Optimal transmission methodology for QoS provision of multi-hop cellular network

Abstract

In this paper, we present a framework of link capacity analysis for optimal transmission over uplink MCN (Multi-hop Cellular Network) environments. An overlaid architecture is employed as the network topology, i.e., single-hop transmission over the inner region and multi-hop transmission over the outer region. In particular, we analyzed the gain that accrued from grafting a relay method onto a conventional, SCN (Single-hop Cellular Network) and investigated the conditions for optimal performance through the numerical results. At high-user density, a MCN exhibits a much more reliable transmission than the SCN. For maximal link capacity, optimal region partitioning is approximately accomplished at the normalized cell radius of 0.6 in most of cases for region division. Finally, the link capacity can be improved 1.2–1.8 times better than the SCN when the number of relay hops is 1.6 and the half-duplex mechanism is used. In addition, the proposed MCN scheme demonstrates an effective reduction in transmission power relative to the SCN.

Appendix

1.1 Interference from the multi-hop network

In this section, the means and variances of the interference terms $I_{0_{m},SN}$ and I _BS,ON in (9) and (12) are derived, which are caused by the multi-hop transmission of other MSs in R _o .

1.1.1 The means of $I_{0_{m},SN}$ and I _BS,ON

The mean of $I_{0_{m},SN}$ in (15) is given by $m_{I_{0_{m},SN}}$

$$ \begin{aligned} =& \sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}E[X_{j_{x}}\, P_{MH}\,G_{j_{x},0_{m}}\,\rho_{R_{o}}]-S_{MH}\\ =&\sum_{x_{j_{x}}=0}^{1}\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}x_{j_{x}} P_{MH}\,c_{1}\,r_{j_{x},0_{m}}^{-l_{1}} P_{X_{j_{x}}}(x_{j_{x}})\,\rho_{R_{o}}-S_{MH} \end{aligned} $$

and the mean of I _BS,ON in (16) is given by $m_{I_{BS,ON}}$

$$ \begin{aligned} =& \sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}E[X_{j_{x}}\, P_{MH}\,G_{j_{x},BS} \,\rho_{R_{o}}]\\ =&\sum_{x_{j_{x}}=0}^{1}\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}x_{j_{x}} P_{MH}\,c_{1} r_{j_{x},BS}^{-l_{1}} \,P_{X_{j_{x}}}(x_{j_{x}}) \rho_{R_{o}}\\ \end{aligned} $$

where $x_{j_{x}}=1$ for the active state and $x_{j_{x}}=0$ for the inactive state according to the RF and pipeline transmission. Since the distance and transmission power between two adjacent nodes at the multi-hop network is relatively short and small not to affect the entire system performance, the path-loss gain $G_{j_{x},0_{m}}$ and $G_{j_{x},BS}$ can be approximated by only the distance factor $r_{j_{x},0_{m}}^{-l_{1}}$ and $r_{j_{x},BS}^{-l_{1}}$ without the use of the shadowing factor.

1.1.2 The variances of $I_{0_{m},SN}$ and $I_{BS,ON}$

The variance of $I_{0_{m},SN}$ in (15) is given by $\sigma_{I_{0_{m},SN}}^{2}$

$$ \begin{aligned} =&V[\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}X_{j_{x}} P_{MH} G_{j_{x},0_{m}} \rho_{R_{o}}] \\ =&\sum_{x_{j_{x}}=0}^{1}\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}x_{j_{x}}^{2} P_{MH}^{2}\,c_{1}^{2} \,r_{j_{x},0_{m}}^{-2l_{1}} P_{X_{j_{x}}}(x_{j_{x}})^{2} \,\rho_{R_{o}}-m_{I_{0_{m},SN}}^{2}\\ \end{aligned} $$

and the variance of I _BS,ON in (16) becomes $\sigma_{I_{BS,ON}}^{2}$

$$ \begin{aligned} =&V[\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}X_{j_{x}}P_{MH}G_{j_{x},BS} \rho_{R_{o}}] \\ =&\sum_{x_{j_{x}}=0}^{1}\sum_{j=0}^{N_{OC}}\sum_{x=1,\;x\in R_{o}}^{A_{R_{o}}}x_{j_{x}}^{2} P_{MH}^{2}\,c_{1}\, r_{j_{x},BS}^{-2l_{1}} P_{X_{j_{x}}}(x_{j_{x}})^{2} \rho_{R_{o}}-m_{I_{BS,ON}}^{2}\\ \end{aligned} $$

