7.3 TCP Congestion Control: AIMD, Slow Start & Thresholds

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📚Module 07: Transport Layer: Congestion Control & ReliabilityTopic 7.3⏱️16 min read

🎯High-Yield For:Semester Exams (All Universities) • Placement Technical Rounds • Engineering Foundations

💡 Core Intuition: The Blind Explorer in an Uncharted Tunnel

🍳 The Everyday Analogy: Navigating in Complete Darkness

Imagine entering a pitch-black cave tunnel whose ceiling height you do not know:

1. Slow Start (Exponential Acceleration): At the cave entrance, you do not crawl forward at 1 centimeter an hour. You start by taking 1 confident stride. If you do not hit your head, you take 2 strides. If clear, you sprint 4 strides, then 8 strides. You double your pace every second because you know you are far below the tunnel ceiling.
2. Threshold Reached (Congestion Avoidance): Once you reach a marked checkpoint where you previously bumped your head (the Slow Start Threshold), sprinting blindly is dangerous. You immediately switch modes: you take exactly one careful additional stride per step ($+1$).
3. Bumping Your Head (Multiplicative Decrease): Suddenly, your forehead hits a low-hanging stalactite! You do not gently slow down by 1 stride; you violently duck down, drop your pace by half (or drop down to 1 step), establish that the new danger height is half of your current speed, and begin cautiously inching forward again.

💻 Bridging to Computer Science

In TCP:

• The sender has zero advance knowledge of the intermediate routers' bottleneck bandwidth or buffer capacity.
• Slow Start quickly ramps up transmission from $1\text{ MSS}$ to probe the link's capacity, doubling the Congestion Window ($\text{cwnd}$) every Round-Trip Time ($RTT$).
• When $\text{cwnd}$ crosses $\text{ssthresh}$ (Slow Start Threshold), TCP switches to Congestion Avoidance (Additive Increase), incrementing $\text{cwnd}$ by only $1\text{ MSS}$ per $RTT$.
• When packet loss occurs, TCP slashes its transmission window (Multiplicative Decrease), producing the famous AIMD Sawtooth Waveform.

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📑Table of Contents

• 🪟 Dual-Window Governance: The Effective Sender Window
• 🚀 Phase 1: Slow Start (Exponential Acceleration)
  • The Algorithm: Slow Start Mechanics
• 🛡️ Phase 2: Congestion Avoidance (Additive Increase)
  • The Algorithm: Congestion Avoidance Mechanics
• 📉 Phase 3: Multiplicative Decrease & The Sawtooth Waveform
  • The Multiplicative Decrease Rule
• ⚖️ Mathematical Proof of Fairness: The Chiu-Jain Convergence Plane
  • The Chiu-Jain Convergence Geometry
  • Vector Trajectories of Rate Adjustment Policies
  • The Proof of Convergence
• 🏭 Real-World Production Context: Modern Linux TCP & Bufferbloat
  • 1. Initial Congestion Window Evolution (IW10)
  • 2. The Bufferbloat Crisis & BBR
• 📐 Numerical Applications & Worked Examples
  • Example 1: Full TCP Tahoe Recovery Trace
  • Example 2: cwnd State Across Successive Transmissions
  • Example 3: Effective Transmit Window with Receiver Limitation
• 🎯 Exam & Interview Pitfall Check

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🪟 Dual-Window Governance: The Effective Sender Window

At any instant, a TCP sender is constrained by two entirely independent physical boundaries:

1. Flow Control Boundary ($\text{rwnd}$): Dictated by the destination receiver to prevent socket buffer overflow.
2. Congestion Control Boundary ($\text{cwnd}$): Maintained internally by the sender to prevent intermediate router queue exhaustion.

Constraint Dimension	Flow Control Boundary ($\text{rwnd}$)	Congestion Control Boundary ($\text{cwnd}$)
Origin & Feedback	Communicated dynamically by the receiving host in TCP header ACK packets	Estimated and maintained locally by the sending host's OS kernel state machine
Target Protected	Protects the destination host's application socket receive buffer	Protects intermediate network routers, switch queues, and physical links
Constraint Dynamic	Static physical cap based on allocated user-space buffer memory	Dynamic elastic boundary probing transit capacity via loss signals
Governing Policy	Sliding Window Protocol (RFC 793, RFC 1323)	AIMD, Slow Start, Congestion Avoidance (RFC 5681)

The maximum volume of unacknowledged data the sender is permitted to inject into the transit path is strictly governed by the Effective Sender Window:

$\mathbf{\text{Send Window} = \min\big(\text{rwnd}, \; \text{cwnd}\big)}$

• If $\text{rwnd} < \text{cwnd}$: The bottleneck is the Receiver's CPU/RAM.
• If $\text{cwnd} < \text{rwnd}$: The bottleneck is the Intermediate Network Fabric.

