Do two data packets collide?

Question

Two data packets each grab a shared transmission line at a random instant, uniform over a $100$ -unit time window, independently. Each packet occupies the line for $40$ units. If their transmission intervals overlap, they collide and both must resend.

What is the probability the two packets collide?

Accepted Answer

Let X, Y \sim U(0, 100) be the two start times. The transmissions overlap (collide) iff X - Y \le 40. In the 100 	imes 100 square, the complement (no collision) is two congruent right triangles where the two starts differ by more than 40, each with legs 100 - 40 = 60: P(	ext{no collision}) = \frac{2 \cdot \frac{1}{2}(60)^2}{100^2} = \frac{60^2}{100^2} = \left(\frac{3}{5}ight)^2 = \frac{9}{25} = 0.36. Therefore P(	ext{collision}) = 1 - 0.36 = 0.64. Equivalently, with hold time w = 40 and window T = 100, P(	ext{collision}) = 1 - (1 - w/T)^2 = 1 - (0.6)^2 = 0.64. Because the transmission l…

Do two data packets collide?

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