poisson
Computes the second-order loss function in the second factorial moment form, as in Zipkin, Foundations of Inventory Management (2000):
G2(x) = (1/2) * Emax(X - x, 0) * max(X - x - 1, 0)
At x = 0 this evaluates to 0.5 * mean^2 (because for Poisson EX(X-1) = mean^2), NOT 0.5 * (mean^2 + mean).
Note: some references define second-order loss as 0.5 * Emax(X - x, 0)^2 instead. The two conventions differ by mean/2 at x = 0. KSL uses the factorial-moment form throughout.
Return
The loss function value, (1/2) Emax(X-x,0) * max(X-x-1,0)
Parameters
x
The value to be evaluated
mean
of the distribution
recursive
true indicates that the recursive logarithmic algorithm should be used