Mobility analysis of CoMP-based ultra-dense networks with stochastic geometry methods

Abstract

This paper conducts mobility analysis in Coordinated multipoint (CoMP) based ultra-dense networks (UDNs) where channel state information (CSI) is outdated due to feedback delay. To depict the impact of mobility on CoMP-based UDNs, related analyses are carried out from two perspectives. For one thing, we define CoMP handover probability as the probability that the serving cluster doesn’t remain the best candidate during the movement and further give its theoretical expression with stochastic geometry methods. For another, coverage probability is evaluated by considering the effect of outdated CSI caused by mobility. Furthermore, to capture the comprehensive effect of mobility on network performance, we propound the effective coverage probability (ECP) incorporating the above two effects. Numerical results illustrate that with the increase of users’ velocity, CoMP handover probability increases while coverage probability decreases but can be compensated by relatively larger cluster size schemes or denser access points deployment. Also, our proposed performance metric ECP reveals the tradeoff between CoMP handover probability and coverage probability, which depends on cluster size and network sensitivity to handover failure.

Appendices

Appendix 1

From Fig. 2, considering a typical mobile user initially connected to ${\varPhi _c}$ with distances $\left\{ {{r_1},{r_2} \ldots {r_n}} \right\}$ and moving to a new location ${O_2}$ with distances $\left\{ {{R_1},{R_2} \ldots {R_n}} \right\}$, $A{P_i}$ is able to become the farthest one among the previous cluster when all the other APs stay in the region ${\varUpsilon _n}$ (${\varUpsilon _n}\mathop = \limits ^\varDelta \odot {O_1}\backslash {S_1}$) whose boundary is restricted by

$$\begin{aligned} {R_i} > {R_j},1 \le j \le n,j \ne i \end{aligned}$$

(23)

where ${R_k} = {\left( {{r_k}^2 + {v^2} - 2v{r_k}\cos {\omega _k}} \right) ^{\frac{1}{2}}},k = 1,2 \ldots n$ and the conditions in (23) is denoted as $\mathbb {C}{_i}$ for clarity. Based on the boundary conditions, ${p_{f\_i}}$ can be derived as (24), where $I\left( \mathbb {C}{_i} \right)$ is the index function that $\mathbb {C}{_i}$ values 1 if the conditions holds or 0 otherwise.

$$\begin{aligned}{p_{f\_i}}&= \int \limits _{0 < {w_n} < 2\pi } { \ldots \int \limits _{0 < {w_1} < 2\pi }{\int \limits _{0 < {r_1} \ldots < {r_n} < \infty } {I\left( \mathbb {C}{_i} \right) } } } f\left( {{r_1}\ldots {r_n}} \right) \\&\quad \times\, f\left( {{w_1}} \right) \ldots f\left( {{w_n}} \right) d{r_1} \ldots d{r_n}d{w_1} \ldots d{w_n} \end{aligned}$$

(24)

where $f\left( {{w_i}} \right) = \frac{1}{{2\pi }},i = 1,2 \ldots n$ and the joint distance distribution of $\left\{ {{r_1},{r_2} \ldots {r_n}} \right\}$ is $f\left( {{r_1},{r_2}, \ldots ,{r_n}} \right) = {e^{ - \pi {\lambda _a}{r_n}^2}}{\left( {2\pi {\lambda _a}} \right) ^n}{r_1}{r_2} \ldots {r_n}$.

Denote the ranges of ${\omega _i},{r_i},i = 1,2 \ldots n$ restricted by ${R_i} > {R_j},j = 1,2 \ldots n,i \ne j$ as ${D_{ij}}$ and two different cases are considered as follows:

Case 1

When $i < j$, the boundaries can be determined by

$$\begin{aligned}{R_i}&> {R_j}\\ \quad &\Rightarrow {r_i}^2 + {v^2} - 2v{r_i}\cos {\omega _i}> {r_j}^2 + {v^2} - 2v{r_j}\cos {\omega _j}\\ \quad &\Rightarrow {r_i}^2 - 2v\cos {\omega _i}{r_i} - \left( {{r_j}^2 - 2v{r_j}\cos {\omega _j}} \right) > 0. \end{aligned}$$

(25)

