Video placement and delivery in edge caching networks: Analytical model and optimization scheme

Abstract

How to place and deliver the videos through edge caching networks is a hot and open issue on releasing the heavy loads of backhaul link caused by explosive growth of data traffic and frequent requests. To address this issue, a joint video caching and transmission scheme is studied. In specific, the successful transmission probability, energy efficiency and system average delay are first derived based on the stochastic geometry theory. A video caching optimization problem is then formulated to balance the energy efficiency and system average delay. Considering the high complexity of the problem solution, we propose a genetic-based video placement and delivery algorithm to obtain a near-optimal solution. The accuracy of the proposed algorithm is finally validated by simulation results.

Appendices

Appendix A: Proof of Theorem 1

When cache is missed, the video is transmitted to user by the remote server through the return link and a SBS closest to the user. Let m denote the SBS closest to the typical user. Assuming t is the transmitted video symbol from SBS, the received signal y₁ can be written as

$$ {y_{1}} = \sqrt P \cdot {h_{m}} \cdot r_{m}^{- \alpha /2} \cdot t + \sum\limits_{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}} {\sqrt P } \cdot {h_{i}} \cdot r_{i}^{- \alpha /2} \cdot t $$

(33)

where h_m and h_i are the channel gain from the SBS closest to user and all SBSs from Φ∖{m} respectively, r_m and r_i are the distance from the SBS closest to user and all SBSs from Φ∖{m}, respectively, so the SIR of typical user can be written as

$$ SIR = \frac{{P \cdot {{\left| {{h_{m}}} \right|}^{2}} \cdot r_{m}^{- \alpha }}}{{\sum\limits_{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}} P \cdot {{\left| {{h_{i}}} \right|}^{2}} \cdot r_{i}^{- \alpha }}} = \frac{{{X_{1}}}}{{{I_{1}}}} $$

(34)

According to the definition of STP, let $\theta _{n,l}^{z,nocache} = {2^{B{R_{n,l}}/\left ({1 - {\rho _{bh}}} \right )W}} - 1,z \in \{ GU,WU\}$, we have

$$ \begin{array}{@{}rcl@{}} STP_{n,l}^{z,nocache} = {\int}_{0}^{\infty} {{f_{r}}\left( x \right) \cdot } \mathbb{P}\left\{ {SIR > \theta_{n,l}^{z,nocache}|{r_{m}} = x} \right\} \\dx,z \in \left\{ {GU,WU} \right\} \end{array} $$

(35)

where $\mathbb {P}\left \{ {SIR > \theta _{n,l}^{z,nocache}|{r_{m}} = x} \right \}$ is the STP under r_m = x, ${f_{r}}\left (x \right )$ is the probability density function (PDF) of the distance r_m.

$$ \begin{array}{@{}rcl@{}} &&\mathbb{P}\left\{ {SIR > \theta_{n,l}^{z,nocache}|{r_{m}} = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {\frac{{P \cdot {{\left| {{h_{m}}} \right|}^{2}} \cdot r_{m}^{- \alpha }}}{{{I_{1}}}} > \theta_{n,l}^{z,nocache}|{r_{m}} = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {{{\left| {{h_{m}}} \right|}^{2}} > \theta_{n,l}^{z,nocache} \cdot {I_{1}} \cdot {P^{- 1}} \cdot {x^{\alpha} }|{r_{m}} = x} \right\} \hfill \\ &&\mathop {\text{ = }}\limits^{(a)} {\mathbb{E}_{{I_{1}}}}\left[ {\exp \left( { - \theta_{n,l}^{z,nocache} \cdot {I_{1}} \cdot {P^{- 1}} \cdot {x^{\alpha} }} \right)} \right] \hfill \\ &&{\text{ = }}{\mathcal{L}_{{I_{1}}}}\left( {\theta_{n,l}^{z,nocache} \cdot {P^{- 1}} \cdot {x^{\alpha} }} \right) \hfill \end{array} $$

(36)

where step (a) holds since ${\left | {{h_{m}}} \right |^{2}}$ obeys the exponential distribution with parameter 1, ${{\mathscr{L}}_{{I_{1}}}}\left ({\zeta _{n,l}^{z,nocache}} \right )$ is the Laplace transform of the interference generated by the HPPP-distribution SBSs that are farther than that in Φ∖{m}, which is calculated as

