Solution of Home Work 5

Pg81-10) A disadvantage of a broadcast subnet is the capacity wasted when multiple hosts attempt to access the

channel at the same time. As a simplistic example, suppose that time is divided into discrete slots, with

each of the n hosts attempting to use the channel with probability p during each slot. What fraction of the

slots are wasted due to collisions?

Solution:

Distinguish n+2 events. Events 1 through n consist of the corresponding host successfully attempting to use

the channel, i.e., without a collision. The probability of these events is p(1-p)^n-1. Event n+1 is an idle channel,

with probability (1-p)ⁿ. Event n+2 is a collision. Since these n+2 events are exhaustive, their probabilities must

sum to unity. The probability of a collision, which is equal to the fraction of slots wasted, is then just

1 - np(1-p)^n-1 - (1-p)ⁿ.