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strict nature of these conditions a bit, the approximation of a Poisson arrival
process can be quite good.
c. Empirically based distributions . To this point our discussion has been about
theoretically based distributions. In these distributions we assume a theoreti-
cal model of uncertainty, such as a Poisson or Normal distribution, but often
decision makers can collect and record empirical data and develop distribu-
tions based on this observed behavior. Our distribution of colorful stones is
such a case. We do not assume a theoretical model of the distribution of col-
ors; we have collected empirical data, in our case, through sampling, which
leads us to a particular distribution.
In the following section we will concentrate on an introduction to the Poisson dis-
tribution. It will be somewhat complex to create an approach that will allow Poisson
arrivals.
8.3.3 Modeling Arrivals with the Poisson Distribution
Earlier, I mentioned that arrivals in simulations are often modeled with a Poisson
distribution. When we employ the Poisson to describe arrivals, we refer to this as
a Poisson Arrival Process . The arrivals can be any physical entity, for example,
autos seeking service at an auto repair facility, bank clients at an ATM, etc. Arrivals
can also be more abstract, like failures or ﬂaws in a carpet manufacturing process
or questions at a customer help desk. The arrivals can occur in time (auto arrivals
during the day) or in space (location of manufacturing ﬂaws on a large area of
household carpet). So how do we sample from a Poisson distribution to produce
arrivals?
Exhibit 8.10 serves as an example of how we will manage the arrival data for
the Autohaus simulation we will perform later in the chapter; an advanced model-
ing example that is essentially a discrete event simulation. Describing a process as
having Poisson arrivals requires that a number of assumptions should be maintained:
1. Probability of observing a single event over a small interval of time or within a
small space is proportional to the length of time or area of the interval
2. Probability of simultaneous events occurring in time and space is virtually zero
3. Probability of an event is the same for all intervals of time and in all areas of
space
4. Events are independent of one another, so the occurrence of an event does not
affect another
Although it may be difﬁcult to adhere to all these assumptions for a particular
system, the Poisson’s usefulness is apparent by its widespread use. For our purposes,
the distribution will work quite well.
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