fj_xmax_hyperexp.h File Reference

Expected maximum of K i.i.d. More...

#include "line/api/fj/fj_types.h"
#include "line/num/number.h"
#include "line/util/error.h"

Include dependency graph for fj_xmax_hyperexp.h:

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Namespaces
namespace	line
namespace	line::fj

Functions
template<class T>
T	line::fj::fj_xmax_hyperexp (unsigned K, const T &p1, const T &mu1, const T &mu2)
	Expected maximum of K i.i.d.

Detailed Description

Expected maximum of K i.i.d.

two-phase hyperexponential service times.

Templated port of matlab/src/api/fj/fj_xmax_hyperexp.m.

X_K^max = sum_{n=1..K} (-1)^{n+1} sum_{m=0..n} C(n,m) p1^m p2^{n-m} / (m mu1 + (n-m) mu2)

from the inclusion-exclusion expansion of 1 - F(x)^K. Every term is rational, so the sum is exact in the field – which is the point, because it alternates: the terms grow like 2^K while the result stays O(log K/mu), so in double the answer is destroyed by cancellation somewhere around K = 25 and is pure noise by K = 40.

REFERENCE DEFECT: FJ_xmax.fj_xmax_hyperexp in jar/src/main/java/jline/api/fj/FJ_xmax.java drops the p1^m p2^{n-m} factor from the inner sum, so the JAR computes a different quantity and ignores p1 entirely except in its validation. MATLAB is ground truth and is what this port follows.

Definition in file fj_xmax_hyperexp.h.