Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Equation 3: European Put Option with EV Variability
f
( x;
µ , σ )
Gumbel
1
= e
σ
where
−(
x−µ
)
σ
µ = location parameter
σ = scale parameter
and where
e
−(
x−µ
)
σ
−e
x is the random variable of interest
This formulation may be useful in situations in which speed variability is due entirely to a rare
event that occurs within a relatively short interval. For example, this might apply to a normally
uncongested country road where an accident occurs, causing speed reductions (delays) that
represent an EV distribution. Sufficient data would be needed to estimate the location, scale, and
shape parameters for the EV distribution. Conceptually, the valuation of unreliability would then
proceed in a manner completely analogous to the manner described earlier where speeds are
distributed log-normally.
An Illustrative Example of the European Put Option
A more-typical case occurs when the events that precipitate unreliability are known to be
distributed EV, but there is no record of associated traffic metrics. This can be because insufficient
data is collected or because speed is not a sufficient measure of the impact of the events on
network reliability. In this case, the event distribution is estimated from information regarding the
event, rather than from a roadway performance metric such as speed. This means that a second
step can link the events to travel-time performance.
This situation is illustrated using the European EV put formulation in a setting of avalanche
closures. Avalanches have characteristics of rare events, in that for most days and months in the
winter, no avalanches occur, but periodically, avalanches of varying intensity and extensiveness
occur. If an avalanche event is distributed EV, then Equation 3 can provide us with the certaintyequivalent
value of closure duration under various conditions of control or mitigation of the
avalanche impacts.
The other relevant feature of this example is that the traffic count data, though collected with high
frequency in the vicinity of the avalanche; do not fully convey the traffic unreliability associated
with the avalanche. Specifically, without considerable additional effort, one cannot determine the
useful traffic performance metrics associated with the avalanche. Thus, the unreliability valuation
exercise must be broken into two pieces. First, the duration of avalanches can be analyzed in an
options framework to derive a certainty-equivalent delay from the (presumed) EV-distributed
duration data. Thus, highly variable and rare-event data can be reduced to a deterministic indicator.
Second, a separate study (not performed here) linking closure duration to traffic delay can then be
applied to the certainty equivalent closure duration for monetizing the benefits of a strategy or
treatment.
The illustrative example that is presented here is of an avalanche closure for Snoqualmie Pass on I-
90 in the State of Washington. The rare event nature of a road closure caused by an avalanche is
determined by examining the number of hours of pass closure per month (for December through
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-6