1.2 Interference from the single-hop network

In this section, the means and variances of the interference terms $I_{0_{m},ON}$ and I _BS,SN in (8) and (10) are derived, which are caused by the single-hop transmission of other MSs in R _i .

1.2.1 The mean and variance of $I_{0_{m},ON}$

$I_{0_{m},ON}$ can be divided into two components. One is the intra-cell interference (K = 0) and the other is the inter-cell interference (K ≠ 0), which are given by

$$ I_{sc}=\sum_{j=0}^{N_{OC}} \sum_{x=1}^{A_{R_{i}}} I_{j_x,0;BS}\cdot \rho_{R_{i}}, \quad I_{oc}=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}} I_{j_x,K;BS}\cdot \rho_{R_{i}}. $$

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Therefore, the inter-network interference $I_{0_{m},ON}$ in (8) is given by

$$ I_{0_{m},ON}= I_{sc}+I_{oc}=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}} I_{j_x,K;0_{m}}\cdot \rho_{R_{i}}. $$

(24)

Using the criterion in (3), the interference term $I_{j_x,K;0_{m}}$ in (23) is then expressed by

$$ I_{j_x,K;0_{m}} = S_{UL} \left(\frac{r_{j_x,K}} {r_{j_x,0_{m}}}\right)^{l_{1}} 10^{(\xi_{j_x,0_{m}}-\xi_{j_x,K})/10} $$

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where $r_{j_{x},K}$ and $\xi_{j_{x},K}$ are the distance and the shadowing factor between the xth MS in the jth cell and the Kth BS. In addition, $r_{j_{x},0_{m}}$ and $\xi_{j_{x},0_{m}}$ are the distance and the shadowing factor between the xth MS in the jth cell and the mth MS in the 0th BS.

The mean of $I_{j_x,K;0_{m}}$ in (24) can be written by

$$E\left[I_{j_x,K;0_{m}}\right]=E\left[E\left[I_{j_x,K;0_{m}}|K=k,\xi_{j_x,k}=\xi_{0}\right]\right].$$

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The probability $P_{j_x,K}(K=k,\xi_{j_x,k}=\xi_{0})$ is then

$$ P_{j_x,K}(k,\xi_{0})= P_{j_x,K}(k|\xi_{0}) P(\xi_{0}). $$

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Let S ^D be the initial pilot power from each BS. Assume that the pilot signal of the kth BS is the strongest for the MS at the “x” position of the jth cell. Then,

$$ S_{k,\;j_x}^{D}=S^{D}\cdot r_{k,\;j_x}^{-l_{1}}\cdot10^{\frac{\xi_{k,\;j_x}}{10}}\geqq S^{D}\cdot r_{i,\;j_x}^{-l_{1}}\cdot10^{\frac{\xi_{i,\;j_x}}{10}} $$

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where 0 $\leqq i \leqq N_{OC}$ and i ≠ k. Set $\xi_{k\;,j_x} = \xi_{j_x,\;k} =\xi_{0}, \xi_{i,\;j_x} = \xi_{j_x,\;i}$ and $r_{k,\;j_x} =r_{j_x,\;k}$ . Then,

$$ \frac{r_{k,\;j_x}^{l_{1}}} {r_{i,\;j_x}^{l_{1}}}10^{\frac{\xi_{i,\;j_x}-\xi_{0}}{10}}\leq 1 \,\,\Rightarrow\,\,\xi_{0}-\xi_{i,\;j_x}\geq 10\log_{10}\left(\frac{r_{k,\;j_x}}{r_{i,\;j_x}}\right)^{l_{1}}. $$

