Skip to main content

Advertisement

SpringerLink
Log in

Performance Analysis of the Content Dissemination Mechanism with Proactive Content Fetching and Full-Duplex D2D Communication: an Evolutionary Perspective

  • Published:
Mobile Networks and Applications Aims and scope Submit manuscript

Abstract

Device-to-device (D2D) communication has been regarded as a promising technique for content dissemination in the fifth generation wireless network. To improve the efficiency of content dissemination, in this paper, we propose a content dissemination mechanism with proactive content fetching (PCF) and full-duplex (FD) D2D communication, referred to as PCF-FD mechanism. This mechanism is characterized by the proactive content requests of partial users that don’t request contents originally. FD communication is utilized to establish more D2D links and increase the area spectral efficiency (ASE). Based on stochastic geometry, we derive the recursive formula of content caching probability from the evolutionary perspective. In addition, the expressions of successful offloading probability (SOP), ASE and average energy consumption are also obtained. Computer experiments show that the analytical results match well with simulation results and our scheme outperforms the conventional scheme without PCF and FD communication by 45% in terms of SOP in a scenario with small user density and light load.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Notes

  1. Once the cache is filled to capacity, the proposed mechanism terminates naturally and some cache eviction strategies are need, which are out of the scope of this paper.

  2. The application of the PCF-FD mechanism is content dissemination in densely populated areas, such as crowds of users in a mall requesting multimedia contents from video sharing websites. In this scenario, PRs are unaware of the content caching distribution {pf} and content popularity distribution {qf}. Since all the content transmissions (including BS content transmissions and D2D content transmissions) are scheduled by BSs, BSs have the full knowledge of the caching state of each content in all the devices, and they can estimate the content caching distribution {pf} at each time period. Thus, the content caching distribution {pf} and content popularity distribution {qf} can be broadcast from BSs to PRs, and PRs send proactive content requests according to Eq. 2.

  3. “N” means “no”.

  4. The duration of a frame is assumed to be 1s in this paper. Note that we don’t take into account multicast transmission for the same file requests [37]. Thus the time slots are partitioned based on the number of devices served by a busy UE, instead of the number of different contents transmitted by a busy UE.

References

  1. Andrews JG, Buzzi S, Choi W, et al (2014) What will 5G be? IEEE J Selected Areas Commun 32(6):1065–1082. https://doi.org/10.1109/JSAC.2014.2328098

    Article  Google Scholar 

  2. Golrezaei N, Molisch AF, Dimakis AG, Caire G (2013) Femtocaching and device-to-device collaboration: a new architecture for wireless video distribution. IEEE Commun Mag 51(4):142–149. https://doi.org/10.1109/MCOM.2013.6495773

    Article  Google Scholar 

  3. Baştug E, Bennis M, Debbah M (2014) Living on the edge: the role of proactive caching in 5G wireless networks. IEEE Commun Mag 52(8):82–89. https://doi.org/10.1109/MCOM.2014.6871674

    Article  Google Scholar 

  4. Wang X, Chen M, Taleb T, et al (2014) Cache in the air: exploiting content caching and delivery techniques for 5G systems. IEEE Commun Mag 52(2):131–139. https://doi.org/10.1109/MCOM.2014.6736753

    Article  Google Scholar 

  5. Liu D, Chen B, Yang C, Molisch AF (2016) Caching at the wireless edge: design aspects, challenges, and future directions. IEEE Commun Mag 54(9):22–28. https://doi.org/10.1109/MCOM.2016.7565183

    Article  Google Scholar 

  6. Li L, Zhao G, Blum RS (2018) A survey of caching techniques in cellular networks: research issues and challenges in content placement and delivery strategies. IEEE Communications Surveys & Tutorials. https://doi.org/10.1109/COMST.2018.2820021

  7. Wu D, Zhou L, Cai Y, Qian Y (2018) Collaborative caching and matching for D2D content sharing. IEEE Wirel Commun 25(3):43–49. https://doi.org/10.1109/MWC.2018.1700325

    Article  Google Scholar 

  8. Feng L, Zhao P, Zhou F, et al. (2018) Resource allocation for 5G D2D multicast content sharing in social-aware cellular networks. IEEE Commun Mag 56(3):112–118. https://doi.org/10.1109/MCOM.2018.1700667

    Article  Google Scholar 

  9. Asadi A, Wang Q, Mancuso V (2014) A survey on device-to-device communication in cellular networks. IEEE Commun Surv Tutor 16(4):1801–1819. https://doi.org/10.1109/COMST.2014.2319555

    Article  Google Scholar 

  10. Wang X, Sheng Z, Yang S, Leung VCM (2016) Tag-assisted social-aware opportunistic device-to-device sharing for traffic offloading in mobile social networks. IEEE Wirel Commun 23(4):60–67. https://doi.org/10.1109/MWC.2016.7553027

    Article  Google Scholar 

  11. Bai B, Wang L, Han Z, et al. (2016) Caching based socially-aware D2D communications in wireless content delivery networks: a hypergraph framework. IEEE Wirel Commun 23(4):74–81. https://doi.org/10.1109/MWC.2016.7553029

    Article  Google Scholar 

  12. Hu J, Yang L -L, Yang K, Hanzo L (2016) Socially aware integrated centralized infrastructure and opportunistic networking: a powerful content dissemination catalyst. IEEE Commun Mag 54(8):84–91. https://doi.org/10.1109/MCOM.2016.7537181

    Article  Google Scholar 

  13. Machado K, Boukerche A, Cerqueira E, Loureiro AAF (2017) A socially-aware in-network caching framework for the next generation of wireless networks. IEEE Commun Mag 55(12):38–43. https://doi.org/10.1109/MCOM.2017.1700244

    Article  Google Scholar 

  14. Luan Z, Qu H, Zhao J, Chen B (2016) Robust digital non-linear self-interference cancellation in full duplex radios with maximum correntropy criterion. China Commun 13(9):53–59. https://doi.org/10.1109/CC.2016.7582296

    Article  Google Scholar 

  15. Ji M, Caire G, Molisch AF (2016) Wireless device-to-device caching networks: basic principles and system performance. IEEE Jon Selected Areas Commun 34(1):176–189. https://doi.org/10.1109/JSAC.2015.2452672

    Article  Google Scholar 

  16. Elayoubi SE, Masucci AM, Roberts J, Sayrac B (2017) Optimal D2D content delivery for cellular network offloading. Mob Netw Appl 22(6):1033–1044. https://doi.org/10.1007/s11036-017-0821-1

    Article  Google Scholar 

  17. Chen B, Yang C, Molisch AF (2017) Cache-enabled device-to-device communications: offloading gain and energy cost. IEEE Trans Wirel Commun 16(7):4519–4536. https://doi.org/10.1109/TWC.2017.2699631

    Article  Google Scholar 

  18. Chen Z, Pappas N, Kountouris M (2017) Probabilistic caching in wireless D2D networks: cache hit optimal versus throughput optimal. IEEE Commun Lett 21(3):584–587. https://doi.org/10.1109/LCOMM.2016.2628032

