A static/opportunistic hybrid-scheduling scheme for MIMO wireless networks

Abstract

Multiple Input Multiple Output (MIMO) based Spatial Time Division Multiple Access (STDMA) Wireless Mesh Networks (WMNs) have attracted extensive research attention. However, there are two problems in existing studies: (1) the employed MIMO link rate models are not suitable for a MIMO link of a practical STDMA WMN, and (2) the designed scheduling algorithms usually cannot take advantage of the multi-user diversity in a WMN. In this paper, we develop an analytical model for determining the MIMO link rate of an STDMA WMN. Based on a node-based slot assignment and scheduling algorithm (Chen and Lea in IEEE Trans Veh Technol 62(1):272–283, 2013), we propose a static/opportunistic hybrid scheduling framework that can exploit multi-user diversity and channel fading. The performance evaluation shows that the proposed framework has 33–46 % throughput gain over the prior joint routing and time slot assignment schemes for MIMO WMNs.

Appendices

Appendix 1: the derivation of (1)

To study our scheme, we need to derive the SINR of a data stream in S.

• $X_t^S(k)$ the signals transmitted from t in slot k with node pattern S. $X_t^S(k)$ is an $n_t^S\times 1$ vector with $E(X_t^S(k)(X_t^S(k))^*)={p_t}I_{n_t}^S/{n_t^S}$, where $M^*$ is the conjugate transpose of matrix (vector) M and $I_n$ is an $n\times n$ identity matrix. Furthermore, different transmitters will transmit independent data streams to their receivers, and $E(X_p^S(k)(X_q^S(k))^*)=0_{n_p^S\times n_q^S}$ with $p\ne q$. Here $0_{a\times b}$ represents an $a\times b$ matrix with all entries equal to 0;

• $H_{r,t}^S(k)$ the channel response function from transmitter t to receiver r in slot k;

• $Y_r^S(k)$: the received signal at r from all of its transmitters in slot k.

To simplify the notations, we drop S and k in the notations below and use $T_r$ for $T_r^S(k)$ and $Y_r$ for $Y_r^S$. $Y_r$ can then be expressed as

$$\begin{aligned} Y_r=\sum \limits _{t\in T_r}H_{r,t}X_t+\sum \limits _{t'\in \left\{ T \setminus T_r\right\} }H_{r,t'}X_{t'}+\varsigma 1_{m\times 1}. \end{aligned}$$

(10)

Here $\varsigma$ is the thermal noise at each receiving antenna and $1_{j\times 1}$ is a $j\times 1$ vector with all elements equal to 1. We assume a rich scattering environment and a block-wise Rayleigh distributed flat fading channel [30]. Therefore, $H_{r,t}$ consists of three parts: path loss, shadow fading and fast fading. Hence $H_{r,t}=\sqrt{l_{r,t}^{-\alpha }\cdot 10^{\frac{f_{r,t}}{10}}}\cdot \left( \begin{array}{ccc} \pi _{1,1} &{} \cdots &{} \pi _{1,n_t} \\ \vdots &{} \ddots &{} \vdots \\ \pi _{m,1} &{} \cdots &{} \pi _{m,n_t} \end{array} \right)$. Note that $l_{r,t}$ is the distance between t and r, $\alpha$ is a constant which usually varies from 2 to 4 and $f_{r,t}$ is the shadow fading effect and is modeled as a log-normal distributed random variable (r.v.). In a mesh backbone network, since all the nodes are stable, it is assumed that $f_{r,t}$ does not change with time. Likewise, the fast fading components ($\pi _{i,j}$) are identically independently distributed (i.i.d.) complex Gaussian random variables with mean and variance equal to 0 and 1 respectively.