Let ${g_1}\left( {{r_i}} \right) = {r_i}^2 - 2v\cos {\omega _i} \bullet {r_i} - \left( {{r_j}^2 - 2v{r_j}\cos {\omega _j}} \right)$, then we can further solving the quadratic inequality of ${g_1}\left( {{r_i}} \right) > 0$ as

$$\begin{aligned} \mathop {{D_{ij}}}\limits _{(i < j)} = \left\{ \begin{array}{l} {\omega _i} \in ({\varphi _{ij}},2\pi - {\varphi _{ij}}),\mathrm{{ }}{r_i} \in (0, + \infty )\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if{\ }{r_j}< 2v\cos {\omega _j}\\ {\omega _i} \in \left( 0,{\varphi _{ij}}\right) \cup \left( 2\pi - {\varphi _{ij}},2\pi \right) ,{r_i} \in (0,{x_{ij}}^{(1)}) \cup ({x_{ij}}^{(2)}, + \infty )\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if{\ }{r_j} < 2v\cos {\omega _j}\\ {\omega _i} \in \left( {0,2\pi } \right) ,{r_i} \in \left( {x_{ij}}^{(2)}, + \infty \right) \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if{\ }{r_j} > 2v\cos {\omega _j} \end{array} \right. \end{aligned}$$

(26)

with  ${x_{ij}}^{(1),(2)} = v\cos {\omega _p} \mp \sqrt{{{(v\cos {\omega _p})}^2} + ({r_l}^2 - 2v{r_l}\cos {\omega _l})}$, ${\varphi _{ij}} = \arccos (\sqrt{\frac{{2v{r_l}\cos {\omega _l} - {r_l}^2}}{{{v^2}}}} )$ with $l = \max (i,j),p = \min (i,j)$.

Case 2

When $i > j$, the boundaries are restricted by

$$\begin{aligned} &{R_i}> {R_j}\\ &\quad \Rightarrow {r_i}^2 + {v^2} - 2v{r_i}\cos {\omega _i} > {r_j}^2 + {v^2} - 2v{r_j}\cos {\omega _j}\\ &\quad \Rightarrow {r_j}^2 - 2v\cos {\omega _j}{r_j} - \left( {{r_i}^2 - 2v{r_i}\cos {\omega _i}} \right) < 0. \end{aligned}$$

(27)

Then we can further solving the quadratic inequality of ${g_1}\left( {{r_j}} \right) < 0$ as

$$\begin{aligned} \mathop {{D_{ij}}}\limits _{(i> j)} = \left\{ \begin{array}{l} {\omega _j} \in \left( {0,2\pi } \right) ,{r_j} \in \left(0,{x_{ij}}^{(2)}\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if{\ }{r_i} > 2v\cos {\omega _i}\\ {\omega _j} \in (0,{\varphi _{ij}}) \cup (2\pi - {\varphi _{ij}},2\pi ),{r_j} \in \left({x_{ij}}^{(1)},{x_{ij}}^{(2)}\right)\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ if{\ }{r_i} < 2v\cos {\omega _i} \end{array} \right. \end{aligned}$$

(28)

Note that the ranges of small subscript ${r_i},{w_i},i = 1,2 \ldots n$ are derived by that of with large subscript ${r_j},{w_j},j = 1,2 \ldots n,j \ne i$. In detail, ${r_i},{w_i}$ is restricted by the ranges of ${r_{i + 1}},{w_{i + 1}} \ldots {r_n},{w_n}$ according to (26) and the ranges of ${r_i},{w_i}$ confine ${r_{i - 1}},{w_{i - 1}} \ldots {r_1},{w_1}$ based on (28).

Then based on the boundary conditions $\mathbb {C}{_i}$ and conditioned on no handoffs, (8) can further derived as

$$\begin{aligned} {p_{f\_i}} = \int \limits _{{D_{i1}} \cap {D_{i2}} \cdots \cap {D_{in}}} {{{\left( {\frac{1}{{2\pi }}} \right) }^n}f\left( {{r_1}, \ldots ,{r_n}} \right) }d{r_1}d{w_1}d{r_2}d{w_2} \ldots d{r_n}d{w_n} \end{aligned}$$

(29)

where ${D_{ii}} = \left\{ \begin{array}{l} {r_1} < {r_2} \cdots < {r_n}\\ {w_k} \in \left[ {0,2\pi } \right] ,k = 1,2 \ldots n \end{array} \right.$are the default constraints.

