NETWORK PROBLEM SET Due Date 6/28/2014 Problem 1 Consider a packet-switched network of N nodes connected by the following topologies: Star: one central node (hub) and all other nodes are attached to the hub. Loop: each node connects to two other nodes to form a closed loop. Fully connected: Each node if directly connected to all other nodes 1. For each case above, give the average number of hops between stations. 2. Prove that for any fixed set of node locations, the minimum possible total length of a tree interconnecting all nodes is less than the minimum possible length of any ring interconnecting all nodes. Problem 2 A parent on travel wishes to establish a conversation with his young son over a communication network. The parent was concerned that the communication will not be of good quality, if they are too far of each other. Assume that the speed of sound is approximately 330 meters/second. Also, assume that the network transmits the sound at the speed of light in cable, 2.3 x 10 8 meters/second. 1. Find the maximum possible distance between the parent and his child if they are to interact in real-time, in the sense of experiencing the same delay in hearing each other as if they were 10 meters apart. 2. Other than the distance, what other factors can impact the quality of the sound? Problem 3 1. Suppose the size of an uncompressed text file is 1 megabyte. a. How long does it take to download the file over a 32 kilobit/second modem? b. How long does it take to take to download the file over a 1 megabit/second modem? 2. Suppose data compression is applied to the 1 megabyte text file. Assuming a compression ratio of 1:6: a. How long does it take to download the compressed file over a 32 kilobits/second modem? b. How long does it take to take to download the compressed file over a 1 megabits/second modem?
3. A scanner has a resolution of 600 x 600 pixels/square inch. Assume that an 8-inch x 10- inch image is scanned and transmitted over the network: a. If scanning uses 8 bits/pixel, how long does it take to send the scanned image over a 32 kilobits/second modem? b. If scanning uses 24 bits/pixel, how long does it take to send the scanned image over a 1 megabits/second modem? Problem 4 Consider a communication network with the following parameters: N = Number of hops between two given end systems, L = Message length in bits, B = Data rate, in bits per second (bps), on all links, P = Packet size H = Overhead (header) bits per packet, S = Call setup time, and D = Propagation delay. 1. Assuming that the queuing delay and processing overhead are negligible, derive an expression the end to end delay for: a. A circuit switching network, and b. A packet switching network 2. Assuming that N=4, L=3200, B=9600, P=1024, H=1, S=0.2, and D=0.001 and ignoring the queuing delay and processing overhead, compute the end to end delay for: a. Circuit switching, and b. Packet switching network Problem 5 Consider a city with a population of 1 million. Assume that at any given time of the day, only 1% of the people in the city are using the phone. 1. If a voice call requires 64 Kbps, what is the total bit rate generated, any given time of the day? Assume now that the people use a high-quality videoconferencing terminal, which requires 1.4 Mbps per videoconferencing call. 2. What is the total bit rate generated by the videoconferencing terminals, at any given time of the day?
