Resource Allocation Problem

Hi Folks,

This is an interesting problem.
I have been thinking about how to do it for a few days.

I would like to calculate a probable time to arrive at work based on a resource list and the number of initial customer cases.

Each morning we come into work, there is a list (from 0 to M) of employees/resources who will handle customer cases.
R1=John
R2=Steve
R3=Frank
R4=Joel
etc

There is an initial number of cases to be handled immediately i.e. Cinitial (from 0 to N), and then new cases come in at random intervals.

Each case gets assigned to a resource based on the order of the resource list. When a resource has finished handling a case, they go back onto the bottom of the list.

So if Cinitial = 2 cases, then John and Steve will be occupied handling the cases immediately at 9am. Frank and Joel will be waiting around for new cases to randomly arrive.

I am able to get statistics for each day - the number of initial cases at 9am, the times the new cases come in (and therefore the times resources 0 to M) are needed, and the total number of new cases during the day.

What would be interesting to calculate - based on Cinitial, what time is most likely for R(x) to be needed? (R(x) could sleep in a little and not need to get to work at 9am).

Then we ask the question sometime during the day instead of in the morning - there was Cinitial cases at the start of the day, Cnew cases have come in, it is now Time t>9am , what is the probable time R(x) is needed?

Your ideas are appreciated on how to work out this problem / what models are needed.