    Article  Google Scholar 

  19. Chen B, Yang C, Xiong Z (2017) Optimal caching and scheduling for cache-enabled D2D communications. IEEE Commun Lett 21(5):1155–1158. https://doi.org/10.1109/LCOMM.2017.2652440

    Article  Google Scholar 

  20. Malak D, Al-Shalash M, Andrews JG (2016) Optimizing content caching to maximize the density of successful receptions in device-to-device networking. IEEE Trans Commun 64(10):4365–4380. https://doi.org/10.1109/TCOMM.2016.2600571

    Google Scholar 

  21. Afshang M, Dhillon HS, Chong PHJ (2016) Fundamentals of cluster-centric content placement in cache-enabled device-to-device networks. IEEE Trans Commun 64(6):2511–2526. https://doi.org/10.1109/TCOMM.2016.2554547

    Article  Google Scholar 

  22. Giatsoglou N, Ntontin K, Kartsakli E, et al (2017) D2D-aware device caching in mmWave-cellular networks. IEEE J Selected Areas Commun 35(9):2025–2037. https://doi.org/10.1109/JSAC.2017.2720818

    Article  Google Scholar 

  23. Sciancalepore V, Giustiniano D, Banchs A, Hossmann-Picu A (2016) Offloading cellular traffic through opportunistic communications: analysis and optimization. IEEE J Selected Areas Commun 34(1):122–137. https://doi.org/10.1109/JSAC.2015.2452472

    Article  Google Scholar 

  24. Wang S, Wang X, Huang J, et al (2016) The potential of mobile opportunistic networks for data disseminations. IEEE Trans Veh Technol 65(2):912–922. https://doi.org/10.1109/TVT.2015.2401605

    Article  Google Scholar 

  25. Wang S, Wang X, Cheng X, et al. (2017) Fundamental analysis on data dissemination in mobile opportunistic networks with Lévy mobility. IEEE Trans Veh Technol 66(5):4173–4187. https://doi.org/10.1109/TVT.2016.2597969

    Google Scholar 

  26. Li Y, Kaleem Z, Chang K (2016) Interference-aware resource-sharing scheme for multiple D2D group communications underlaying cellular networks. Wirel Pers Commun 90(2):749–768. https://doi.org/10.1007/s11277-016-3203-2

    Article  Google Scholar 

  27. Kaleem Z, Li Y, Chang K (2016) Public safety users’ priority-based energy and time-efficient device discovery scheme with contention resolution for ProSe in third generation partnership project long-term evolution-advanced systems. IET Commun 10(15):1873–1883. https://doi.org/10.1049/iet-com.2016.0029

    Article  Google Scholar 

  28. Vo N -S, Duong TQ, Tuan HD, Kortun A (2018) Optimal video streaming in dense 5G networks with D2D communications. IEEE Access 6:209–223. https://doi.org/10.1109/ACCESS.2017.2761978

    Article  Google Scholar 

  29. Yin C, Nguyen HT, Kundu C, et al (2018) Secure energy harvesting relay networks with unreliable backhaul connections. IEEE Access 6:12074–12084. https://doi.org/10.1109/ACCESS.2018.2794507

    Article  Google Scholar 

  30. Sexton C, Kaminski NJ, Marquez-Barja JM et al (2017) 5G: adaptable networks enabled by versatile radio access technologies. IEEE Commun Surv Tutor 19(2):688–720. https://doi.org/10.1109/COMST.2017.2652495

    Article  Google Scholar 

  31. Tam HHM, Tuan HD, Nasir AA, et al. (2017) MIMO energy harvesting in full-duplex multi-user networks. IEEE Trans Wirel Commun 16(5):3282–3297. https://doi.org/10.1109/TWC.2017.2679055

    Article  Google Scholar 

  32. Nguyen V -D, Duong TQ, Tuan HD, et al. (2017) Spectral and energy efficiencies in full-duplex wireless information and power transfer. IEEE Trans Commun 65(5):2220–2233. https://doi.org/10.1109/TCOMM.2017.2665488

    Article  Google Scholar 

  33. Naslcheraghi M, Ghorashi SA, Shikh-Bahaei M (2017) Performance analysis of inband FD-D2D communications with imperfect SI cancellation for wireless video distribution. In: 2017 8th international conference on the network of the future (NOF), pp 176-181. https://doi.org/10.1109/NOF.2017.8251246

  34. Ahlehagh H, Dey S (2014) Video-aware scheduling and caching in the radio access network. IEEE/ACM Trans Netw 22(5):1444–1462. https://doi.org/10.1109/TNET.2013.2294111

    Article  Google Scholar 

  35. Yang C, Yao Y, Chen Z, Xia B (2016) Analysis on cache-enabled wireless heterogeneous networks. IEEE Trans Wirel Commun 15(1):131–145. https://doi.org/10.1109/TWC.2015.2468220

    Article  Google Scholar 

  36. Blaszczyszyn B, Giovanidis A (2015) Optimal geographic caching in cellular networks. In: Proceedings of IEEE international conference on communications (ICC), pp 3358–3363. https://doi.org/10.1109/ICC.2015.7248843

  37. Cui Y, Jiang D (2017) Analysis and optimization of caching and multicasting in large-scale cache-enabled heterogeneous wireless networks. IEEE Trans Wirel Commun 16(1):250–264. https://doi.org/10.1109/TWC.2016.2622236

    Article  MathSciNet  Google Scholar 

  38. Chen Y, Ding M, Li J, et al. (2017) Probabilistic small-cell caching: performance analysis and optimization. IEEE Trans Veh Technol 66(5):4341–4354. https://doi.org/10.1109/TVT.2016.2606765

    Google Scholar 

  39. Liu D, Yang C (2017) Caching policy toward maximal success probability and area spectral efficiency of cache-enabled HetNets. IEEE Trans Commun 65(6):2699–2714. https://doi.org/10.1109/TCOMM.2017.2680447

    Article  Google Scholar 

  40. Vu TX, Chatzinotas S, Ottersten B, Duong TQ (2018) Energy minimization for cache-assisted content delivery networks with wireless backhaul. IEEE Wireless Commun Lett 7(3):332–335. https://doi.org/10.1109/LWC.2017.2776924

    Article  Google Scholar 

  41. Ren G, Qu H, Zhao J, et al (2017) A distributed user association and resource allocation method in cache-enabled small cell networks. China Commun 14(10):95–107. https://doi.org/10.1109/CC.2017.8107635

    Article  Google Scholar 

  42. Jiang J, Zhang S, Li B, Li B (2016) Maximized cellular traffic offloading via device-to-device content sharing. IEEE J Selected Areas Commun 34(1):82–91. https://doi.org/10.1109/JSAC.2015.2452493

    Article  Google Scholar 

  43. Wang S, Zhang Y, Wang H, et al. (2018) Large scale measurement and analytics on social groups of device-to-device sharing in mobile social networks. Mob Netw Appl 23(2):203–215. https://doi.org/10.1007/s11036-017-0927-5

    Article  MathSciNet  Google Scholar 

  44. Wang Z, Sun L, Zhang M, et al. (2017) Propagation- and mobility-aware D2D social content replication. IEEE Trans Mob Comput 16(4):1107–1120. https://doi.org/10.1109/TMC.2016.2582159

