Package jline.api.qsys

Class Qsys_hh1_lindley

java.lang.Object
jline.api.qsys.Qsys_hh1_lindley
public final class Qsys_hh1_lindley extends Object
Conditional waiting-time moments of the Hl/Hn/1 Lindley recursion.

Hyperexponential primitives are mixtures of exponentials, so conditioning on the arrival phase i and the service phase j reduces one Lindley step to the M/M/1 step of Qsys_mm1_lindley at rates lambda[i] and mu[j], and the conditional moment is the corresponding mixture 

   E[W_{n+1}^m | W_n] = sum_i sum_j pa[i] ps[j] E_{ij}[W_{n+1}^m | W_n].

Phases are drawn independently for each customer, which is what makes the mixture exact rather than an approximation; a Markov-modulated arrival stream would not decompose this way.

Note that the variance is not the corresponding mixture of the per-phase variances, because the phase is itself random: it is recovered here from the first two mixed raw moments, which adds the between-phase spread of the means.

Port of MATLAB qsys_hh1_lindley.m. Verified against 4e6 Monte Carlo replications to 2e-3 relative error for m = 1, 2.

Reference: S. Palomo, J. Pender, "Learning the Tandem Network Lindley Recursion", Proc. Winter Simulation Conference, 2021, theorem 3.

Since:
LINE 3.1.0

  • Method Details

    • qsys_hh1_lindley

      public static QsysLindleyResult qsys_hh1_lindley(double[] lambda, double[] pa, double[] mu, double[] ps, double[] Wn)
      Conditional mean and variance of the next waiting time.
      Parameters:
      lambda - arrival phase rates, positive
      pa - arrival phase probabilities, nonnegative and summing to 1
      mu - service phase rates, positive
      ps - service phase probabilities, nonnegative and summing to 1
      Wn - current waiting times, finite and nonnegative
      Returns:
      the conditional moments up to order 2

    • qsys_hh1_lindley

      public static QsysLindleyResult qsys_hh1_lindley(double[] lambda, double[] pa, double[] mu, double[] ps, double[] Wn, int mmax)
      Conditional raw moments of the next waiting time.
      Parameters:
      lambda - arrival phase rates, positive
      pa - arrival phase probabilities, nonnegative and summing to 1
      mu - service phase rates, positive
      ps - service phase probabilities, nonnegative and summing to 1
      Wn - current waiting times, finite and nonnegative
      mmax - highest moment order, at least 1; raised to 2 so the variance is always available
      Returns:
      the conditional moments