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\!\!{\mathcal{L}_{{I_{1}}}}\left( {\zeta_{n,l}^{z,nocache}} \right) = {\mathbb{E}_{{I_{1}}}}\left[ {\exp \left( { \!- \sum\limits_{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}} {\zeta_{n,l}^{z,nocache} \cdot P} \cdot {{\left| {{h_{i}}} \right|}^{2}} \cdot x_{i}^{- \alpha }} \right)} \right] \hfill \\ &&\!\!\!\!\!= {\mathbb{E}_{\Phi ,{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}}{\prod} {\exp \left( { - \zeta_{n,l}^{z,nocache} \cdot P \cdot {{\left| {{h_{i}}} \right|}^{2}} \cdot x_{i}^{- \alpha }} \right)} } \right] \hfill \\ &&\!\!\!\!\!= {\mathbb{E}_{\Phi} }\left[ {\underset{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}}{\prod} {{\mathbb{E}_{{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\exp \left( { - \zeta_{n,l}^{z,nocache} \cdot P \cdot {{\left| {{h_{i}}} \right|}^{2}} \cdot x_{i}^{- \alpha }} \right)} \right]} } \right] \hfill \\ &&\!\!\!\!\!\mathop = \limits^{(b)} {\mathbb{E}_{\Phi} }\left[ {\underset{{b_{i}} \in {\Phi} \backslash \left\{ m \right\}}{\prod} {\frac{1}{{1 + \zeta_{n,l}^{z,nocache} \cdot P \cdot x_{i}^{- \alpha }}}} } \right] \hfill \\ &&\!\!\!\!\!\mathop = \limits^{(c)} \exp \left( { - 2\pi \cdot \lambda \int\limits_{x}^{\infty} {\frac{{\zeta_{n,l}^{z,nocache} \cdot P \cdot {\psi^{- \alpha }}}}{{1 + \zeta_{n,l}^{z,nocache} \cdot P \cdot {\psi^{- \alpha }}}}\psi d\psi } } \right) \hfill \\ &&\!\!\!\!\!= \exp \left( { - \pi \cdot \lambda \cdot {x^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }; - \theta_{n,l}^{z,nocache}} \right) - 1} \right)} \right) \hfill \end{array} $$

(37)

where step (b) can be obtained from that ${\left | {{h_{i,j}}} \right |^{2}}$ obeys the exponential distribution with parameter 1, and step (c) holds due to the probability generating functional (PGFL) of HPPP. The PDF of the distance r_m is

$$ {f_{{r_{m}}}}(x) = 2\pi \lambda x \cdot \exp \left( { - \pi \lambda {x^{2}}} \right) $$

(38)

Combining Eqs. 36, 37 and 38 to Eq. 35, we obtain Theorem 1.

Appendix B: proof of Theorem 2

When cache is missed, the SIR of WU is shown as follows

$$ SIR = \frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot {\chi^{- \alpha /2}}} \right|}^{2}}}}{{\sum\limits_{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}} {P \cdot {h_{j}} \cdot {\chi^{- \alpha /2}}} + \sum\limits_{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}} P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }}} = \frac{{{X_{2}}}}{{{I_{2}}}} $$

(39)

where Φ₃ is the set consist of three SBSs closest to WU. Let $\theta _{n,l}^{WU,nocache}$ denote the threshold of SIR of worst-case users, there holds

$$ STP_{n,l}^{WU,nocache} = {\int}_{0}^{\infty} {{f_{\chi} }\left( x \right) \cdot \mathbb{P}\left\{ {SIR > \theta_{n,l}^{WU,nocache}|\chi = x} \right\}} $$

(40)

where $\mathbb {P}\left \{ {SIR > \theta _{n,l}^{WU,nocache}|\chi = x} \right \}$ is the STP under χ = x,

$$ \begin{array}{@{}rcl@{}} &&\mathbb{P}\left\{ {SIR > \theta_{n,l}^{WU,nocache}|\chi = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {\frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot r_{i}^{- \alpha /2}} \right|}^{2}}}}{{{I_{2}}}} > \theta_{n,l}^{WU,nocache}|\chi = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {{h_{i}} \cdot r_{i}^{- \alpha /2}} } \right|}^{2}} > \theta_{n,l}^{WU,nocache} \cdot {I_{2}} \cdot {P^{- 1}}|\chi = x} \right\} \hfill \\ &&= {\mathbb{E}_{{I_{2}}}}\left[ {\exp \left( { - \theta_{n,l}^{WU,nocache} \cdot {I_{2}} \cdot {P^{- 1}}/{{\text{x}}^{- \alpha }}} \right)} \right] \hfill \\ &&{\text{ = }}{\mathcal{L}_{{I_{2}}}}\left( {\theta_{n,l}^{WU,nocache} \cdot {P^{- 1}}/{{\text{x}}^{- \alpha }}} \right) \hfill \end{array} $$