    Article  Google Scholar 

  45. Zhou H, Leung VCM, Zhu C, et al. (2017) Predicting temporal social contact patterns for data forwarding in opportunistic mobile networks. IEEE Trans Veh Technol 66(11):10372–10383. https://doi.org/10.1109/TVT.2017.2740218

    Article  Google Scholar 

  46. Ali KS, ElSawy H, Alouini M -S (2016) Modeling cellular networks with full-duplex D2D communication: a stochastic geometry approach. IEEE Trans Commun 64(10):4409–4424. https://doi.org/10.1109/TCOMM.2016.2601912

    Article  Google Scholar 

  47. Breslau L, Cao P, Fan L et al (1999) Web caching and Zipf-like distributions: evidence and implications. In: Proceedings of IEEE conference on computer communications (INFOCOM), pp 126–134. https://doi.org/10.1109/INFCOM.1999.749260

  48. Zheng K, Yang Z, Zhang K, et al. (2016) Big data-driven optimization for mobile networks toward 5G. IEEE Netw 30(1):44–51. https://doi.org/10.1109/MNET.2016.7389830

    Article  Google Scholar 

  49. Zeydan E, Baştug E, Bennis M, et al. (2016) Big data caching for networking: moving from cloud to edge. IEEE Commun Mag 54(9):36–42. https://doi.org/10.1109/MCOM.2016.7565185

    Article  Google Scholar 

  50. Zhou B, Cui Y, Tao M (2016) Stochastic content-centric multicast scheduling for cache-enabled heterogeneous cellular networks. IEEE Trans Wirel Commun 15(9):6284–6297. https://doi.org/10.1109/TWC.2016.2582689

    Article  Google Scholar 

  51. Malak D, Al-Shalash M, Andrews JG (2018) Spatially correlated content caching for device-to-device communications. IEEE Trans Wirel Commun 17(1):56–70. https://doi.org/10.1109/TWC.2017.2762661

    Article  Google Scholar 

  52. Lee S, Huang K (2012) Coverage and economy of cellular networks with many base stations. IEEE Commun Lett 16(7):1038–1040. https://doi.org/10.1109/LCOMM.2012.042512.120426

    Article  Google Scholar 

  53. Andrews JG, Baccelli F, Ganti RK (2011) A tractable approach to coverage and rate in cellular networks. IEEE Trans Commun 59(11):3122–3134. https://doi.org/10.1109/TCOMM.2011.100411.100541

    Article  Google Scholar 

  54. Haenggi M (2012) Stochastic geometry for wireless networks. Cambridge University Press, New York

    Book  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by National Natural Science Foundation of China under Grant 61531013 and National Science and Technology Major Project under Grant 2018ZX03001016.

Author information

Authors and Affiliations

  1. School of Electronic and Information Engineering, Xi’an Jiaotong University, Xi’an, 710054, China

    Hua Qu, Gongye Ren, Jihong Zhao, Youwei Shi & Zhenjie Tan

  2. Suzhou Caiyun Network Technologies Co., Ltd, Suzhou, 215000, China

    Hua Qu & Jihong Zhao

  3. School of Telecommunication and Information Engineering, Xi’an University of Posts and Telecommunications, Xi’an, 710061, China

    Jihong Zhao

Authors
  1. Hua Qu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gongye Ren
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jihong Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Youwei Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zhenjie Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gongye Ren.

Appendix

Appendix

1.1 A.1 Proof of Proposition 2

For a reference CR located at the origin, the probability that it requests f and it is in state CR-d1 is \(p_{\text {cr,d,1},{f}}=q_{f} (1-p_{f}) p_{\text {in},{f}} p_{\bar {\text {t}},-f}\). On this condition, the probability that this CR successfully receives f is

$$ p_{\text{s}|(\text{cr,d,1},{f})} = {\int}_{0}^{\infty} \mathbb{P}\left[\gamma_{\text{cr,d,1},{f}}\geq\gamma_{0}|r\right] f_{r_{f}|(\text{cr,d,1},{f})}(r) dr, $$
(24)

where γcr,d,1,f is the SINR of this CR, \(f_{r_{f}|(\text {cr,d,1},{f})}(r)\) is the PDF of the distance from this reference CR to the nearest UE caching f conditioned on that the CR is in state CR-d1. According to conditional probability formula, we have

$$ f_{r_{f}|(\text{cr,d,1},{f})}(r) = \left\{\begin{array}{lllllllllll} \frac{f_{r_{f}}(r)}{p_{\text{in},{f}}}, & 0 \leq r \leq R_{\text{d}}, \\ 0, &\text{otherwise}, \end{array}\right. $$
(25)

where \(f_{r_{f}}(r)= 2\pi \lambda _{\text {UE},{f}}re^{-\pi \lambda _{\text {UE},{f}}r^{2}}\) is the PDF of the distance from this reference CR to the nearest UE caching f [53]. Thus, the probability that a CR is in state CR-d1 and successfully receives its requested content is given by

$$ \begin{array}{lllllllllll} p_{\text{cr,d,1,s}} &=& \sum\limits_{f = 1}^{F} p_{\text{cr,d,1,s},{f}} = \sum\limits_{f = 1}^{F} p_{\text{s}|(\text{cr,d,1},{f})} p_{\text{cr,d,1},{f}} \\ &=& \sum\limits_{f = 1}^{F} q_{f} (1-p_{f}) p_{\bar{\text{t}},-f} {\int}_{0}^{R_{\text{d}}} \mathbb{P}\\&&\left[\gamma_{\text{cr,d,1},{f}}\geq\gamma_{0}|r\right] 2\pi\lambda_{\text{UE},{f}}re^{-\pi\lambda_{\text{UE},{f}}r^{2}} dr. \end{array} $$
(26)

The SINR of the reference CR in state CR-d1 can be written as \(\gamma _{\text {cr,d,1},{f}}=\frac {P_{\text {UE}}hr^{-\alpha }}{I_{f,r}+I_{-f,r}+\sigma _{\text {d}}^{2}}\), where \(I_{f,r}\triangleq {\sum }_{x_{i}\in {\Phi }_{\text {UE,b},{f}}\setminus \{x_{0}\}} P_{\text {UE}}h_{i}\|x_{i}\|^{-\alpha }\) is the interference from the busy UEs transmitting f, x0 denotes the location of the associated UE of the reference CR, and \(I_{-f,r}\triangleq {\sum }_{x_{j}\in {\Phi }_{\text {UE,b},\textit {-f}}} P_{\text {UE}}h_{j}\|x_{j}\|^{-\alpha }\) is the interference from the busy UEs that transmit contents except f. h models the small-scale Rayleigh fading and it follows exponential distribution with unit mean, i.e., h ∼ exp(1). ΦUE,b,f denotes the PPP comprised of the busy UEs that transmit f, and ΦUE,b,-f denotes the PPP comprised of the busy UEs that don’t transmit f.

