Online Routing in Multi-hop Wireless Networks

Abstract

In this chapter, we investigate the routing problem in multi-hop wireless networks, whereby link qualities are unknown and time-varying due to fast-fading wireless channels.

Appendix

The proof of Theorem 7.1 in Sect. 7.3 is based on some auxiliary results and Theorem 6.9 of [CL06], reproduced below as Theorem 7.3.

Theorem 7.3

Let $\varPhi (U)=\psi (\sum _{i=1}^{N}\phi (u_i))$, where $U=(u_1,\ldots ,u_N)$. Consider a selection strategy that selects action $I_t$ at time t according to distribution $P_t = \{p_{1,t}, p_{2,t},\ldots , P_{N,t}\}$, whose elements $p_{i,t}$ are defined as:

$$\begin{aligned} \begin{aligned} p_{i,t}=(1-\gamma _t)\frac{\phi ^{\prime }(R_{i,t-1})}{\sum _{k=1}^{N}\phi ^{\prime }(R_{i,t-1})}+\frac{\gamma _t}{N}, \end{aligned} \end{aligned}$$

(7.10)

where $R_{i,t-1}=\sum _{s=1}^{t=1}(g(s,i)-g(s,I_s))$.

If the following assumptions hold:

• 1. $\sum _{t=1}^{n}\frac{1}{\gamma ^2_t}=o(\frac{n^2}{\ln {n}})$

• 2. For all vector $V_t=(v_{1,t},\ldots ,v_{n,t})$ with $\mid {v_{i,t}}\mid \le \frac{N}{\gamma _t}$, we have

  $$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\psi (\phi (n))}\sum _{t=1}^{n}C(V_t)=0, \end{aligned} \end{aligned}$$

  (7.11)

  where

  $$\begin{aligned} C(V_t)=\sup _{U\in {R^N}}\psi ^{\prime }\left( \sum _{i=1}^{N}\phi (u_i)\right) \sum _{i=1}^{N}\phi ^{\prime \prime }(u_i)v^2_{i,t}. \end{aligned}$$

• 3. For all vectors $U_t=(u_{1,t},\ldots ,u_{n,t})$, with $u_{i,t}\le {t}$,

  $$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\psi (\phi (n))}\sum _{t=1}^{n}\gamma _t\sum _{i=1}^{N}\nabla _i\varPhi (U_t)=0, \end{aligned} \end{aligned}$$

  (7.12)

• 4. For all vectors $U_t=(u_{1,t},\ldots ,u_{n,t})$, with $u_{i,t}\le {t}$,

  $$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{\ln {n}}{\psi (\phi (n))}\sqrt{\sum _{t=1}^{n}\frac{1}{\gamma ^2_t}\left( \sum _{i=1}^{N}\nabla _i\varPhi (U_t)\right) ^2}=0, \end{aligned} \end{aligned}$$

  (7.13)

Then the selection strategy satisfies:

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{n}\left( \max _{i=1,\ldots ,N}\sum _{t=1}^{n}g(t,i)-\sum _{t=1}^{n}g(t,I_t)\right) =0. \end{aligned} \end{aligned}$$

(7.14)

Proof of Theorem 7.1

In order to prove Theorem 7.1, we show that the selected parameters $\gamma _t=t^{-\frac{1}{3}}$ and $\eta _t=\frac{\gamma ^3_t}{N^2}$ satisfy (1)–(4) in Theorem 7.3.

• 1. For $\gamma _t=t^{-\frac{1}{3}}$, we have

  $$\begin{aligned} \begin{aligned} \sum _{t=1}^{n}\frac{1}{\gamma ^2_t}=\sum _{t=1}^{n}t^{\frac{2}{3}}=H_n\left[ \frac{-2}{3}\right] . \end{aligned} \end{aligned}$$

  (7.15)

  Then,

  $$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{\ln {n}}{n^2}\sum _{t=1}^{n}\gamma ^2_t=\lim _{n\rightarrow \infty }\frac{\ln {n}}{n^2}H_n\left[ \frac{-2}{3}\right] =0. \end{aligned} \end{aligned}$$

  (7.16)

• 2. For $\psi (x)=\frac{1}{\eta _t}\ln {x}$ and $\phi (x)=\exp (\eta _t{x})$, we obtain

  $$\begin{aligned} \begin{aligned} C(V_t)=sup\left( \eta _t\sum _{i=1}{N}v^2_{i,t}\right) =\frac{\eta _tN^3}{\gamma ^2_t}. \end{aligned} \end{aligned}$$

  (7.17)

  Hence,

  $$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty }\frac{1}{\psi (\phi (n))}\sum _{t=1}^{n}C(V_t)&=\lim _{n\rightarrow \infty }\frac{1}{n} \sum _{t=1}^{n}t^{-\frac{1}{3}}\\&=\lim _{n\rightarrow \infty }\frac{1}{n}H_n\left[ \frac{1}{3}\right] =0. \end{aligned} \end{aligned}$$

  (7.18)

• 3. For $\varPhi (U)=\frac{1}{\eta _t}\ln \left( \sum _{i=1}^{N}\exp (\eta _t{u_i})\right)$, $\nabla _i\phi (U_t)$ is given by:

  $$\begin{aligned} \begin{aligned} \nabla _i\phi (U_t)=\frac{\exp (\eta _t{u_i})}{\sum _{i=1}^{N}\exp (\eta _tu_i)}. \end{aligned} \end{aligned}$$

  (7.19)

  Therefore,

  $$\begin{aligned} \begin{aligned}&\lim _{n\rightarrow \infty }\frac{1}{\psi (\phi (n))}\sum _{t=1}^{n}\gamma _t \sum _{i=1}^{N}\nabla _i\phi (U_t)=\\&\lim _{n\rightarrow \infty }\frac{1}{n}\sum _{t=1}^{n}t^{-\frac{1}{3}}\sum _{i=1}{N}\frac{\exp (\eta _tu_i)}{\sum _{i=1}^{N}\exp (\eta _tu_i)}=\\&\lim _{n\rightarrow \infty }\frac{1}{n}H_n\left[ \frac{1}{3}\right] =0. \end{aligned} \end{aligned}$$

  (7.20)

• 4. By substituting (7.19) into (7.13), we obtain (4).

Therefore, (7.14) ensures since all assumptions (1)–(4) are held, which completes the proof of Theorem 7.1.