Package jline.api.pfqn.sens

Class Pfqn_sens_mvaldmx

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

  • Method Details

    • pfqn_sens_mvaldmx

      public static Ret.pfqnSensMvaldmx pfqn_sens_mvaldmx(Matrix lambda, Matrix D, Matrix N, Matrix Z, Matrix mu, Matrix S)
      Exact second moments (variances and covariances) of the queue lengths of a mixed open/closed product-form queueing network with limited load-dependent service rates. This is the load-dependent and mixed counterpart of Pfqn_sens_mva, which is restricted to closed load-independent models.

      The method is the moment analysis of Akyildiz and Strelen. Their Theorem 1, equation (11), states that multiplying a queue-length moment by one further factor Q_jT costs one derivative with respect to a parameter y_j that scales the service demands s_ir of the classes r in T at station j:

         E[Q_jT^k ...] = d/dy_j E[Q_jT^(k-1) ...]|_{y_j=1}
                   + nbar_jT E[Q_jT^(k-1) ...]

      Taking k=2 and T={s} gives the second moment, hence

         Cov[n(i,r),n(j,s)] = d nbar(i,r) / dy_(j,s) |_{y=1}

      which is evaluated here by forward-mode differentiation of the mixed load-dependent MVA of Bruell-Balbo-Afshari, i.e. of exactly the recursion implemented by Pfqn_mvaldmx. The differentiated equations are (13) for the residence times, (15)-(17) for the conditional marginal probabilities, (18) for the throughputs, (19) and (24)-(31) for the effective capacities (delegated to Pfqn_sens_ldmx_ec), (32) for the closed-class queue lengths and (33) for the open-class ones.

      Because a demand-scaling parameter perturbs the whole network through the closed-class throughputs, the derivatives must be propagated for every parameter, so the cross-station covariances come out at no extra cost and are returned in QCovFull. This is unlike Pfqn_sens_mva, whose cheaper same-station recursion cannot reach them.

      Reference: 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. The closed load-independent case reduces to E. de Souza e Silva and R. R. Muntz, IEEE Trans. Computers 37(9):1125-1129, 1988, which Pfqn_sens_mva implements directly.

      Open classes are supported: an open class contributes to the load Lo(i) that drives the effective capacities, and equation (21) supplies the corresponding dLo/dy. The moments of an open class are those of its queue length at a station, which is finite even though its population is infinite.

      Parameters:
      lambda - arrival rate vector (1 x R); must be zero on closed classes
      D - service demand matrix (M x R)
      N - population vector (1 x R); Inf entries denote open classes
      Z - think time vector (1 x R)
      mu - load-dependent rate matrix (M x sum(N)), limited load dependence
      S - number of servers per station (M x 1); accepted for signature compatibility with pfqn_mvaldmx, which likewise does not read it: the multiserver behaviour is carried entirely by mu
      Returns:
      the base measures and their exact second moments

    • pfqn_sens_mvaldmx

      public static Ret.pfqnSensMvaldmx pfqn_sens_mvaldmx(Matrix lambda, Matrix D, Matrix N, Matrix Z)