According to the definition of states of users, the probability that an arbitrary UE being busy, pUE,b, is

$$ \begin{array}{lllllllllll} p_{\text{UE,b}} \!= &\!\rho p_{\text{cr,s,2}} + \rho p_{\text{cr,d,2}} + \rho p_{\text{cr,c,2}} + (1-\rho)\theta p_{\text{pr,s,2}}\!\! \\ &\!+ (1-\rho)\theta p_{\text{pr,d,2}} + (1-\rho)\theta p_{\text{pr,f,2}}\\& \!+ (1-\rho)(1-\theta)p_{\text{nr,b}}. \end{array} $$
(27)

Based on the results in Table 3, after some mathematical manipulations, we can conclude pUE,b = pnr,b. Thus, the density of busy UEs is λUE,b = pUE,bλUE = pnr,bλUE. For an arbitrary UE, the probability that it transmits f is (1 − uf,0)pf, and the density of ΦUE,b,f is λUE,b,f = (1 − uf,0)pfλUE according to the thinning property of PPP. The density of ΦUE,b,-f can be easily obtained as λUE,b,−f = λUE,b − λUE,b,f.

Conditioned on given r, we have

$$ \begin{array}{lllllllllll} & \mathbb{P}\left[\gamma_{\text{cr,d,1},{f}}\geq\gamma_{0}|r\right] \\ =& \mathbb{E}_{I_{f,r},I_{-f,r}} \left[ \mathbb{P} \left[ \left. \frac{P_{\text{UE}}hr^{-\alpha}}{I_{f,r}+I_{-f,r}+\sigma_{\text{d}}^{2}} \geq \gamma_{0} \right| r,I_{f,r},I_{-f,r} \right] \right] \\ =& \mathbb{E}_{I_{f,r},I_{-f,r}}\left[\mathbb{P}\left[\left.h\geq\gamma_{0} r^{\alpha}P_{\text{UE}}^{-1}\left( I_{f,r}+I_{-f,r}+\sigma_{\text{d}}^{2}\right)\right|r,I_{f,r},I_{-f,r}\right]\right]\\ =& \mathbb{E}_{I_{f,r},I_{-f,r}} \left[\exp\left( -\gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\left( I_{f,r}+I_{-f,r}+\sigma_{\text{d}}^{2}\right)\right)\right]\\ =& \exp\left( -\gamma_{0} r^{\alpha} \sigma_{\text{d}}^{2} P_{\text{UE}}^{-1}\right) \mathcal{L}_{I_{f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) \mathcal{L}_{I_{-f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right), \end{array} $$
(28)

where \(\mathcal {L}_{I_{f,r}}(s)\) and \(\mathcal {L}_{I_{-f,r}}(s)\) are the Laplace transform of If,r and I−f,r, respectively. The last line comes from the independence of If,r and I−f,r. Given the expression of If,r, we have

$$ \begin{array}{lllllllllll} \mathcal{L}_{I_{f,r}}(s) &= \mathbb{E}_{I_{f,r}}\left[e^{-sI_{f,r}}\right]\\ &= \mathbb{E}_{{\Phi}_{\text{UE,b},{f}},\{h_{i}\}} \left[\exp\left( -s\left( \sum\limits_{x_{i}\in{\Phi}_{\text{UE,b},{f}}\setminus\{x_{0}\}} P_{\text{UE}}h_{i}\|x_{i}\|^{-\alpha}\right)\right)\right]\\ &= \mathbb{E}_{{\Phi}_{\text{UE,b},{f}}} \left[\prod\limits_{x_{i}\in{\Phi}_{\text{UE,b},{f}}\setminus\{x_{0}\}} \mathbb{E}_{h_{i}}\left[\exp\left( -sP_{\text{UE}}h_{i}\|x_{i}\|^{-\alpha}\right)\right]\right]\\ &= \mathbb{E}_{{\Phi}_{\text{UE,b},{f}}} \left[\prod\limits_{x_{i}\in{\Phi}_{\text{UE,b},{f}}\setminus\{x_{0}\}} \frac{1}{1+sP_{\text{UE}}\|x_{i}\|^{-\alpha}}\right]\\ &= \exp\left( -2\pi\lambda_{\text{UE,b},{f}}{\int}_{r}^{\infty} \frac{sP_{\text{UE}}v^{-\alpha}}{1+sP_{\text{UE}}v^{-\alpha}}vdv \right). \end{array} $$
(29)

Let \(s=\gamma _{0} r^{\alpha } P_{\text {UE}}^{-1}\), and we have

$$ \begin{array}{lllllllllll} \mathcal{L}_{I_{f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) &= \exp\left( -2\pi\lambda_{\text{UE,b},{f}}{\int}_{r}^{\infty} \frac{ \gamma_{0} r^{\alpha} v^{-\alpha} }{1+\gamma_{0} r^{\alpha} v^{-\alpha}}vdv\right) \\ &= \exp\left( -\frac{2\pi}{\alpha-2}\gamma_{0}\lambda_{\text{UE,b},{f}}r^{2} {~}_{2}F_{1}\left( 1,1-\frac{2}{\alpha};2-\frac{2}{\alpha};-\gamma_{0}\right) \right)\\ &= \exp\left( -\pi\lambda_{\text{UE,b},{f}}r^{2}A_{1}(\alpha,\gamma_{0})\right), \end{array} $$
(30)

where \(A_{1}(\alpha ,\gamma _{0})\triangleq \frac {2}{\alpha -2}\gamma _{0}{~}_{2}F_{1}\left (1,1-\frac {2}{\alpha };2-\frac {2}{\alpha };-\gamma _{0}\right )\). In a similar way, we can obtain

$$ \mathcal{L}_{I_{-f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) = \exp\left( -\pi\lambda_{\text{UE,b},\textit{-f}}r^{2}A_{2}(\alpha,\gamma_{0})\right), $$
(31)

where \(A_{2}(\alpha ,\gamma _{0})\triangleq \frac {2}{\alpha }\gamma _{0}^{2/\alpha }B\left (1-\frac {2}{\alpha },\frac {2}{\alpha }\right )\). Substituting Eqs. 30 and 31 into Eq. 28, and then plugging Eq. 28 into Eq. 26, Eq. 6 is derived.

Following the same steps, we can get the similar form of pcr,d,2,s as Eq. 26, i.e.,

$$ \begin{array}{lllllllllll} & p_{\text{cr,d,2,s}} = \sum\limits_{f = 1}^{F} p_{\text{cr,d,2,s},{f}} = \sum\limits_{f = 1}^{F} p_{\text{s}|(\text{cr,d,2},{f})} p_{\text{cr,d,2},{f}} \\ =& \sum\limits_{f = 1}^{F} q_{f} (1-p_{f}) \left( 1-p_{\bar{\text{t}},-f}\right) {\int}_{0}^{R_{\text{d}}} \mathbb{P}\left[\gamma_{\text{cr,d,2},{f}}\geq\gamma_{0}|r\right] 2\pi\lambda_{\text{UE},{f}}re^{-\pi\lambda_{\text{UE},{f}}r^{2}} dr. \end{array} $$
(32)

