cme.h File Reference

Concentrated matrix exponentials, and the two-moment fit built on them. More...

#include <algorithm>
#include <cmath>
#include <cstddef>
#include <vector>
#include "line/api/mam/iltcme_table.h"
#include "line/api/mam/map_moment.h"
#include "line/num/number.h"
#include "line/util/error.h"
#include "line/util/matrix.h"

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Classes
struct	line::mam::CmeRepresentation< T >
	The unit-mean (alpha, A) form of a CME, with the SCV it attains. More...

Namespaces
namespace	line
namespace	line::mam

Functions
std::vector< std::size_t >	line::mam::cme_supported_orders ()
	Every phase count 2n+1 the vendored table realizes, ascending.
const iltcme::CmeEntry &	line::mam::cme_table_entry (std::size_t order)
	The most concentrated table entry realizing order phases.
double	line::mam::cme_min_scv (std::size_t order)
	The minimal SCV a CME of this order attains.
template<class T>
CmeRepresentation< T >	line::mam::cme_representation (std::size_t order)
template<class T>
Map< T >	line::mam::me_to_map (const std::vector< T > &alpha, const Matrix< T > &A)
	Assemble the renewal (D0, D1) of a matrix-exponential law (alpha, A).
template<class T>
Map< T >	line::mam::dist_fit_me (double mean, double scv, std::size_t maxPhases=0)
	Two-moment matrix-exponential fit for 0 < scv < 1, a port of dist_fit_me.m.

Detailed Description

Concentrated matrix exponentials, and the two-moment fit built on them.

Templated port of matlab/src/lang/processes/CME.m and dist_fit_me.m (mirrored by jline.lang.processes.CME / MEFit and the native Python fit_me_mean_scv).

A CME of order 2n+1 is the matrix-exponential law whose squared coefficient of variation is numerically minimal for that order, from the tables of Horvath, Horvath and Telek. Its SCV decays as O(1/n^2), so it reaches far below the Erlang bound 1/order that binds any phase-type of the same order: at 101 phases a CME reaches 3.9e-4 where Erlang-101 stops at 9.9e-3. The unit-mean density with n harmonics is f(x) = mu1 e^{-mu1 x} ( c + sum_k a_k cos(k w mu1 x) + b_k sin(k w mu1 x) ), which is alpha exp(A x) (-A e) for the block-diagonal A = blkdiag( -mu1, mu1 [-1, -k w; k w, -1], k = 1..n ).

The coefficients come from the SAME vendored iltcme table the CME inverse Laplace transform reads (api/mam/iltcme_table.h), so the two share one source of truth; a TU using this header must link src/api/mam/iltcme_table.cpp.

IT IS NOT A PHASE-TYPE. The off-diagonal entries of A are not rates – the rotation blocks carry a negative one – so a CTMC assembled from a CME does not describe the model. sn_is_phasetype is the test every consumer applies, and sn_nonmarkov_toph tags the result ME rather than PH on the strength of it.

ARITHMETIC: transcendental. The two-moment fit takes a square root and the table itself is a double table, so this does not instantiate under Rational.

Definition in file cme.h.