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🚀 Phase 1: Slow Start (Exponential Acceleration)

When a TCP connection begins, the sender does not know whether the path can support $10\text{ Kbps}$ or $10\text{ Gbps}$. Injecting a giant burst of packets immediately could cause instant buffer overflow.

Slow Start vs Congestion Avoidance: Two Modes of cwnd Growth

Contrasting rapid exponential bandwidth discovery against cautious linear capacity probing

Discovery Phase

Slow Start Phase (Exponential)

🚀

Dominant Architecture / Domaincwnd < ssthresh

• •Initial Window: Starts at 1 MSS (or modern IW10).
• •Per-ACK Rule: cwnd increases by +1 MSS for every arriving ACK.
• •Per-RTT Rule: cwnd doubles every round trip: cwnd(k) = 2^k MSS.
• •Rapidly discovers order of magnitude of link capacity.

"Double window every RTT until hitting the slow start threshold."

Sustained Probing

Congestion Avoidance Phase (Additive)

🛡️

Dominant Architecture / Domaincwnd >= ssthresh

• •Triggered when cwnd reaches or exceeds ssthresh.
• •Per-ACK Rule: cwnd increases by (1 / cwnd) MSS for each arriving ACK.
• •Per-RTT Rule: cwnd increases by exactly +1 MSS per round trip time.
• •Cautiously probes for incremental spare bandwidth without triggering loss.

"Inching forward by 1 MSS per RTT to respect link capacity limits."

The Algorithm: Slow Start Mechanics

1. Initialization: Set $\text{cwnd} = 1\text{ MSS}$ (historically $1\text{ MSS}$, modern Linux defaults to $10\text{ MSS}$).
2. Per-ACK Increment Rule: For every individual incoming valid acknowledgment:

$\text{cwnd} \leftarrow \text{cwnd} + 1\text{ MSS}$

3. Per-RTT Growth Dynamic:
   • Round 0: Sender transmits $1\text{ MSS}$.
   • Receiver sends 1 ACK $\implies \text{cwnd}$ becomes $1 + 1 = 2\text{ MSS}$.
   • Round 1: Sender transmits $2\text{ MSS}$.
   • Receiver sends 2 ACKs $\implies \text{cwnd}$ becomes $2 + 1 + 1 = 4\text{ MSS}$.
   • Round 2: Sender transmits $4\text{ MSS}$.
   • Receiver sends 4 ACKs $\implies \text{cwnd}$ becomes $4 + 4 = 8\text{ MSS}$.

After $k$ round-trip times ($RTT$s) with no packet loss:

$\mathbf{\text{cwnd}(k) = 2^k \text{ MSS}}$

Despite its name ("Slow Start"), the growth is actually exponential! It is called "slow" only in comparison to naive blasting of maximum window size at connection startup.

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🛡️ Phase 2: Congestion Avoidance (Additive Increase)

Exponential doubling cannot continue indefinitely without overflowing router buffers. TCP introduces a governor called the Slow Start Threshold ($\text{ssthresh}$) (initially set to a large value, typically $64\text{ KB}$ or the maximum receiver window).

Architecture Flow

The TCP Congestion State Transition Lifecycle

How the transport layer navigates exponential growth, linear probing, and multiplicative recovery

🚀Startup

Slow Start

Exponential growth: cwnd doubles every RTT until ssthresh.

→

cwnd >= ssthresh

📈Steady State

Congestion Avoidance

Additive increase: cwnd increases by +1 MSS per RTT.

→

Packet Loss Event

💥Detection

Multiplicative Decrease

Halve ssthresh: ssthresh = max(2, cwnd / 2).

→

Timeout Recovery

🔄Reset

Re-Enter Slow Start

Set cwnd = 1 MSS (Tahoe) and ramp back up to new threshold.

