Package jline.api.pfqn.nc

Class Pfqn_clw_lld

java.lang.Object

jline.api.pfqn.nc.Pfqn_clw_lld

public final class Pfqn_clw_lld
extends Object

Computes g(K) of a multichain closed product-form network with limited load-dependent (LLD) stations and (optionally) infinite-server delay by numerically inverting its p-dimensional generating function (Bertozzi-McKenna eqs. 2.17/2.23) G(z) = exp(sum_j rho_{j0} z_j) prod_i F_i(sum_j rho_{ji} z_j) where F_i is the transform of the station factor of queue i (eq. 2.16) with load-dependent rate scalings S_i(k) = mu(i,k). For an LLD queue, S_i(k) = c_i constant for k >= l_i, and F_i is the rational function (eq. 2.19) F_i(x) = [c_i + sum_{n=1}^{l_i-1} (c_i - S_i(n)) / prod_{k=1}^n S_i(k) x^n] / (c_i - x), analytic except for a simple pole at x = c_i. Multiserver and load-independent queues are special cases. Since g(K) depends on S_i(k) only for k <= sum(K), general load-dependent input is truncated to LLD at sum(K) without loss of exactness. g(K) is recovered by p nested one-dimensional lattice-Poisson inversions (CLW eq. 2.3) with restrictive static scaling adapted from CLW eqs. 5.41-5.46 (each queue normalized by its pole c_i, simple pole) and log-domain recovery (eq. 7.1). Cost is prod_j 2 l_j K_j contour points, each of cost O(sum_i l_i); practical for moderate populations and few chains.

Method Summary

Modifier and Type	Method	Description
static Ret.pfqnNc	pfqn_clw_lld(Matrix L, Matrix N)
static Ret.pfqnNc	pfqn_clw_lld(Matrix L, Matrix N, Matrix Z)
static Ret.pfqnNc	pfqn_clw_lld(Matrix L, Matrix N, Matrix Z, Matrix mu)
static Ret.pfqnNc	pfqn_clw_lld(Matrix L, Matrix N, Matrix Z, Matrix mu, Matrix lpar, Matrix gampar)

Methods inherited from class java.lang.Object

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

Method Details

pfqn_clw_lld

public static Ret.pfqnNc pfqn_clw_lld(Matrix L,
 Matrix N)

pfqn_clw_lld

public static Ret.pfqnNc pfqn_clw_lld(Matrix L,
 Matrix N,
 Matrix Z)

pfqn_clw_lld

public static Ret.pfqnNc pfqn_clw_lld(Matrix L,
 Matrix N,
 Matrix Z,
 Matrix mu)

pfqn_clw_lld

public static Ret.pfqnNc pfqn_clw_lld(Matrix L,
 Matrix N,
 Matrix Z,
 Matrix mu,
 Matrix lpar,
 Matrix gampar)

Parameters:

L - (q' x p) single-server relative traffic intensities, L(i,j)=rho_{ji}.

N - (1 x p or p x 1) closed-chain population vector K.

Z - (1 x p) aggregate infinite-server relative intensities rho_{j0}; null = 0.

mu - (q' x n) load-dependent rate scalings mu(i,k) = S_i(k); if fewer than sum(K) columns are given the last column is extended (LLD assumption); null = ones (all queues load-independent).

lpar - (1 x p) inner lattice parameters l_j; null = 1,2,2,3,3,...

gampar - (1 x p) aliasing parameters gamma_j; null = 11,13,13,15,15,...

Returns:

normalization constant g(K) (Inf on overflow) and its natural log.