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Probability

Probability. Review. Given. N = population X = sample size P = probability of success/event Q = 1-P, probability of failure/non event. Wanted. Probability of X successes in population of N where member event occurs with probability P. Solution. BiCoef (N,X) P X Q (N-X) N!

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Probability

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  1. Probability Review

  2. Given • N = population • X = sample size • P = probability of success/event • Q = 1-P, probability of failure/non event

  3. Wanted Probability of X successes in population of N where member event occurs with probability P

  4. Solution BiCoef(N,X) PX Q(N-X) N! --------------- PXQ(N-X) X! (N-X)!

  5. Example • What is probability of having 4 heads after flipping a coin 6 times? • N = 6 • X = 4 • P = ½ • Q = 1 – P 6! 30 ------- (.5)6 = ----- 0.0156 = 0.234 or 23.4% 4!2! 2

  6. Example • What is probability of having 7 heads after flipping a coin 7 times with a trick coin? • N = 6 • X = 4 • P = .9 • Q = 1 – P 7! ------- (.9)7 (.1)0 = (.9)7 = .478 7!0!

  7. Problems Solver Guide

  8. Similar to Chapter 1 Problem 12 A packet switch receives a packet and determines the outbound link to which the packet should be forwarded. When the packet arrives, one other packet is halfway done being transmitted on this outbound link and 480 other packets are waiting to be transmitted. Packets are transmitted in order of arrival.  Suppose all packets are 64 bytes and the link is 40 Gbps. What is the queuing delay for the packet?

  9. Similar to Chapter 1 Problem 12 What is the queuing delay for the packet? Queingdelay = ((L * (Q+1) – B*L) / (R)= 6.15 usec

  10. Similar to Chapter 1 Problem 16 Consider a router buffer preceding an outbound link. In this problem you will use Little's formula, a famous formula from queuing theory. Let N denote the average number of packets in the buffer plus the packet being transmitted. Let a denote the rate of packets arriving ". " at the link. Let d denote the average total delay (i.e., the queuing delay plus the transmission delay) experienced by a packet. Little's formula is N = a · d. Suppose that on average, the buffer contains 360 packets (in addition to the packet currently being transmitted) and the average queuing delay is 300 msec The links transmission rate is 60 packets/sec. Using Little's formula, what is the average packet arrival rate, assuming there is no packet loss?

  11. Similar to Chapter 1 Problem 16 Little's formula is N = a · d. Using Little's formula, what is the average packet arrival rate, assuming there is no packet loss? T=1/R a = N / (Q+T)= 1137 packets/sec

  12. Similar to Chapter 1 Problem 24 Suppose you would like to urgently deliver 4 Tbytes of data from Boston to Los Angeles. You have available a 2.4 Gbps dedicated link for data transfer. Would you prefer to transmit the data via this link or instead use FedEx overnight delivery? Explain.

  13. Similar to Chapter 1 Problem 24 Suppose you would like to urgently deliver 4 Tbytes of data from Boston to Los Angeles. You have available a 2.4 Gbps dedicated link for data transfer. Would you prefer to transmit the data via this link or instead use FedEx overnight delivery? Explain. 4E12*8bits / 2.4E9bps  3.704 hrs to transmit

  14. Similar to Chapter 1 Problem 25 Suppose two host, A and B, are separated by 10,000 km and are connected by a direct link of R = 512 kbps. Suppose the propagation speed over the link is 2.5·108 meters/sec.  a. Calculate the bandwidth delay product, R· dprop b. Consider sending a file of 48 Tbits from Host A to Host B. Suppose the file is sent continuously as one large message. What is the maximum number of bits that will be in the link at any given time? c. Provide an interpretation of the bandwidth delay product.  d. What is the width (in meters) of a bit in the link? e. Derive a general expression for the width of a bit in terms of the propagation speed, s, the transmission rate, R, and the length of the link m.

  15. Similar to Chapter 1 Problem 25 a. Calculate the bandwidth delay product: R*D/S = 20.48 kbits b. Max bits in link: 20.48 kbits c. bandwidth delay product is the (max) number of bits that could be in the link d. Width of bit: D/(R*D/S)= S/R = 488.281 m e. Width of bit = S/R

  16. Similar to Chapter 1 Problem 26 Suppose two host, A and B, are separated by 10,000 km and are connected by a direct link, R. Suppose the propagation speed over the link is 2.5·108 meters/sec. For what value of R is the width of a bit as long as the length of the link?

  17. Similar to Chapter 1 Problem 26 Suppose two host, A and B, are separated by 10,000 km and are connected by a direct link, R. Suppose the propagation speed over the link is 2.5·108 meters/sec. For what value of R is the width of a bit as long as the length of the link? R= (S / width_of_bit) = S/D = 25 bps

  18. Similar to Chapter 1 Problem 28 Suppose two host, A and B, are separated by 100 km and are connected by a direct link of R = 56 kbps and will send a file of 350 Gbytes. Suppose the propagation speed over the link is 2.5·108meters/sec.  a. How long does it take to send the file, assuming it is sent continuously? b. Suppose not the file is broken into packets of length 1.45 kbits. Suppose that each packet is acknowledged by the receiver and the transmission time of the acknowledgment is negligible. Finally assume that the sender cannot send a packet until the preceding one is acknowledged. How long does it take to send the file? c. Compare the results from (a) and (b).

  19. Similar to Chapter 1 Problem 28 a. How long does it take to send the file, assuming it is sent continuously? Transmission delay + Propagation Delay= F/R+D/S= 350E9bytes / 56kbps +100km/2.5E8mps= 19.29 months b. Suppose now the file is broken into packets of length 1.45 kbits. Suppose that each packet is acknowledged by the receiver and the transmission time of the acknowledgment is negligible. Finally assume that the sender cannot send a packet until the preceding one is acknowledged. How long does it take to send the file? NumPackets*2*Prop delay + Transmission delay (F/P)*2D/S+ F/R = 19.29 months + 25.7 min c. Compare the results from (a) and (b).

  20. Similar to Chapter 1 Problem 29 Suppose there is a 10 Mbps microwave link between a geostationary satellite and its base station on Earth. Every minute the satellite takes a digital photo and sends it to the base station. Assume a propagation speed of 2.4·108meters/sec. a. What is the propagation delay of the link? b. What is the bandwidth delay product, R·dprop? c. Let x denote the size of the photo. What is the minimum value of x for the microwave link to be continuously transmitting?

  21. Similar to Chapter 1 Problem 29 a. What is the propagation delay of the link? D/S= 150 msec b. What is the bandwidth delay product, R·dprop?R*D/S=1.5 Mbits c. Let x denote the size of the photo. What is the minimum value of x for the microwave link to be continuously transmitting? 1min*R = 600 Mbits or 75 Mbytes

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