Being different from γcr,d,1,f, the SINR of a CR in state CR-d2 has the form \(\gamma _{\text {cr,d,2},{f}}=\frac {P_{\text {UE}}hr^{-\alpha }}{I_{f,r}+\hat {I}_{-f,r}+\zeta P_{\text {UE}}+\sigma _{\text {d}}^{2}}\), where \(\hat {I}_{-f,r}\triangleq {\sum }_{x_{j}\in {\Phi }_{\text {UE,b},\textit {-f}}\setminus \{o\}} P_{\text {UE}}h_{j}\|x_{j}\|^{-\alpha }\) and ζ is the residual SI fraction after SI cancelation. Since the reference CR in state CR-d2, which is located at the origin o, is included in ΦUE,b,-f, the interference generated by other busy UEs that transmit contents except f is calculated by \(\hat {I}_{-f,r}\). Although \(\hat {I}_{-f,r}\) is different from I−f,r, we can conclude \(\mathcal {L}_{\hat {I}_{-f,r}}(s)=\mathcal {L}_{I_{-f,r}}(s)\) according to the Slivnyak’s theorem [54]. Thus, we have

$$ \begin{array}{lllllllllll} &\mathbb{P}\left[\gamma_{\text{cr,d,2},{f}}\geq\gamma_{0}|r\right]\\ =& \exp\left( -\gamma_{0}r^{\alpha}\zeta-\gamma_{0} r^{\alpha} \sigma_{\text{d}}^{2} P_{\text{UE}}^{-1}\right) \mathcal{L}_{I_{f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) \mathcal{L}_{\hat{I}_{-f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) \\ =& \exp\left( -\gamma_{0}r^{\alpha}\zeta-\gamma_{0} r^{\alpha} \sigma_{\text{d}}^{2} P_{\text{UE}}^{-1}\right) \mathcal{L}_{I_{f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right) \mathcal{L}_{I_{-f,r}}\left( \gamma_{0} r^{\alpha} P_{\text{UE}}^{-1}\right). \end{array} $$
(33)

Plugging Eq. 33 into Eq. 32 and pcr,d,2,s is derived as Eq. 7.

1.2 A.2 Proof of Theorem 1

We proof Theorem 1 by applying the law of total probability. At a certain time period, there are three kinds of UEs: CRs, PRs and NRs. For a CR, if it caches content f at the t-th time period, it also caches f at the next time period. If it doesn’t cache f, the probability that it stores f at the next time period is qf. Thus, given \({p_{f}^{t}}\) and qf, the probability that a CR caches f at the (t + 1)-th time period is \(p_{f}^{t + 1}={p_{f}^{t}}+\left (1-{p_{f}^{t}}\right )q_{f}\). For a PR, the transition of caching states of f is more complex. For the sake of clarity, we list all the events for a PR with regard to f and their corresponding probabilities at the t-th time period in Table 6. According to Table 6, for a PR, the caching probability of f at the (t + 1)-th time period is \(p_{f}^{t + 1}={p_{f}^{t}}+p_{\text {pr,d,1,s},{f}}^{t}+p_{\text {pr,d,2,s},{f}}^{t}\). For a NR, its caching state of f doesn’t change at the next time period and thus we have \(p_{f}^{t + 1}={p_{f}^{t}}\).

Table 6 All the events for a PR with regard to f at the t-th time period and their probabilities
Full size table

At the t-th time period, the probabilities that a typical UE takes the roles of CR, PR and NR are ρ, (1 − ρ)𝜃t and (1 − ρ)(1 − 𝜃t), respectively. According to the law of total probability, the recursive formula of content caching probability is given by Eq. 10.

1.3 A.3 Proof of Theorem 2

From the perspective of receivers, Td is given by

$$ \begin{array}{lllllllllll} T_{\text{d}} = & \sum\limits_{f = 1}^{F} \left( \rho p_{\text{cr,d,1},{f}} \lambda_{\text{UE}} \mathbb{E} \left[\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{cr,d,1},{f}}\right)\right]\right.\\ &+ \rho p_{\text{cr,d,2},{f}} \lambda_{\text{UE}} \mathbb{E} \left[\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{cr,d,2},{f}}\right)\right] \\ & + (1-\rho)\theta p_{\text{pr,d,1},{f}} \lambda_{\text{UE}} \mathbb{E} \left[\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{pr,d,1},{f}}\right)\right]\\ &+\left. (1-\rho)\theta p_{\text{pr,d,2},{f}} \lambda_{\text{UE}} \mathbb{E} \left[\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{pr,d,2},{f}}\right)\right] \right), \end{array} $$
(34)

which is measured in unit nat. Strictly speaking, the number of devices served by a busy device, Nb, is correlated with the distribution of SINR in a complex manner. For simplicity, we ignore this correlation and regard them as independent variables as in [22] and [37]. Based on this assumption, taking \(\mathbb {E} \left [\frac {W_{\text {d}}}{N_{\text {b}}}\ln \left (1 + \gamma _{\text {cr,d,1},{f}}\right )\right ]\) as an example, we have

$$ \begin{array}{lllllllllll} \mathbb{E} \left[\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{cr,d,1},{f}}\right)\right] &= \sum\limits_{n = 1}^{\infty} \mathbb{P}[N_{\text{b}}=n] \mathbb{E}\left[\left.\frac{W_{\text{d}}}{N_{\text{b}}}\ln\left( 1 + \gamma_{\text{cr,d,1},{f}}\right)\right|N_{\text{b}}=n\right] \\ &\overset{(\text{a})}{\approx} \sum\limits_{n = 1}^{\infty} \mathbb{P}[N_{\text{b}}=n] \mathbb{E}\left[\frac{W_{\text{d}}}{n}\ln\left( 1 + \gamma_{\text{cr,d,1},{f}}\right)\right]\\ &\overset{(\text{b})}{\approx} \sum\limits_{n = 1}^{2} \mathbb{P}[N_{\text{b}}=n] \mathbb{E}\left[\frac{W_{\text{d}}}{n}\ln\left( 1 + \gamma_{\text{cr,d,1},{f}}\right)\right]\\ &= \omega W_{\text{d}} \mathbb{E}\left[\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)\right], \end{array} $$
(35)

where \(\omega \triangleq {\sum }_{n = 1}^{2} \frac {1}{n} \mathbb {P}\left [N_{\text {b}} = n\right ]\). (a) follows from the independence of Nb and γcr,d,1,f, and (b) comes from the fact that \(\mathbb {P}\left [N_{\text {b}} = n\right ]\) is very small when n > 2.

