Question

Recall the macroscopic description of TCP throughput. In the period of time from when the connection's rate varies from \(W /(2 \cdot R T T)\) to \(W / R T T\), only one packet is lost (at the very end of the period). a. Show that the loss rate (fraction of packets lost) is equal to $$ L=\text { loss rate }=\frac{1}{\frac{3}{8} W^{2}+\frac{3}{4} W} $$ b. Use the result above to show that if a connection has loss rate \(L\), then its average rate is approximately given by $$ =\frac{1.22 \cdot M S S}{R T T \sqrt{L}} $$

Short answer

The loss rate is \( L = \frac{1}{\frac{3}{8} W^2 + \frac{3}{4} W} \), leading to an average rate of \( \frac{1.22 \times MSS}{RTT \times \sqrt{L}} \).

Step-by-step solution

Understand TCP Throughput Formula

The exercise involves TCP throughput and relates to network performance. TCP adjusts its window size, growing until a packet is lost. The throughput can be approximated by considering time until packet loss and packet loss rate. Throughput is modeled as a function of window size \( W \).

Analyze Period Dynamics

The given period is when TCP's window size \( W \) varies from \( \frac{W}{2 \cdot RTT} \) to \( \frac{W}{RTT} \). During this time, one packet is lost, indicating a cycle of TCP's congestion window dynamics (additive increase and loss).

Calculate Number of Packets Sent

During the period from \( \frac{W}{2 \cdot RTT} \) to \( \frac{W}{RTT} \), the number of packets sent can be calculated using arithmetic series formula since the window increases by one packet per round trip time. This total can be approximated as \( \frac{3}{8} W^2 + \frac{3}{4} W \).

Derive Loss Rate (Part a)

The loss rate \( L \) is the fraction of packets lost: one packet is lost and the total number of packets sent is \( \frac{3}{8} W^2 + \frac{3}{4} W \). Therefore, \( L = \frac{1}{\frac{3}{8} W^2 + \frac{3}{4} W} \).

Use Loss Rate for Average Rate (Part b)

Once \( L \) is expressed in terms of \( W \), use it to find the average rate. The TCP throughput, approximated by \( \frac{1.22 \times MSS}{RTT \times \sqrt{L}} \), links loss rate with average rate.

Key concepts

Loss Rate

The concept of "Loss Rate" is pivotal in understanding TCP throughput. Essentially, loss rate is the fraction of packets lost during a given transmission period. In the context of TCP, as packets traverse the network, some might not reach their destination. This can be due to various factors, such as network congestion or errors on the line.
In this exercise, we find that the loss rate, \( L \), for a TCP connection can be derived from the size of the congestion window, \( W \). When the window varies from \( \frac{W}{2 \cdot RTT} \) to \( \frac{W}{RTT} \), the total number of packets sent can be represented mathematically as \( \frac{3}{8} W^2 + \frac{3}{4} W \). During this time, only one packet is lost at the end of the cycle. Hence, the actual loss rate is determined by this formula:

• \( L = \frac{1}{\frac{3}{8} W^2 + \frac{3}{4} W} \)

This equation shows that the loss rate is inversely proportional to the square of the congestion window size and the additional terms. Therefore, a large window size implies a low loss rate, which typically leads to better throughput in TCP communication.

Congestion Window

The congestion window, often referred to as "cwnd", is a critical factor in TCP transmission. It determines the number of packets that can be sent before receiving an acknowledgment from the receiver. This helps to manage flow control and prevent network congestion.
The dynamics of the congestion window in TCP follow a cycle of additive increase and multiplicative decrease. During periods of stability, the window size incrementally increases until it reaches a threshold where packet loss occurs. This loss signals congestion, prompting TCP to reduce the window size.
In the given exercise, the congestion window varies from \( \frac{W}{2 \cdot RTT} \) to \( \frac{W}{RTT} \), indicating phases of gradual growth. Once a packet is lost, the congestion control mechanism dramatically decreases the window size. This repeating cycle plays a crucial role in TCP's ability to rapidly adjust to changing network conditions and maintain an efficient throughput.

Packet Loss

Packet loss occurs when transmitted data packets fail to reach their intended destination. In the context of TCP, packet loss is an essential signal used by the protocol to gauge the state of the network, particularly regarding congestion.
A single lost packet in a cycle of congestion window growth can lead to a multiplicative decrease, which reduces the window size to alleviate potential congestion in the network. This is TCP’s inherent strategy to control congestion and ensure data flows smoothly.
It's important to understand that packet loss affects both the current and future data flow, with implications for throughput, latency, and any potential retransmissions. The formula derived in this exercise highlights the impact of packet loss on calculated throughput, emphasizing that fewer packets lost generally results in more stable and higher throughput.

Round Trip Time (RTT)

Round Trip Time (RTT) is a fundamental network metric that represents the time taken for a signal to travel from the source to the destination and back again. RTT is vital for TCP as it's used to estimate how long acknowledgments should be expected and to adjust the sending rate accordingly.
TCP relies heavily on RTT to determine optimal sending times; shorter RTTs mean faster acknowledgments and potentially higher throughput, while longer RTTs suggest latency, requiring adjustments in timing or window size to maintain efficiency.

• In this exercise, the range between \( \frac{W}{2 \cdot RTT} \) and \( \frac{W}{RTT} \) illustrates the period during which the congestion window increases before experiencing packet loss.

RTT influences TCP's decisions, ensuring data is sent at a pace that balances speed and network reliability, ultimately impacting overall throughput.