(41)

where ${{\mathscr{L}}_{{I_{2}}}}\left ({\zeta _{n,l}^{WU,nocache}} \right )$ is the Laplace transform of the interference generated by the HPPP-distribution SBSs that are farther than that in Φ∖Φ_K, which is calculated as

$$ \begin{array}{@{}rcl@{}} &&{\mathcal{L}_{{I_{2}}}}\left( {\zeta_{n,l}^{WU,nocache}} \right) = {\mathbb{E}_{{I_{3}}}}\left[ {\exp \left( { - \zeta_{n,l}^{WU,nocache} \cdot \left( {\sum\limits_{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}} {P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot {\chi^{- \alpha }}} + \sum\limits_{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}} {P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} } \right)} \right)} \right] \hfill \\ &&= {\mathbb{E}_{{{\Phi}_{K}},{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,nocache} \cdot P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot {\chi^{- \alpha }}} \right)} \cdot \underset{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,nocache} \cdot P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} \right)} } \right] \hfill \\ &&= {\left( {\frac{1}{{1 + \theta_{n,l}^{WU,nocache}}}} \right)^{2}} \cdot {\mathbb{E}_{{{\Phi}_{K}},{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,nocache} \cdot P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} \right)} } \right] \hfill \\ &&= {\left( {\frac{1}{{1 + \theta_{n,l}^{WU,nocache}}}} \right)^{2}} \cdot \exp \left( { - \pi \cdot \lambda \cdot {\chi^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }; - \theta_{n,l}^{WU,nocache}} \right) - 1} \right)} \right) \hfill \end{array} $$

(42)

The joint pdf of the link distances is

$$ {f_{\chi} }\left( x \right) = 2{\pi^{2}}{\lambda^{2}}{x^{3}}\exp (- \lambda \pi {x^{2}}) $$

(43)

Combining Eqs. 41, 42 and 43 to Eq. 40, we obtain Theorem 2.

Appendix C: proof of Theorem 3

Similarly, when cache is hit, the SIR of GU is expressed as

$$ SIR = \frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot r_{i}^{- \alpha /2}} \right|}^{2}}}}{{\sum\limits_{{b_{j}} \in {\Phi} \backslash {{\Phi}_{K}}} P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot r_{j}^{- \alpha }}} = \frac{{{X_{3}}}}{{{I_{3}}}} $$

(44)

Let $\theta _{n,l}^{GU,sch}$ denote the threshold of SIR of general users, we discuss the STP in two cases.

• Case 1: when K = 1, we have the following conclusion according to Appendix A

  $$ \begin{array}{@{}rcl@{}} STP_{n,l}^{GU,sch} = 1/{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }; - \theta_{n,l}^{GU,sch}} \right), \\ sch \in \left\{ {CSVC,CV,CMQL} \right\} \\ \end{array} $$

  (45)

• Case 2: when $K \geqslant 2$, there holds

  $$ \begin{array}{@{}rcl@{}} STP_{n,l}^{GU,sch} &=& {\int}_{0}^{{x_{2}}} {{\int}_{{x_{1}}}^{{x_{3}}} { \cdot \cdot \cdot } } {\int}_{{x_{K - 1}}}^{\infty} {f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) \cdot \mathbb{P}\left\{ {SIR > \theta_{n,l}^{GU,sch}|r = x} \right\} \\&&d{x_{1}}d{x_{2}} \cdot \cdot \cdot d{x_{K}} \end{array} $$

  (46)

  where $\mathbb {P}\left \{ {SIR > \theta _{n,l}^{GU,sch}|\boldsymbol {r = x}} \right \}$ is the STP under r = x,

