In a G/G/1 queue, the interarrival and service time are IID random variables with mean 1/lambda y 1/mu respectively (lambda is the rate for arrivals and mu is the rate for the service). My question es: Why the arrivals and departures times for G / G / 1 queue have mean exponential?
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The $\lambda$ and $\mu$ have nothing to do with the exponential distribution. It just turns out that these two parameters are most often used as the parameters of the the exponentially distributed inter-arrival time and service time.
Let $A$ be the inter-arrival time and let $B$ be the service time in the $G/G/1$ queue. Then the mean inter-arrival time is $\mathbb{E}[A] = 1/\lambda$ and the mean service time is $\mathbb{E}[B] = 1/\mu$. For example, let the inter-arrival time $A$ have a Uniform distribution on $[0,1]$. Then, $\mathbb{E}[A] = 1/2$ and $\lambda = 2$. This does not mean that $A$ has an exponential distribution with parameter $2$ ($A$ has a Uniform distribution on $[0,1]$ like we just assumed it to have).