The Algorithm: Congestion Avoidance Mechanics

Once $\text{cwnd} \ge \text{ssthresh}$, TCP switches to Congestion Avoidance:

1. Per-ACK Increment Rule: For every incoming valid acknowledgment:

$\text{cwnd} \leftarrow \text{cwnd} + \frac{1}{\text{cwnd}} \text{ MSS}$

2. Per-RTT Growth Dynamic:
   • During one $RTT$, the sender dispatches $\text{cwnd}$ segments.
   • As all $\text{cwnd}$ segments are acknowledged: $\Delta \text{cwnd} = \text{cwnd} \times \left(\frac{1}{\text{cwnd}}\right) = 1\text{ MSS}$
   • The window increases by exactly $1\text{ MSS}$ per $RTT$:

$\mathbf{\text{cwnd}(k + 1) = \text{cwnd}(k) + 1\text{ MSS}}$

This linear, cautious pacing allows TCP to steadily probe for newly available bandwidth without shocking the network.

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📉 Phase 3: Multiplicative Decrease & The Sawtooth Waveform

When an intermediate router drops a packet due to buffer exhaustion, the sender detects packet loss.

The Multiplicative Decrease Rule

Regardless of whether loss is detected via Timeout or 3 Duplicate ACKs, the Slow Start Threshold is immediately cut in half:

$\mathbf{\text{ssthresh} \leftarrow \max\left(2\text{ MSS}, \; \left\lfloor \frac{\text{cwnd}}{2} \right\rfloor\right)}$

In classic TCP Tahoe:

• After setting $\text{ssthresh} = \text{cwnd} / 2$, the sender drops $\text{cwnd}$ back to $1\text{ MSS}$: $\text{cwnd} \leftarrow 1\text{ MSS}$
• The sender re-enters Slow Start, exponentially doubling $\text{cwnd}$ up to the new $\text{ssthresh}$, and then switches to linear Congestion Avoidance.

Phase / State	Operational Condition	Rate of Change	Algorithmic Behavior & Waveform Trajectory
Phase 1: Slow Start	$\text{cwnd} < \text{ssthresh}$	Exponential: doubles per $RTT$	Rapidly escalates window ($1 \to 2 \to 4 \to 8\text{ MSS}$) to probe pipe capacity. Steep exponential rising edge.
Phase 2: Congestion Avoidance	$\text{cwnd} \ge \text{ssthresh}$	Additive: $+1\text{ MSS}$ per $RTT$	Linear upward slope ($8 \to 9 \to 10 \dots 16\text{ MSS}$). Cautiously probes spare bandwidth without queue collapse.
Congestion / Loss Event	Packet loss detected (e.g. at $16\text{ MSS}$)	Instantaneous threshold cut	Buffer overflow occurs. Threshold halved: $\text{ssthresh} \leftarrow \max(2, \lfloor 16 / 2 \rfloor) = 8\text{ MSS}$.
Tahoe Collapse	Post-loss recovery	Floor reset: $\text{cwnd} \leftarrow 1\text{ MSS}$	Immediate vertical plunge down to $1\text{ MSS}$. Re-enters Slow Start up to $8\text{ MSS}$, repeating the cycle.

This cycle of linear ramp-up followed by sudden downward cuts produces the famous AIMD Sawtooth Waveform.

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⚖️ Mathematical Proof of Fairness: The Chiu-Jain Convergence Plane

Why does TCP use Additive Increase Multiplicative Decrease (AIMD) rather than other combinations like Additive Increase Additive Decrease (AIAD) or Multiplicative Increase Multiplicative Decrease (MIMD)?

In 1989, Dah-Ming Chiu and Raj Jain published a seminal mathematical paper proving that only AIMD satisfies both optimal efficiency and distributive fairness among competing flows sharing a bottleneck link.

Geometric Axes of the Chiu-Jain Phase Plane

• Flow State Vector: $\mathbf{x} = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$, where $x_1$ and $x_2$ represent the transmission rates of User 1 and User 2.
• The Efficiency Boundary: The line $x_1 + x_2 = C$ with slope $-1$. Any operating point $\mathbf{x}$ below this line underutilizes the physical bottleneck; points above it overload router queues and induce loss.
• The Fairness Boundary: The ray $x_1 = x_2$ radiating from the origin at $45^\circ$. All points along this line represent equal bandwidth allocation.
• The Optimal Operating Point: The intersection of the Efficiency Line and Fairness Line: $\mathbf{x}^* = \left(\frac{C}{2}, \; \frac{C}{2}\right)$.

