I am trying to calculate the average wait time for a customer in a queue where there is a single server. I have the distribution for the arrival probability of the customer at time (t) as
$F(t) = 1 - e^{-t/200}$
And it's given that the customers arrive one at a time.
I also have the beta distributions for the processing time of the customer. I can calculate the average and max processing time of the customers using the beta distributions. Now, I need to calculate the average and max waiting time of customer (time the customer has to wait before they reach the server).
(avg/max)Wait = (number of customers in the queue) * ((avg/max) * processing time for a customer)
I can calculate the second part of the equation using the beta distribution given to me. But how do I calculate the average number of customers in the queue?
Use Little's Law.
$$L = \lambda W$$
where,
$L$ is the long-term average number of customers in the stationary system,
$λ$ is the long-term average effective arrival rate,
$W$ is the average time that a customer spends in the system