Appendix 2

According to (5), a handoff does not occur if there is no other AP closer than the present farthest AP to the user, hence,

$$\begin{aligned}{g_i}&= \int \limits _{0 \,< {w_n} \,< 2\pi } \,{ \ldots \,\int \limits _{0 \,< {w_1} \,< 2\pi }\,{\int \limits _{0 \,< {r_1} \cdots \,< {r_n} < \infty } \,{{e^{ - {\lambda _a}\,{S_2}}}}{I\left( \mathbb {C}{_i} \right) } } } \,f\left( {{r_1}\,\ldots {r_n}} \,\right) \\&\quad \times \, f\left( {{w_1}} \right) \,\ldots \, f\left( {{w_n}} \right) d{r_1} \ldots d{r_n}d{w_1} \,\ldots \, d{w_n} \end{aligned}$$

(30)

where $I\left( \mathbb {C}{_i} \right)$ is solved in Lemma 1 and the “dangerous area” can be calculated by geometry knowledge as

$$\begin{aligned} {S_2}&= {S_{circle{O_2}}} - {S_{lap}}\\&= \pi {R_i}^2 - \left( {{S_{\sec torA\,{O_1}\,B}} \,+ \,{S_{\sec torA\,{O_2}\,B}} - {S_{quadrangleA\,{O_1}\,B\,{O_2}}}} \right) \\&= \pi {R_i}^2 - \left( {{\beta _i}{r_n}^2 \,+ \,{\varphi _i}{R_i}^2 - v{r_n}\sin {\beta _i}} \right) \\&=\left( {\pi - {\varphi _i}} \right) {R_i}^2- {\beta _i}{r_n}^2 \,+\, v{r_n}\sin {\beta _i} \end{aligned}$$

(31)

with ${\varphi _i} = \pi \, - \,{\beta _i} - {\sin ^{ - 1}}\left( {v\sin {\beta _i}/{R_i}} \right)$ and ${\beta _i} = {\cos ^{ - 1}}\left( {{r_n}^2 + {v^2} - {R_i}^2} \right) /2v{r_n}$.

Combining $I\left( \mathbb {C}{_i} \right)$ and the condition that no AP appears in ${S_2}$, we can further derive ${g_i}$ as

$$\begin{aligned} {g_i} = \int \limits _{{D_{i1}} \cap {D_{i2}} \cdots \cap {D_{in}}} {{{\left( {\frac{1}{{2\pi }}} \right) }^n}{{e^{ - {\lambda _a}{S_2}}}}f\left( {{r_1}, \ldots ,{r_n}} \right) }d{r_1}d{w_1}d{r_2}d{w_2} \ldots d{r_n}d{w_n}. \end{aligned}$$

(32)

Finally, the overall non-handover probability can be got by simply adding the mutually exclusive n cases as (9).

Appendix 3

Since ${h_{i,m}}$ is zero and complex Gaussian distribution, i.e., ${h_{i,m}} \sim {{\mathcal {C}}}{{\mathcal {N}}}\left( {0,{\mu } } \right)$, ${\chi _i} = {\chi _{i,m}} = {\left| {{h_{i,m}}} \right| ^2}$(subscript m is removed for simplicity) contributes to Rayleigh distribution, i.e., ${\chi _i} \sim \exp \left( {\frac{1}{\mu }} \right)$. Thus, the CDF and PDF of ${\chi _i}$ can be expressed respectively as

$$\begin{aligned} {f_{{\chi _i}}}\left( y \right) = \frac{1}{\mu }{e^{ - \frac{y}{\mu }}},\end{aligned}$$

(33a)

$$\begin{aligned} {F_{{\chi _i}}}\left( y \right) = 1 - {e^{ - \frac{y}{\mu }}}. \end{aligned}$$

(33b)

In turn, the CDF of ${\phi _i}, i = 1,2 \ldots n$ can be derived as

$$\begin{aligned} {F_{{\phi _i}}}\left( z \right) = \mathbb {P}\left( {{\phi _i} \le z} \right) = \mathbb {P}\left( {\left( {\mathop {\max }\limits _{i = 1,2 \ldots M} {{\left| {{h_i}} \right| }^2}} \right) \le z} \right) \mathop = \limits ^{\left( a \right) } {F_{{\chi _i}}}^M\left( z \right) \end{aligned}$$

(34)

where (a) follows that the sub-channels are independent from each other. Then the PDF can be derived from the concept of probability theory, yielding

$$\begin{aligned} {f_{{\phi _i}}}\left( z \right)&= \frac{{d{F_{{\phi _i}}}\left( z \right) }}{{dz}} = M{f_{{\chi _i}}}\left( z \right) {F_{{\chi _i}}}{\left( z \right) ^{M - 1}} = M\frac{1}{\mu }{e^{ - \frac{z}{\mu }}}{\left( {1 - {e^{ - \frac{z}{\mu }}}} \right) ^{M - 1}}\\&= \frac{M}{\mu }{e^{ - \frac{z}{\mu }}}{\sum \limits _{k = 0}^{M - 1} {\left( {\begin{array}{c} {M - 1}\\ k \end{array}} \right) \left( { - {e^{ - \frac{z}{\mu }}}} \right) } ^k}\\ &=\mathrm{{ }}\frac{M}{\mu }{\sum \limits _{k = 0}^{M - 1} {\left( {\begin{array}{c} {M - 1}\\ k \end{array}} \right) \left( { - 1} \right) } ^k}{e^{ - \frac{{\left( {k + 1} \right) z}}{\mu }}}. \end{aligned}$$

