Package jline.api.pfqn.nc

Class Pfqn_rgfmc

java.lang.Object
jline.api.pfqn.nc.Pfqn_rgfmc
public final class Pfqn_rgfmc extends Object
Harrison and Coury, "On the asymptotic behaviour of closed multiclass queueing networks", Perf. Eval. 47:131-138, 2002, Thm 1, as algorithmised by Harrison and Lee, "A new recursive algorithm for computing generating functions in closed multi-class queueing networks", IEEE MASCOTS 2004, eqs. (4)-(5).

THINK TIMES ARE NOT IN EITHER PAPER. Both write the generating function as the RATIONAL prod_i (1 - rho_i z)^-m_i, every node a load-independent single server. An infinite server multiplies it by the ENTIRE exp(sum_r Z_r z_r), which breaks the step Thm 1 rests on: G_n(z') = -sum_i r_i holds only because the residues of n(z)/d(z) sum to zero when deg d >= deg n + 2 (Bertozzi and McKenna, SIAM Review 35(2):239-268, 1993, fact (IV), p. 246), and an exponential numerator does not decay at infinity. The delay is therefore carried by their own repair, eqs. (3.19)-(3.21): only the first k_r+1 Taylor coefficients of exp(Z_r z_r) can reach the coefficient of z_r^k_r, so replacing the exponential by that polynomial is EXACT and leaves a rational integrand. The price is that the eliminated class's population re-enters the term count, which is the population-insensitivity Harrison-Lee Sec. 4 advertises; the class kept for the base case pays nothing.

The elimination is exact in exact arithmetic but is an ALTERNATING sum over residues, so near-coincident loads over an eliminated class destroy significance. The worst cancellation ratio is tracked and the routine REFUSES past maxcancel rather than returning a confidently wrong lG.

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static final class 
    Pfqn_rgfmc.Result
    Normalizing constant and its logarithm.

  • Field Summary

    Fields
    Modifier and Type
    Field
    Description
    static final double
    DEF_MAXCANCEL
    Default nats of cancellation tolerated before refusing.
    static final int
    DEF_MAXTERMS
    Default cap on residue terms carried between eliminations.
    static final double
    DEF_TOL
    Default relative tolerance for calling two affine forms proportional.

  • Method Summary

    Modifier and Type
    Method
    Description
    static Pfqn_rgfmc.Result
    pfqn_rgfmc(Matrix L, double[] N, double[] Z)
    Exact multiclass normalizing constant by recursion on generating functions.
    static Pfqn_rgfmc.Result
    pfqn_rgfmc(Matrix L, double[] N, double[] Z, double tol, int maxterms, double maxcancel)
     

    Methods inherited from class java.lang.Object

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

  • Field Details

    • DEF_TOL

      public static final double DEF_TOL
      Default relative tolerance for calling two affine forms proportional.
      See Also:

    • DEF_MAXTERMS

      public static final int DEF_MAXTERMS
      Default cap on residue terms carried between eliminations.
      See Also:

    • DEF_MAXCANCEL

      public static final double DEF_MAXCANCEL
      Default nats of cancellation tolerated before refusing.
      See Also:

  • Method Details

    • pfqn_rgfmc

      public static Pfqn_rgfmc.Result pfqn_rgfmc(Matrix L, double[] N, double[] Z)
      Exact multiclass normalizing constant by recursion on generating functions.
      Parameters:
      L - service demand matrix (M x R)
      N - population vector (R), nonnegative integers
      Z - think time vector (R)
      Returns:
      the normalizing constant and its logarithm

    • pfqn_rgfmc

      public static Pfqn_rgfmc.Result pfqn_rgfmc(Matrix L, double[] N, double[] Z, double tol, int maxterms, double maxcancel)
      Parameters:
      L - service demand matrix (M x R)
      N - population vector (R), nonnegative integers
      Z - think time vector (R)
      tol - relative tolerance for calling two affine forms proportional
      maxterms - cap on residue terms carried between eliminations
      maxcancel - nats of cancellation tolerated before refusing
      Returns:
      the normalizing constant and its logarithm