Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Valuing Reliability for Rare Events
In this appendix, options theory from financial economics is once again applied to the
problem of unreliability, this time in the context of rare events and the decision to invest
given the probability of a low probability, but high consequence event. Some events that
influence the performance of the highway network are considered to be rare events. For
example, an important challenge for highway agencies is how to mitigate interruptions in
service due to road closure by avalanche, flooding, or bridge failure. Valuing unreliability
generated by rare events is influenced by the following considerations:
• The occurrence of rare events is not believed to be accurately characterized by
random draws from a normally distributed variable. The rare-event distributions
are more mathematically complex, making it more difficult to perform the
necessary option value calculations.
• Longer time periods often must be examined to properly identify the parameters
of the statistical distributions that characterize the event probabilities.
• Transportation network unreliability may not be directly associated with rare
events because a long history of system performance data may not exist.
• The impacts caused by rare events may be more complex than those that affect the
variability of speed on a few network segments. Some segments may be closed
for extended periods of time. Thus, the travel delays that occur may result in
travel path diversions, rather than simply changing the variability of speed on the
segments that are affected.
• The time horizon during which the transportation agency may make plans for
dealing with rare events is often much longer than the time horizon that the
agency has to deal with a recurring event.
These considerations do not always arise. Events that are not normally distributed may
contribute to unreliability on a regular basis. However, most rare events are associated
with phenomena such as severe flooding, rare weather events, structural failures, and
other events that occur infrequently.
Using Extreme Value Functions to Characterize Rare Events
A class of distributions known as extreme value (EV) distributions is thought to better
represent the incidence of rare events than normal distributions. Extreme value
distributions tend to have probability density functions that are quite asymmetric. Much
of the weight of the distribution is clustered at low outcome values. In addition, EV
distributions tend to have long tails that are fatter than the upper tail of a normal
distribution. These EV distributions represent a situation in which, on most days, an event
does not occur that affects reliability in a significant way. However, on rare occasions, an
event does occur that affects reliability in a dramatic fashion.
The extreme value distributions used to characterize rare events are referred to as Type I,
II, and III distributions. They are also known as Gumbel, Fréchet, and Weibull
distributions, respectively. Each of these distributions is a variation of the Generalized
Extreme Value (GEV) distribution, differing only in their parameter values, but having
very different shapes:
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-2