For \(\mathbb {E}\left [\ln \left (1+\gamma _{\text {cr,d,1},{f}}\right )\right ]\), we have

$$ \begin{array}{lllllllllll} \mathbb{E}\left[\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)\right] &= {\int}_{0}^{R_{\text{d}}} \mathbb{E}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)\right|r\right] f_{r_{f}|(\text{cr,d,1},{f})} (r) dr \\ &= \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} \underbrace{ {\int}_{0}^{R_{\text{d}}} re^{-\pi\lambda_{\text{UE},{f}}r^{2}} \mathbb{E}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)\right|r\right] dr}_{Z_{1}(f)}. \end{array} $$
(36)

Since \(\mathbb {E}\left [X\right ]={\int }_{0}^{\infty } \mathbb {P}\left [X>t\right ] dt\) for random variable X > 0, so

$$ \begin{array}{lllllllllll} & \mathbb{E}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)\right|r\right]\\ =& {\int}_{0}^{\infty} \mathbb{P}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,1},{f}}\right)>t\right|r\right] dt \\ =& {\int}_{0}^{\infty} \mathbb{P}\left[\left.\gamma_{\text{cr,d,1},{f}}>e^{t}-1\right|r\right] dt \\ =& {\int}_{0}^{\infty} \exp\left( -(e^{t}-1)r^{\alpha}\sigma_{\text{d}}^{2}P_{\text{UE}}^{-1} - \pi\lambda_{\text{UE,b},{f}}r^{2} A_{1}(\alpha,e^{t}-1) \right.\\ & \left.- \pi\lambda_{\text{UE,b},\textit{-f}}r^{2} A_{2}(\alpha,e^{t}-1) \right) dt. \end{array} $$
(37)

Similarly, \(\mathbb {E}\left [\ln \left (1+\gamma _{\text {cr,d,2},{f}}\right )\right ]\) is derived as

$$\begin{array}{@{}rcl@{}} &&\mathbb{E}\left[\ln\left( 1+\gamma_{\text{cr,d,2},{f}}\right)\right] = \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} \\&&\times\underbrace{ {\int}_{0}^{R_{\text{d}}} re^{-\pi\lambda_{\text{UE},{f}}r^{2}} \mathbb{E}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,2},{f}}\right)\right|r\right] dr}_{Z_{2}(f)}, \end{array} $$
(38)

where

$$ \begin{array}{lllllllllll} &\mathbb{E}\left[\left.\ln\left( 1+\gamma_{\text{cr,d,2},{f}}\right)\right|r\right] \\ =& {\int}_{0}^{\infty} \exp\left( -(e^{t}-1)r^{\alpha}\zeta -(e^{t}-1)r^{\alpha}\sigma_{\text{d}}^{2}P_{\text{UE}}^{-1} \right.\\&- \pi\lambda_{\text{UE,b},{f}}r^{2} A_{1}(\alpha,e^{t}-1) \\ & \left.- \pi\lambda_{\text{UE,b},\textit{-f}}r^{2} A_{2}(\alpha,e^{t}-1) \vphantom{P_{\text{UE}}^{-1}}\right) dt. \end{array} $$

\(\mathbb {E}\left [\ln \left (1+\gamma _{\text {pr,d,1},{f}}\right )\right ]\) has the same form as Eq. 36 and \(\mathbb {E}\left [\ln \left (1+\gamma _{\text {pr,d,2},{f}}\right )\right ]\) has the same form as Eq. 38. Combining Eqs. 34, 35, 36 and 38, we have

$$ \begin{array}{rllllllllll} T_{\text{d}}\! \approx & \!\sum\limits_{f = 1}^{F} \left( \omega\rho p_{\text{cr,d,1},{f}} \lambda_{\text{UE}} W_{\text{d}} \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} Z_{1}(f)\right.\\ &\!+ \omega\rho p_{\text{cr,d,2},{f}} \lambda_{\text{UE}} W_{\text{d}} \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} Z_{2}(f) \\ &\!+ \omega(1-\rho)\theta p_{\text{pr,d,1},{f}} \lambda_{\text{UE}} W_{\text{d}} \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} Z_{1}(f)\\ &\left.\!+ \omega(1-\rho)\theta p_{\text{pr,d,2},{f}} \lambda_{\text{UE}} W_{\text{d}} \frac{2\pi\lambda_{\text{UE},{f}}}{p_{\text{in},{f}}} Z_{2}(f) \right)\\ \!=&\! 2\pi\omega\lambda_{\text{UE}} W_{\text{d}} \sum\limits_{f = 1}^{F} \left( \lambda_{\text{UE},{f}} (1-p_{f}) \left( \rho q_{f} + (1-\rho) \theta q_{\text{pr},{f}} \right) \right.\\ & \left.\times \left( p_{\bar{\text{t}},-f}Z_{1}(f) + \left( 1-p_{\bar{\text{t}},-f}\right)Z_{2}(f) \right) \right). \end{array} $$
(39)

Substituting Eq. 39 into Eq. 12, we have Eq. 15.

1.4 A.4 Proof of Proposition 3

According to the file request protocol in Fig. 1, the CRs that don’t succeed in receiving contents via D2D communication are attached to the nearest BS to retrieve these files. Hence, the density of CRs that are eventually associated with BSs, denoted by λc, is

$$\begin{array}{@{}rcl@{}} \lambda_{\text{c}} &=& (p_{\text{cr,c,1}} + p_{\text{cr,c,2}} + p_{\text{cr,d,1}} - p_{\text{cr,d,1,s}} \\&&+ p_{\text{cr,d,2}} - p_{\text{cr,d,2,s}} ) \rho \lambda_{\text{UE}}. \end{array} $$
(40)

At the beginning of the PCF-FD mechanism, no files are cached in devices and all the requests have to be handled by BSs. In this case, λc is large and plenty of BSs must be active to accommodate these requests. After a number of time periods, a part of requests can be accomplished without the assistance of BSs, and the number of users that connect BSs decreases. Accordingly, the density of active BSs decreases. Based on this fact, it is unreasonable to assume that all the BSs transmit signals at all time. According to the result in [52], the probability that a typical BS is active is

$$\begin{array}{@{}rcl@{}} p_{\text{BS,b}} &=& 1 - {\int}_{0}^{\infty} { e^{- \lambda_{\text{c}} x } g \left( x;\lambda_{\text{BS}} \right) dx }\\& \approx& 1 - \left( 1 + \frac{\lambda_{\text{c}}}{3.5\lambda_{\text{BS}}} \right)^{-3.5}. \end{array} $$
(41)

So the density of active BSs is λBS,b = pBS,bλBS and the PPP of active BSs is denoted by ΦBS,b.

The average energy consumption of a content transmission from BSs is obtained as

$$ \begin{array}{lllllllllll} &\mathbb{E} [E_{\text{BS}}] = \mathbb{E} \left[ \frac{P_{\text{BS}} S}{R_{\text{BS}}} \right] = {\int}_{0}^{\infty} { \mathbb{E} \left[ \left. \frac{P_{\text{BS}} S}{R_{\text{BS}}} \right| r \right] f_{r_{\text{BS}}}(r) dr } \\ =& {\int}_{0}^{\infty} { {\int}_{0}^{\infty} { \mathbb{P} \left[ \left. \frac{P_{\text{BS}} S}{R_{\text{BS}}} > t \right| r \right] dt } f_{r_{\text{BS}}}(r) dr} \\ =& {\int}_{0}^{\infty} { {\int}_{0}^{\infty} { \mathbb{P} \left[ \left. \frac{ P_{\text{BS}} h r^{-\alpha} }{ I_{\text{BS},\textit{r}} + \sigma_{\text{c}}^{2} } < 2^{\frac{ P_{\text{BS}} S }{ W_{\text{c}} t } } - 1 \right| r \right] dt } f_{r_{\text{BS}}}(r) dr } \\ =& {\int}_{0}^{\infty} { {\int}_{0}^{\infty} { \mathbb{E}_{I_{\text{BS},\textit{r} } } \left[ \mathbb{P} \left[ \left. h < \varphi_{\text{BS}}(t) \left( I_{\text{BS},\textit{r} } + \sigma_{\text{c}}^{2} \right) r^{\alpha} P_{\text{BS}}^{-1} \right| r,I_{\text{BS},\textit{r} } \right] \right] dt } f_{r_{\text{BS}}}(r) dr } \\ =& {\int}_{0}^{\infty} { {\int}_{0}^{\infty} { \left( 1 - \exp \left( - \varphi_{\text{BS}}(t) r^{\alpha} \sigma_{\text{c}}^{2} P_{\text{BS}}^{-1} \right) \mathcal{L}_{I_{\text{BS},\textit{r} } } \left( \varphi_{\text{BS}}(t) r^{\alpha} P_{\text{BS}}^{-1} \right) \right) dt } f_{r_{\text{BS}}}(r) dr }, \end{array} $$
(42)

