Gateway Placement for Throughput Optimization in Wireless Mesh Networks

Abstract

In this paper, we address the problem of gateway placement for throughput optimization in multi-hop wireless mesh networks. Assume that each mesh node in the mesh network has a traffic demand. Given the number of gateways to be deployed (denoted by k) and the interference model in the network, we study where to place exactly k gateways in the mesh network such that the total throughput is maximized while it also ensures a certain fairness among all mesh nodes. We propose a novel grid-based gateway deployment method using a cross-layer throughput optimization, and prove that the achieved throughput by our method is a constant times of the optimal. Simulation results demonstrate that our method can effectively exploit the available resources and perform much better than random and fixed deployment methods. In addition, the proposed method can also be extended to work with multi-channel and multi-radio mesh networks under different interference models.

Appendix

Lemma 1

Under fPrIM model, consider the active fraction α(e) ∈ [0,1] of each link. A sufficient condition that this α is schedulable is, for each e , ${\alpha}(e)+\sum_{e' \in {\textbf I}_{1}(e)} {\alpha}(e') \le 1$ . A necessary condition that this α is schedulable is, for each e , ${\alpha}(e)+\sum_{e' \in {\textbf I}_{1}(e)} {\alpha}(e') \le C_1$ , where $C_{1} = \lceil \frac{2\pi}{\arcsin \frac{\gamma-1}{2\gamma}} \rceil$.

Proof

The sufficient condition comes directly from the correctness of Algorithm 1 which gives a valid link-channel schedule. Thus, we will only concentrate on the correctness of the necessary condition.

To prove that any valid interference-free link scheduling ${\mathcal S}$ under fPrIM must satisfy that ${\alpha}(e)+\sum_{e' \in {\textbf I}_{1}(e)} {\alpha}(e') \le C_1$ for each e, we only need to prove that for all incoming neighboring links of link e there are at most $\lceil \frac{2\pi}{\arcsin \frac{\gamma-1}{2\gamma}} \rceil$ links that can be scheduled at any same time slot. Recall that here ${\textbf I}_{1}(e)$ is the set of incoming links of e that interfere e.

Consider any communication link ${\textbf {L}}_{i,j}$, where v _j is the receiver. Consider two links ${\textbf {L}}_{p,q}$ and ${\textbf {L}}_{s,t}$ that are ${\textbf {L}}_{i,j}$’s incoming links in conflict graph F _G , where v _q and v _t are the receivers. We now prove that if $\angle v_qv_jv_t \le \arcsin \frac{\gamma-1}{2\gamma}$, then link ${\textbf {L}}_{p,q}$ interferes with link ${\textbf {L}}_{s,t}$. This will complete the proof of this lemma.

Draw two rays v _j v _a , v _j v _b emanated from node v _j such that $\angle v_av_jv_b = \arcsin \frac{\gamma-1}{2\gamma}$ and v _q , v _t are in the cone as shown in Fig. 7a. Without loss of generality, we assume that $\Vert {v_j-v_q} \Vert \ge \Vert {v_j-v_t} \Vert$. Draw a circle $\cal C$ centered at v _j with radius $\Vert {v_j-v_q} \Vert$. Let u ₁ u ₂ be the line passing v _q that is tangent to circle $\cal C$ and u ₁, u ₂ are the intersections of this line with line v _j v _a , v _j v _b respectively. Since $\angle u_1v_jv_q \le \arcsin \frac{\gamma-1}{2\gamma}$, we have

$$\Vert {u_1\!-\!v_q} \Vert\! \le\! \Vert {v_j\!-\!v_q} \Vert\cdot \frac{\gamma-1}{2\gamma}\! \le \!2 r_p \cdot \frac{\gamma-1}{2\gamma}\!=\!r_p \cdot \frac{\gamma-1}{\gamma}.$$

Thus, $\Vert {v_p-u_1} \Vert \le \Vert {v_p-v_q} \Vert + \Vert {u_1-v_q} \Vert \le r_p \cdot \frac{1}{\gamma} +r_p \cdot$ $\frac{\gamma-1}{\gamma} =r_p$. Similarly, $\Vert {v_p\!-\!u_2} \Vert \le r_p$. Following we prove that node v _p interferes with v _t by cases.

Figure 7

Links in a small neighborhood will interfere with each other in fixed protocol interference model (fPrIM)

Case 1

v _p u ₁ u ₂ v _j is a convex quadrangle as shown in Fig. 7a. In this case, v _t is either inside triangle v _p v _j u ₂ or triangle v _p u ₁ u ₂. Since both $\Vert {v_p-u_1} \Vert$, $\Vert {v_p-u_2} \Vert$ and $\Vert {v_p\!-\!v_j} \Vert$ are not greater than r _p , we have $\Vert {v_p\!-\!v_t} \Vert\! \le\! r_p$.

Case 2

v _j is inside $\triangle u_1u_2v_p$ as shown in Fig. 7b. In this case, v _t is inside triangle $\triangle u_1u_2v_p$. Then it is easy to show that $\Vert {v_p-v_t} \Vert \le \max\{\Vert {v_p-u_1} \Vert,\Vert {v_p-u_2} \Vert\} \le r_p$.

Case 3

v _p is inside $\triangle u_1u_2v_j$ as shown in Fig. 7c. In this case, v _t is inside one of the three triangles: $\triangle u_1u_2v_p$, $\triangle u_1v_jv_p$, $\triangle u_2v_jv_p$. Similarly, we have $\Vert {v_p-v_t} \Vert \le r_p$.

Obviously, the above three cases covers all possible situations. This proves that link ${\textbf {L}}_{p,q}$ interferes with ${\textbf {L}}_{s,t}$.□

For easy reading, we summarize all used notations in this paper in Table 7.

Table 7 Notation used