Service variability parameter of the Robust Queueing Theory (RQT) framework. More...

#include <cstddef>
#include <string>
#include "line/api/qsys/qsys_types.h"
#include "line/num/number.h"
#include "line/util/error.h"

Include dependency graph for qsys_gigk_rqt_gamma.h:

Namespaces
namespace	line
namespace	line::qsys

Functions
template<class T>
T	line::qsys::qsys_gigk_rqt_gamma (const T &rho, const T &mu, const T &Gamma_a, const T &sigma_s, std::size_t k, const T &alpha_a, const std::string &regime="independent")
	Service variability parameter of the Robust Queueing Theory (RQT) framework.

Detailed Description

Service variability parameter of the Robust Queueing Theory (RQT) framework.

Templated port of matlab/src/api/qsys/qsys_gigk_rqt_gamma.m, cross-checked against jar/src/main/java/jline/api/qsys/Qsys_gigk_rqt_gamma.java.

The adaptation of Section 7.1 turns the first two moments into an uncertainty set:

Gamma_s = (2 (theta0 + theta1 sigma_s^2/k + theta2 Gamma_a^2 rho^2 k))^((a-1)/a)

Gamma_a k^((a-1)/a),

with (theta0,theta1,theta2) regressed so that the worst-case system time of Theorem 3 approximates the MEAN system time of the corresponding stochastic queue. The arrival side needs no adaptation: Gamma_a = sigma_a for an external renewal stream. Since the last term cancels Gamma_a at alpha = 2, the adaptation acts on the sum Gamma_a + Gamma_s/k^(1/alpha) that Theorem 3 reads.

THE FACTOR 2 IS NOT IN THE PRINTED FORMULA and is restored here. Section 7.1 states that the functional form is motivated by Kingman's bound, which the alpha=2 bound of Theorem 3 reproduces when (Gamma_a+Gamma_s)^2 = 2(sigma_a^2+sigma_s^2); the published thetas are all near unity, i.e. corrections to that bound rather than a substitute for its factor 2. Dropping the factor puts M/M/1 about 40% BELOW its exact mean system time at rho = 0.9, contradicting the errors of at most 9.5% that Tables 2-3 report; restoring it gives +4.7%.

CAUTION: the form is not dimensionally homogeneous, since theta0 is an additive constant on a scale of variances, so it is only valid in the time unit the regression was run in. It is evaluated here in units of the mean service time, 1/mu = 1, and converted back.

ARITHMETIC. A real exponent makes this transcendental.

Reference: C. Bandi, D. Bertsimas, N. Youssef (2015). Robust Queueing Theory. Operations Research 63(3), 676-700, Section 7.1 and Table 1.

Definition in file qsys_gigk_rqt_gamma.h.

Namespaces

Functions

Detailed Description