  $$ \begin{array}{@{}rcl@{}} &&\mathbb{P}\left\{ {SIR > \theta_{n,l}^{GU,sch}|r = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {\frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot r_{i}^{- \alpha /2}} \right|}^{2}}}}{{{I_{3}}}} > \theta_{n,l}^{GU,sch}|r = x} \right\} \hfill \\ &&= \mathbb{P}\!\left\{ \!{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {{h_{i}} \cdot r_{i}^{- \alpha /2}} } \right|}^{2}} \!\!>\! \theta_{n,l}^{GU,sch} \cdot {I_{3}} \cdot {P^{- 1}}|r = x} \right\} \hfill \\ &&\mathop {\text{ = }}\limits^{(d)} {\mathbb{E}_{{I_{2}}}}\left[ {\exp \left( { - \theta_{n,l}^{GU,sch} \cdot {I_{3}} \cdot {P^{ - 1}}/\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {x_{i}^{- \alpha }} } \right)} \right] \hfill \\ &&{\text{ = }}{\mathcal{L}_{{I_{2}}}}\left( {\theta_{n,l}^{GU,sch} \cdot {P^{- 1}}/\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {x_{i}^{ - \alpha }} } \right) \hfill \end{array} $$

  (47)

  where step (d) holds since ${\left | {\sum \limits _{{b_{i}} \in {{\Phi }_{K}}} {{h_{i}} \cdot x_{i}^{- \alpha /2}} } \right |^{2}}$ obeys the exponential distribution with parameter $1/\sum \limits _{{b_{i}} \in {{\Phi }_{K}}} {x_{i}^{- \alpha }}$, ${{\mathcal L}_{{I_{3}}}}\left ({\zeta _{n,l}^{GU,sch}} \right )$ is the Laplace transform of the interference generated by the HPPP-distribution SBSs that are farther than that in Φ∖Φ_K, which is calculated as

  $$ \begin{array}{@{}rcl@{}} &&{\mathcal{L}_{{I_{3}}}}\left( {\zeta_{n,l}^{GU,sch}} \right) = {\mathbb{E}_{{I_{3}}}}\left[ {\exp \left( { - \sum\limits_{{b_{j}} \in {\Phi} \backslash {{\Phi}_{K}}} {\zeta_{n,l}^{GU,sch} \cdot P} \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot x_{j}^{- \alpha }} \!\right)} \!\!\right] \hfill \\ &&= {\mathbb{E}_{{{\Phi}_{K}},{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{j}} \in {\Phi} \backslash {{\Phi}_{K}}}{\prod} {\exp \left( { - \zeta_{n,l}^{GU,sch} \cdot P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot x_{j}^{- \alpha }} \right)} } \right] \hfill \\ &&= {\mathbb{E}_{{{\Phi}_{K}}}}\left[ {\underset{{b_{j}} \in {\Phi} \backslash {{\Phi}_{K}}}{\prod} {{\mathbb{E}_{{{\left| {{h_{j}}} \right|}^{2}}}}\left[ {\exp \left( { - \zeta_{n,l}^{GU,sch} \cdot P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot x_{j}^{- \alpha }} \right)} \right]} } \right] \hfill \\ &&= {\mathbb{E}_{{{\Phi}_{K}}}}\left[ {\underset{{b_{i}} \in {\Phi} \backslash {{\Phi}_{K}}}{\prod} {\frac{1}{{1 + \zeta_{n,l}^{GU,sch} \cdot P \cdot x_{j}^{- \alpha }}}} } \right] \hfill \\ &&= \exp \left( { - 2\pi \cdot \lambda \int\limits_{{x_{K}}}^{\infty} {\frac{{\zeta_{n,l}^{GU,sch} \cdot P \cdot {\psi^{- \alpha }}}}{{1 + \zeta_{n,l}^{GU,sch} \cdot P \cdot {\psi^{- \alpha }}}}\psi d\psi } } \right) \hfill \\ &&= \exp \left( { \!- \pi \cdot \lambda\! \cdot {x_{K}^{2}}\!\left( \! {{~}_{2}{F_{1}}\!\left( \! { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }, - \frac{{\theta_{n,l}^{GU,sch}}}{{\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {{{\left( {\frac{{{x_{i}}}}{{{x_{K}}}}} \right)}^{- \alpha }}} }}} \!\right) - 1\!} \right)} \!\right) \hfill\!\!\! \\ \end{array} $$

  (48)

The PDF of the distance from K SBSs nearest to typical users is

$$ {f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) = {\left( {2\pi \lambda } \right)^{K}}\exp \left( { - \pi \lambda {x_{K}^{2}}} \right){\prod}_{k = 1}^{K} {{x_{k}}} $$

(49)

Obviously, when $K \geqslant 2$, the calculation of the successful transmission probability involves multidimensional integral, which is difficult to calculate. Especially when K is large, according to [38], the nearest SBS to the typical user can be regarded as a uniform distribution in the circle with the typical user as the center and the radius r_K. Therefore, joint distribution functions of the nearest K SBSs to typical users can be approximately written

$$ {f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) \approx \frac{{2{{\left( {\pi \lambda } \right)}^{K}}}}{{\left( {K - 1} \right)!}} \cdot x_{K}^{2K - 1} \cdot \exp \left( { - \pi \lambda {x_{K}^{2}}} \right){\prod}_{k = 1}^{K - 1} {\frac{{2{x_{k}}}}{{{x_{K}^{2}}}}} $$

(50)

Combining Eqs. 48, 49 and 50 to Eq. 46, we obtain Theorem 3.

