Assuming that packet transmissions and retransmission can both be described as a Poisson process, calculate the probability that a data packet transmi

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Assuming that packet transmissions and retransmission can both be described as a Poisson process, calculate the probability that a data packet transmission in an S-ALOHA system will experience a collision with another user. Assume the total traffic rate λt = 10 packets/sec and the packet duration τ = 10 msec.
1. 0.9
2. 0.09
3. 0.009
4. 9.0

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Correct Answer - Option 2 : 0.09

The formula to calculate probability in Poisson Process is:

$P\left( {X = x} \right) = \frac{{{{\left( {\lambda t} \right)}^x}}}{{x!}}{e^{ - \lambda t}}$

Where:

x = number of success

t = time duration

= Average number of success during the time duration

Calculation:

Given that, In an S-ALOHA data packet transmission system,

x = experience a collision with another user = 1

t = packet duration

= 10 msec = 10 × 10-3 sec

λ = total traffic rate during packet transmission

= 10 packets/sec

$\therefore \lambda t = 10 \times 10 \times {10^{ - 3}} = {10^{ - 1}} = 0.1$

$\therefore P\left( {X = 1} \right) = \frac{{{{\left( {0.1} \right)}^1}.{e^{ - 0.1}}}}{{1!}}$

$= \frac{{0.090}}{1} = 0.090$