Calculation of packet jitter for non-poisson traffic

Abstract

The packet delay variation, commonly called delay jitter, is an important quality of service parameter in IP networks especially for real-time applications. In this paper, we propose the exact and approximate models to compute the jitter for some non-Poisson FCFS queues with a single flow that are important for recent IP network. We show that the approximate models are sufficiently accurate for design purposes. We also show that these models can be computed sufficiently fast to be usable within some iterative procedure, e.g., for dimensioning a playback buffer or for flow assignment in a network.

Appendix: A Notation

For the sake of completeness, we recall the definitions of the distributions used in the paper.

1.1 A.1 Gamma

The cdf is the regularized gamma function

$$F_{R}(x, k, \theta) = \frac{1}{\Gamma(k)} {\int}_{0}^{x /\theta} t^{k - 1} e^{-t} dt . $$

The parameters are related to the mean m and variance v = s ² of the distribution by

$$\begin{array}{@{}rcl@{}} m & = & k \theta \end{array} $$

(69)

$$\begin{array}{@{}rcl@{}} v & = & k \theta^{2} \end{array} $$

(70)

which we can solve to get

$$\begin{array}{@{}rcl@{}} \theta & =& \frac{v}{m} \end{array} $$

(71)

$$\begin{array}{@{}rcl@{}} k & = &\frac{m^{2}}{ v}. \end{array} $$

(72)

1.2 A.2 Pareto

The Pareto Type I P(x; x _m , α) with scale x _m and shape α has a density function given by

$$ f_{R}(x) = \left\{\begin{array}{ll} \alpha \displaystyle\frac{ x_{m}^{\alpha}} {x^{\alpha+1}} & \text{if \(x \geq x_{m}\)} \\ 0 & \text{otherwise.} \end{array}\right. $$

(73)

The Pareto parameters α and x _m are related to the mean m and variance v = s ² by

$$\begin{array}{@{}rcl@{}} m & =& \left\{\begin{array}{ll} x_{m}\displaystyle\frac{\alpha} {\alpha-1} & \text{if \(\alpha > 1\)} \\ \infty & \text{if \(\alpha \leq 1\)} \end{array}\right. \end{array} $$

(74)

$$\begin{array}{@{}rcl@{}} v & =& \left\{\begin{array}{lll} {x_{m}^{2}} \displaystyle\frac{\alpha}{(\alpha-1)^{2} (\alpha-2)}& \text{if \(\alpha > 2\)} \\ \infty & \text{if \(\alpha \leq 2.\)} \end{array}\right. \end{array} $$

(75)

For α > 2, we can solve (74–75) to get

$$\begin{array}{@{}rcl@{}} \alpha &=& 1 + \sqrt{1+(m^{2} / v )} \end{array} $$

(76)

$$\begin{array}{@{}rcl@{}} x_{m} &=& m \frac{\alpha-1}{\alpha} . \end{array} $$

(77)

1.3 A.3 Normal

We denote the pdf of the standard normal distribution

$$ \phi(x) = \frac{1}{ \sqrt{2 \pi}} \exp(- x^{2} / 2 ) $$

(78)

and the corresponding cdf

$$ {\Phi}(x) = \frac{1}{ \sqrt{2 \pi}} {\int}_{- \infty}^{x} \exp(- t^{2} / 2 ) dt. $$

(79)

The error function is given by

$$ \text{erf}(x) = \frac{2}{\sqrt{\pi}} {{\int}_{0}^{x}} e^{- t^{2}} dt $$

(80)

and we have the relation

$$ {\Phi} (x) = \frac{1}{2}+ \frac{1}{2} \operatorname{erf} \left( \frac{x}{ \sqrt{2}} \right) . $$

(81)

The pdf of a normally distributed random variables X with location μ and scale σ is given by

$$ f_{X}(x; \mu, \sigma) = \frac{1}{\sigma} \phi\left( \frac{ x - \mu} { \sigma} \right) $$

(82)

and the corresponding cdf by

$$ F_{X}(x; \mu, \sigma) = {\Phi} \left( \frac{ x - \mu}{\sigma} \right) . $$

(83)

1.4 A.4 Truncated Normal

The pdf of the truncated normal is that of a normal random variable X with location and scale parameters μ and σ and truncated to the interval [0, ∞], is given by

$$ f_{X}(x; \mu, \sigma) = Nt(x; m, s) = \frac{1}{ \sigma {\Phi} \left( {\mu}/{\sigma} \right)} \phi \left( \frac{x - \mu}{\sigma} \right) $$

(84)

which is simply a gaussian distribution with support [0, ∞] with the appropriate normalization. The mean m and variance v = s ² are given by

$$\begin{array}{@{}rcl@{}} m & =& \mu + \sigma A \end{array} $$

(85)

$$\begin{array}{@{}rcl@{}} v & =& \sigma^{2} \left( 1 - B \right) \end{array} $$

(86)

where we have defined

$$\begin{array}{@{}rcl@{}} A & =& \frac{\phi\left( \frac{\mu}{\sigma} \right)} {\Phi\left( \frac{\mu}{\sigma} \right)} \\ B & =& A \left( A -\frac{\mu}{\sigma} \right). \end{array} $$

Note that the parameters μ and σ need not exist for an arbitrary choice of m and s.

1.5 A.5 Log-Normal

This is the distribution of a random variable X such that logX is normally distributed. The pdf for location μ and scale σ is given by

$$ f_{X}(x; \mu, \sigma) = \frac{1}{x\sqrt{2\pi}\sigma}\ exp \left( -\frac{\left( \ln x-\mu\right)^{2}}{2\sigma^{2}} \right) $$

(87)

The mean m and variance v = s ² are given by

$$\begin{array}{@{}rcl@{}} m &=& e^{\mu+\sigma^{2}/2} \\ v &=& (e^{\sigma^{2}} -1) e^{2\mu+\sigma^{2}} \end{array} $$

which can be inverted to give

$$\begin{array}{@{}rcl@{}} \mu & =&\ln\left( \frac{m^{2}}{\sqrt{v+m^{2}}}\right) \\ \sigma & =&\sqrt{\ln\left( 1+\frac{v}{m^{2}}\right)} \end{array} $$