Appendix D: Proof of Theorem 4

Similarly, when cache is hit, the SIR of WU is expressed as

$$ SIR = \frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot {\chi^{- \alpha /2}}} \right|}^{2}}}}{{\sum\limits_{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}} {P \cdot {h_{j}} \cdot {\chi^{- \alpha /2}}} + \sum\limits_{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}} P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }}} = \frac{{{X_{4}}}}{{{I_{4}}}} $$

(51)

Let $\theta _{n,l}^{WU,sch}$ denote the threshold of SIR of worst-case users, the STP of WU is showed as follows

$$ STP_{n,l}^{WU,sch} = {\int}_{0}^{\infty} {{f_{\chi} }\left( x \right) \cdot } \mathbb{P}\left\{ {SIR > \theta_{n,l}^{WU,sch}|\chi = x} \right\}dx $$

(52)

where $\mathbb {P}\left \{ {SIR > \theta _{n,l}^{WU,sch}|\chi = x} \right \}$ is the STP under χ = x,

$$ \begin{array}{@{}rcl@{}} &&\mathbb{P}\left\{ {SIR > \theta_{n,l}^{WU,sch}|\chi = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {\frac{{{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {\sqrt P } \cdot {h_{i}} \cdot r_{i}^{- \alpha /2}} \right|}^{2}}}}{{{I_{4}}}} > \theta_{n,l}^{WU,sch}|\chi = x} \right\} \hfill \\ &&= \mathbb{P}\left\{ {{{\left| {\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {{h_{i}} \cdot r_{i}^{- \alpha /2}} } \right|}^{2}} > \theta_{n,l}^{WU,sch} \cdot {I_{4}} \cdot {P^{- 1}}|\chi = x} \right\} \hfill \\ &&= {\mathbb{E}_{{I_{4}}}}\left[ {\exp \left( { - \theta_{n,l}^{WU,sch} \cdot {I_{4}} \cdot {P^{- 1}}/K \cdot {{\text{x}}^{- \alpha }}} \right)} \right] \hfill \\ &&{\text{ = }}{\mathcal{L}_{{I_{4}}}}\left( {\theta_{n,l}^{WU,sch} \cdot {P^{- 1}}/K \cdot {{\text{x}}^{- \alpha }}} \right) \hfill \end{array} $$

(53)

where ${{\mathscr{L}}_{{I_{4}}}}\left ({\zeta _{n,l}^{WU,sch}} \right )$ is the Laplace transform of the interference generated by the HPPP-distribution SBSs that are farther than that in Φ∖Φ_K, which is calculated as

$$ \begin{array}{@{}rcl@{}} &&{\mathcal{L}_{{I_{4}}}}\left( {\zeta_{n,l}^{WU,sch}} \right) = {\mathbb{E}_{{I_{4}}}}\left[ {\exp \left( { - \zeta_{n,l}^{WU,sch} \cdot \left( {\sum\limits_{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}} {P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot {\chi^{- \alpha }}} + \sum\limits_{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}} {P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} } \right)} \right)} \right] \hfill \\ &&= {\mathbb{E}_{{{\Phi}_{K}},{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{j}} \in {{\Phi}_{3}}\backslash {{\Phi}_{K}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,sch} \cdot P \cdot {{\left| {{h_{j}}} \right|}^{2}} \cdot {\chi^{- \alpha }}} \right)} \cdot \underset{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,sch} \cdot P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} \right)} } \right] \hfill \\ &&= {\left( {1 + \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right)^{K - 3}} \cdot {\mathbb{E}_{{{\Phi}_{K}},{{\left| {{h_{i}}} \right|}^{2}}}}\left[ {\underset{{b_{u}} \in {\Phi} \backslash {{\Phi}_{3}}}{\prod} {\exp \left( { - \zeta_{n,l}^{WU,sch} \cdot P \cdot {{\left| {{h_{u}}} \right|}^{2}} \cdot r_{u}^{- \alpha }} \right)} } \right] \hfill \\ &&= {\left( {1 + \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right)^{K - 3}} \cdot \exp \left( { - \pi \cdot \lambda \cdot {\chi^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }; - \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right) - 1} \right)} \right) \hfill \end{array} $$

