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I am analyzing the occurrence of emergencies in a given area and the application of queueing theory to determine the resources an emergency service should have ready in order to answer to emergency calls.

For basic analysis, the Erlang B formula gives the probability of an emergency occurring while no resources are available. Erlang B uses the offered load, the product of the arrival rate of an emergency and the time the resource is bound to the emergency.

Given data from dispatch centers the arrival rate can either be counted (e.g. for each hour of a day) or the inter arrival time can be calculated. Both are empirical values which I determine for each hour, weekday combination of a given year. Both measures are also interchangeable (in an ideal setup). For example, an arrival rate of 2 per hour equals an inter arrival time of 0.5 hour or 30 minutes. So, you can calculate one from the other. However, from the empirical perspective I observe differences in the arrival rates and inter arrival times.

An example: For each Sunday (52 in total in the given year) and the hour from 9AM to 10AM I got the following values:

  • Total events: 428
  • Arrival rate from total events: 428 / 52 = 9.269 arrivals per hour
  • Mean inter arrival time: 409.206 seconds
  • Arrival rate from mean inter arrival time: 3600 seconds / 409.206 seconds = 8.798 arrivals per hour

The difference between the rates does give different values for the Erlang B and thus does have an impact on the analysis. Depending on the chosen threshold the number of servers (resources) needed to answer every emergency directly varies. It's no big variation, but can lead to a difference of resources.

In this particular example:

  • chosen quality threshold: 99.9 %, which is (1-Erlang B) * 100
  • needed resources from total events arrival rate: 3
  • needed resources for inter arrival time arrival rate: 4

The question now is, which result is the more appropriate and why? Is it just pure reason or is there a scientific base for the more appropriate result?

Final thoughts: The arrival rate from total events is a value derived from a directly counted integer quantity. Small variations in this quantity might have a bigger impact on the arrival rate than small variations in the inter arrival time measured in seconds. However, the arrival rate derived from the mean inter arrival time is a value calculated from the mean of the observed times, which for the given example are mostly 52 values (occurrences of weekday, hour combination in a year). Intuitively, the inter arrival time option seems more appropriate to me.

Finally, one might think about larger time intervals. For now, the 1 hour intervals are used because of the variation in the arrival rates over a day (24 hour analysis does not capture peeks e.g. at noon). This might involve analyzing the data for clusters of similar arrivals or inter arrival times.

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It turns out that the count of the events gives the more reliable results. The reason seems to be the method I have calculated the mean inter arrival times for each weekday / hour combination wich are merely the means of the inter arrival times that happen to lie in the given combination. By doing this I shortened the interval by a random time.

As an example:

If you count 4 events in an hour with inter arrival times of 18 minutes (e1 to e2), 6 minutes (e2 to e3) and 12 minutes (e3 to e4), the sum of the inter arrival times is 36 minutes and the mean is 12 minutes. However, the total duration of the interval is 60 minutes, not 36 minutes, thus resulting in an error when one tries to use the measured inter arrival times.

In addition, the method I used also used the inter arrival time to the next event in the following our in the current hour, leading to additional error.

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