In Pure ALOHA, we established that collisions are inevitable when multiple stations share a channel without coordination. But to understand ALOHA's efficiency—or more accurately, its inefficiency—we need to answer a deceptively subtle question:

Exactly when can a collision occur?

The answer lies in a concept called the vulnerable period—the time window during which a transmitted frame is susceptible to being destroyed by overlapping transmissions. This concept is so fundamental that understanding it is prerequisite to any meaningful analysis of random access protocols.

At first glance, you might think: "A collision happens when two stations transmit at the same time." But this intuition is dangerously incomplete. Frames don't need to start at the same time to collide—they merely need to overlap. And this overlap requirement creates a vulnerability window that is surprisingly larger than most newcomers expect.

Before we can understand the vulnerable period, we must precisely define what constitutes a collision in Pure ALOHA.

The Collision Condition:

A collision occurs when any portion of one frame overlaps with any portion of another frame at the receiver. Even a single bit of overlap is sufficient to corrupt both frames, rendering them unreadable.

This definition has profound implications:

Why Complete Destruction?

In radio systems, when two signals arrive at a receiver simultaneously, they interfere with each other. The receiver sees a superposition—a jumbled mix of both signals—and cannot extract either original message. Unlike some physical phenomena where signals might partially survive, digital frames are an all-or-nothing affair: any corruption invalidates the entire frame.

Receiver-Centric View:

It's crucial to understand that collisions are defined at the receiver, not the transmitter. Two stations might begin transmission at different absolute times, but their frames collide if they arrive at the receiver during overlapping intervals. In systems with minimal propagation delay (like a single room), transmitter and receiver timing are nearly identical. In systems with significant propagation delay (like satellite links or the original ALOHAnet), this distinction becomes more important.

For simplicity, our analysis typically assumes negligible propagation delay, meaning transmission start times directly determine collision occurrence.

Now we can precisely define the vulnerable period:

Definition:

The vulnerable period for a frame is the total time interval during which the transmission of another frame would result in a collision.

Let's think about this from the perspective of a frame we're trying to transmit successfully—call it Frame X. Frame X starts transmission at time $t_0$ and completes at time $t_0 + T$, where T is the frame transmission time.

Question: During what time interval must no other frame begin transmitting to ensure Frame X succeeds?

Answer: No other frame can begin transmitting from time $(t_0 - T)$ to time $(t_0 + T)$.

This gives us a vulnerable period of 2T—twice the frame transmission time.

Why 2T? The Two-Frame Intuition:

Let's trace through the logic carefully:

Case 1: Another frame starts BEFORE Frame X (but after $t_0 - T$)

Suppose Frame Y starts at time $t_0 - T + \epsilon$ (a small time after $t_0 - T$). Frame Y finishes at time $t_0 + \epsilon$. Since Frame X starts at $t_0$ and Frame Y ends at $t_0 + \epsilon$, Frame Y's final portion overlaps with Frame X's initial portion → Collision

Case 2: Another frame starts DURING Frame X's transmission

If Frame Y starts anywhere between $t_0$ and $t_0 + T$, it clearly overlaps with Frame X → Collision

Case 3: Another frame starts AFTER Frame X ends

If Frame Y starts at or after $t_0 + T$, Frame X has already completed. No overlap → No collision

Combining these cases: the danger zone extends from $(t_0 - T)$ to $(t_0 + T)$, a total duration of 2T.

A timeline diagram makes the vulnerable period crystal clear. Consider the following scenarios:

Timeline Representation:

Time →

     t₀-T        t₀         t₀+T        t₀+2T
       |          |           |           |
       |← T →|← Frame X →|    |
       |          |           |           |
       |←───── Vulnerable Period ─────→|
                   (2T total)

Any frame that begins during the vulnerable period window [t₀-T, t₀+T) will collide with Frame X.

Scenario Analysis:

Let's visualize several collision and success scenarios:

Scenario A: Frame Y starts at t₀ - 0.8T (COLLISION)

Time →
       t₀-T        t₀         t₀+T
         |          |           |
    |←── Frame Y ──→|  
             |←── Frame X ──→|
                ↑
           Overlap zone

Frame Y ends at t₀ + 0.2T, overlapping with Frame X's first 0.2T → Collision

Scenario B: Frame Y starts at t₀ + 0.5T (COLLISION)

Time →
         t₀         t₀+T       t₀+2T
          |           |           |
    |←── Frame X ──→|
               |←── Frame Y ──→|
           ↑
       Overlap zone

Frame Y begins mid-transmission of Frame X → Collision

Scenario C: Frame Y starts at exactly t₀ + T (NO COLLISION)

Time →
         t₀         t₀+T       t₀+2T
          |           |           |
    |←── Frame X ──→|
                    |←── Frame Y ──→|
                    ↑
              No overlap

Frame Y begins exactly as Frame X ends → Success (both frames)

Let's formalize the vulnerable period concept mathematically, which will set up our efficiency derivation.

