Question

In an M/M/1 queuing system, the number of arrivals in an interval of length T is a Poisson random variable (i.e the probability of there being n arrivals in an interval of length T is \(\frac{{{e^{ - \lambda T}}{{\left( {\lambda T} \right)}^n}}}{{n!}}\) . The probability density function f(t) of the inter-arrival time is given by
1. \({\lambda ^2}\left( {{e^{ - {\lambda ^2}t}}} \right)\)
2. \(\frac{{{e^{ - {\lambda ^2}t}}}}{{{\lambda ^2}}}\)
3. λe^-λt
4. \(\frac{{{e^{ - \lambda t}}}}{\lambda }\)

Answer 1

Correct Answer - Option 3 : λe^-λt

Probability of n arrivals in time 'T' = \(\frac{{{e^{ - \lambda T}}{{\left( {\lambda T} \right)}^n}}}{{n!}}\)

So, probability density function of inter arrival time = λe^-λt