Allocation Scheme	Additive / Multiplicative Mechanics	Phase Plane Vector Behavior	Fairness Convergence Proof
AIAD (Additive Inc / Additive Dec)	Increase: $\mathbf{x} + \alpha\mathbf{1}$, Decrease: $\mathbf{x} - \beta\mathbf{1}$	Moves parallel along a $45^\circ$ line back and forth	❌ Does Not Converge: Trapped in a limit cycle; difference $(x_1 - x_2)$ never decreases.
MIMD (Multiplicative Inc / Multiplicative Dec)	Increase: $\alpha\mathbf{x}$, Decrease: $\beta\mathbf{x}$	Moves along rays passing through the origin	❌ Does Not Converge: Ratio $x_1 / x_2$ remains fixed; unfair allocations remain permanently unfair.
AIMD (Additive Inc / Multiplicative Dec)	Increase: $\mathbf{x} + \alpha\mathbf{1}$, Decrease: $\beta\mathbf{x}$	Additive steps shift trajectory at $45^\circ$; multiplicative cuts pull toward origin	✅ Provably Converges: Repeated AIMD cycles pull the operating state asymptotically toward $(C/2, C/2)$.

The Chiu-Jain Convergence Geometry

Consider two flows with throughputs $x_1$ and $x_2$ sharing a bottleneck pipe of capacity $C$:

• Efficiency Line: $x_1 + x_2 = C$. (Points below the line underutilize the link; points above cause packet loss).
• Fairness Line: $x_1 = x_2$ (a $45^\circ$ line passing through the origin).
• The Goal: Drive any arbitrary starting point $(x_1, x_2)$ toward the Optimal Intersection Point $(C/2, C/2)$.

Vector Trajectories of Rate Adjustment Policies

1. Additive Increase ($+1, +1$):
   • Both users increase throughput by the identical constant amount: $(x_1 + \alpha, \; x_2 + \alpha)$.
   • The vector moves at a $45^\circ$ angle parallel to the Fairness Line.
   • It moves closer to the efficiency line without changing the difference $(x_1 - x_2)$.
2. Multiplicative Decrease ($\times 0.5, \times 0.5$):
   • Both users reduce throughput proportionally: $(\beta x_1, \; \beta x_2)$ where $\beta = 0.5$.
   • The vector moves directly toward the origin $(0, 0)$.
   • Because the line to the origin makes a steeper angle for the user with higher throughput, the absolute gap $|x_1 - x_2|$ shrinks by $50\%$!

The Proof of Convergence

• Additive Increase pushes the state toward the efficiency line parallel to the fairness axis.
• Multiplicative Decrease pulls the state toward the origin, halving the disparity between the two flows.
• Together, each AIMD cycle acts as a spiral convergence that inexorably drives competing flows to the intersection of fairness and efficiency:

$\lim_{k \to \infty} \big|x_1(k) - x_2(k)\big| = 0 \quad \text{and} \quad \lim_{k \to \infty} \big(x_1(k) + x_2(k)\big) = C$

Under MIMD or AIAD, the trajectory forms closed loops or diverges, failing to provide distributed fairness.

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🏭 Real-World Production Context: Modern Linux TCP & Bufferbloat

1. Initial Congestion Window Evolution (IW10)

In RFC 2581 (1999), TCP's initial window was strictly $1\text{ to }2\text{ MSS}$ ($1.4\text{ to }2.8\text{ KB}$).

• In 2013, Google published measurements demonstrating that slow-start latency dominated web page retrieval times.
• RFC 6928 updated the standard, setting $\text{IW} = 10\text{ MSS}$ ($\approx 14.6\text{ KB}$).
• This enables typical web requests (HTTP headers, small CSS/JS files) to complete in a single $RTT$, eliminating multiple round-trip slow-start stalls.

2. The Bufferbloat Crisis & BBR

Traditional AIMD assumes that packet loss is the only signal of congestion.

