I am trying to understand how to approach a problem involving a Poisson Process queue with a deterministic service time. We have that the mean rate of arrival time is your standard $\lambda$ customers a minute (exponential) and we have a service time that is fixed $u$ minutes. We are then told to assume that the first customer arrives with no queue.
I am having trouble figuring out the next parts which are: Find the density function for the sojourn time of the first customer (I think I have an answer but would like reassurance). The expected sojourn time of the first customer as well as the standard deviation of this. We are then asked to do this again for the second customer (Expected sojourn of second customer, standard deviation of this as well as the density function).
Everything I have read focusses on the general and long term means but I have not been able to find a clear explanation of what to do here. Any help will be greatly appreciated.