Package jline.api.lossn

Class Lossn_asy

java.lang.Object

jline.api.lossn.Lossn_asy

public final class Lossn_asy
extends Object

Computes the fluid limit of a loss network by convex duality.

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	Lossn_asy.LossnAsyResult	Result of lossn_asy(Matrix, Matrix, Matrix).

Method Summary

Modifier and Type	Method	Description
static Lossn_asy.LossnAsyResult	lossn_asy(Matrix nuVec, Matrix Amat, Matrix cVec)	Computes the fluid limit of a loss network by convex duality.

Methods inherited from class java.lang.Object

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

Method Details

lossn_asy

public static Lossn_asy.LossnAsyResult lossn_asy(Matrix nuVec,
 Matrix Amat,
 Matrix cVec)

Computes the fluid limit of a loss network by convex duality.

Scales offered loads nu and capacities C together, nu -> N nu and C -> N C, and returns the limit of the class loss probabilities as N grows (Kelly 1986; Ross 1995, Theorem 5.13). In this limit link j blocks independently with probability a_j, and the a_j solve the complementary slackness conditions

   sum_k A(j,k) nu_k prod_i (1-a_i)^A(i,k) = C_j  if a_j > 0,
                                           <= C_j  if a_j = 0,
 

the route loss being 1 - prod_j (1-a_j)^A(j,k). With y_j = -log(1-a_j) these are the optimality conditions of the strictly convex problem

   minimize  sum_k nu_k exp(-sum_j A(j,k) y_j) + sum_j C_j y_j,  y >= 0,
 

solved here by cyclic coordinate descent, each coordinate by Newton's method on a concave increasing function. The result needs no summation over states and costs O(J R) per sweep, so it scales to any capacity; it reports zero loss on links that are underloaded at the fluid scale, so it is accurate in heavy (overloaded) traffic and for large C. A and C need not be integral.

References: F. P. Kelly, "Blocking probabilities in large circuit-switched networks", Adv. Appl. Probab. 18, 1986. F. P. Kelly, "Loss Networks", Ann. Appl. Probab. 1(3), 1991. K. W. Ross, "Multiservice Loss Models for Broadband Telecommunication Networks", Springer, 1995, Sec. 5.8.

Parameters:

nuVec - offered load of route (class) r (1xR)

Amat - capacity requirement of link j for route r (JxR, nonnegative)

cVec - capacity of link j (Jx1, nonnegative)

Returns:

QLen, Loss, the link blocking a and the sweep count