Problem Set

Nyquist theorem. A noiseless channel has bandwidth 8000 Hz. What is the maximum data rate with 16 signal levels? With 256 signal levels?
Shannon capacity. A channel has bandwidth 4 MHz and SNR = 24 dB. Compute the maximum error-free data rate. If 64 signal levels are used with a Nyquist-based scheme, is the channel being used within its theoretical limit?
Nyquist vs Shannon. A channel has $H = 3000$ Hz and $\mathrm{SNR} = 31$ (about 15 dB). What is the maximum number of signal levels $V$ that can be used reliably?
Hamming code. Encode the data bits $d_1 d_2 d_3 d_4 = 0110$ using Hamming(7,4). If bit 5 of the transmitted codeword is flipped, show how the receiver detects and corrects the error.
CRC computation. Compute the CRC for the message 10110010 using the generator polynomial $G(x) = x^3 + x + 1$ (binary 1011). Write the transmitted codeword.
CRC verification. The receiver gets the codeword 101101110 and the generator is 1011. Perform modulo-2 division to determine whether the frame was received correctly.
ALOHA throughput. A slotted ALOHA system has 4 stations, each transmitting with probability $p = 0.2$ in each slot. Compute the throughput $S$ and compare it to the theoretical maximum.
CSMA/CD minimum frame. A 1 Gbps Ethernet has a maximum segment length of 100 m and propagation speed $2 \times 10^8$ m/s. What is the minimum frame size? How does it compare to the standard Ethernet minimum of 64 bytes?
Switching latency. A 1000-byte frame passes through 5 store-and-forward switches on 100 Mbps links (ignore propagation delay). Compute the total latency. Repeat for cut-through switching.
Subnetting. Given the network 172.16.0.0/16Create subnets to support 100 hosts, 50 hosts, 25 hosts, and 10 hosts using VLSM. List the network address, usable range, and broadcast for each.
Route aggregation. Aggregate the following routes into the most specific supernet: 198.51.100.0/24``198.51.101.0/24``198.51.102.0/24``198.51.103.0/24.
IPv6 addressing. Expand the IPv6 address 2001:db8::1 to its full 128-bit form. How many /64 subnets does a /56 prefix provide? How many /128 addresses per /64?
Dijkstra”s algorithm. Given the network topology below, find the shortest path tree from router S to all destinations:
```
S --2-- A --1-- B
|         |         |
5         3         2
|         |         |
C --4-- D --1-- E
```
Construct the routing table at S.
Distance vector convergence. Three routers X, Y, Z are connected: X—Y (cost 1), Y—Z (cost 1), X—Z (cost 5). Initially, X’s route to Z goes through Y. If link Y—Z fails, trace the count-to-infinity problem for 4 iterations. Explain how split horizon with poisoned reverse prevents it.
TCP congestion control. Given MSS = 1000 bytes, initial ssthresh = 8000 bytes. Trace cwnd through: slow start for 3 RTTs, then 2 RTTs of congestion avoidance, then a timeout. What is the value of ssthresh after the timeout?
RTT estimation. Using $\alpha = 1/8$ , $\beta = 1/4$ And measured RTTs of 100 ms, 120 ms, 80 ms, compute $\mathrm{RTT_s}$ , $\mathrm{RTT_d}$ And RTO after each measurement (starting from $\mathrm{RTT_s} = \mathrm{RTT_d} = 0$ ).
DNS resolution. A client at 192.168.1.100 wants to resolve www.example.com. Describe the complete resolution process, including: the recursive query to the local resolver, the iterative queries to root and TLD servers, and the role of caching. How many round trips are needed on a cold cache?
HTTP performance. A web page contains 1 HTML document (10 KB), 5 CSS files (20 KB total), 10 images (500 KB total), and 2 JavaScript files (100 KB total). Compare the total time to load the page over HTTP/1.0 (6 parallel TCP connections), HTTP/1.1 (1 persistent connection, no pipelining), and HTTP/2 (1 connection with multiplexing). Assume RTT = 50 ms and bandwidth = 10 Mbps. Ignore processing time.

