Package jline.api.npfqn

Class Npfqn_dps_morrison

java.lang.Object

jline.api.npfqn.Npfqn_dps_morrison

public class Npfqn_dps_morrison
extends Object

Two-term heavy-usage asymptotic approximation for a closed queueing network with one infinite-server (think) station and one discriminatory processor-sharing (DPS) station, after J.A. Morrison, "Asymptotic analysis of a large closed queueing network with discriminatory processor sharing", Queueing Systems 9 (1991) 191-214.

The network is NOT product-form, so nothing here computes a normalizing constant: the method expands the GENERATING FUNCTION of the balance equations. The substitution P(n) = <w,n> f(n) clears the DPS denominator and turns the balance recursion into a linear PDE with affine coefficients (Morrison eq. 2.5); rescaling z = 1 - xi/sqrt(N) and expanding in powers of N^(-1/2) leaves a degenerate leading operator whose kernel is the functions of the similarity variable eta, and the solvability condition along its characteristic gives an ODE for the amplitude (eq. 2.20). RESULT 1 (eq. 4.11) and RESULT 2 (eq. 4.17) are the two-term approximations returned here.

Scaling. Morrison writes K_j = N b_j and lambda_j = N r_j g_j with N large and usage rho = sum_j b_j/g_j = 1 - a/sqrt(N). N is bookkeeping only: b, g and a all move with it and the approximation is invariant, so this routine fixes N = 1, i.e. b = N_pop, g = Z./S, r = 1./Z and a = 1 - rho. Accuracy is governed by the PHYSICAL regime -- large populations with rho near 1 -- and not by any choice made here. rho > 1 is admissible: it is the saturated regime of Morrison's appendix A.

This is the Java port of the MATLAB npfqn_dps_morrison; the two agree to double precision on the parity tests.

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static class	Npfqn_dps_morrison.Result	Mean queue lengths, sojourn times and throughputs, with the intermediate constants.

Constructor Summary

Constructors

Constructor	Description
Npfqn_dps_morrison()

Method Summary

Modifier and Type	Method	Description
static Npfqn_dps_morrison.Result	npfqn_dps_morrison(Matrix N, Matrix Z, Matrix S, Matrix w)	Evaluates Morrison's two-term approximation.

Methods inherited from class java.lang.Object

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

Constructor Details

Npfqn_dps_morrison

public Npfqn_dps_morrison()

Method Details

npfqn_dps_morrison

public static Npfqn_dps_morrison.Result npfqn_dps_morrison(Matrix N,
 Matrix Z,
 Matrix S,
 Matrix w)

Evaluates Morrison's two-term approximation.

Parameters:

N - per-class populations, 1 x K, finite and positive

Z - per-class mean think times, 1 x K, finite and positive

S - per-class mean DPS service times, 1 x K, finite and positive

w - per-class DPS weights, 1 x K, finite and positive

Returns:

the mean performance measures and the intermediate constants