Thus, $P_{j_x,\;K}(k|\xi_{0})$ in (27) yields $P_{j_x,\;K}(k|\xi_{0}) = Pr[S_{k,\;j_x}^{D}=\max(S_{i,\;j_x}^{D})|\xi_{0}]$

$$ \begin{aligned} =&\prod_{i=0, i \neq k}^{N_{OC}}P \left [(\xi_{0}-\xi_{i,\;j_x})\geq10\log_{10} \left (\frac{r_{k,\;j_x}} {r_{i,\;j_x}}\right)^{l_{1}}\right]\\ =& \prod_{i=0, i \neq k}^{N_{OC}}\left[1-Q \left (\frac{\xi_{0}-10\log_{10}(\frac{r_{k,\;j_x}}{r_{i,\;j_x}})^{l_{1}}}{\sigma_{i,\;j_x}}\right)\right] \\ \end{aligned} $$

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where $\sigma_{i,j_x}$ is set to a constant σ₀. W.r.t. the 0th BS, the following condition must be satisfied:

$$ \psi\left(\frac{r_{k,j_x}^{l_{1}}} {r_{0,j_x}^{l_{1}}}10^\frac{{\xi_{0,j_x}}-\xi_{0}}{10}\right) =\left\{\begin{array}{ll}1 & \frac{r_{k,j_x}^{l_{1}}}{r_{0,j_x}^{l_{1}}}10^{\frac{\xi_{0,j_x}-\xi_{0}}{10}}< 1 \\ 0 & \frac{r_{k,j_x}^{l_{1}}}{r_{0,j_x}^{l_{1}}}10^{\frac{\xi_{0,j_x}-\xi_{0}}{10}}\geq 1.\end{array}\right. $$

The conditional expectation in (26) is $E[I_{j_x,K;0_{m}}|k,\xi_{0}]$

$$ =\int\limits_{\xi_{j_{x},0_{m}}}\int\limits_{\xi_{0,\;j_x}} \left(\frac{r_{j_x,k}} {r_{j_x,\;0_{m}}}\right)^{l_{1}}\,10^{\frac{\left(\xi_{j_{x},\;0_{m}}-\xi_{0}\right)} {10}}\,\psi \left (\frac{r_{k,\;j_x}^{l_{1}}}{r_{0,\;j_x}^{l_{1}}} 10^{\frac{\xi_{0,\;j_{x}} -\xi_{0}}{10}}\right) f_{\xi_{0,\;j_x}}(\xi)\,f_{\xi_{j_x,\;0_{m}}}(\xi)\,d\xi_{0,\;j_{x}}\,d\xi_{j_{x},\;0_{m}}. $$

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Thus, $E[I_{j_x,K;0_{m}}]$ , $V[I_{j_x,\;K;0_{m}}]$ are obtained by (27) and (30):

$$ \begin{aligned} E[I_{j_x,\;K;0_{m}}] =& \int_{\xi_{j_x,0}=\xi_0} \sum_{k=0}^{N_{OC}}E[I_{j_x,K;0_{m}}|k,\xi_{0}] P_{j_x,K}(k,\xi_{0}) d\xi_0, \\ VAR[I_{j_x,\;K;0_{m}}]=&E[I^{2}_{j_x,K;0_{m}}]-E[I_{j_x,K;0_{m}}]^{2},\quad\quad \hbox{where} \\ E[I^{2}_{j_x,\;K;0_{m}}]=&\int_{\xi_{j_x,k}=\xi_0} \sum_{k=0}^{N_{OC}}E[I^{2}_{j_x,K;0_{m}}|k,\xi_{0}] P_{j_x,K}(k,\xi_{0}) d\xi_0, \cr E[I^{2}_{j_x,K;0_{m}}|k,\xi_{0}]=&\int_{\xi_{j_{x},\;0_{m}}}\int_{\xi_{0,j_x}} \left(\frac{r_{j_x,k}} {r_{j_x,0_{m}}}\right)^{2l_{1}}10^{\frac{\left(\xi_{j_{x},0_{m}}-\xi_{0}\right)} {5}}\psi \left (\frac{r_{k,j_x}^{l_{1}}}{r_{0,j_x}^{l_{1}}} 10^{\frac{\xi_{0,j_{x}} -\xi_{0}}{10}}\right) \cr&f_{\xi_{0,j_x}}(\xi) f_{\xi_{j_x,0_{m}}}(\xi)d\xi_{0,j_{x}}d\xi_{j_{x},0_{m}}. \\ \end{aligned} $$