Furthermore, let $H_r=[ H_{r,t_1},\cdots , H_{r,t_n} ]$ and ${\varPhi }_r=[ X_{t_1}^T]$ with $t_j \in T_r$. Here $M^T$ is the transpose of matrix M. In case all the nodes in $T_r$ are transmitting to node r, it is required that $\sum _{\lbrace j \mid t_j\in T_r\rbrace } n_{t_j} \le m$. Hence (10) can be rewritten as $Y_r=H_r {\varPhi }_r+\sum \limits _{t'\in \lbrace T\setminus T_r\rbrace }H_{r,t'}X_{t'}+\varsigma 1_{m\times 1}.$ When the streams arrive at receiver r, ZF (zero forcing) is applied. Let $M^\dagger$ denote the pseudo-inverse of matrix M, i.e., $M^\dagger =(M^*M)^{-1}M^*$. After ZF, the received signal becomes

$$\begin{aligned} H_r^\dagger Y_r={\varPhi }_r+H_r^\dagger \left( \sum \limits _{t'\in \left\{ T \setminus T_r\right\} }H_{r,t'}X_{t'}+\varsigma 1_{m\times 1} \right) . \end{aligned}$$

(11)

Performing the pseudo-inverse of $H_r$ requires that all the columns of $H_r$ be linearly independent. This condition can usually be satisfied for two reasons. First, the number of rows (i.e., the number of receiving antennas) is $\ge$ the number of columns (i.e., the number of data streams). Second, each entry (each element in the matrix) of $H_r$ is independent. Hence we will ignore the case when the columns in $H_r$ are linearly dependent.

Since $n_t$ antennas will transmit $n_t$ different data streams, the transmitting power at an antenna is ${p_t}/{n_t}$. At receiver r, the SINR, denoted by $\omega _{r,t}^i$, for the ith stream from node t

$$\begin{aligned} \omega _{r,t}^i=\frac{{\frac{p_t}{n_t}\cdot l_{r,t}^{-\alpha } \cdot 10^{\frac{f_{r,t}}{10}}}/{\left( (H_{r}^* H_{r})^{-1}\right) _{i,i}}}{\sum \limits _{t' \in \left\{ T \setminus T_r \right\} }{\left( \frac{p_{t}}{n_{t'}}\cdot l_{r,t'}^{-\alpha } \cdot 10^{\frac{f_{r,t'}}{10}}H_{r}^\dagger H_{r,t'}H_{r,t'}^* (H_{r}^\dagger )^* \right) _{i,i}}/{\left( (H_{r}^* H_{r})^{-1}\right) _{i,i}}+\kappa }. \end{aligned}$$

(12)

Here $(\cdot )_{i,i}$ is the ith entry of a matrix, and $\kappa$ is the power of the thermal noise $\varsigma$. Let $p_{r,t}={p_t}/{n_t}\cdot l_{r,t}^{-\alpha } \cdot 10^{{f_{r,t}}/{10}}$ and $\triangle =m-\sum _{t \in T_r}n_t+1$.

Our goal is to derive the CDF of $\omega _{r,t}^i$ when ZF is used at the receiver side. Although this was also intended in [23], its assumption that nodes follow a homogeneous PPP (Poisson Point Process) on the plane does not apply to a mesh network where nodes’ locations are fixed. In the following, we show how to derive the CDF for $\omega _{r,t}^i$ in a mesh network.

According to [23], the numerator and each element in the denominator of (12) can be modelled as gamma distributed random variables

$$\begin{aligned} z_t&={p_{r,t}}/{\left( (H_{r}^* H_{r})^{-1} \right) _{i,i}} \sim {\varGamma }(\triangle ,p_{r,t}) \quad t \in T_r \end{aligned}$$

(13a)

$$\begin{aligned} z_{t'}&={\left( p_{r,t'}H_{r}^\dagger H_{r,t'}H_{r,t'}^*(H_{r}^\dagger )^*\right) _{i,i}}/{\left( (H_{r}^*H_{r})^{-1} \right) _{i,i}} \sim {\varGamma }(n_{t'},p_{r,t'} ), \quad t' \in \left\{ T \setminus T_r\right\} . \end{aligned}$$

(13b)

Since nodes are situated at different locations, all the elements in $H_{r,t'}$ can be assumed to be independent from each other. Moreover, similar to lemma 1 of [23], the distribution of $z_{t'}$ is only dependent on $H_{r,t'}$ but not on $H_r$. Hence the denominators ($z_{t'}$) in (12) are also assumed to be independent.