(54)

The joint PDF of the link distances is

$$ {f_{\chi} }\left( x \right) = 2{\pi^{2}}{\lambda^{2}}{x^{3}}\exp (- \lambda \pi {x^{2}}) $$

(55)

Combining Eqs. 53, 54 and 55 to Eq. 52, we obtain Theorem 4.

Appendix E: Proof of Theorem 5

The number of video retransmissions is defined as the expectation of the reciprocal of successful transmission probability, i.e.

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left\{ {1/STP_{n,l,K}^{GU,sch}} \right\} = \mathbb{E}\left\{ {\mathcal{L}{{\left( {\zeta_{n,l}^{GU,sch}} \right)}^{- 1}}} \right\} \hfill \\ &&= {\int}_{0}^{{x_{2}}} {{\int}_{{x_{1}}}^{{x_{3}}} { \cdot \cdot \cdot } } {\int}_{{x_{K - 1}}}^{\infty} {{f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) \cdot } \mathcal{L}{\left( {\zeta_{n,l}^{GU,sch}} \right)^{- 1}}d{x_{1}}d{x_{2}} \cdot \cdot \cdot d{x_{K}} \hfill \\ &&= {\int}_{0}^{{x_{2}}} {{\int}_{{x_{1}}}^{{x_{3}}} { \cdot \cdot \cdot } } {\int}_{{x_{K - 1}}}^{\infty} {{f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) \cdot } \exp \left( {\pi \cdot \lambda \cdot {x_{K}^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }, - \frac{{\zeta_{n,l}^{GU,sch} \cdot P}}{{x_{K}^{\alpha} }}} \right) - 1} \right)} \right)d{x_{1}}d{x_{2}} \cdot \cdot \cdot d{x_{K}} \hfill \\ &&\approx {{\int}_{0}^{1}} {{{\int}_{0}^{1}} { \cdot \cdot \cdot } } {\int}_{0}^{\infty} {{f_{\boldsymbol{r}}}\left( \boldsymbol{x} \right) \cdot } \exp \left( {\pi \cdot \lambda \cdot {x_{K}^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }, - \frac{{\zeta_{n,l}^{GU,sch} \cdot P}}{{x_{K}^{\alpha} }}} \right) - 1} \right)} \right)d{x_{1}}d{x_{2}} \cdot \cdot \cdot d{x_{K}} \hfill \\ &&= {{\int}_{0}^{1}} { \cdot \cdot \cdot {{\int}_{0}^{1}} {{{\left( {{\text{2}} - {~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }, - \frac{{\theta_{n,l}^{GU,sch} \cdot P}}{{\sum\limits_{{b_{i}} \in {{\Phi}_{K}}} {v_{i}^{- \alpha }} }}} \right)} \right)}^{- 1}}d{v_{1}} \cdot \cdot \cdot d{v_{K - 1}}} } \hfill \end{array} $$

(56)

$$ \begin{array}{@{}rcl@{}} &&\mathbb{E}\left\{ {1/STP_{n,l,K}^{WU,sch}} \right\} = \mathbb{E}\left\{ {\mathcal{L}{{\left( {\zeta_{n,l}^{WU,sch}} \right)}^{- 1}}} \right\} \hfill \\ &&= {\int}_{0}^{\infty} {{f_{\chi} }\left( x \right) \cdot } \mathcal{L}{\left( {\zeta_{n,l}^{WU,sch}} \right)^{- 1}}dx \hfill \\ &&= {\int}_{0}^{\infty} {{f_{\chi} }\left( x \right) \cdot {{\left( {1 + \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right)}^{3 - K}} \cdot \exp \left( {\pi \cdot \lambda \cdot {x^{2}}\left( {{~}_{2}{F_{1}}\left( { - \frac{2}{\alpha },1;1 - \frac{2}{\alpha }; - \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right) - 1} \right)} \right)dx} \hfill \\ &&= {\left( {1 + \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right)^{3 - K}}/{\left( {1 - \frac{{\theta_{n,l}^{WU,sch}}}{K}} \right)^{2}} \hfill \end{array} $$

(57)

Theorem 5 is proved.