Oblivious routing in wireless mesh networks

Abstract

As new network applications have arisen rapidly in recent years, it is becoming more difficult to predict the exact traffic pattern of a network. In consequence, a routing scheme based on a single traffic demand matrix often leads to a poor performance. Oblivious routing (Racke in Proceedings of the 43rd annual IEEE symposium on foundations of computer science 43–52, 2002) is a technique for tackling the traffic demand uncertainty problem. A routing scheme derived from this principle intends to achieve a predicable performance for a set of traffic matrixes. Oblivious routing can certainly be an effective tool to handle traffic demand uncertainty in a wireless mesh network (WMN). However, a WMN has an additional tool that a wireline network does not have: dynamic bandwidth allocation. A router in a WMN can dynamically assign bandwidth to its attached links. This capability has never been exploited previously in works on oblivious routing for a spatial time division multiple access (STDMA) based WMN. Another useful insight is that although it is impossible to know the exact traffic matrix, it is relatively easy to estimate the amount of the traffic routed through a link when the routing scheme is given. Based on these two insights, we propose a new oblivious routing framework for STDMA WMNs. Both analytical models and simulation results are presented in this paper to prove that the performance—in terms of throughput, queue lengths, and fairness—of the proposed scheme can achieve significant gains over conventional oblivious routing schemes for STDMA based WMNs.

Appendices

Appendix 1: The derivation of property 1

Property 1

$\left( \sum _{\left\{ e \mid e \in E_p^S \right\} } {\sum _{\left\{ f \mid f \in F \right\} }y_{f,e}^Sh_f}/{c_e^S}\right) -\theta ^S\le 0$ for every $H \in W'$ if and only if there exists $\left\{ w_f^{S,p} \mid S\in N,p\in S,f\in F \right\}$, $\left\{ \pi _f^{S,p}\mid S\in N,p\in S, f\in F\right\}$, $\left\{ \phi _{T,v}^{S,p} \mid S,T\in N,p \in S, v \in T \right\}$, and $\left\{ \sigma _{f,v}^{S,p} \mid S\in N, p\in S, f\in F,v\in V\right\}$, with the following conditions:

$$\sum _{\left\{ f \mid f \in F \right\} }\left( w_f^{S,p}b_f-\pi _f^{S,p}d_f\right) \le 0, \quad \forall S\in N, p\in S, f\in F;$$

(16a)

$$\sum _{\left\{ v \mid v \in T \right\} } \phi _{T,v}^{S,p}-\theta ^S\le 0, \quad \forall S,T\in N, p\in S;$$

(16b)

$$\sum _{\left\{ e \mid e \in E_p^S \right\} }\frac{y_{f,e}^S}{c_e^S}-\sigma _{f,src_f}^{S,p}-w_f^{S,p}+\pi _f^{S,p}\le 0,\quad \forall S\in N, p\in S, f\in F;$$

(16c)

$$\sigma _{f,v}^{S,p}-\sigma _{f,u}^{S,p}-\phi _{f,v}^{S,p}/c_e^T\le 0, \quad \forall S,T\in N, p\in S,f\in F,tx_e=v,rx_e=u$$

(16d)

Proof

The proof is similar to the one in [3]. For a given $\left\{ y_{f,e}^S \mid f\in F, e\in E \right\}$, and any node $p\in S$, the following LP cauculates the maximum value of $\sum _{\left\{ e,f \mid e \in E_p^S, f\in F \right\} } {y_{f,e}^Sh_f}/{c_e^S}-\theta _S$.

$${\text {max}}\sum \limits _{\left\{ e,f\mid e \in E_p^S, f\in F \right\} } {y_{f,e}^Sh_f}/{c_e^S}-\theta _S$$

(17a)

$${\text {s.t.}}\sum \limits _{\left\{ (S,e) \in NT_{f,p} \right\} } z_{f,e}^S-\sum \limits _{\left\{ (S,e) \in NR_{f,p} \right\} } z_{f,e}^S = 0\quad p \ne src_f,dst_f$$

