Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Equation 2 - Gumbel Probability Density Function
−r T −t
P( I,t)= e ( ) ⎧
⎛
I( e −h−1 ξ − e − H −1 ξ
)− V 0 ( 1 − µ + σ ξ) ( e − H −1 ξ − e −h−1 ξ
)− σ ξ Γ 1 − ⎞ ⎫
⎨
( ξ,h−1 ξ , H −1 ξ
)
⎝
⎜
⎠
⎟ ⎬
⎩⎪
⎭⎪
where
H = 1 + ξ ⎛
σ 1− I ⎞
− µ
⎝
⎜ V 0 ⎠
⎟
h = 1 + ξ ( σ 1− µ )
µ = the location parameter of the estimated EV function
σ = the scale parameter of the estimated EV function
ξ = the shape parameter of the estimated EV function
Γ( )= the incomplete gamma distribution
The location, scale, and shape parameters of EV distributions are estimated by special
fitting procedures applied to the random variable. The parameter, x, represents the event
frequency or system performance data that are distributed. This could include such as
speed (delay) data for a long time period. The procedures for estimating the parameters
are available in Stata® and similar, comprehensive statistical software packages or in
standalone software such as MathWave®. The location and scale parameters for the
Gumbel distribution can be estimated using the Gumbel distribution fitting options from
these statistical software packages or by using standalone software.
Valuing Unreliability for Processes Characterized by Extreme Value
Distributions
Valuing options when the value of the process of interest follows an EV distribution is
conceptually similar to the process described earlier for log-normally distributed values.
However, the mathematics is more complicated and the role of time in the methodology
is more pronounced because the option’s life occurs over a longer period of time.
The precise formulation of the valuation formula depends on the data available and the
unreliability process being examined. In the case in which the speed metric is a
distributed EV, and the valuation at the end of the option life is appropriate (similar to the
log-normal case), a closed form of a European put formulation is available (Markose and
Alentorn, 2005). The value of the put can be calculated for a European put option with a
strike price (speed guarantee) of I and a life (evaluation interval) of t using Equation 3.
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-5