line::mmdp Namespace Reference

Classes
struct	MmdpPair
	The (Q, R) pair of an MMDP. More...

Functions
template<class T>
bool	mmdp_isfeasible (const Matrix< T > &Q, const Matrix< T > &R, double tol=1e-10)
	True when (Q, R) is a valid MMDP.
template<class T>
std::vector< T >	mmdp_rates (const Matrix< T > &R)
	The per-state rates, i.e.
template<class T>
T	mmdp_mean_rate (const Matrix< T > &Q, const Matrix< T > &R)
	Stationary mean flow rate, pi diag(R).
template<class T>
T	mmdp_scv (const Matrix< T > &Q, const Matrix< T > &R)
	SCV of the RATE under the stationary law: Var[r] / E[r]^2.
template<class T>
MmdpPair< T >	mmdp_from_map (const Matrix< T > &D0, const Matrix< T > &D1)
	The MMDP of a MAP: Q = D0 + D1, R = diag(row sums of D1).
template<class T>
MmdpPair< T >	mmdp2 (const T &r0, const T &r1, const T &sigma0, const T &sigma1)
	The two-state MMDP, in the reference's own parameterization.
template<class T>
T	mmdp2_mean_rate (const T &r0, const T &r1, const T &sigma0, const T &sigma1)
	The closed-form mean rate of a two-state MMDP.
template<class T>
T	mmdp2_scv (const T &r0, const T &r1, const T &sigma0, const T &sigma1)
	The closed-form SCV of a two-state MMDP.

Function Documentation

mmdp2()

template<class T>

MmdpPair< T > line::mmdp::mmdp2	(	const T &	r0,
		const T &	r1,
		const T &	sigma0,
		const T &	sigma1 )

The two-state MMDP, in the reference's own parameterization.

sigma0 is the rate out of state 0 and sigma1 the rate out of state 1, so the generator is [[-s0, s0], [s1, -s1]] and the stationary law is (s1, s0)/(s0+s1) – the chain spends LONGER in the state it leaves more slowly, which is why the weights look swapped.

Definition at line 170 of file mmdp.h.

References line::InputError::InputError(), line::Matrix< T >::Matrix(), mmdp2(), line::mmdp::MmdpPair< T >::Q, and line::mmdp::MmdpPair< T >::R.

Referenced by mmdp2().

mmdp2_mean_rate()

template<class T>

T line::mmdp::mmdp2_mean_rate	(	const T &	r0,
		const T &	r1,
		const T &	sigma0,
		const T &	sigma1 )

The closed-form mean rate of a two-state MMDP.

Definition at line 188 of file mmdp.h.

References mmdp2_mean_rate().

Referenced by mmdp2_mean_rate().

mmdp2_scv()

template<class T>

T line::mmdp::mmdp2_scv	(	const T &	r0,
		const T &	r1,
		const T &	sigma0,
		const T &	sigma1 )

The closed-form SCV of a two-state MMDP.

Definition at line 194 of file mmdp.h.

References mmdp2_scv().

Referenced by mmdp2_scv().

mmdp_from_map()

template<class T>

MmdpPair< T > line::mmdp::mmdp_from_map	(	const Matrix< T > &	D0,
		const Matrix< T > &	D1 )

The MMDP of a MAP: Q = D0 + D1, R = diag(row sums of D1).

The row sum of D1 is the total ARRIVAL rate out of a phase, so the fluid analogue flows at exactly the rate at which that phase would be generating arrivals. The MAP's phase process is unchanged, which is why Q is its full generator.

Definition at line 143 of file mmdp.h.

References line::Matrix< T >::cols(), line::InputError::InputError(), line::Matrix< T >::Matrix(), mmdp_from_map(), line::mmdp::MmdpPair< T >::Q, line::mmdp::MmdpPair< T >::R, and line::Matrix< T >::rows().

Referenced by mmdp_from_map().

mmdp_isfeasible()

template<class T>

bool line::mmdp::mmdp_isfeasible	(	const Matrix< T > &	Q,
		const Matrix< T > &	R,
		double	tol = 1e-10 )

True when (Q, R) is a valid MMDP.

Q must be a generator – square, non-positive diagonal, non-negative off-diagonal, rows summing to zero – and R must be DIAGONAL with non-negative entries. The diagonality is not a storage convention: a non-diagonal R would make the flow rate depend on a transition rather than on the state, which is a different process.

Definition at line 58 of file mmdp.h.

References line::Matrix< T >::cols(), mmdp_isfeasible(), and line::Matrix< T >::rows().

Referenced by mmdp_isfeasible().

mmdp_mean_rate()

template<class T>

T line::mmdp::mmdp_mean_rate	(	const Matrix< T > &	Q,
		const Matrix< T > &	R )

Stationary mean flow rate, pi diag(R).

Definition at line 88 of file mmdp.h.

References line::Matrix< T >::cols(), line::mc::ctmc_solve(), line::InputError::InputError(), mmdp_mean_rate(), and line::Matrix< T >::rows().

Referenced by mmdp_mean_rate().

mmdp_rates()

template<class T>

std::vector< T > line::mmdp::mmdp_rates

(

const Matrix< T > &

R

)

The per-state rates, i.e.

the diagonal of R.

Definition at line 80 of file mmdp.h.

References mmdp_rates(), and line::Matrix< T >::rows().

Referenced by mmdp_rates().

mmdp_scv()

template<class T>

T line::mmdp::mmdp_scv	(	const Matrix< T > &	Q,
		const Matrix< T > &	R )

SCV of the RATE under the stationary law: Var[r] / E[r]^2.

A one-state process has no variability and returns zero. A mean rate of zero leaves the ratio undefined, and the reference returns infinity rather than dividing – that is a signal the caller can test, where a NaN is not.

Definition at line 109 of file mmdp.h.

References line::Matrix< T >::cols(), line::mc::ctmc_solve(), line::InputError::InputError(), mmdp_scv(), and line::Matrix< T >::rows().

Referenced by mmdp_scv().