(17b)

$$\sum \limits _{\left\{ (S,e) \in NT_{f,p} \right\} } z_{f,e}^S-\sum \limits _{\left\{ (S,e) \in NR_{f,p} \right\} } z_{f,e}^S = h_f \quad p=src_f$$

(17c)

$$\sum \limits _{\left\{ e,f \mid e\in E_p^S, f \in F \right\} }{y_{f,e}^S}/{c_e^S}\le \epsilon ^S$$

(17d)

$$\sum \limits _{\left\{ S \mid S \in N \right\} } \epsilon ^S=1$$

(17e)

$$d_f\le h_f \le b_f, z_{f,e}^S, \epsilon ^S, h_f \ge 0$$

(17f)

Here, $\left\{ \epsilon ^S\mid S \in N \right\}$ and $\left\{ z_{f,e}^S \mid S\in N,e\in E,f\in F\right\}$ is the optimal slot assignment and routing scheme for traffic demand $\left\{ h_f\mid f\in F\right\}$. We then take the dual of (17), and get:

$${\text {max}}\sum \limits _{\left\{ f \mid f\in F\right\} } \left( w_f^{S,p}b_f-\pi _f^{S,p}d_f\right)$$

(18a)

$${\text {s.t.}}(14\text{a}{-}14\text{d})$$

(18b)

According to duality theory, $\sum _{\left\{ e \in E_p^S, f\in F \right\} } {y_{f,e}^Sh_f}/{c_e^S}-\theta _S \le 0$ if and only if (18a)$\le 0$. $\square$

Appendix 2: The derivation of property 2

Property 2

The derived oblivious ratio $(or^\star)$ of the proposed node based scheme is upper bounded by the oblivious ratio of the link based scheme.

Proof

To prove the property, we first show the formulation of the link based scheme. Let L be the set of link patterns, and $\tilde{c}_e^Q$ be the average transmission rate of link e when link pattern Q is activated. Note that, $\tilde{c}_e^Q$ can also be derived from (1) and (2). Similar to (8), the oblivious ratio for the link based scheme ($or_{link}^*$) can be derived from the following LP:

$${\text {min}}\sum \limits _{\left\{ Q \mid Q \in L \right\} } \rho ^Q$$

(19a)

$${\text {s.t.}}\sum \limits _{\left\{ e \mid e\in ET_p \right\} } x_{f,e}-\sum \limits _{\left\{ e \mid e\in ER_p \right\} } x_{f,e} = 0 \quad \forall p \ne src_f,dst_f$$

(19b)

$$\sum \limits _{\left\{ e \mid e\in ET_p \right\} } x_{f,e}-\sum \limits _{\left\{ e \mid e\in ER_p\right\} } x_{f,e} = 1\quad \forall p=src_f$$

(19c)

$$\sum \limits _{\left\{ f \mid f \in F \right\} } \omega _f^eb_f-\pi _f^ed_f \le 0$$

(19d)

$$\sum \limits _{\left\{ l \mid l \in Q \right\} } \phi _l^e\tilde{c}_l^Q-\sum \limits _{\left\{ l \mid l \in Q \right\} }\rho ^Q\tilde{c}_l^Q \le 0$$

(19e)

$$x_{f,e}-\sigma _{f,src_f}^e-\omega _f^e+\pi _f^e \le 0$$

(19f)

$$\sigma _{f,v}^e-\sigma _{f,u}^e-\phi _l^e \le 0\quad e=(v,u)$$

(19g)

$$x_{f,e}, \rho ^Q, \omega _f^e, \pi _f^e, \phi _l^e \ge 0$$

(19h)