• Modern router vendors installed massive multi-megabyte DRAM buffers to prevent packet drops.
• Instead of dropping packets, routers buffer thousands of segments, inflating $RTT$s from $20\text{ ms}$ to $2,000\text{ ms}$—a pathology known as Bufferbloat.
• In response, Google developed BBR (Bottleneck Bandwidth and RTT): a model-based congestion control algorithm that tracks bottleneck bandwidth and minimum propagation delay ($RTprop$), capping in-flight data at the Bandwidth-Delay Product ($BDP$) without needing to induce packet loss.

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📐 Numerical Applications & Worked Examples

Example 1: Full TCP Tahoe Recovery Trace

Problem Statement:
A TCP connection using TCP Tahoe experiences a timeout when its congestion window reaches $\text{cwnd} = 32\text{ KB}$. The Maximum Segment Size is $MSS = 2\text{ KB}$. The Round-Trip Time of the path is $RTT = 100\text{ ms}$. Assume receiver window $\text{rwnd}$ is unlimited.

1. Find the new value of $\text{ssthresh}$ and $\text{cwnd}$ immediately after the timeout.
2. Trace the value of $\text{cwnd}$ round-by-round until it reaches $32\text{ KB}$ again.
3. Compute the total time (in milliseconds) required for the connection to recover back to $32\text{ KB}$.

Step-by-Step Solution:

1. Identify Given Parameters:

   • $\text{cwnd}_{\text{loss}} = 32\text{ KB} = 16\text{ MSS}$
   • $MSS = 2\text{ KB}$
   • $RTT = 100\text{ ms}$

2. Compute New Threshold & Initial Recovery Window: $\text{ssthresh} = \frac{\text{cwnd}_{\text{loss}}}{2} = \frac{16\text{ MSS}}{2} = \mathbf{8\text{ MSS}} \; (16\text{ KB})$ $\text{cwnd} = \mathbf{1\text{ MSS}} \; (2\text{ KB}) \quad \text{(Reset to Slow Start under Tahoe)}$

3. Trace Window Progression Round-by-Round:

RTT Round	Phase	cwnd at Start of Round	Window Growth Dynamic	cwnd at End of Round
Round 1	Slow Start	$1\text{ MSS}$ ($2\text{ KB}$)	Doubles ($1 \to 2$)	$2\text{ MSS}$ ($4\text{ KB}$)
Round 2	Slow Start	$2\text{ MSS}$ ($4\text{ KB}$)	Doubles ($2 \to 4$)	$4\text{ MSS}$ ($8\text{ KB}$)
Round 3	Slow Start	$4\text{ MSS}$ ($8\text{ KB}$)	Doubles ($4 \to 8$)	$8\text{ MSS}$ ($16\text{ KB}$)
Round 4	Congestion Avoidance	$8\text{ MSS}$ ($16\text{ KB}$)	Linear ($+1\text{ MSS}$)	$9\text{ MSS}$ ($18\text{ KB}$)
Round 5	Congestion Avoidance	$9\text{ MSS}$ ($18\text{ KB}$)	Linear ($+1\text{ MSS}$)	$10\text{ MSS}$ ($20\text{ KB}$)
...	...	...	...	...
Round 11	Congestion Avoidance	$15\text{ MSS}$ ($30\text{ KB}$)	Linear ($+1\text{ MSS}$)	$16\text{ MSS}$ ($32\text{ KB}$)

4. Calculate Total Elapsed Recovery Time:
   • Slow Start Phase: $3\text{ RTTs}$ (reaches threshold of $8\text{ MSS}$ at end of Round 3).
   • Congestion Avoidance Phase: Needs to advance from $8\text{ MSS}$ to $16\text{ MSS}$. $\text{Rounds in CA} = 16 - 8 = 8\text{ RTTs}$
   • Total RTTs required: $\text{Total Rounds} = 3 + 8 = \mathbf{11\text{ RTTs}}$
   • Total elapsed time: $\text{Time} = 11 \times 100\text{ ms} = \mathbf{1100\text{ ms}} \; (1.1\text{ seconds})$

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Example 2: cwnd State Across Successive Transmissions

Problem Statement:
Consider an instance of TCP's AIMD algorithm where the window size at the start of the slow start phase is $2\text{ MSS}$, and the threshold $\text{ssthresh}$ at the start of the first transmission is $8\text{ MSS}$. Assume that a timeout occurs during the fifth transmission. Find the congestion window size at the end of the sixth transmission under TCP Tahoe.

