The respective Z-score with X = 67 is

Z = (X - µ)/

Z = (67 – 55)/15

Z = 0.8

Using Z-tables, the probability is

P [-2.2 < Z < 0.8] = P [Z < 0.8] – P [Z < -2.2]

                              = 0.7881- 0.0139

                              = 0.7742

c. The respective Z-score with X = 85 is

Z = (X - µ)/

Z = (85 – 55)/15

Z = 2

Using Z-tables, the probability is

P [Z > 2] = 1 – P [Z 2]

               = 1 – 0.9773

               = 0.0227

d. The respective Z-score with X = 97 is

Z = (X - µ)/

Z = (97 – 55)/15

Z = 2.8

Using Z-tables, the probability is

P [Z > 2.8] = 1 – P [Z 2.8]

                  = 1 – 0.9974

                  = 0.0026

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