Package jline.api.lossn

Class Lossn_kr_asy

java.lang.Object

jline.api.lossn.Lossn_kr_asy

public final class Lossn_kr_asy
extends Object

Normal approximation of multirate single-link blocking for large C.

Nested Class Summary

Nested Classes

Modifier and Type	Class	Description
static final class	Lossn_kr_asy.LossnKrAsyResult	Result of lossn_kr_asy(Matrix, Matrix, double).

Method Summary

Modifier and Type	Method	Description
static Lossn_kr_asy.LossnKrAsyResult	lossn_kr_asy(Matrix rho, Matrix b, double C)	Normal approximation of multirate single-link blocking for large C.

Methods inherited from class java.lang.Object

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

Method Details

lossn_kr_asy

public static Lossn_kr_asy.LossnKrAsyResult lossn_kr_asy(Matrix rho,
 Matrix b,
 double C)

Normal approximation of multirate single-link blocking for large C.

In the critically loaded regime, where C and the offered loads grow together with sum_k b_k rho_k / C -> 1, the normalized occupancy of a stochastic knapsack converges to a truncated multivariate normal law (Kelly 1991; Ross 1995, Theorem 2.8) and blocking is asymptotically proportional to the call size (Ross 1995, Corollary 2.7). For a knapsack of fixed capacity C the resulting approximation is

      alpha   = (C - sum_k b_k rho_k) / sqrt(C),
      sigma^2 = sum_k b_k^2 rho_k / C,
      delta   = phi(alpha/sigma) / (sigma Phi(alpha/sigma)),
      B_k     = b_k delta / sqrt(C),
 

with phi and Phi the standard normal density and distribution function. B_k is capped at 1. The approximation is accurate when |alpha| is small (roughly below 1); far from critical loading use Lossn_kr for the exact values, or Lossn_asy for the fluid (Kelly) limit.

References: K. W. Ross, "Multiservice Loss Models for Broadband Telecommunication Networks", Springer, 1995, Sec. 2.7. M. I. Reiman, "A critically loaded multirate link with trunk reservation", Queueing Systems 9, 1991.

Parameters:

rho - offered load of class k (1xK, nonnegative)

b - bandwidth units held by a class-k call (1xK, nonnegative)

C - link capacity (positive)

Returns:

approximate blocking B, and alpha, sigma, delta