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likelihood of these conditions. This is often required in computer modeling, but in the
case of collision risk, this will often involve some conditions that are relatively likely
combined with others that are orders of magnitude less likely. The analytic approach
provides advantages in dealing with combinations of events with a wide range of
probabilities.
Some of the disadvantages of probabilistic models are:
1. Limited applicability - Often the problem is far too complex to allow a closed-form
solution, or in order to make the problem mathematically tractable, simplifying
assumptions have to made that reduce the validity or applicability of the result.
APPROACHES TO COLLISION RISK ANALYSIS
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2. Over simplification - Mathematical equations are generally gross simplifications of
reality. If the simplifications eliminate details that are unimportant, this is good; if they
cause the desired result to be incorrect, that is a serious drawback. The same can be
said about Monte Carlo simulation, but there is a better opportunity in the latter to add
complications if they are found to be necessary.
3. Invisibility - Monte Carlo models afford the opportunity to produce graphical or hardcopy
intermediate output, allowing the developer and user to observe intermediate
results to insure that the model behaves realistically. This is not so easy to do with
closed-form models.
4. Dependencies - Analytical models often assume that relationships are straight-forward,
whereas they may depend on combinations of factors. For example, there are simple
analytic models that are based on average values of the independent variables, whereas
relationships might vary non-linearly with these variables. Complex dependencies can
be incorporated more easily in a Monte Carlo model, but it is often possible to handle
dependencies in a probabilistic model. Conversely, dependencies are sometimes
ignored in Monte Carlo models. Consideration of dependencies in probabilistic models
is discussed later in this paper.
Some problems that beset both types of models are:
1. Data unavailability - Both models require data that are often not available.
2. Invalidity - Both types of models are usually not sufficiently validated, if validated at
all. To properly validate a model often requires a large effort and special collection of
field data independent of the data used to construct the model. The validation should
include conditions similar to those for which the model is to be exercised. For
example, the FAA and other organizations have, over the years, built many delay
models. Most, if not all, of these have been given a cursory validation, at best. The
validation is made under current conditions, and then the model is often used to predict
delay under much greater traffic loading and for changed operating conditions, where
completely new bottlenecks might, in actuality, appear. If this is true of delay, which is
measurable and an everyday occurrence, how much more must it be true of collision
risk, which is hard to measure and, fortunately, very rarely observed?
One cannot say that probabilistic models are better than Monte Carlo Models, or viceversa.
Both have advantages and disadvantages. Due to the immense complexity and
lack of knowledge of the subject, all are but crude approximations of reality and are
likely to produce incorrect results if they are built with incorrect assumptions, contain
programming or mathematical errors, are fed incorrect data, and/or are used beyond
their range of applicability. In the case of collision risk, when one is dealing with
combinations of events and conditions that vary widely in their likelihood, the
probabilistic model has an advantage, if it can be constructed with sufficient realism.
SEPARATION SAFETY MODELING
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5.3.2 Examples of Probabilistic, Collision Risk Models
The Gas Law Model
The simplest analytical collision risk model that has actually been used in FAA studies is
the so-called Gas Law Model, which is of the form:
N = C * X2
Where
N = the expected number of midair collisions per year
C = a constant
X = the number of aircraft in an area in a given time period
This has been used to justify why a certain device, such as a radar system, is needed in a
particular terminal area. The assumption behind this model is that aircraft are completely
randomly placed and on random headings, like gas molecules in a bottle. The parameter C
was developed by looking at collision statistics in a large number of terminal areas over a
large number of years. The annual operations count at the primary airport was used as the
parameter X.
This produced very desirable results, in that the radar system was justified. However,
　
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