Question

Consider a LAN with four nodes S₁, S₂, S₃ and S₄. Time is divided into fixed - size slots, and a node can begin its transmission only at the beginning of a slot. A collision is said to have occurred if more than one node transmit in the same slot. The probabilities of generation of a frame in a time slot by S₁, S₂, S_3, and S₄ are 0.1, 0.2, 0.3 and 0.4, respectively. The probability of sending a frame in the first slot without any collision by any of these four stations is ________.

Answer 1

Data:

S₁ = 0.1, S2 = 0.2, S3 = 0.3 and S4 = 0.4

Formula:

Probability of successful transmission in first slot = P (at a time only one node transfers)

\(P(T) = S_1.\bar S_2.\bar S_3.\bar S_4 + \bar S_1.S_2.\bar S_3.\bar S_4 + \bar S_1.\bar S_2.S_3.\bar S_4 + \bar S_1.\bar S_2.\bar S_3.S_4\)

Calculation:

P(T) = 0.1 × 0.8 × 0.7 × 0.6 + 0.9 × 0.2 × 0.7 × 0.6 + 0.9 × 0.8 × 0.3 × 0.6 + 0.9 × 0.8 × 0.7 × 0.6

∴ P(T) = 0.4404

The probability of sending a frame in the first slot without any collision by any of these four stations is 0.4404