On the convolution of Pareto and gamma distributions

Abstract

It is well established that the burst and idle times for on/off traffic are modeled by the Pareto and gamma distributions, respectively. Thus, the inter arrival times between on-traffic (off-traffic) is the convolution of Pareto and gamma random variables. In this note, we derive exact expressions for the probability density function of the inter arrival times. A computer program and tabulations of the associated percentage points are also provided.

Introduction

A variable of primary importance in computer networks is the inter arrival time between on-traffic or that between off-traffic, see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. It is well established that the popular models for the burst and idle times of on/off traffic are the Pareto and gamma distributions, respectively. This is because the Pareto distributions are associated with heavy tails while the gamma distributions are associated with short tails.

In this note, we derive the exact probability distribution of the inter arrival time, denoted by a random variable, R say. Let X and Y be independent random variables representing the burst and idle times, respectively. Then, R = X + Y. We take X to have the Pareto distribution specified by the probability density function (pdf):

$$f_X(x)=\frac{{\mathit{ac}}^a}{(x+c{)}^{a+1}}$$

for x > 0, a > 0 and c > 0. We take Y to have the gamma distribution specified by the pdf:

$$f_Y(y)=\frac{y^{\alpha -1}\exp (-y/\lambda )}{{\lambda }^{\alpha }\Gamma (\alpha )}$$

for y > 0, α > 0 and λ > 0.

Assuming X and Y are independent random variables distributed according to (1), (2), respectively, we derive various expressions for the pdf of R = X + Y. These are given in Section 2. A computer program and a tabulation of the associated percentage points are provided in Section 3. The calculations involve several special functions, including the incomplete gamma function defined by

$$\gamma (a\text{,}x)={\int }_0^x{t}^{a-1}\exp (-t)\mathrm{d}t\text{,}$$

the confluent hypergeometric function defined by

$${}_1{F}_1(\alpha \text{;}\beta \text{;}x)=\sum _{k=0}^{\infty }\frac{(\alpha {)}_k}{(\beta {)}_k}\frac{x^k}{k!}\text{,}$$

the Gauss hypergeometric function defined by

$${}_2{F}_1(\alpha \text{,}\beta \text{;}\gamma \text{;}x)=\sum _{k=0}^{\infty }\frac{(\alpha {)}_k(\beta {)}_k}{(\gamma {)}_k}\frac{x^k}{k!}$$

and, the Appell hypergeometric series defined by

$${\Phi }_1(\alpha \text{,}\beta \text{,}\gamma \text{;}x\text{,}y)=\sum _{k=0}^{\infty }\sum _{l=0}^{\infty }\frac{(\alpha {)}_{k+l}(\beta {)}_k}{(\gamma {)}_{k+l}}\frac{x^k{y}^l}{k!l!}\text{,}$$

where (f)_k = f(f + 1) ⋯ (f + k − 1) denotes the ascending factorial. The properties of the above special functions can be found in Prudnikov et al. [21] and Gradshteyn and Ryzhik [22].