Problem 6 Suppose two friends want to communicate secretly using the following code: If the bit in the original message has value 1, then the bit is replaced with the secret code word 111; If the bit in the original message has value 0, then the bit is replaced with the secret code word 000; 1. The receiver takes the three received bits and decides which bit was sent by taking the majority vote of the three bits. The channel has a bit error rate p = 10-3. Assuming that bit errors occur at random and independently of each other, find the probability that the receiver makes a decoding error. Assume now that instead of using the majority vote rule, the sender adds two check bits to a group of 2n information bits. The first check bit is the parity check of the first n bits, and the second check bit is the parity check of the second n bits. 2. What error patterns can the receiver detect using the above code? 3. Assuming that the channel has a bit error rate p = 10-3, and that bit errors occur at random and independently of each other, find the probability that a failure remains undetected? Problem 7 1. Using the CRC-CCITT polynomial (x 16 + x 12 + x 5 + 1), generate the 16-bit CRC code for a message consisting of a 1 followed by 15 0s. 2. Let G(x) = x 5 + x 4 + x + 1 be the polynomial generator and M = 11100011. Find the CRC for this message. 3. Let G(x) = x 5 + x 4 + x 2 + 1 be the polynomial generator, T = 101000110101110 be the message received by the destination node. Would the receiver accept the message? Why? 4. In a CRC error-detecting scheme, choose G(x) = x 4 + x + 1. a. Encode the bits 10010011011. b. Suppose the channel introduces an error pattern 10001000000000 (i.e., the channel flips a bit from 0 to 1 or 1 to 0 in position 1 and 5). Show the message that is received. Can the error be detected? c. Repeat (b) with error pattern 10001100000000. Show the message received. Can the error bit be detected? Problem 8 Consider the use of 1000-bit frames on a 1-Mbps satellite channel with 270-ms delay. We define the throughput, T, of a channel as follows as the number of bits transmitted per second. We assume that the channel is error free and the processing time of frames and acknowledgments is negligible. What is the maximum throughput (bits per second) for the following flow control schemes? 1. Stop-and-wait flow control? 2. Sliding window flow control with a window size of 7?
3. Sliding window flow control with a window size of 127? 4. Sliding window flow control with a window size of 255? Problem 9 1. Given the two main reasons for why Ethernet must specify a minimum frame size First, it makes it easier to distinguish between valid frame and garbage. Secondly, if the frame is too short, the transmission will be finished before the collision signal returns (the worst case takes 2tao where "tao" is the propagation delay). That is, the frame has to be longer than 2tao. See details on page 277. Make sure you understand this). 2. Consider building a CSMA/CD network running at 1 Gbps over a 1-km cable with no repeaters. The signal speed in the cable is 200,000 km/sec. What is the minimum frame size? Problem 10 1. The IP header checksum only verifies the integrity of IP header. Discuss the pros and cons of doing the checksum on the header part versus on the entire packet. 2. Identify the range of IPv4 addresses spanned by Class A, Class B, and Class C. 3. What are all the possible subnet masks for the Class C address space? List all the subnet masks in dotted- decimal notation, and determine the number of hosts per subnet supported for each subnet mask. 4. Assuming a Classless address space, perform CIDR aggregation on the following /24 IP addresses: 128.56.24.0/24, 128.56.25.0/24 and 128.56.26.0/24. 5. Perform CIDR aggregation on the following /24 IP addresses: 200.96.86.0/24; 200.96.87.0/24; 200.96.88.0/24; 200.96.89.0/24. Problem 11 1. A small organization has a Class C address for seven networks each with 24 hosts. What is an appropriate subnet mask? 2. A university has one Class B address and 150 LANs with 100 hosts in each LAN. a. Design an appropriate subnet addressing scheme. b. Design an appropriate CIDR addressing scheme. Problem 12 1. Suppose an application layer entity wants to send an L-byte message to its peer process, using an existing TCP connection. The TCP segment consists of the message plus 20 bytes of header. The segment is encapsulated into an IP packet that has an additional 20 bytes of header. The IP packet in turn encapsulated inside an Ethernet frame that has 18 bytes of header and trailer. What percentage of the transmitted bits in the physical layer correspond to message information, if L = 1000 bytes?
2. Suppose that the TCP entity receives a 1.5 megabyte file from the application layer and that the IP layer is willing to carry blocks of maximum size 1500 bytes. Calculate the amount of overhead incurred from segmenting the file into packet-sized units. 3. Consider a TCP that implements an extension to allow for window sizes much larger than 64 KB. Suppose that the extended TCP is used over a 1-Gpbs link with a latency of 99 ms to transfer a 10-MB file. Assuming that (i) TCP receiver s window is 1 MB, (ii) TCP sends 1-KB packets, (iii) no congestion and (iv) no lost packets, how many RTTs does it take until slow start opens the send window to 1 MB?