(35)

Then according to the concept of probability theory, the PDF of ${{\tilde{\phi }} _i},i = 1,2 \ldots n$ can be attained as

$$\begin{aligned} {f_{{{{\tilde{\phi }} }_i}}}\left( x \right)&= \int _0^\infty {{f_{{{{\tilde{\phi }} }_i},{\phi _i}}}\left( {x,y} \right) } dy = \int _0^\infty {{f_{{{{\tilde{\phi }} }_i}\left| {{\phi _i}} \right. }}\left( {x\left| y \right. } \right) {f_{{\phi _i}}}\left( y \right) } dy \\&= \int _0^\infty {\frac{{{f_{{{{\tilde{\chi }} }_i},{\chi _i}}}\left( {x,y} \right) }}{{{f_{{\chi _i}}}\left( y \right) }}{f_{{\varphi _i}}}\left( y \right) } dy. \end{aligned}$$

(36)

In the following, ${f_{{{{\tilde{\chi }} }_i},{\chi _i}}}\left( {x,y} \right)$ can be obtained by considering the special case of ${m_B} = 1$ (the fading parameter of Nakagami-m model) in [40, Eq.(6.2)]

$$\begin{aligned} {f_{{{{\tilde{\chi }} }_i},{\chi _i}}}\left( {x,y} \right) = \frac{1}{{\left( {1 - \rho } \right) {\mu ^2}}}{e^{ - \frac{{x + y}}{{\mu \left( {1 - \rho } \right) }}}} \times {I_0}\left( {\frac{{2\sqrt{\rho xy} }}{{\mu \left( {1 - \rho } \right) }}} \right) \end{aligned}$$

(37)

where ${I_0}\left( \bullet \right)$ denotes the $0_{th}$ order modified Bessel function of first kind [36, Eq. (8.406.3)] and can be written as

$$\begin{aligned} {I_0}\left( z \right) = {j^{ - n}}{J_n}\left( {jz} \right) \left| {_{n = 0}} \right. = {J_0}\left( {jz} \right) . \end{aligned}$$

(38)

Then ${f_{{{{\tilde{\phi }} }_i}}}\left( x \right)$ can be got first by substituting (38) into (37) and then by plugging (33a), (35) and (37) into (36) as

$$\begin{aligned}{f_{{{{\tilde{\phi }} }_i}}}\left( x \right) \\ = \frac{M}{{\left( {1 - \rho } \right) {\mu ^2}}}{e^{ - \frac{x}{{\mu \left( {1 - \rho } \right) }}}}{\sum \limits _{k = 0}^{M- 1} {\left( {\begin{array}{c} {M-1}\\ k \end{array}} \right) \left( { - 1} \right) }^k}\int _0^\infty {{e^{ - \frac{{1 + k - k\rho }}{{\mu \left( {1 - \rho } \right) }}y}}{J_0}\left( {2 \,\times \, j\frac{{\sqrt{\rho x} }}{{\mu \left( {1 - \rho } \right) }} \bullet \sqrt{y} } \right) } dy. \end{aligned}$$

(39)

Using [36, Eq. (6.643.4)], (39) can be further derived as

$$\begin{aligned} {f_{{{{\tilde{\phi }} }_i}}}\left( x \right) &= {\sum \limits _{k = 0}^{M-1} {\left( {\begin{array}{c} {M-1}\\ k \end{array}} \right) \left( { - 1} \right) }^k}\frac{M}{{\mu \left( {1 + k - k\rho } \right) }}{e^{ - \frac{{1 + k}}{{\mu \left( {1 + k - k\rho } \right) }}x}}\\&\quad \times \, L_0^0\left( { - \frac{{\rho x}}{{\mu \left( {1 - \rho } \right) \left( {1 + k - k\rho } \right) }}} \right) . \end{aligned}$$

(40)

Since $L_0^0\left( x \right) = 1$ [36, Eq. (8.970.1)], we can finally get

$$\begin{aligned} {f_{{{{\tilde{\phi }} }_i}}}\left( x \right) \mathrm{{ = }}M{\sum \limits _{k = 0}^{M-1} {\left( {\begin{array}{c} {M-1}\\ k \end{array}} \right) \left( { - 1} \right) } ^k}{\eta _k}{e^{ - {\eta _k}\left( {1 + k} \right) x}} \end{aligned}$$

(41)

where ${\eta _k} = \frac{1}{{\mu \left( {1 + k - k\rho } \right) }}$.