pfqn_grnmol.h File Reference

Normalizing constant by the closed-form Grundmann-Moeller rule. More...

#include <cstddef>
#include <vector>
#include "line/num/number.h"
#include "line/util/error.h"
#include "line/util/matrix.h"

Include dependency graph for pfqn_grnmol.h:

Go to the source code of this file.

Namespaces
namespace	line
	Conservation laws of a layered queueing network, enumerated from its structure.
namespace	line::pfqn

Functions
template<class T>
T	line::pfqn::pfqn_grnmol (const Matrix< T > &L, const std::vector< int > &N)
	Normalizing constant by the closed-form Grundmann-Moeller rule.

Detailed Description

Normalizing constant by the closed-form Grundmann-Moeller rule.

Templated port of matlab/src/api/pfqn/pfqn_grnmol.m, which applies the rule directly rather than through the successive-degree recursion of pfqn_cub:

G = [ (sum N + M - 1)! / prod_r N_r! ] * sum_{i=0}^{S} w_i H_i w_i = 2^{-2S} (-1)^i c_i^{2S+1} / (i! (i + c_i)!), c_i = 2(S-i) + M H_i = sum_{|b| = S-i} prod_r ( ((2b+1)/c_i)' L(:,r) )^{N_r}

with b ranging over the M-vectors of non-negative integers summing to S-i, which is what the reference's call to matlab/util/sprod.m enumerates. The point (2b+1)/c_i is barycentric by construction, since sum_m (2b_m+1) = c_i.

DEGREE. pfqn_grnmol.m sets S = ceil((sum(N)-1)/2), the value pfqn_cub.m uses, so the rule is exact for every total population, odd or even. (An earlier reference wrote ceil(sum(N)-1)/2, a half-integer S for even sum(N) that made the function uncallable there; MATLAB has been corrected and this port follows it, accepting even populations.)

ARITHMETIC. Every ingredient – the barycentric points, the binomial-like weights, the integer powers – is rational, and the factorial prefactor is formed as an exact factorial rather than through gammaln. The routine is therefore EXACT in rational arithmetic and is deliberately left ungated: at the reference degree it returns the exact normalizing constant, and checking that against pfqn_ca is the sharpest test the rule admits.

Definition in file pfqn_grnmol.h.