Question

Consider a modification to TCP's congestion control algorithm. Instead of additive increase, we can use multiplicative increase. A TCP sender increases its window size by a small positive constant \(a(0

Short answer

The relationship is that the time to increase the window size from \(W/2\) to \(W\) is constant regardless of \(W\), determined by \(n = \frac{\log 2}{\log (1+a)}\).

Step-by-step solution

Understanding TCP Congestion Window

In the TCP protocol, the congestion window (cwnd) determines the number of packets that can be sent without receiving an acknowledgment (ACK). Traditionally, TCP uses an additive increase approach to manage this window, but in this exercise, we are using multiplicative increase with parameter \(a\).

Formulate the Congestion Window Increase

When an ACK is received, the window size is increased by multiplying the current window by a small constant \(a\). Thus, each ACK results in the window size becoming \((1+a)\) times its current size. So if the initial window size is \(W/2\), each successful ACK increases the window size multiplicatively.

Calculate Time for Windows Growth

Each ACK increases the window size by a factor of \((1+a)\). To grow the window from \(W/2\) to \(W\), we have the equation: \(\left(1+a\right)^n \frac{W}{2} = W\). Solving for \(n\), \(\left(1+a\right)^n = 2\). Thus, \(n = \frac{\log 2}{\log (1+a)}\), which is independent of \(W\).

Determine Functional Relationship with Loss Rate

The loss rate \(L\) is typically characterized in TCP by how often the network experiences packet loss. Under this modified algorithm, the time to increase from \(W/2\) to \(W\) is constant as derived, while normal TCP behavior depends on the congestion window size and loss events itself, which makes the functional relationship less straightforward than traditional methods.

Conclude Fixed Time to Reach Maximum Window Size

With the multiplicative increase method used here, regardless of the TCP's throughput or the current window \(W\), the time to increase from \(W/2\) to \(W\) is constant. Hence, the functional relationship focuses on supplying the independence of this growth from specific loss rate \(L\).

Key concepts

Multiplicative Increase

In TCP congestion control, the window size typically increases using an additive approach where the window grows by a fixed size with each acknowledgment (ACK) received. However, in this modified approach, we use a **multiplicative increase**. Instead of adding, the current congestion window is multiplied by a constant factor, denoted as \( a \), where \( 0 < a < 1 \).

This means each valid ACK results in the window becoming \( (1 + a) \) times its previous size. Suppose the congestion window begins at a size of \( W/2 \); after the first valid ACK, it becomes \( (1 + a) \times \frac{W}{2} \). With each successive ACK, this process continues to increase the window size at an exponential rate.

The fundamental difference between multiplicative and traditional additive increase is that multiplicative increase can potentially grow the congestion window much faster, which is beneficial when the network conditions can accommodate larger numbers of packets being sent.

Packet Loss Rate

The **packet loss rate** (\( L \)) in TCP congestion control represents how frequently packets are lost during transmission over a network.

In conventional TCP, packet loss prompts a reduction in the congestion window size to prevent congestion and potential network overload. In this modified congestion mechanism using multiplicative increase, we understand that as the congestion window increases, more packets are sent, possibly leading to higher packet losses.

Nevertheless, in this scenario, the time taken to expand the congestion window from \( W/2 \) to \( W \) remains constant, irrespective of \( L \). This is due to the steady exponential growth of the window, calculated using \( (1+a)^n = 2 \), which doesn't depend on packet losses directly. Consequently, even when the packet loss rate impacts overall throughput, it does not affect the time taken for window expansion within this control paradigm.

Throughput Analysis

**Throughput** in a TCP context refers to the rate at which data is successfully received at the destination. With our modified congestion control using multiplicative increase, the ability of TCP to manage throughput effectively is key.

In conventional TCP, throughput is often limited by packet loss rates, the congestion control mechanism's responsiveness, and network capacity. With the multiplicative increase method, the inherent exponential growth aids in potentially increasing throughput due to faster expansion of the congestion window. However, this is contingent on the network's ability to handle the surge in data flow without additional packet loss.

Despite concerns about packet losses, the constant time to increase the congestion window suggests that throughput can be optimized over constant intervals.

• This design assumes that the window can be doubled rapidly, potentially improving data transfer rates.
• It relies on network stability; sudden increases in throughput could otherwise lead to added congestion and packet drops.

Congestion Window

The **congestion window** (\( cwnd \)) is a cornerstone of TCP's ability to manage data flow and network congestion. It dictates the number of packets that can be sent before awaiting an acknowledgment. In traditional TCP, the congestion window increases linearly under an additive approach, receiving a steady increment upon each successful ACK.

Here, with a multiplicative increase, the congestion window grows more rapidly, potentially leading to substantial increases in the amount of outstanding data. This growth pattern occurs because the window size is multiplied by \( (1 + a) \) upon every valid ACK.

One significant insight from our exercise is that the time for the congestion window to grow from \( W/2 \) to \( W \) remains fixed, as deduced by the equation \( (1+a)^n = 2 \). Thus, irrespective of initial value \( W \), this time threshold is consistent, allowing for predictable expansion dynamics in network environments.

• It implies a structured and predictable increase in the congestion window size.
• Ensures a steady flow adapted to network capacity, assuming minimal unexpected loss occurrences.