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Subject:

Expected distribution for random events of finite duration

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Tue, 28 Sep 2004 15:15:19 -0500

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 ```(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 Ph: (262) 634-9100 FAX: (262) 681-1133 email: [log in to unmask] web: http://www.a2q.com The A2Q Method (tm) -- What do you want to improve today? ```