Package jline.api.pfqn.sens

Class Pfqn_sens_mva

java.lang.Object
jline.api.pfqn.sens.Pfqn_sens_mva
public final class Pfqn_sens_mva extends Object

  • Method Details

    • pfqn_sens_mva

      public static Ret.pfqnSensMva pfqn_sens_mva(Matrix L, Matrix N, Matrix Z, Matrix mi)
      Exact second moments (variances and per-station covariances) of the queue lengths of a closed product-form (BCMP) queueing network. The moments are obtained by an MVA-type recursion evaluated on the same population lattice as pfqn_mva, so no derivative of the model is ever formed and the cost is O(M*R^2) per lattice point rather than the O(M^2*R^2) of the differentiated-MVA algorithm used by Pfqn_sens.

      The recursion is obtained by differentiating the Reiser-Lavenberg MVA equation Q(j,v|N) = X(v|N) * L(j,v) * (mi(j) + Qtot(j|N-e_v)) with respect to the visit ratio theta(i,k) of class k at station i and rescaling. Writing W(k,i;v,j|N) = Cov[n(i,k),n(j,v)] at population N,

         W(k,i;v,j|N) = Q(j,v|N) * ( Q(i,k|N-e_v) - Q(i,k|N) )
                + [i==j & k==v] * Q(j,v|N)
                + X(v|N) * L(j,v) * sum_t W(k,i;t,j|N-e_v)

      with W(.|0) = 0. This routine evaluates the same-station case i==j, which is self-contained: the inner sum then only involves same-station terms, so a single scalar Ssum(j,k|N) = sum_t W(k,j;t,j|N) carried along the lattice closes the recursion. The cross-station case i!=j is not self-contained (it couples every station pair) and costs as much as the full Jacobian, so it is left to Pfqn_sens.

      Setting i==j and mi==1 reproduces Corollary 1 of the reference below, i.e. its equations (2.9a) for the variance and (2.10) for the covariance; the station multiplicity mi cancels identically because X(v|N)*L(j,v)*(mi(j)+Qtot(j|N-e_v)) = Q(j,v|N) is the MVA equation for any mi. The equivalent statement for an infinite-server station, equation (2.9b) of the reference, is recovered automatically because LINE folds the delay into the think time Z, which enters only through X(v|N) and carries no queue-length moment of its own.

      Reference: E. de Souza e Silva and R. R. Muntz, "Simple Relationships Among Moments of Queue Lengths in Product Form Queueing Networks", IEEE Trans. Computers 37(9):1125-1129, 1988 (Theorems 1-2 and Corollary 1). The underlying identity Cov[n(i,k),n(j,v)] = theta(i,k) * dQ(j,v)/dtheta(i,k) is the k=2 case of Theorem 1 of I. F. Akyildiz and J. C. Strelen, "Moment Analysis for Load-Dependent Mixed Product Form Queueing Networks", IEEE Trans. Communications 39(6):828-832, 1991.

      Restricted to closed populations. Mixed and load-dependent models are handled by Pfqn_sens_mvaldmx. For a station of multiplicity mi(i)>1, which LINE treats as mi(i) identical replicas sharing the demand row L(i,:), the moments returned are those of the aggregate queue length over the replicas.

      Parameters:
      L - service demand matrix (M x R), L(i,r) = visits_ir / rate_ir
      N - population vector (1 x R)
      Z - think time vector (1 x R), null or empty for zeros
      mi - station server multiplicity (1 x M), null for ones
      Returns:
      the base measures and their exact second moments

    • pfqn_sens_mva

      public static Ret.pfqnSensMva pfqn_sens_mva(Matrix L, Matrix N, Matrix Z)

    • pfqn_sens_mva

      public static Ret.pfqnSensMva pfqn_sens_mva(Matrix L, Matrix N)