Step-by-Step Solution:

1. Initial State:
   • $\text{cwnd}_1 = 2\text{ MSS}$, $\text{ssthresh} = 8\text{ MSS}$.
2. Transmission 1 (Slow Start): Transmits $2\text{ MSS}$. ACKs received $\implies \text{cwnd}_2 = 2 \times 2 = 4\text{ MSS}$.
3. Transmission 2 (Slow Start): Transmits $4\text{ MSS}$. ACKs received $\implies \text{cwnd}_3 = 4 \times 2 = 8\text{ MSS}$ (Hits threshold!).
4. Transmission 3 (Congestion Avoidance): Starts at $\text{cwnd}_3 = 8\text{ MSS}$. Grows linearly $\implies \text{cwnd}_4 = 8 + 1 = 9\text{ MSS}$.
5. Transmission 4 (Congestion Avoidance): Starts at $\text{cwnd}_4 = 9\text{ MSS}$. Grows linearly $\implies \text{cwnd}_5 = 9 + 1 = 10\text{ MSS}$.
6. Transmission 5: Transmits with $\text{cwnd}_5 = 10\text{ MSS}$. Timeout occurs!
   • New threshold: $\text{ssthresh}_{\text{new}} = \lfloor 10 / 2 \rfloor = 5\text{ MSS}$.
   • Reset window: $\text{cwnd} = 1\text{ MSS}$.
7. Transmission 6 (Slow Start): Transmits $1\text{ MSS}$. ACK received $\implies \text{cwnd} = 1 \times 2 = \mathbf{2\text{ MSS}}$.

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Example 3: Effective Transmit Window with Receiver Limitation

Problem Statement:
A TCP sender has an estimated congestion window of $\text{cwnd} = 16\text{ KB}$. The receiver advertises $\text{rwnd} = 10\text{ KB}$. The sender has transmitted bytes up to $42,000$ ($\text{LastByteSent} = 42,000$) and received an ACK for byte $36,000$ ($\text{LastByteAcked} = 36,000$). How many additional bytes can the sender transmit immediately?

Step-by-Step Solution:

1. Determine Effective Window: $\text{Window} = \min(\text{rwnd}, \; \text{cwnd}) = \min(10\text{ KB}, \; 16\text{ KB}) = 10\text{ KB} = 10,240\text{ bytes}$
2. Calculate Data Currently in Flight: $\text{In Flight} = \text{LastByteSent} - \text{LastByteAcked} = 42,000 - 36,000 = 6,000\text{ bytes}$
3. Calculate Usable Transmit Capacity: $\text{Usable Capacity} = \text{Window} - \text{In Flight} = 10,240 - 6,000 = \mathbf{4,240\text{ bytes}}$

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🎯 Exam & Interview Pitfall Check

Core Conceptual Questions

• Why does cwnd increase by 1 MSS per RTT in Congestion Avoidance if the formula is cwnd = cwnd + 1/cwnd?: The formula cwnd = cwnd + 1/cwnd is applied per incoming acknowledgment. During one RTT, the sender has cwnd packets outstanding, which generates cwnd individual ACKs. Summing the increment across all ACKs in that RTT yields: $\text{Total Increment} = \text{cwnd} \times (1 / \text{cwnd}) = 1\text{ MSS}$.
• Why does AIMD converge to fair bandwidth sharing while AIAD does not?: In Additive Increase Additive Decrease (AIAD), both the increase and decrease vectors move parallel to the $45^\circ$ fairness line. Consequently, the initial throughput disparity $(x_1 - x_2)$ is preserved forever. Only Multiplicative Decrease cuts the absolute gap by $50\%$ on every loss event, driving competing flows toward equal bandwidth allocation.

Common Interview Traps

• Trap: Confusing Per-ACK Updates with Per-RTT Updates: In Slow Start, cwnd doubles every RTT, but it increments by 1 MSS per ACK. If an exam question asks: "If cwnd is 8 MSS and 1 ACK arrives in slow start, what is the new cwnd?", the answer is $9\text{ MSS}$ (NOT 16 MSS!). It only becomes 16 MSS after all 8 ACKs return.
• Trap: Assuming ssthresh is Halved After Resetting cwnd: The threshold must ALWAYS be computed from the current window at the moment of loss ($\text{ssthresh} = \text{cwnd} / 2$). Resetting $\text{cwnd} = 1\text{ MSS}$ happens after calculating $\text{ssthresh}$.

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