Hence the portion of traffic routed through link e in link pattern Q (denoted as $\tilde{y}_{f,e}^Q$) is $\tilde{y}_{f,e}^Q=\tilde{c}_e^Q\rho ^Q/(\sum _{Q_1\in L}\tilde{c}_e^{Q_1}\rho ^{Q_1})\cdot x_{f,e}$, and $\tilde{Y}=\left\{ \tilde{y}_{f,e}^Q\right\}$. Let $\tilde{\omega }_{f}^{Q,e}=\omega _f^e\rho ^Q/(\sum _{Q_1\in L}\tilde{c}_e^{Q_1}\rho ^{Q_1})$ $\tilde{\pi }_f^{Q,e}=\pi _f^{Q,e}\rho ^Q/(\sum _{Q_1\in L}\tilde{c}_e^{Q_1}\rho ^{Q_1})$ and ${\tilde{\phi }}_l^{Q,e}=\phi _l^e\rho ^Q/(\sum _{Q_1\in L}{\tilde{c}}_e^{Q_1}\rho ^{Q_1})$. It is obvious that by replacing $\omega _f^e, \pi _f^e, \phi _l^e$ with $\omega _f^{Q,e}, \pi _f^{Q,e}, \phi _l^{Q,e}$ (19d–19d) can still be satisfied. This implies that a link-based scheme is a special case of a node-based scheme by restricting that each node in a node pattern be associated with only one link. Hence the oblivious ratio of a node-based scheme will never be worse than a link-based scheme. $\square$

Appendix 3: The derivation of (9)

To justify whether a new node pattern will improve the master problem or not, we first take the dual of (8):

$${\text {min}}\sum \limits _{\left\{ f \mid f \in F \right\} } \lambda _f^{src_f}$$

(20a)

$${\text {s.t.}}\sum \limits _{\left\{ T \mid T\in N, p\in S \right\} } \varpi _T^{S,p} \le 1$$

(20b)

$$\lambda _f^u-\lambda _f^v-{\eta _f^{S,p}}/{c_e^S}\le 0\quad e=(v,u)$$

(20c)

$$\sum \limits _{\left\{ f, e \mid f \in F, e\in E_v^T \right\} } {\delta _{f,T,e}^{S,p}}/{c_e^T}-\varpi _T^{S,p} \le 0$$

(20d)

$$\zeta ^{S,p}d_f \le \eta _f^{S,p}\le \zeta ^{S,p}b_f$$

(20e)

$$\sum \limits _{\left\{ e\in E_v^T \right\} }\delta _{f,T,e}^{S,p}-\sum \limits _{\left\{ e\in E_v^T\cap ER_v\right\} }\delta _{f,T,e}^{S,p}=0\quad v\ne src_f,dst_f$$

(20f)

$$\sum \limits _{\left\{ e\in E_v^T \right\} }\delta _{f,T,e}^{S,p}-\sum \limits _{\left\{ e\in E_v^T\cap ER_v\right\} }\delta _{f,T,e}^{S,p}=\lambda _f^v\quad v=src_f$$

(20g)

$$\varpi _T^{S,p}, \eta _f^{S,p}, \delta _{f,T,e}^{S,p}, \zeta ^{S,p}, \lambda _f^v \ge 0$$

(20h)

When a new node pattern is introduced, we first construct the following LP:

$${\text {min}}\sum \limits _{\left\{ T, p \mid T \in N, p \in S \right\} } \varpi _T^{S,p}$$

(21a)

$${\text {s.t.}}(20\text{c}{-}20\text{g})$$

(21b)

$$\varpi _T^{S,p}, \eta _f^{S,p}, \delta _{f,T,e}^{S,p}, \zeta ^{S,p} \ge 0$$

(21c)

Here $\left\{ \lambda _f^u\right\}$ is derived from (20). As long as the optimal value of (21) is larger than 1, the new node pattern will violate the constraints in (20). According to the duality theory, by adding this node pattern into the master problem, the result of the master problem will be improved. $rp^S=1-\min _{\left\{ T, p\mid T\in N, p\in S\right\} }\varpi _T^{S,p}$ is called the reduced cost of node pattern S.

Appendix 4: Proof the convergence of $E\left( {\hat{\epsilon }}_e^S(H,x)\right)$

Here we will prove that when M is large enough the long term average value of ${\hat{\epsilon }}_e^S(H,x)$ can be arbitrarily close to $\epsilon _e^S(H)$ by probability. We first make the following assumptions:

• Traffic arrivals can be modeled as a wide sense stationary process with finite mean and variance (this can be easily achieved by setting a limit to the maximum amount of instant traffic for each flow at its source node). Moreover, let $R_f(\tau )$ be the auto-correlation function of traffic flow f, $|R_f (\tau )|\le c_0$ for all $\tau$, and $\lim _{\tau \rightarrow \infty }R_f(\tau )=0$. Here, $c_0$ is a constant;

• Traffic arrivals of different flows are independent.

