I have a service node with a queue The arrival rate in this service node is exponential(0.4) The service time is exponential(0.2) Running a simulation i've calculated the average population of the system (service node + queue) (using welford's algorithm) and its standard deviation. Can I calculate the average time in the system using Little's law W = average population / arrival rate ? I need the standard deviation of the time in the system too, there is an easy way to calculate it?
2026-03-26 11:16:42.1774523802
mean and std deviation from little's law
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Yes, $W$ gives you the mean sojourn time of a customer in the system. For the service time, since it is given to be $\sim \exp (\mu)(\mu=0.4)$ the std is just $1/\mu$.
From your description it seems to be an $M/M/1$ queue. Then once a customer arrives in the system it spends a time which consists of $\exp(\mu)$ in the service node and waiting time $W_q$ which is the residual waiting time for the customer in front. You can show with some effort that $W\sim \exp(\mu-\lambda)$ and hence can find the std of $W$. See this for reference.