Package jline.api.qsys

Class Qsys_gigk_rqt_gamma

java.lang.Object
jline.api.qsys.Qsys_gigk_rqt_gamma
public final class Qsys_gigk_rqt_gamma extends Object

  • Method Summary

    Modifier and Type
    Method
    Description
    static double
    qsys_gigk_rqt_gamma(double rho, double mu, double Gamma_a, double sigma_s, int k)
    Service variability parameter under the service-distribution-independent regime and the finite-variance tail coefficient alpha = 2.
    static double
    qsys_gigk_rqt_gamma(double rho, double mu, double Gamma_a, double sigma_s, int k, double alpha_a, String regime)
    Service variability parameter of the RQT framework, obtained from the first two moments by the adaptation of Section 7.1, 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), where (theta0,theta1,theta2) are regressed so that the worst-case system time of Theorem 3 approximates the MEAN system time of the corresponding stochastic queue.

    Methods inherited from class java.lang.Object

    clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

  • Method Details

    • qsys_gigk_rqt_gamma

      public static double qsys_gigk_rqt_gamma(double rho, double mu, double Gamma_a, double sigma_s, int k, double alpha_a, String regime)
      Service variability parameter of the RQT framework, obtained from the first two moments by the adaptation of Section 7.1, 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), where (theta0,theta1,theta2) are 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.

      Parameters:
      rho - traffic intensity lambda/(k*mu)
      mu - service rate of each server, which sets the time unit
      Gamma_a - variability parameter of the arrival uncertainty set
      sigma_s - standard deviation of the service time
      k - number of servers
      alpha_a - effective arrival tail coefficient in (1,2]
      regime - adaptation regime of Table 1: "independent" (service distribution unknown), "normal" or "pareto"
      Returns:
      the service variability parameter Gamma_s

    • qsys_gigk_rqt_gamma

      public static double qsys_gigk_rqt_gamma(double rho, double mu, double Gamma_a, double sigma_s, int k)
      Service variability parameter under the service-distribution-independent regime and the finite-variance tail coefficient alpha = 2.
      Parameters:
      rho - traffic intensity lambda/(k*mu)
      mu - service rate of each server
      Gamma_a - variability parameter of the arrival uncertainty set
      sigma_s - standard deviation of the service time
      k - number of servers
      Returns:
      the service variability parameter Gamma_s