Analysis of Optimal Carrier Sensing Range in Wireless Ad Hoc Networks with Order Dependent Capture Capability

Abstract

IEEE 802.11 wireless ad hoc network performance in distributed coordination function mode is limited by hidden and exposed terminals problem. Request-to-send/clear-to-send based hand shake reduces the problem to some extent, but the network performance is a function of nodal carrier sense (CS) range and interference range. While a large CS range compared to the interference range can reduce the collision related throughput loss, it has a negative impact of increased exposed terminals. Via experimental studies it was recently demonstrated that, the effect of interference to a reception process differs depending on arrival order of the desired signal and interfering signal. In view of this frame arrival order dependent capture (ODC) capability of receivers, in this paper we investigate the optimal choice of CS range and explore the possibility of maximizing the network performance. Via mathematical analysis, supported by extensive network simulations, we demonstrate the network performance benefit of ODC dependent optimal CS range. The distinctive characteristic of ODC diminishes at higher data rates, and as a result the performance gain with optimal CS range reduces. Nevertheless, at low-to-moderately-high data rates, the performance gain is shown to be quite significant.

Appendix: Derivation of Different Areas

1.1 Area of Hidden Region for RTS Frames: $A_{\varOmega }$

$A_{\varOmega }$ refers to the area of the horizontal-dashed region in Fig. 3. Let $d_0$ and $d_1$ be defined as: $d_0 = \frac{R_c}{K+1} \text{ and } d_1 = \frac{R_c}{K-1}$. Then, the hidden area for RTS frames can be obtained as:

$$\begin{aligned} A_{\varOmega }(d) = \left\{ \begin{array} {l@{\quad }c@{\quad }l} 0 &{} \text{ for } &{} 0 \le d \le d_0 \\ \theta _2(d) {R_i^2(d)} - \theta _1(d) R_c^2 + 2A_{\bigtriangleup (n_0RP_1)} &{} \text{ for } &{} d_0 < d \le \min (d_1, R_t) \\ \pi \left( {R_i^2(d)} - R_c^2\right) &{} \text{ for } &{} d_1 < d \le R_t, \end{array} \right. \end{aligned}$$

(26)

where, $\theta _1(d)$, $\theta _2(d)$, and $A_{\bigtriangleup (n_0RP_1)}$ are derived as follows:

$$\begin{aligned} \theta _1(d) = \cos ^{-1}\left( \frac{R_c^2 + d^2 - R_i^2(d)}{2 R_c d}\right) \text{ and } \theta _2(d) = \pi - \cos ^{-1}\left( \frac{d^2 + {R_i^2(d)} - R_c^2}{2 d R_i(d)}\right) . \end{aligned}$$

Denoting $s(d) = \frac{R_c + R_i(d) + d}{2}$, we get

$$\begin{aligned} A_{\bigtriangleup (n_0RP_1)} = \sqrt{s(d) (s(d)-R_c) (s(d)-R_i(d)) (s(d) - d)}. \end{aligned}$$

1.2 Area of Non-hidden Region for RTS Frames Covered by the Transmission Range of Sender: $A_{\varPsi }$

Refer to Fig. 5. Let $d_2 = \frac{R_t}{K+1} \text{ and } d_3 = \frac{R_t}{K-1}.$ Then,

$$\begin{aligned} A_{\varPsi }(d) = \left\{ \begin{array}{l@{\quad }c@{\quad }l} \pi {R_i^2(d)} &{} \text{ for } &{} 0 \le d \le d_2 \\ \pi {R_i^2(d)} - \theta _4(d) {R_i^2(d)} + \theta _3(d) R_t^2 \\ \ \ \ - 2\sqrt{s(d) (s(d)-R_t) (s(d)-R_i(d)) (s(d) - d)} &{} \text{ for } &{} d_2 < d \le \min (d_3, R_t) \\ \pi R_t^2 &{} \text{ for } &{} d_3 < d \le R_t \end{array} \right. \nonumber \\ \end{aligned}$$

(27)

where, $s(d) = \frac{R_t + R_i(d) + d}{2}$ and $\theta _3(d)$ and $\theta _4(d)$ are calculated as follows:

$$\begin{aligned} \theta _3(d) = \cos ^{-1}\left( \frac{R_t^2 + d^2 - R_i^2(d)}{2 R_t d}\right) \text{ and } \theta _4(d) = \pi - \cos ^{-1}\left( \frac{d^2 + {R_i^2(d)} - R_t^2}{2 d R_i(d)}\right) , \end{aligned}$$

1.3 Area of Hidden Region for Data Frames: $A_{\varTheta }$

Referring to Fig. 9, $A_{\varTheta } = A_{\varOmega } - A_{\varPhi }$, where $A_{\varPhi }$ ($= A_\chi$, discussed in Sect. 4.2.2) is non-zero only if $d > R_c - R_t$ and its derivation is similar to that of $A_{\varOmega }$ in (26). The hidden area for the data frames $A_{\varTheta }$ has two sub-regions $A_{\varTheta _i}$ and $A_{\varTheta _o}$. Let $d_c = \frac{R_c}{K}$. The area of $A_{\varTheta _i}$ can be determined as:

$$\begin{aligned} A_{\varTheta _i} = \left\{ \begin{array}{l@{\quad }c@{\quad }l} A_{\varTheta } &{} \text{ for } &{} d \le d_c \\ \pi R_c^2 - A_{\varUpsilon } &{} \text{ for } &{} d > d_c, \end{array} \right. \end{aligned}$$

(28)

where $A_{\varUpsilon }$ is the area of intersection of two circles having same radius $R_c$ whose centers are $d$ distance apart. It is given by: $A_{\varUpsilon } = 2R_c^2 \cos ^{-1}\left( \frac{d}{2R_c}\right) - \frac{1}{2} d \sqrt{4R_c^2-d^2}$. Finally, $A_{\varTheta _o}$ is derived as:

$$\begin{aligned} A_{\varTheta _o} = \left\{ \begin{array}{l@{\quad }c@{\quad }l} 0 &{} \text{ for } &{} d \le d_c, \\ A_{\varTheta } - A_{\varTheta _i} &{} \text{ for } &{} d > d_c. \end{array} \right. \end{aligned}$$

(29)