Hint: Total data = 630 KB = 5.04 Mb. Transmission time = 5.04 / 10 = 0.504 s.

HTTP/1.0: 16 objects, 6 connections. Each connection requires 1 RTT for TCP handshake + 1 RTT for HTTP request/response. With 6 parallel connections: 3 round trips for the TCP + HTTP per batch = 150 ms per batch, with 3 batches (6 + 6 + 4 objects) = 3 $\times$ 150 + 504 = 954 ms.
HTTP/1.1: 1 connection, sequential requests. 1 RTT for handshake + 16 RTTs for requests (serial) + 504 ms = 1 $\times$ 50 + 16 $\times$ 50 + 504 = 1354 ms.
HTTP/2: 1 RTT for handshake, all requests multiplexed. 1 $\times$ 50 + 504 = 554 ms.

Firewall rules. A company has a web server at 203.0.113.10A mail server at 203.0.113.20And an internal network 10.0.0.0/24. Write a set of packet filtering rules that: (a) allows external HTTP/HTTPS to the web server, (b) allows external SMTP to the mail server, (c) allows internal users to access any external service, (d) blocks all other inbound traffic.
RSA encryption. Given primes $p = 5$ , $q = 11$ And public exponent $e = 3$ : (a) Compute $n$ , $\phi(n)$ And the private key $d$ . (b) Encrypt the message $m = 7$ . (c) Decrypt the ciphertext to verify.
TCP throughput bound. A TCP connection over a satellite link has RTT = 600 ms and bandwidth = 50 Mbps. The receiver advertises rwnd = 1 MB. If cwnd grows to 2 MB during slow start, what is the maximum achievable throughput? What is the BDP, and is the window large enough to fill the pipe?
CDMA orthogonality. Four stations share a channel using CDMA with chip codes: $C_1 = (+1, -1, +1, +1)$ , $C_2 = (+1, +1, -1, +1)$ , $C_3 = (+1, +1, +1, -1)$ $C_4 = (-1, +1, +1, +1)$ . Station 1 sends bit 1, station 2 sends bit 0, station 3 sends bit 1, station 4 is silent. Compute the combined signal and show that each receiver correctly recovers its station’s bit.

Common Pitfalls

Confusing throughput and latency. Latency: time for a single packet to travel. Throughput: rate of data delivery. Fix: Total time = latency + (file size / throughput).
Wrong TCP flow control vs congestion control. Flow control: receiver-side (window to prevent buffer overflow). Congestion control: sender-side (avoid network congestion). Fix: Flow control: receiver advertises window. Congestion control: slow start, congestion avoidance, fast retransmit.
Confusing routing and forwarding. Routing: building the routing table (network-layer process). Forwarding: looking up the next hop (per-packet). Fix: Routing algorithms: distance vector, link state. Forwarding: match destination IP to routing table entry.

Worked Examples

Example 1: Subnetting

Problem. An organisation has IP address 192.168.1.0/24. It needs 8 subnets. Design the subnetting.

Solution. Borrow 3 bits ( $2^3 = 8$ subnets). New mask: /27 (255.255.255.224). Subnets: 192.168.1.0/27, 192.168.1.32/27, …, 192.168.1.224/27. Each subnet has 30 usable hosts.

$\blacksquare$

Example 2: TCP handshake

Problem. Describe the TCP three-way handshake.

Solution. Client sends SYN (seq = x). Server responds SYN-ACK (seq = y, ack = x + 1). Client sends ACK (ack = y + 1). Connection established.

$\blacksquare$

Summary

OSI and TCP/IP models; each layer has specific functions and protocols.
TCP: reliable, connection-oriented; flow control (sliding window), congestion control (slow start, AIMD).
IP addressing and subnetting: CIDR notation, variable-length subnet masking.
Routing: distance vector (RIP), link state (OSPF), path vector (BGP).

Cross-References

Topic	Site	Link
[Networks]	A-Level	View
[Networks]	IB	View
[Networks]	DSE	View
[Networks]	University	View

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