map_pnt.h File Reference

Counting probabilities of a MAP: P_n(t), the matrix whose (i,j) entry is the probability of n arrivals in [0,t) ending in phase j, given phase i at 0. More...

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

Include dependency graph for map_pnt.h:

Go to the source code of this file.

Namespaces
namespace	line
namespace	line::mam

Functions
template<class T>
std::vector< Matrix< T > >	line::mam::map_pntbisect (const Map< T > &m, std::size_t na, const T &t)
	P_0(t) .
template<class T>
std::vector< Matrix< T > >	line::mam::map_pnt (const Map< T > &m, std::size_t na, const T &t, long M=-1)
	P_0(t) .
template<class T>
std::vector< Matrix< T > >	line::mam::map_pntquad (const Map< T > &m, std::size_t na, const T &t)
	The same counting probabilities by NUMERICAL INTEGRATION, map_pntquad.
template<class T>
Matrix< T >	line::mam::map_pntiter (const Map< T > &m, std::size_t na, const T &t, long M=-1)
	The reference's entry point: only the highest count is returned.

Detailed Description

Counting probabilities of a MAP: P_n(t), the matrix whose (i,j) entry is the probability of n arrivals in [0,t) ending in phase j, given phase i at 0.

Port of matlab/lib/kpctoolbox/map/map_pntiter.m (and its map_pntbisect helper), mirrored by jline.api.mam.Map_pntiter and the native Python map_pntiter.

The method is uniformization. With tau = max_i |D0(i,i)|, K = D0/tau + I and K1 = D1/tau, the number of uniformization steps in [0,t) is Poisson(tau t), and conditioning on it gives

V(0,0) = I, V(0,k) = V(0,k-1) K, V(n,k) = V(n,k-1) K + V(n-1,k-1) K1, P_n(t) = sum_{k=0..N} w_k V(n,k), w_k = e^{-tau t} (tau t)^k / k!,

truncated at the N for which the Poisson tail falls below machine epsilon. map_pntiter evaluates this at t/2^M and then squares M times through the discrete convolution P_n <- sum_{j=0..n} P_j P_{n-j}, which is exact because the counts over disjoint intervals add and the phase at the split is summed over by the matrix product.

REFERENCE DEFECT, NOT REPRODUCED (found 2026-08-01, verified in MATLAB R2025a). map_pntbisect.m weights V(n,k) by w_n instead of w_k – the Poisson weight of the ARRIVAL count rather than of the uniformization step count – and never propagates V(0,k), leaving it zero for every k >= 1. Both are wrong and neither is visible on a Poisson process, because there K = 0 collapses V(n,k) to delta(n,k) and the two weights coincide on the only surviving term. On an Erlang-2 MAP of mean 1 at t = 1.3 the reference returns, against the identities any counting law must satisfy:

P_0(t) vs exp(D0 t) max |diff| = 1.93e-1 sum_n P_n(t) vs exp((D0+D1) t) max |diff| = 4.97e-1 sum_n n pie P_n(t) e vs lambda t 0 against 1.3

The version here satisfies all three to round-off, and the tests assert them. map_pntiter and map_pntquad have NO CALLER in any codebase, so the defect is latent and changes no published result; fixing MATLAB, the JAR and Python is tracked separately.

ARITHMETIC: transcendental, for the Poisson weights.

Definition in file map_pnt.h.