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

   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",
   address = "Halifax, NS, Canada",