Multi-hop Delay Performance in Wireless Mesh Networks

Abstract

Wireless Mesh Network (WMN) technology is an attractive solution to meet the demand of broadband network access anywhere and anytime. In order to effectively support delay-sensitive applications such as video streaming and interactive gaming in a WMN, it is crucial to develop feasible methodologies and techniques for accurately analyzing, predicting and guaranteeing end-to-end delay performance over multi-hop wireless communication paths. In this paper, we extend the link-layer effective capacity model and derive a lower bound of delay-bound violation probability, or complementary cumulative distribution function, over multi-hop wireless connections. A fluid traffic model with cross traffic and a Rayleigh fading channel with additive Gaussian noise and Doppler spectrum are considered in our study. The average multi-hop delay and jitter performance bounds are also obtained. Analytical results are verified by extensive computer simulations under different traffic load and wireless channel conditions. We find that multi-hop delay performance is much more sensitive to traffic load and maximum Doppler rate than traffic correlation.

Appendices

Appendix A: Derivation of Eq. 5

The delay values for all the hops on an h-hop routing path are assumed as independent and identically distributed (i.i.d.) random variables, so the CDF of delay performance over this h-hop path can be derived as

$$\begin{array}{*{20}l}F_h(x) & = {\displaystyle Prob\left\{\sum_{i=1}^{h}D_i\leq x\right\}}\\ & = (1-\gamma)F_{h-1}(x)+\int_{0}^{x}F_{h-1}(x-D_{h})f(D_{h})dD_h\\ & = (1-\gamma)^2F_{h-2}(x)+2(1-\gamma)\int_{0}^{x}F_{h-2}(x-D_{h-1})f(D_{h-1})dD_{h-1}+\int_{0}^{x}\int_{0}^{x-D_h}F_{h-2}(x-D_h-D_{h-1}) \cdot f(D_{h-1})f(D_h)dD_{h-1}dD_h\\ & = (1-\gamma)^3F_{h-3}(x) +3(1-\gamma)^2\int_{0}^{x}F_{h-3}(x-D_{h-2})f(D_{h-2})dD_{h-2} +3(1-\gamma)\int_{0}^{x}\int_{0}^{x-D_{h-1}}F_{h-3}(x-D_{h-1}-D_{h-2}) \cdot f(D_{h-2})f(D_{h-1})dD_{h-2}dD_{h-1} +\int_{0}^{x}\int_{0}^{x-D_h}\int_{0}^{x-D_h-D_{h-1}}F_{h-3}(x\!-\!D_h\!-\!D_{h-1}\!-\!D_{h-2}) \cdot f(D_{h-2})f(D_{h-1})f(D_h)dD_{h-2}dD_{h-1}dD_{h}\\ & \vdots\\ & = {\displaystyle\sum_{j=1}^{h}\left(\begin{matrix} h-1 \\ h-j \end{matrix} \right)(1-\gamma)^{h-j}} \cdot Prob\left\{ D_i\!>\!0,i\!=\!2,3,\ldots j; D_1\!+\!D_2\!+\ldots+\!D_j\!\leq\! x \right\},\end{array}$$

(13)

where

$$\begin{array}{*{20}l}\\&Prob\left\{ D_i>0,i=2,3,\ldots j; D_1+D_2+\ldots+D_j\leq x \right\}\\ & = \int_{0}^{x}\,Prob \left\{D_2+D_3+\cdots+D_j\leq x-D_1\right\} \cdot f_{D_1}(D_1) dD_1\\ & =\int_{0}^{x}\,\int_{0}^{x-D_1}\,Prob \left\{D_3+D_4+\cdots+D_j \leq x-D_1-D_2\right\} \cdot f_{D_1}(D_2)\,f_{D_1}(D_1)\,dD_2 \,dD_1 \\ &\vdots \\ & = {\displaystyle\int_{0}^{x}\, \int_{0}^{x-D_1} \, \cdots \, \int_{0}^{x-\sum_{i=1}^{j-2}D_i}\, Prob\left\{D_j\leq x-\sum_{i=1}^{j-1}\:x_i \right\}} \cdot \left(\prod_{i=1}^{j-1} \:f_{D_1}(D_i)\right)\,dD_{j-1}\cdots \,dD_2 \,dD_1 \\ &\vdots \\ & ={\displaystyle \gamma^{j-1}-\gamma^{j-1} e^{-\theta x}\left(\sum_{i=1}^{j-1}\,\frac{(\theta x)^{i-1}}{\left(i-1\right)!}+\frac{\gamma(x\theta)^{j-1}}{(j-1)!}\right)}. \end{array}$$

(14)

Substituting Eq. 14 into Eq. 13, we obtain

$$\begin{array}{*{20}l}F_h(x) &= Prob\left\{\sum_{i=1}^{h}D_i\leq x\right\}\\ & =\sum_{j=1}^{h}\left(\begin{matrix} h-1 \\ h-j \end {matrix}\right)(1-\gamma)^{h-j}\ \gamma^{j-1} \cdot \left[1- e^{-\theta x}\left(\sum_{i=1}^{j-1}\,\frac{(\theta x)^{i-1}}{\left(i-1\right)!}+\frac{\gamma(x\theta)^{j-1}}{(j-1)!}\right) \right]. \end{array}$$

(15)

Appendix B: Derivation of Eqs. 8 and 9

The delay values D _i (1 ≤ i ≤ h) over an h-hop routing path are assumed as i.i.d. random variables, so the mean and standard deviation (jitter) of multi-hop delay performance can be derived as

$$\begin{array}{*{20}l}E\left[\sum_{i=1}^{h}D_i\right] & = \sum_{i=1}^{h}\,E[D_i]\\ & = h\, \int_{-\infty}^{\infty}D_i f(D_i)d(D_i)\\ & = h\,\int_{0}^{\infty}\,D_i\,\gamma\theta \,e^{-\theta\,D_i}d(D_i)\\ & = \frac{\gamma h}{\theta}, \end{array}$$

(16)

and

$$\begin{array}{*{20}l}\sqrt{Var\left(\sum_{i=1}^{h}D_i\right)} & = \sqrt{\sum_{i=1}^{h}\, Var[D_i]}\\ & = \sqrt{h\,(E[D_i^2]-E[D_i]^2)}\\ & = \sqrt{h\,\left(\int_{0}^{\infty}D_i^2\,f(D_i)d(D_i)-\left(\frac{\gamma}{\theta}\right)^2\right)}\\ & = \sqrt{h \,\left(\frac{2\gamma}{\theta^2}-\left(\frac{\gamma}{\theta}\right)^2\right)}\\ & = \frac{\sqrt{(2\gamma-\gamma^2)h}}{\theta}. \end{array}$$

(17)