Let $v_t={z_t}/{\gamma \kappa }$ and $v_{t'}={z_{t'}}/{\kappa }$, where $v_t \sim {\varGamma }( \triangle , {p_{r,t}}/{\gamma \kappa })$ and $v_{t'} \sim {\varGamma }( n_{t'}, {p_{r,t'}}/{\kappa } )$. According to [32], $Pr ( \sum\nolimits_{t' \in \{ T {\setminus} T_r \} }v_{t'} \le y )= b\sum\nolimits_{k=0}^{\infty }\delta _k( 1-\sum\nolimits_{i=0}^{\rho _r+k-1}\frac{y^i}{\theta _1^ii!}e^{-\frac{y}{\theta _1}} )$. Since $\lim _{y\rightarrow \infty }Pr ( \sum\nolimits_{t'\in \lbrace T\setminus T_r \rbrace } v_{t'} \le y )\rightarrow b\sum\nolimits_{k=0}^\infty \delta _k$,we can obtain $b\sum _{k=0}^{\infty }\delta _k=1$. From (12-13)

$$\begin{aligned} Pr\left( \omega _{r,t}^i \ge \gamma \right)&=\int _0^\infty Pr\left( \sum \limits _{t'\in \left\{ T \setminus T_r \right\} }v_{t'}\le y \mid v_0=y-1 \right) Pr\left( v_0=y-1 \right) dy \nonumber \\&=1-e^{-\frac{1}{\theta _0}}\sum \limits _{p=0}^{\triangle -1}\frac{b}{\theta _0^p p!}\sum \limits _{k=0}^{\infty }\delta _k\left[ 1-\frac{u^{-(p+1)}d_{p,k}^{(p)}}{{\varGamma }(\triangle -p)\theta _0^{\triangle -p}}\right] . \end{aligned}$$

(14)

Appendix 2: the selection of $k^*$

According to (14), even though there is an infinite number of terms, as k grows larger and larger, $\sum _{k=0}^\infty \delta _kd_{p.k}^{(p)}$ shrinks to 0. One possible way of determining $k^*$ is:

$$\begin{aligned}&\max \limits _{0 \le p \le \triangle -1}\left\{ \delta _kd_{p,k}^{(p)}-\delta _{k+1}d_{p,k+1}^{(p)}\right\} \le \varepsilon .\nonumber \\&\delta _kd_{p,k}^{(p)}-\delta _{k+1}d_{p,k+1}^{(p)}<\delta _kd_{p,k}^{(p)}=\delta _k\sum _{i=0}^{\rho _r+k-1}\frac{(p+i)!}{i!(u\theta _1)^i}<\delta _k\sum _{i=0}^\infty \frac{(\triangle +i-1)^i}{i!}v^i=\delta _k\frac{(\triangle -1)!}{(1-v)^\triangle }\le \varepsilon . \end{aligned}$$

(15)

Moreover, $\delta _k$ will increase first. After it reaches the peak point, it tends to decrease to 0. Let $k_{max}=\arg \max _k \delta _k$. $k^*$ becomes:

$$\begin{aligned} k^*=\arg \max \limits _{k \ge k_{\max }}\left\{ {\varepsilon (1-v)^\triangle }/{(\triangle -1)!}\ge \delta _k \right\} . \end{aligned}$$

(16)

Appendix 3: maximum throughput scaling parameter of the link-based scheme

To compute the maximum throughput scaling parameter of the link-based scheme, we can use the following formula:

$$\begin{aligned} \text {max}\quad&\zeta \end{aligned}$$

(17a)

$$\begin{aligned} \text {s.t.}\quad&\sum \limits _{ \left\{ e \mid e \in E,tx_e=v \right\} } x_{f,e} -\sum \limits _{\left\{ e \mid e \in E,rx_e=v \right\} } x_{f,e}=0 \quad \forall f \in F, v\ne src_f,dst_f, \end{aligned}$$