where \(\varphi _{\text {BS}}(t) \triangleq 2^{\frac { P_{\text {BS}} S }{ W_{\text {c}} t }} - 1\). According to Eq. 30, we have

$$ \mathcal{L}_{I_{\text{BS},\textit{r} } } \left( \varphi_{\text{BS}}(t) r^{\alpha} P_{\text{BS}}^{-1} \right) = \exp \left( - \pi \lambda_{\text{BS,b}} r^{2} A_{1} \left( \alpha,\varphi_{\text{BS}}(t)\right) \right). $$
(43)

Plugging Eq. 43 and \(f_{r_{\text {BS}}}(r) = 2 \pi \lambda _{\text {BS}} r e^{- \pi \lambda _{\text {BS}} r^{2} }\) into Eq. 42, Eq. 16 is obtained.

1.5 A.5 Proof of Proposition 4

When a CR is in state CR-d1, the average energy consumption for receiving a content is given by

$$ \begin{array}{lllllllllll} \mathbb{E} \left[ E_{\text{cr,d,1}} \right] &= \sum\limits_{f = 1}^{F} p_{f|(\text{cr,d,1})} \left( p_{\text{s}|(\text{cr,d,1},{f})} \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,1},{f}} } \right| \gamma_{\text{cr,d,1},{f}} \geq \gamma_{0} \right] \right.\\ & \left.+ \left( 1 - p_{\text{s}|(\text{cr,d,1},{f})} \right) \left( \mathbb{E} \left[ \frac{ P_{\text{UE}} T_{0} }{ N_{\text{b}} } \right] + \mathbb{E} [E_{\text{BS}}] \right) \right), \end{array} $$
(44)

where \(p_{f|(\text {cr,d,1})} = \frac {p_{\text {cr,d,1},{f}}}{p_{\text {cr,d,1}}}\) is the probability that a CR requests content f conditioned on that it is in state CR-d1, \(p_{\text {s}|(\text {cr,d,1},{f})} = \frac {p_{\text {cr,d,1,s},{f}}}{p_{\text {cr,d,1},{f}}}\) is the probability that a CR successfully receives a content conditioned on that it is in state CR-d1 and requests f. \(\mathbb {E} \left [ \left . \frac { P_{\text {UE}} S }{ R_{\text {cr,d,1},{f}} } \right | \gamma _{\text {cr,d,1},{f}} \geq \gamma _{0} \right ]\) is the mean consumed energy for a CR in state CR-d1 when it successfully receives f from another device, and Rcr,d,1,f denotes the data rate when a CR is in state CR-d1 and requests f. \(\mathbb {E} \left [ \frac { P_{\text {UE}} T_{0} }{ N_{\text {b}} } \right ] + \mathbb {E} [E_{\text {BS}}]\) is the mean consumed energy when the D2D transmission is failed. With TDMA, the busy UE serves each of its receivers with duration 1/Nb in a frame.Footnote 4 After T0 frames, the failed D2D transmission is detected and the request is delivered to the BSs.

Let \(X_{f} \triangleq \frac { P_{\text {UE}} S }{ R_{\text {cr,d,1},{f}} }\) and \(x_{0} = \frac { P_{\text {UE}} S }{ W_{\text {d}} \log _{2} (1 + \gamma _{0} ) }\), then we have

$$ \begin{array}{lllllllllll} \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,1},{f}} } \right| \gamma_{\text{cr,d,1},{f}} \geq \gamma_{0} \right] &= \mathbb{E} [ X_{f} | X_{f} \leq x_{0} ] \\ &= {\int}_{0}^{R_{\text{d}}} { \mathbb{E} [ X_{f} | X_{f} \leq x_{0},r ] f_{r_{f} | (\text{cr,d,1},{f}) } (r) dr } \\ &= {\int}_{0}^{R_{\text{d}}} { \mathbb{E} [ X_{f} | X_{f} \leq x_{0},r ] \frac{ f_{r_{f}} (r) }{ p_{\text{in},{f}} } dr }. \end{array} $$
(45)

Since Xf is positive, according to the definition of conditional expectation, we have

$$ \begin{array}{lllllllllll} & \mathbb{E} [ X_{f} | X_{f} \leq x_{0},r ] \\ =& {\int}_{0}^{\infty} { x f_{X_{f}} (x | X_{f} \leq x_{0},r ) dx } = \frac{1}{ F_{X_{f}} (x_{0} | r ) } {\int}_{0}^{x_{0}} { x f_{X_{f}} (x | r ) dx } \\ =& \frac{1}{ F_{X_{f}} (x_{0} | r ) } \left( \left. \left[ - x (1 - F_{X_{f}} (x | r ) ) \right] \right|_{0}^{x_{0}} + {\int}_{0}^{x_{0}} { (1 - F_{X_{f}} (x | r ) ) dx } \right) \\ =& \frac{1}{ F_{X_{f}} (x_{0} | r ) } \left( - x_{0} (1 - F_{X_{f}} (x_{0} | r ) ) + {\int}_{0}^{x_{0}} { (1 - F_{X_{f}} (x | r ) ) dx } \right) \\ =& - \frac{ x_{0} \mathbb{P} [ X_{f} > x_{0} | r ] }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } + \frac{1}{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } {\int}_{0}^{x_{0}} { \mathbb{P} [ X_{f} > x | r ] dx }. \end{array} $$
(46)

According to the derivation in Eqs. 28, 30 and 31, we have

$$ \begin{array}{rllllllllll} \mathbb{P} [ X_{f} \leq x | r ] =& \mathbb{P} [ \gamma_{\text{cr,d,1},{f}} \geq \varphi_{\text{UE}} (x) | r ] \\ =& \exp \left( - \varphi_{\text{UE}} (x) r^{\alpha} \sigma_{\text{d}}^{2} P_{\text{UE}}^{-1} - \pi \lambda_{\text{UE,b},{f}} r^{2} A_{1} (\alpha,\varphi_{\text{UE}} (x) ) \right.\\ &\left.- \pi \lambda_{\text{UE,b},-{f}} r^{2} A_{2} (\alpha,\varphi_{\text{UE}} (x) ) \right), \end{array} $$
(47)

where \(\varphi _{\text {UE}} (x) \triangleq 2^{\frac { P_{\text {UE}} S }{ W_{\text {d}} x } } - 1\). When x = x0, φUE(x0) = γ0.