Setup:

• Let T = frame transmission time (constant for all frames)
• Let frame arrivals follow a Poisson process with rate G frames per time T (i.e., on average G frame attempts occur per frame-time)
• Consider Frame X starting at time t₀

Success Condition:

Frame X succeeds if and only if:

1. No other frame started in the interval $(t_0 - T, t_0)$ — the 'backward' vulnerable portion
2. No other frame starts in the interval $(t_0, t_0 + T)$ — the 'forward' vulnerable portion

Combined: No frame (other than Frame X) can start in the interval $(t_0 - T, t_0 + T)$

Probability Calculation:

Since arrivals follow a Poisson process with rate G per time T:

• The expected number of arrivals in time interval 2T is: $\lambda \cdot 2T = G \cdot \frac{2T}{T} = 2G$

• The probability of zero arrivals (other than Frame X itself) in time 2T is:

$$P(\text{no collision}) = P(0 \text{ other arrivals in } 2T) = e^{-2G}$$

The 2G Factor:

The factor of 2 in the exponent ($e^{-2G}$ rather than $e^{-G}$) comes directly from the vulnerable period being 2T rather than T. This factor of 2 is what distinguishes Pure ALOHA from Slotted ALOHA, where synchronization reduces the vulnerable period to exactly T.

Key Equation:

$$P(\text{successful transmission}) = e^{-2G}$$

This is the foundation for deriving Pure ALOHA's throughput, which we'll complete in the next page.

Understanding the details of collision mechanics illuminates why 2T is the exact vulnerable period—no more, no less.

Boundary Conditions:

Let's examine the exact boundaries of the vulnerable period:

At exactly $t_0 - T$: If Frame Y starts at exactly $t_0 - T$, it ends at exactly $t_0$. Frame X starts at $t_0$. Do they collide?

In the mathematical idealization, the frames are exactly adjacent—Frame Y's last bit and Frame X's first bit occupy adjacent instants with no gap and no overlap. In this idealized model: no collision.

In practice: any timing jitter would likely cause overlap, so this boundary is treated conservatively.

Just after $t_0 - T$ (e.g., at $t_0 - T + \epsilon$): Frame Y ends at $t_0 + \epsilon$, overlapping with Frame X's first $\epsilon$ time → collision.

This analysis confirms: the vulnerable period is the open interval $(t_0 - T, t_0 + T)$, with length exactly 2T.

Multi-Frame Collisions:

While we've analyzed pairwise collisions, in reality, multiple frames can be involved:

• If 3 frames overlap, all 3 are destroyed
• The vulnerable period analysis still applies: each frame requires its 2T window to be clear of all other frames
• Under Poisson arrivals, the probability of multi-frame pileups increases with load

Propagation Delay Effects:

In systems with non-negligible propagation delay τ:

• The vulnerable period can increase beyond 2T
• Different stations experience different vulnerable windows due to their positions
• For satellite systems with τ >> T, the analysis becomes significantly more complex

For our analysis, we assume τ ≈ 0, so vulnerable period = 2T exactly.

The 2T vulnerable period is not just a theoretical concern—it has profound implications for protocol design and efficiency.

The Fundamental Efficiency Limit:

Because each frame 'occupies' 2T of vulnerable time while only transmitting for time T, there's inherent inefficiency:

• Even with perfect traffic scheduling, at most T/2T = 50% of the time can be productively used
• With random arrivals, the situation is worse (as we'll rigorously prove in the next page)
• The maximum throughput of Pure ALOHA is approximately 18.4% (we'll derive this exactly)

Design Implications:

Let's develop intuition for how the vulnerable period translates into collision probability across different load levels.

The Exponential Decay:

Success probability $P_s = e^{-2G}$ is an exponentially decaying function of offered load G:

Exponential Decay Intuition:

The exponential form arises from Poisson statistics: for a frame to succeed, no other frames can start in the 2T window. Each potential interferer is an independent event, and the probability of all of them 'not happening' multiplies, yielding the exponential.

Throughput Curve Shape:

The throughput S = G × e^{-2G} (which we'll derive fully next page) has a distinctive shape:

1. Rising phase (G < 0.5): As load increases from 0, throughput increases—more attempts mean more successes, even though collision rate is rising

2. Peak (G = 0.5): Maximum throughput occurs when the marginal benefit of another transmission attempt exactly equals the marginal cost of increased collisions

3. Falling phase (G > 0.5): Collision rate increases faster than traffic, so actual throughput decreases even as more attempts are made

This non-monotonic behavior is characteristic of contention-based protocols and creates the possibility of throughput collapse under overload.

Practical Implications:

Real Pure ALOHA systems must:

1. Monitor channel utilization and throttle new transmissions if load rises
2. Design backoff algorithms that don't allow offered load G to grow unbounded
3. Consider admission control to prevent too many active stations
4. Accept that practical throughput will be well below the theoretical maximum

The vulnerable period concept is the key to understanding all of these design constraints.

The vulnerable period is the conceptual cornerstone for understanding Pure ALOHA efficiency. Let's consolidate:

What's Next:

With the vulnerable period firmly understood, we're now ready to derive Pure ALOHA's throughput equation and prove that maximum efficiency is exactly 1/(2e) ≈ 18.4%. In the next page, we'll complete this classic derivation and understand why Pure ALOHA can never exceed this fundamental limit.