This paper derives equations. Assume the Internet operates as $$M$$/Pareto/$$1$$/$$K$$ queues, the flow blocking probability is $$P_B = 1-\dfrac{\Gamma(K)}{m+\lambda\Gamma(K)}$$, where $$\lambda$$ is the arrival rate, $$m$$ is the first busy period (i.e. mean of Pareto), $$\Gamma(K)=\dfrac{m}{1-\rho}(P_0+P_1+...+P_{K-1})$$ is the $$k$$-th busy period, and $$P_j$$ denotes the probability of having $$j$$ customers in the $$M$$/Pareto/$$1$$/$$K$$ queue.

The generating function for $$P_j$$ is obtained from the M/G/1 analysis:

$P(z)=\sum_{j=0}^\infty P_j z^j = (1-\rho)(1-z)\dfrac{B(\lambda(1-z))}{B(\lambda(1-z))-z}$

with $$B(s)$$ is the Laplace transform of the Pareto pdf $$b(t)=\alpha t^{-(\alpha+1)}$$, where $$t>1$$, $$1<\alpha<2$$, mean equals to $$m=\alpha/(\alpha-1)$$. However, using this method is computationally intensive and the paper proposed the following approximation:

\begin{align*} p &= \left.\dfrac{dP(z)}{dz}\right|_{z=1} \\ L &= p-\rho \\ x &= (L-p)/L \\ g &= \dfrac{\rho(1-x)}{x(1-\rho)} \\ y &= 1+\dfrac{gx(1-x|{K-1})}{1-x+gx|K} \\ P_B &= \dfrac{gx|K}{y} \end{align*}

## Bibliographic data

@inproceedings{
title = "Internet Flow Blocking Probability Calculation",
author = "Helen Y Tang and Steed J Huang and H.-M. Fred Chen",
booktitle = "Proc. Canadian Conference on Electrical and Computer Engineering",
volume = "2",
pages = "659--663",
month = "Mar 7-10",
year = "2000",
}