Combining Eqs. 45, 46 and 47, we have

$$ \begin{array}{lllllllllll} & \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,1},{f}} } \right| \gamma_{\text{cr,d,1},{f}} \geq \gamma_{0} \right] \\ =& {\int}_{0}^{R_{\text{d}}} { \left( - \frac{ x_{0} \mathbb{P} [ X_{f} > x_{0} | r ] }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } + \frac{ 1 }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } {\int}_{0}^{x_{0}} { \mathbb{P} [ X_{f} > x | r ] dx } \right) \frac{ f_{r_{f}} (r) }{ p_{\text{in},{f}} } dr } \\ =& - \frac{x_{0}}{p_{\text{in},{f}}} {\int}_{0}^{R_{\text{d}}} { \frac{ \mathbb{P} [ X_{f} > x_{0} | r ] }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } f_{r_{f}} (r) dr } + \frac{1}{p_{\text{in},{f}}} {\int}_{0}^{R_{\text{d}}} { \!\!\!\! {\int}_{0}^{x_{0}} { \frac{ \mathbb{P} [ X_{f} > x | r ] }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } f_{r_{f}} (r) dx } dr } \\ =& x_{0} - \frac{1}{p_{\text{in},{f}}} {\int}_{0}^{R_{\text{d}}} { \!\!\!\! {\int}_{0}^{x_{0}} { \frac{ \mathbb{P} [ X_{f} \leq x | r ] }{ \mathbb{P} [ X_{f} \leq x_{0} | r ] } f_{r_{f}} (r) dx } dr } \\ =& x_{0} - \frac{1}{p_{\text{in},{f}}} {\int}_{0}^{R_{\text{d}}} { \!\!\!\! {\int}_{0}^{x_{0}} { \frac{e^{Q(f,r,\varphi_{\text{UE}} (x))}}{e^{Q(f,r,\gamma_{0})}} 2 \pi \lambda_{\text{UE},{f}} r e^{- \pi \lambda_{\text{UE},{f}} r^{2} } dx } dr } \\ =& x_{0} - \frac{ 2 \pi \lambda_{\text{UE},{f}} }{ p_{\text{in},{f}} } {\int}_{0}^{R_{\text{d}}} { r e^{- \pi \lambda_{\text{UE},{f}} r^{2} - Q(f,r,\gamma_{0}) } {\int}_{0}^{x_{0}} { e^{Q(f,r,\varphi_{\text{UE}} (x))} dx } dr }, \end{array} $$
(48)

where

$$ \begin{array}{lllllllllll} Q(f,r,\varphi_{\text{UE}} (x)) \triangleq & - \varphi_{\text{UE}} (x) r^{\alpha} \sigma_{\text{d}}^{2} P_{\text{UE}}^{-1} \\&- \pi \lambda_{\text{UE,b},{f}} r^{2} A_{1}(\alpha,\varphi_{\text{UE}} (x) ) \\ &- \pi \lambda_{\text{UE,b},-{f}} r^{2} A_{2}(\alpha,\varphi_{\text{UE}} (x) ), \end{array} $$
(49)

and Q(f,r,γ0) = Q(f,r,φUE(x0)).

Thus, we have

$$ \begin{array}{rllllllllll} \mathbb{E} \left[ E_{\text{cr,d,1}} \right] =& \sum\limits_{f = 1}^{F} \frac{p_{\text{cr,d,1},{f}}}{p_{\text{cr,d,1}}} \left( p_{\text{s}|(\text{cr,d,1},{f})} \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,1},{f}} } \right| \gamma_{\text{cr,d,1},{f}} \geq \gamma_{0} \right] \right.\\ & \left.+ \left( 1 - p_{\text{s}|(\text{cr,d,1},{f})} \right) \left( \mathbb{E} \left[ \frac{ P_{\text{UE}} T_{0} }{ N_{\text{b}} } \right] + \mathbb{E} [E_{\text{BS}}] \right) \right) \\ \approx & \frac{1}{p_{\text{cr,d,1}}} \sum\limits_{f = 1}^{F} \left( p_{\text{cr,d,1,s},{f}} \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,1},{f}} } \right| \gamma_{\text{cr,d,1},{f}} \geq \gamma_{0} \right] \right.\\ & \left.+ \left( p_{\text{cr,d,1},{f}} - p_{\text{cr,d,1,s},{f}} \right) (\omega P_{\text{UE}} T_{0} + \mathbb{E} [E_{\text{BS}}] ) \right), \end{array} $$
(50)

where \(\mathbb {E} \left [ \left . \frac { P_{\text {UE}} S }{ R_{\text {cr,d,1},{f}} } \right | \gamma _{\text {cr,d,1},{f}} \geq \gamma _{0} \right ]\) is given by Eq. 48, and \(\mathbb {E} [E_{\text {BS}}]\) is given by Eq. 16. Similarly, \(\mathbb {E} \left [ E_{\text {cr,d,2}} \right ]\) is obtained as

$$ \begin{array}{lllllllllll} \mathbb{E} \left[ E_{\text{cr,d,2}} \right] \approx & \frac{1}{p_{\text{cr,d,2}}} \sum\limits_{f = 1}^{F} \left( p_{\text{cr,d,2,s},{f}} \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,2},{f}} } \right| \gamma_{\text{cr,d,2},{f}} \geq \gamma_{0} \right] \right.\\ & \left.+ \left( p_{\text{cr,d,2},{f}} - p_{\text{cr,d,2,s},{f}} \right) (\omega P_{\text{UE}} T_{0} + \mathbb{E} [E_{\text{BS}}] ) \right), \end{array} $$
(51)

where

$$ \begin{array}{lllllllllll} & \mathbb{E} \left[ \left. \frac{ P_{\text{UE}} S }{ R_{\text{cr,d,2},{f}} } \right| \gamma_{\text{cr,d,2},{f}} \geq \gamma_{0} \right] \\ =& x_{0} - \frac{ 2 \pi \lambda_{\text{UE},{f}} }{ p_{\text{in},{f}} } {\int}_{0}^{R_{\text{d}}} r e^{- \pi \lambda_{\text{UE},{f}} r^{2} - V(f,r,\gamma_{0}) } \\&{\int}_{0}^{x_{0}} { e^{V(f,r,\varphi_{\text{UE}} (x))} dx } dr , \end{array} $$
(52)

\(V(f,r,\varphi _{\text {UE}} (x)) \! \triangleq \! - \varphi _{\text {UE}} (x) r^{\alpha } \zeta + Q(f,r,\varphi _{\text {UE}} (x))\) and V (f,r,γ0) = V (f,r,φUE(x0)). Thus, Proposition 4 is proved.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qu, H., Ren, G., Zhao, J. et al. Performance Analysis of the Content Dissemination Mechanism with Proactive Content Fetching and Full-Duplex D2D Communication: an Evolutionary Perspective. Mobile Netw Appl 24, 532–555 (2019). https://doi.org/10.1007/s11036-018-1155-3

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11036-018-1155-3

Keywords

Search

Navigation