mfq_ld_mean.h File Reference

Stationary mean fluid level E[X] of a first- or second-order level-dependent (multi-regime) Markovian fluid queue, in closed form from the matrix-exponential building blocks. More...

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

Include dependency graph for mfq_ld_mean.h:

Go to the source code of this file.

Classes
struct	line::mam::LevelDependentFluidBlocks< T >
	The matrix-exponential building blocks of a multi-regime fluid queue, the output of mfq_ld_solve. More...

Namespaces
namespace	line
namespace	line::mam

Functions
template<class T>
T	line::mam::mfq_ld_mean (const LevelDependentFluidBlocks< T > &b)
	Stationary mean fluid level of a level-dependent fluid queue.

Detailed Description

Stationary mean fluid level E[X] of a first- or second-order level-dependent (multi-regime) Markovian fluid queue, in closed form from the matrix-exponential building blocks.

Port of matlab/src/api/mam/mfq_ld_mean.m and the BUTools LevelDependentFluidStationaryMean it wraps. Its inputs are exactly the outputs of mfq_ld_solve, and it calls nothing else, so it is complete on its own terms: a caller holding the blocks from any source can use it.

THE DENSITY. Over regime k, that is over the level interval [T(k), T(k+1)], the stationary density is the sum of a forward and a backward matrix exponential,

pi_k(x) = iniF_k exp(KF_k (x - T(k))) cloF_k
        + iniB_k exp(KB_k (T(k+1) - x)) cloB_k,

one anchored at each end of the regime, plus point masses at the K+1 thresholds. E[X] is then the mass contribution sum_j T(j) masses_j e plus the integral of x pi_k(x) over each regime.

THE INTEGRALS. Both regime integrals reduce to J0 = int_0^L exp(M u) du and J1 = int_0^L u exp(M u) du, which are read off ONE matrix exponential of the nilpotent augmentation

A = [ M  I  0 ;  0  0  I ;  0  0  0 ],   W = exp(A L),
J0 = W(1:n, n+1:2n),   J1 = L J0 - W(1:n, 2n+1:3n).

That form is used rather than the obvious M^-1 (exp(M L) - I) because KF and KB are SINGULAR whenever a regime has a zero-drift direction, which is the normal case and not an edge case; the augmentation never inverts anything and is exact for a singular M. The forward integral is anchored at T(k), giving the weight T(k) J0 + J1, and the backward one runs the other way, giving T(k+1) J0 - J1.

ARITHMETIC. Gated on num_traits<T>::has_transcendental because it calls expm, which is a tolerance-controlled Pade approximation in any arithmetic. The rest is finite exact linear algebra.

Definition in file mfq_ld_mean.h.