(apology for any cross posting)

I have a real situation here, but I don't know enough theory behind it.

We have requests for emergency medical service, coming from residents of
a community to the central fire station, where the emergency Medical
Technicians (EMT's) wait.  when called, they respond as fast as possible.

 From a given station, the response time is say, 3.5 minutes, on
average.  If a second call comes in while the crew is on the first call,
then a back-up comes from another station, and the response time is say
8 minutes, on average.  If a third call comes in on top of the first
two, a new back-up station responds, and the response time may get to 15
minutes.

Now, we know the number of calls per year, and the time spent on each
call.  We can work out a histogram for number of calls per week, or time
between each call.  We can work out the distribution for the time spent
on each call.  Suppose at first that the time of day is not a factor -
i.e., calls come in at random times of the day.

I expect (before I review the data) that the time spent on each call
will be logNormal (in continuous time), or Poisson (in integer minutes).
 I expect the number of calls per day to be a Poisson distribution, as well.

1)    My objective is to estimate the number of times per year that two
calls will overlap, and that the number of times 3 calls will overlap at
once.

I could set up a simulation model and see what happened, but I want to
get some idea of what to 'expect' and perhaps some non-simulation
estimates of what will happen, which I can put up against the simulation
as a form of validity check.

What distribution of time between calls should I expect?  and If I knew
this, would it do me any good?

One thought was to assume each call took 1 hr, compute the number of
calls per hour (a small fraction of 1), and assert that the probability
of 2 calls in the same 1 hr period would be p(call) squared.  Does that
make any sense?  Clearly, I would use the real time per 1 call as my
time unit, and if calls in the afternoon were more frequent (they
generally are) I would look only at the pm time frame.

for a simulation I need an average calls per day (or hour), and the
distribution, plus the average time per call, and that distribution.  If
I then run the puppy for a long time, would I not get, from the history,
a frequency of double calls and triple calls?  What other input
information would be needed?

Cheers & hope this makes sense to you,

Jay

--
Jay Warner
Principal Scientist
Warner Consulting, Inc.
4444 North Green Bay Road
Racine, WI 53404-1216
USA

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