Let ${\hat{a}}_f(H,x)=(1-{1}/{M}){\hat{a}}_f(H,x-1)+a_f(H,x)/M$. ${\hat{a}}_f (H,x)$ can be interpreted as the estimated arrival of flow f at time x. The variance of ${\hat{a}}_f (H,x)$, denoted as $V_f$, is:

$$\begin{aligned} V_f&=\left( \sum \limits _{i=0}^{x-1}(1-{1}/{M})^{2i}R_f(0) +\sum \limits _{k=0,i>k}^{x-1}2(1-{1}/{M})^{2(i-k)}R_f(k)\right) /M^2\nonumber \\&\quad \le {2c_0}\left( 1+\sum \limits _{k=1}^{x-1}(1-{1}/{M})^{3k+2}\right) /(2M-1) \end{aligned}$$

(22)

Here, $\lim _{M\rightarrow \infty }V_f\rightarrow 0$. Denote $g_{e}^S(H,x)$ as the amount of traffic arrives to link e from flow f in node pattern S at slot x. Also we can derive the expectation of ${\hat{a}}_{e}^S(H,x)$ as

$$E\left( {\hat{a}}_{e}^S(H,x)\right) =\sum \limits _{\left\{ f \right\} }\left( (1-{1}/{M})^xg_{f,e,0}^S+y_{f,e}^Sh_f\left( 1-(1-\frac{1}{M})^x\right) \right)$$

Here, $g_{f,e,0}^S$ is the initial value of link e for flow f in node pattern S and $\lim _{M\rightarrow \infty }E({\hat{a}}_e^S(H,x))\rightarrow \sum \limits _{\left\{ f \right\} }y_{f,e}^Sh_f$. To simplify the presentation, let $a_e={{\hat{a}}_e^S(H,x)}/{c_e^S}$, $\mu _e=E\left( {{\hat{a}}_e^S(H,x)}/{c_e^S}\right)$, and $\delta =\delta _1+\delta _0$. Therefore, for any $\delta _0$, e, there exists $M\ge M_1>1$, $|{\mu _e}/{\sum _{\left\{ e'\in E_p^S\right\} }\mu _{e'}}-\epsilon _e^S(H)|<\delta _0$. Denote $z(x)={\hat{\epsilon }}_e^S(H,x)-{\mu _e}/{\sum _{\left\{ e'\in E_p^S \right\} }\mu _{e'}}$, $|{\hat{\epsilon }}_e^S(H,x)-\epsilon _e^S(H)|\le \delta _0+|z(x)|$. Let $u_1=a_e(\sum _{\left\{ e'\in E_p^S\right\} }\mu _{e'})-\mu _e(\sum _{\left\{ e'\in E_p^S\right\} }a_{e'})-\delta (\sum _{\left\{ e'\in E_p^S\right\} }a_{e'})(\sum _{\left\{ e''\in E_p^S\right\} }\mu _{e''})$ and $u_2=-a_e(\sum _{\left\{ e'\in E_p^S\right\} }\mu _{e'})+\mu _e(\sum _{\left\{ e'\in E_p^S\right\} }a_{e'})-\delta (\sum _{\left\{ e'\in E_p^S\right\} }a_{e'})(\sum _{\left\{ e''\in E_p^S\right\} }\mu _{e''})$.

Hence, ${\text {Pr}}(|{\hat{\epsilon }}_e^S(H,x)-\epsilon _e^S(H)|\ge \delta )\le {\text {Pr}}(|z(x)|\ge \delta _1)={\text {Pr}}(u_1\ge 0)+{\text {Pr}}(u_2\ge 0)$.