(17b)

$$\begin{aligned}&\sum \limits _{ \left\{ e \mid e \in E,tx_e=src_f \right\} } x_{f,e} -\sum \limits _{\left\{ e \mid e \in E,rx_e=src_f \right\} } x_{f,e} =\zeta h_f \quad \forall f\in F, \end{aligned}$$

(17c)

$$\begin{aligned}&\sum \limits _{\left\{ L \mid L \in LP \right\} } \mu ^L \le 1, \end{aligned}$$

(17d)

$$\begin{aligned}&\sum \limits _{\left\{ f \mid f \in F\right\} } x_{f,e} \le \sum \limits _{ \left\{ L \mid L \in LP, e \in E^L \right\} } c_{e}^L \mu ^L \quad \forall e \in E, \end{aligned}$$

(17e)

$$\begin{aligned}&\mu ^L \ge 0,\ x_{f,e} \ge 0 \quad \forall L \in LP, f\in F, e\in E. \end{aligned}$$

(17f)

Here, L denotes a link pattern, and LP is the set of all the link patterns. Note that, a link pattern determines both the set of activated links and the number of data streams used by each link. $\mu ^L$ is the activation time of link pattern L, $E^L$ is the set of activated links in link pattern L, and $c_e^L$ is the average rate of link e in link pattern L. $c_e^L$ is determined by the product of the average data rate of each data stream (refer to (1-3) in Sect. 3.2) and the number of data streams used for link e in link pattern L. Since the number of link patterns is even larger than that of the node patterns’, the column generation method is used to resolve the issue. Moreover, the column generation method for a link based scheme is a special case of a node based scheme, and similar technics we discussed before can still be used here.

Appendix 4: upper bound of the quantization loss

Given the intended frame length z, the quantization loss $\delta _L$ after frame generation of the node-based scheme is upper bounded by $\frac{c1}{2z}+\frac{n'}{2z+n'}$, where $c_1=\max _{l\in E}\frac{ (\max _{S\in T_{l,2}}c_e^S)\mid T_{l,2}\mid }{2z (\sum _{\lbrace S\mid e\in E_p^S\rbrace }c_e^S\lambda _{p,e}^S )}$ and $T_{l,2}=\lbrace S \mid S\in N',l\in E_p^S,\theta ^Sz<1/2\rbrace$. Denote ${\tilde{x}}_{f,e}$ to be the normalized routing scheme of (4) in Sect. 3.1. In other words, ${\tilde{x}}_{f,e}={x_{f,e}}/{\zeta h_f}$, where r is the maximum throughput scaling parameter derived from (4) in Sect. 3.1. Let $\zeta _z'$ be the maximum throughput scaling parameter after frame generation with intended frame length equal to z. $\zeta _z'$ can be derived from (7) in Sect.  3.1. We then construct the following LP.

$$\begin{aligned} \max\,&\zeta _{2,z} \end{aligned}$$

(18a)

$$\begin{aligned} \text {s.t.}&\sum \limits _{f\in F} {\tilde{x}}_{f,e}h_f\zeta _{2,z}\le \sum \limits _{\left\{ S\mid e\in E_p^S \right\} }c_e^S\tau _{z,p,e}^S. \end{aligned}$$

(18b)

Here $\tau _{z,,p,e}^S={[z\cdot \theta ^S]\cdot \frac{\lambda _{p,e}^S}{\theta ^S}}/{\sum _{S\in N}[z\cdot \theta ^S]}$, and $\tau _{z,p,e}^{S}$ can be obtained after (4) has been resolved. Clearly, $\zeta _{2,z}\le \zeta _z'$. Define $E_1=\left\{ e\mid e\in E, \sum _{f\in F}{\tilde{x}}_{f,e}h_f\zeta =\sum _{\lbrace S\mid e\in E_p^S, \rbrace }c_e^S \tau _{z,p,e}^S\right\}$, and $\zeta _3=\min _{e\in E_1}{\sum _{\lbrace S\mid e\in E_p^S\rbrace }c_e^S\lambda _{p,e}^S}/{\sum _{f\in F}{\tilde{x}}_{f,e}h_f}$. Obviously $\zeta \le \zeta _3$. Hence,

$$\begin{aligned} \delta _L=1-\frac{\zeta _{2,z}}{\zeta }\le \min \limits _{e\in E_1}{\sum \limits _{\left\{ S\mid e\in E_p^S\right\} }c_e^S\left( \lambda _{p,e}^S-\tau _{z,p,e}^S\right) }/{\sum \limits _{\left\{ S \mid e\in E_p^S\right\} }c_e^S\lambda _{p,e}^S}. \end{aligned}$$

