# Probability Review.

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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) PX Q(N-X) N! PX Q(N-X) X! (N-X)!

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

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!

Problems Solver Guide

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? Packet length: 64 bytes Transmission rate: 40 Gbps Current packet transmission: 50% Packets in queue: 480 packets Queuing delay: _________

Similar to Chapter 1 Problem 12
What is the queuing delay for the packet? Queing delay = ((L * (Q+1) – B*L) / (R) = 6.15 usec Packet length: L=64 bytes Transmission rate: R=40 Gbps Current packet transmission: B=50% Packets in queue: Q=480 packets Queuing delay: _________

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? Average packets in buffer: 360 packets Average queuing delay: 300 msec Packet transmission rate: 60 packets/sec Average packet arrival rate: _________

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 Average packets in buffer: N-1 = 360 packets Average queuing delay: Q=300 msec Packet transmission rate: R=60 packets/sec Average packet arrival rate: a=_________

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. Data size: 4 Tbytes Transmission rate: 2.4 Gbps Transmit time: _________

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 hrs to transmit Data size: 4 Tbytes Transmission rate: 2.4 Gbps Transmit time: _________

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. Distance between host: 10,000 km File Size: 6 Tbytes Propagation speed: 250,000,000 mps Link Rate: 512 kbps a. bandwidth delay product: __________ b. bits in the link: c. bandwidth delay product (define): d. Width of a bit: e. Width of a bit (formula):

Similar to Chapter 1 Problem 25
a. Calculate the bandwidth delay product: R*D/S = kbits b. Max bits in link: 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 = m e. Width of bit = S/R Distance between host: D=10,000 km File Size: F=6 Tbytes Propagation speed: S=250,000,000 mps Link Rate: R=512 kbps a. bandwidth delay product: __________ b. bits in the link: c. bandwidth delay product (define): d. Width of a bit: e. Width of a bit (formula):

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? Distance between host: 10,000 km Propagation speed: 250,000,000 mps R: __________

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 Distance between host: D=10,000 km Propagation speed: S=250,000,000 mps R: __________

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). Distance between host: 100 km File Size: 350 Gbytes Propagation speed: 250,000,000 mps Link Rate: 56 kbps Packet Size: 1.45 kbits (a) Transmit time continuous: ____________ (b) Transmit time segmented: (c) Compare:

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= 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 = months min c. Compare the results from (a) and (b). Distance between host: D=100 km File Size: F=350 Gbytes Propagation speed: S=250,000,000 mps Link Rate: R=56 kbps Packet Size: P=1.45 kbits (a) Transmit time continuous: ____________ (b) Transmit time segmented: (c) Compare:

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? Orbit: 36,000 km Propagation speed: 240,000,000 mps Link Rate: 10 Mbps (a) propagation delay: ____________ (b) bandwidth delay product: (c) size of the photo:

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 Orbit: D=36,000 km Propagation speed: S=240,000,000 mps Link Rate: R=10 Mbps (a) propagation delay: ____________ (b) bandwidth delay product: (c) size of the photo: