In most data link protocols that retransmit damaged frames, if the probability that a frame is damaged is p (0 ≤ p < 1) and acknowledgments are never lost, what is the mean number of transmissions required to successfully send one frame?

Introduction / Context:
Many link-layer protocols use ARQ (Automatic Repeat reQuest): a sender retransmits a frame when it is detected as damaged or lost. When acknowledgments are assumed reliable (never lost) and only data frames can be damaged with probability p, the number of transmissions per successfully delivered frame follows a geometric pattern.

Given Data / Assumptions:

• Each attempt to send the frame succeeds with probability (1 − p).
• Failures (damage) occur independently with probability p.
• Acknowledgments are never lost or corrupted.
• Retransmissions continue until success.

Concept / Approach:
Let X be the number of transmissions needed for success. X is a geometric random variable with success probability q = (1 − p). The expected value of a geometric distribution (counting the successful trial) is E[X] = 1 / q = 1 / (1 − p). This captures the average total sends per successfully delivered frame, including necessary retries.

Step-by-Step Solution:
Define success probability: q = 1 − p.Recognize X ~ Geometric(q) (first success model).Compute expectation: E[X] = 1 / q = 1 / (1 − p).Interpretation: if p = 0.1, E[X] ≈ 1 / 0.9 ≈ 1.111… transmissions per successfully delivered frame.

Verification / Alternative check:
Series method: E[X] = Σ_{k≥1} k * p^{k−1} * (1 − p) = (1 − p) * Σ k p^{k−1} = 1 / (1 − p). This matches the standard geometric expectation.

Why Other Options Are Wrong:
1 / p: Grows unbounded as p → 0; contradicts intuition.

1 + p: Linear, underestimates retries for larger p.

1 − p: This is the success probability, not the expectation.

p / (1 − p): Expected number of failures before success; missing the final successful transmission.

Common Pitfalls:
Forgetting to include the successful final transmission; confusing the expected count of failures p/(1−p) with total transmissions 1/(1−p); assuming acknowledgments can also fail, which would change the model.

Final Answer:
1 / (1 − p)