(19)

Denote $T_{e,1}=\lbrace S \mid S\in N',e\in E_p^S,\theta ^Sz\ge \frac{1}{2}\rbrace$, $T_{e,3}=\lbrace S \mid S\in N',e\notin E_p^S\rbrace$ and $n'=|N'|$ ($n'=\mid T_{e,1}\mid +\mid T_{e,2}\mid +\mid T_{e,3}\mid$ is the number of node patterns used in (7)). Therefore,

$$\begin{aligned} \sum \limits _{\left\{ S \mid e\in E_p^S\right\} }c_{e}^S\tau _{z,p,e}^S\ge {\sum \limits _{\left\{ S \mid S\in T_{e,1}\right\} }c_e^S \frac{\tau _{z,p,e}^S}{\theta ^S}(\theta ^Sz-\frac{1}{2})}/{n_z}. \end{aligned}$$

(20)

where $n_z=\sum _{S\in N'}\theta ^Sz\le \sum _{\left\{ S\mid S\in T_{e,1}\cup T_{e,3}\right\} }(\theta ^Sz+\frac{1}{2})+\sum _{\lbrace S\mid S\in T_{e,2}\rbrace }\theta ^Sz\le z+\frac{\mid T_{e,1}\mid }{2}+\frac{\mid T_{e,3}\mid }{2}\le z+\frac{n'}{2}$. Hence $\lambda _{p,e}^S-\tau _{z,p,e}^S\le \lambda ^S(1-\frac{z}{n_z}-\frac{1}{2n_z\theta ^S})\le \lambda _{p,e}^S(1-\frac{z}{n_z})\le \lambda _{p,e}^S \frac{n'}{2z+n'}$. Since for any $e\in T_{e,2}, \tau _{z,p,e}^Sz<\theta ^S z<\frac{1}{2}$, $\sum \limits _{\lbrace S \mid e\in T_{e,2}\rbrace }c_e^S\tau _{z,p,e}^S \le \frac{\left( \max _{S\in T_{e,2}}c_e^S\right) \mid T_{e,2}\mid }{2z}$, $1-\frac{z}{n_z}-\frac{1}{2n_z\theta ^S}<1-\frac{z}{n_z}\le \frac{\frac{\mid T_{e,1}\mid }{2}+\frac{\mid T_{e,3}\mid }{2}+\frac{\mid T_{e,2}\mid }{2}}{z-\frac{\mid T_{e,1}\mid }{2}-\frac{\mid T_{e,3}}{2}-\frac{\mid T_{e,2}\mid }{2}}$. Therefore,

$$\begin{aligned} \delta _L&\le \min \limits _{e\in E}{(\max _{\left\{ S\in T_{e,2}\right\} } c_e^S)\mid T_{e,2}\mid }/{\left( 2z\sum \limits _{\left\{ S \mid e\in E_p^S \right\} }c_e^S\lambda _e^S\right) }+\frac{\sum \limits _{\left\{ S\mid e\in T_{e,1}\right\} }c_e^S \lambda _e^S\frac{n'}{2z+n'} }{\sum \limits _{\left\{ S\mid e\in E_p^S \right\} }c_e^S \lambda _e^S} \le \frac{c1}{2z}+\frac{n'}{2z+n'}. \end{aligned}$$

(21)

Here, $c_1=\max _{e\in E}{(\max _{\lbrace S\in T_{e,2}\rbrace } c_e^S) \mid T_{e,2} \mid }/{(2z\sum _{\left\{ S\mid e\in E_p^S \right\} }c_e^S \lambda _e^S)}$. We can oberve that, $\delta _L$ converges to 0 as z approaches to infinity.