Therefore, the mean and variance of $I_{0_{m},ON}$ are

$$ E[I_{0_{m},ON}]=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}}E[I_{j_{x},K;0_{m}}] \,\rho_{R_{i}},\,\,\,\,V[I_{0_{m},ON}]=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}}V[I_{j_{x},K;0_{m}}] \,\rho_{R_{i}}. $$

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1.2.2 The mean and variance of I _BS,SN

I_BS,SN can be divided into the intra-cell (K = 0) and inter-cell interferences (K ≠ 0) by

$$ I_{sc}= \sum_{j=0}^{N_{OC}} \sum_{x=1}^{A_{R_{i}}} I_{j_x,0;BS}\cdot \rho_{R_{i}} - S_{UL},\,\, \,\,I_{oc}=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}} I_{j_x,K;BS}\cdot \rho_{R_{i}}. $$

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Therefore, the intra-network interference I_BS,SN in (10) is given by

$$ \begin{aligned} I_{BS,SN}=&I_{sc}+I_{oc}=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}} I_{j_x,K;BS}\cdot \rho_{R_{i}} - S_{UL},\,\,\hbox{where}\\ I_{j_x,K;BS} =& S_{UL} \left(\frac{r_{j_x,K}}{r_{j_x,0}}\right)^{l_{1}} 10^{(\xi_{j_x,0}-\xi_{j_x,K})/10}. \\ \end{aligned} $$

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Then, the mean and variance of I_BS,SN can be derived like $E[I_{0_{m},ON}]$ , $V[I_{0_{m},ON}]$

$$ \begin{aligned} E[I_{j_x,K;BS}] =& \int\limits_{\xi_{j_x,k}=\xi_0} \sum_{k=0}^{N_{OC}}E[I_{j_x,K;BS}|k,\xi_{0}] P_{j_x,K}(k,\xi_{0}) d\xi_0.\\ V[I_{j_x,K;BS}]=&E[I^{2}_{j_x,K;BS}]-E[I_{j_x,K;BS}]^{2}\\ \hbox{where}\,\,\,E[I^{2}_{j_x,K;BS}]=& \int_{\xi_{j_x,k}=\xi_0} \sum_{k=0}^{N_{OC}}E[I^{2}_{j_x,K;BS}|k,\xi_{0}] P_{j_x,K}(k,\xi_{0}) d\xi_0 \\ E[I^{2}_{j_x,K;BS}|k,\xi_{0}]=&\int_{\xi_{j_x,0}=\xi} \left(\frac{r_{j_x,k}}{r_{j_x,0}}\right)^{2l_{1}}10^{\frac{(\xi-\xi_{0})}{5}}\psi \left (\frac{r_{k,j_x}^{l_{1}}}{r_{0,j_x}^{l_{1}}} 10^{\frac{\xi-\xi_{0}}{10}}\right) f_{\xi_{j_x,0}}(\xi)d\xi. \\ \end{aligned} $$

Therefore, the mean and variance of $I_{j_{x},K;BS}$ are

$$ E[I_{BS,SN}]=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}}E[I_{j_{x},K;BS}]\cdot\rho_{R_{i}}-S_{UL}, V[I_{BS,SN}]=\sum_{j=0}^{N_{OC}}\sum_{x=1}^{A_{R_{i}}}V[I_{j_{x},K;BS}]\cdot\rho_{R_{i}}. $$

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