pfqn_sens_momldmx.h File Reference

Exact higher queue-length moments (up to order three) of mixed open/closed product-form networks with limited load dependence. More...

#include <cmath>
#include <cstddef>
#include <vector>
#include "line/api/pfqn/pfqn_sens_ldmx_ec.h"
#include "line/api/pfqn/pfqn_sens_mom.h"
#include "line/num/number.h"
#include "line/util/error.h"
#include "line/util/matrix.h"
#include "line/util/population.h"

Include dependency graph for pfqn_sens_momldmx.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>
SensMomResult< T >	line::pfqn::pfqn_sens_momldmx (const std::vector< T > &lambda, const Matrix< T > &D, const std::vector< int > &N, const std::vector< T > &Z, const Matrix< T > &mu, const std::vector< int > &groups)
	Exact moments E[Q], Var[Q], E[Q^2], E[Q^3] and covariances of the grouped queue lengths of a mixed load-dependent product-form network.

Detailed Description

Exact higher queue-length moments (up to order three) of mixed open/closed product-form networks with limited load dependence.

Templated port of matlab/src/api/pfqn/pfqn_sens_momldmx.m. Theorem 1 of Akyildiz and Strelen (IEEE TComm 39(6):828-832, 1991) states that one further factor Q_(j,T) in a moment costs one derivative with respect to a parameter y_j scaling the demands of the classes in T at station j. With m = E[Q_(j,T)]:

Var[Q_jT] = dm/dy_j E[Q_jT^3] = d2m/dy_j^2 + (1 + 3m) dm/dy_j + m^3 Cov[Q_iT,Q_jT'] = dm_iT/dy_(j,T')

The derivatives come from second-order forward-mode differentiation of the Bruell-Balbo-Afshari mixed load-dependent MVA (the recursion of pfqn_mvaldmx and pfqn_sens_mvaldmx), with the effective-capacity second derivatives supplied by pfqn_sens_ldmx_ec. The result struct is the one of pfqn_sens_mom, so the two are interchangeable on closed load-independent input.

Arithmetic. Field-only apart from the skewness, as in pfqn_sens_mom; the primal Pc(0) keeps the reference's eps floor.

Definition in file pfqn_sens_momldmx.h.