On Maximizing Transmission Reliability for Real-Time Routing in Multi-hop Wireless LANs

Abstract

Three factors have been often shown to significantly affect the reliability of real-time transmission in wireless local area networks—transmission rate, power, and packet size. We analyze these factors and determine the optimal combination of factors subject to time constraints that yields the most reliable transmission of real-time data. We propose a cross-layer framework to jointly design the routing and MAC protocol combined with our optimization approach. The approach under a non-real-time routing protocol that produces a path metric is compared with a real-time routing protocol. Additionally, the real-time routing protocol enhances a guaranteed rate with our approach. Our experiments reveal that real-time performance in terms of miss ratio and throughput is significantly increased in lossy link and heavy traffic environments. Miss ratio and average throughput are improved by up to 30% over a state-of-the art routing protocol and 35% over a MAC protocol, respectively.

Appendix: Sketch of Proof for det(H)

Equation 20 shows the determinant of Equation 16 which is 3 by 3 matrix. We have to show that Equation 20 is positive.

$$\begin{aligned} det(H) = {\varGamma } - {\varDelta } + {\varLambda } \end{aligned},$$

(20)

where

$$\begin{aligned} {\varGamma }= & {} \frac{\partial ^2p_e}{\partial l^2}\left( \frac{\partial ^2p_e}{\partial r^2}\cdot \frac{\partial ^2p_e}{\partial p_t^2} - \frac{\partial ^2p_e}{\partial p_t \partial r}\cdot \frac{\partial ^2p_e}{\partial r \partial p_t}\right) \nonumber \\ {\varDelta }= & {} \frac{\partial ^2p_e}{\partial r \partial l}\left( \frac{\partial ^2p_e}{\partial l \partial r}\cdot \frac{\partial ^2p_e}{\partial p_t^2} - \frac{\partial ^2p_e}{\partial p_t \partial r}\cdot \frac{\partial ^2p_e}{\partial l \partial p_t} \right) \nonumber \\ {\varLambda }= & {} \frac{\partial ^2p_e}{\partial p_t \partial l}\left( \frac{\partial ^2p_e}{\partial l \partial r}\cdot \frac{\partial ^2p_e}{\partial r \partial p_t} - \frac{\partial ^2p_e}{\partial r^2} \cdot \frac{\partial ^2p_e}{\partial l \partial p_t }\right) \nonumber \\ \frac{\partial ^2p_e}{\partial l^2}= & {} -64\cdot \alpha \cdot ln\left( 2A\right) \cdot A^{8l} \nonumber \\ \frac{\partial ^2p_e}{\partial r \partial l}= & {} 8\cdot A^{8l-1}\cdot \frac{\partial p_b}{\partial r}\left( 1- lnA\right) \nonumber \\ \frac{\partial ^2p_e}{\partial p_t \partial l}= & {} 8\cdot A^{8l-1}\cdot \frac{\partial p_b}{\partial p_t}\left( 1- lnA\right) \nonumber \\ \frac{\partial ^2p_e}{\partial l \partial r}= & {} 8\cdot \alpha \cdot \frac{\partial p_b}{\partial r} \cdot A^{8l-1}\cdot \left( 1+8l \cdot lnA\right) \nonumber \\ \frac{\partial ^2p_e}{\partial r^2}= & {} -8\cdot \left( 8l-1\right) A^{8l-1}\cdot \left( \frac{\partial p_b}{\partial r}\right) ^2+8l\cdot \alpha \cdot A^{8l-1}\cdot \frac{\partial ^2 p_b}{\partial r^2} \nonumber \\ \frac{\partial ^2p_e}{\partial p_t \partial r}= & {} -8\cdot \left( 8l-1\right) A^{8l-1}\cdot \left( \frac{\partial p_b}{\partial r}\right) ^2+8l \cdot \alpha \cdot A^{8l-1}\cdot \frac{\partial ^2 p_b}{\partial r \partial p_t} \nonumber \\ \frac{\partial ^2p_e}{\partial l \partial p_t}= & {} 8\cdot \alpha \cdot \frac{\partial p_b}{\partial p_t} \cdot A^{8l-1}\cdot \left( 1+8l \cdot lnA\right) \nonumber \\ \frac{\partial ^2p_e}{\partial r \partial p_t}= & {} -8\cdot \left( 8l-1\right) A^{8l-1}\cdot \left( \frac{\partial p_b}{\partial p_t}\right) ^2+8l \cdot \alpha \cdot A^{8l-1}\cdot \frac{\partial ^2 p_b}{\partial p_t\partial r} \nonumber \\ \frac{\partial ^2p_e}{\partial p_t^2}= & {} -8\cdot \left( 8l-1\right) A^{8l-1}\cdot \left( \frac{\partial p_b}{\partial p_t}\right) ^2+8l\cdot \alpha \cdot A^{8l-1}\cdot \frac{\partial ^2 p_b}{\partial p_t^2} \end{aligned},$$

(21)

where $1-p_b = A$.

$$\begin{aligned} \frac{\partial p_b}{\partial r}=\frac{\partial p_b}{\partial p_t}= & {} -C \cdot \exp ^{-B} \end{aligned}$$

(22)

$$\begin{aligned} \frac{\partial ^2 p_b}{\partial r^2}=\frac{\partial ^2 p_b}{\partial p_t \partial r}= & {} -C \cdot \exp ^{-B}\cdot \frac{B}{r} \end{aligned}$$

(23)

$$\begin{aligned} \frac{\partial ^2 p_b}{\partial r \partial p_t}=\frac{\partial ^2 p_b}{\partial p_t^2}= & {} C \cdot \exp ^{-B}\cdot \frac{3log_2^m}{4r(m-1)} \end{aligned}$$

(24)

where $\frac{3log_2m}{4(m-1)}\cdot \frac{E_b}{N_0} = B, \frac{\sqrt{m}-1}{\sqrt{m}log_2\sqrt{m}} \cdot \frac{1}{\sqrt{2}\pi } = C$

Equations 22,23 and 24 are substituted for Eq. 21. A, B, and C are positive. We realize that Eq. 22 $< 0$, Eq.23 $<0$ and Eq. 24 $> 0$. Hence, $\frac{\partial ^2p_e}{\partial r \partial l}\,< 0, \frac{\partial ^2p_e}{\partial p_t \partial l}\,< 0, \frac{\partial ^2p_e}{\partial l \partial r}\,< 0, \frac{\partial ^2p_e}{\partial r^2}\,< 0, \frac{\partial ^2p_e}{\partial l \partial p_t}\,< 0$, and $\frac{\partial ^2p_e}{\partial r \partial p_t}\,<$ 0. We apply these inequalities to Eq. 20. Finally, we obtain as following.

$$\begin{aligned} {\varGamma } = 0,\ {\varDelta } < 0,\ and\ {\varLambda } > 0 \end{aligned}$$

(25)

Therefore, $det(H) > 0$