It is well-accepted that Pareto (heavy-tailed) and Gamma (short-tailed) distributions can be used to model the on-off time of a bursty traffic. If \(X\) is Pareto, i.e. \(X\sim\dfrac{ac^a}{(x+c)^{a+1}}\), and \(Y\) is Gamma, i.e. \(Y\sim\dfrac{y^{\alpha-1}e^{-y/\lambda}}{\lambda^\alpha\Gamma(\alpha)}\), then \(R=X+Y\) models the time between successive on-off cycles.

This paper proposed and proved the solution for the p.d.f. of \(R\) as the convolution of the p.d.f. of Pareto and Gamma.

Bibliographic data

   title = "On the convolution of Pareto and gamma distributions",
   author = "Saralees Nadarajah and Samuel Kotz",
   journal = "Computer Networks",
   volume = "51",
   pages = "3650--3654",
   year = "2007",