The mean and variance of $u_1$ and $u_2$ is

$$\begin{aligned} E(u_1)=&E(u_2)=-\delta _1\left( \sum _{\left\{ e'\in E_p^S\right\} }\mu _{e'}\right) ^2\\ V(u_1)=&\sum \limits _{\left\{ f\in F \right\} }\left( \left( \sum \limits _{\left\{ e'\in E_p^S\right\} }\frac{y_{f,e}^S}{c_e^S}\cdot \mu _{e'}-\frac{y_{f,e'}^S}{c_e^S}\cdot \mu _e\right) -\delta _1\left( \sum \limits _{\left\{ e'\in E_p^S\right\} }\mu _{e'}\right) \left( \sum \limits _{\left\{ e''\in E_p^S\right\} }\frac{y_{f,e''}^S}{c_{e''}^S}\right) \right) V_f\\ V(u_2)=&\sum \limits _{\left\{ f\in F \right\} }\left( \left( \sum \limits _{\left\{ e'\in E_p^S \right\} }\frac{y_{f,e'}^S}{c_{e'}^S}\cdot \mu _e-\frac{y_{f,e}^S}{c_e^S}\mu _{e'}\right) -\delta _1\left( \sum \limits _{\left\{ e'\in E_p^S\right\} }\mu _{e'}\right) \left( \sum \limits _{\left\{ e''\in E_p^S\right\} }\frac{y_{f,e''}^S}{c_{e''}^S}\right) \right) V_f \end{aligned}$$

Set $\delta _1={2\sqrt{\max \left\{ V(u_1),V(u_2)\right\} }}/{E(u_1)}$. According to Chebyshev’s Inequality,

$${\text {Pr}}(|z(x)|\ge \delta _1)={\text {Pr}}(u_1\ge 0)+{\text {Pr}}(u_2\ge 0)\le {V(u_1)}/{(E(u_1))^2}+{V(u_2)}/{(E(u_2))^2}\le 1/2.$$

Furthermore, $\lim \limits _{x\rightarrow \infty }({1}/{x}\sum \limits _{t=0}^{x-1}|z(t)|\ge \delta _1)$ is equivalent to the event $|z(t)|\ge \delta _1$ happens for infinitely many t as $x\rightarrow \infty$. Let $\left\{ t_{k'}\right\}$ denote the sequence of time such that $|z(t_{k'})|\ge \delta _1$. Select a sub-sequence (denoted as $\left\{ t_{k''}\right\}$) of $\left\{ t_{k'}\right\}$ such that the auto-correlation among $|z(t_{k''})|$ is negligible. Clearly, the cardinality of $\left\{ t_{k''}\right\}$ (denoted as m) is also infinite. Hence,

$$\lim \limits _{x\rightarrow \infty }{\text {Pr}}\left( {1}/{x}\sum \limits _{t=0}^{x-1}|z(t)|\ge \delta _1\right) \le \lim \limits _{x\rightarrow \infty }\bigcup \limits _{t_{k''}}{\text {Pr}}(|z(t_{k''})|\ge \delta _1)\le \lim \limits _{m\rightarrow \infty }({1}/{2})^m\rightarrow 0$$

Moreover, since $\lim \limits _{M\rightarrow \infty }V(u_1)\rightarrow 0$, and $\lim \limits _{M\rightarrow \infty }V(u_2)\rightarrow 0$. For arbitrarily small $\delta _1$ and $\delta _0$ ($\delta =\delta _0+\delta _1$) we can always find a large enough $M_2$, such that for any $M\ge M_2>1$,

$$\lim \limits _{x\rightarrow \infty }{\text {Pr}}\left( {1}/{x}\sum \limits _{t=0}^{x-1}|{\hat{\epsilon }}_e^S(H,x)-\epsilon _e^S (H)|\ge \delta \right) \le \lim \limits _{\delta _1\rightarrow 0}\lim \limits _{x\rightarrow \infty }{\text {Pr}}\left( {1}/{x}\sum \limits _{t=0}^{x-1}|z(t)|\ge \delta _1\right) \rightarrow 0$$

(24)

$${\text {Pr}}\left( \left| E\left( {\hat{\epsilon }}_e^S(H,x)\right) -\epsilon _e^S (H)\right| \ge \delta \right) =0$$

(25)