On the Analysis of Content Dissemination with Reactive Content Pushing in Cache-Enabled D2D Networks

Device-to-device (D2D) communication has huge potential to offload ever increasing cellular traffic. As a premise, devices should store as many popular contents as possible. For augmenting the number of cached contents, in this paper, we propose a reactive content pushing (RCP) mechanism in cache-enabled D2D networks, in which some devices overhear the transmitted contents utilizing the broadcast nature of wireless medium. Since no proactive content pushing is involved, it doesn’t generate interference to the D2D links for normal content requests and its implementation is quite simple. By modeling the D2D network by Poisson point process (PPP), we derive the probabilities that users are in the defined states and the law of evolution of content caching distribution. Moreover, based on the PPP model, we derive the expressions of the successful offloading probability, the average delay and the upper bound of average energy consumption of a request. Simulation results demonstrate the accuracy of the theoretical results and the superiority of the RCP mechanism compared with general content dissemination process.

1. \({\bar{\text{X}}}\) (\({\text{X}}\) represents \({\text{C}}\), \({\text{T}}\), \({\text{P}}\) or \({\text{S}}\)) signifies the complementary event of \({\text{X}}\).

This work was supported by National Natural Science Foundation of China under Grant Nos. 61372092 and 61531013.

Authors and Affiliations

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

   Gongye Ren, Hua Qu, Jihong Zhao & Youwei Shi

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

   Jihong Zhao

3. School of Software Engineering, Xi’an Jiaotong University, Xi’an, 710054, China

   Yanpeng Zhang

Corresponding author

Correspondence to Gongye Ren.

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Appendices

1.1 Proof of Proposition 2

According to the file request protocol and the definition of successful D2D transmission, we have

where \(R_{f}\) and \(r_{f}\) are the instantaneous data rate and the distance from the nearest transmitter with \(f\) to the reference receiver. The PDF of \(r_{f}\) is given by \(f_{{r_{f} }} \left( r \right) = 2\pi \lambda_{{{\text{t}},f}} re^{{ - \pi \lambda_{{{\text{t}},f}} r^{2} }}\) (\(r \ge 0\)) [29]. Considering an interference-limited network and neglecting background noise, \(R_{f}\) is given by

where \(h\) is a random variable that follows the exponential distribution with mean 1 for Rayleigh fading, and \(\alpha\) is path loss exponent. \(I_{{\bar{f},r}}\) is the total interference from all the BTs that don’t transmit \(f\) and \(I_{f,r}\) is the total interference from all the other BTs that transmit \(f\). They are all normalized by \(P_{\text{d}}\) and only consider path loss and Rayleigh fading in our model.

Similar to the proof of Theorem 2 in [29], we have

where \(\mathcal{\mathcal{L}}_{{I_{{\bar{f},r}} + I_{f,r} }} \left( \cdot \right)\) denotes the Laplace transform of random variable \(I_{{\bar{f},r}} + I_{f,r}\). Plugging (12) into (10), we have

The proof is completed.□

1.2 Proof of Proposition 3

Equation (5) is derived based on the law of total probability, and we proof it by presenting the probabilities of the following three cases and the content caching probability in each case. These cases correspond to the three roles of users: BTs, ITs and receivers.

Case 1

The probability of a user being a BT at the \(t\)-th time period is \(\left( {1 - \rho } \right)p_{\text{t,b}}^{t}\). For a BT, its cached files remain unchanged, i.e., the content caching probability for content \(f\) at the \(\left( {t + 1} \right)\)-th time period is also \(p_{f}^{t}\).

Case 2

The probability of a user being an IT at the \(t\)-th time period is \(\left( {1 - \rho } \right)\left( {1 - p_{\text{t,b}}^{t} } \right)\). The cached files of an IT at the \(\left( {t + 1} \right)\)-th time period depend on whether certain uncached file is received successfully. In order to derive \(p_{f}^{t + 1}\) for a reference IT concisely, we derive the probabilities of the following events at the \(t\)-th time period at first:

1. 1.

   Event\({\text{C}}\) The reference IT caches \(f\);

2. 2.

   Event\({\text{T}}\) The nearest BT of the reference IT transmits \(f\);

3. 3.

   Event\({\text{P}}\) The content \(f\) is chosen to push from the nearest BT to the reference IT (on the condition that \({\bar{\text{C}}}\) and \({\text{T}}\) occur simultaneously^{Footnote 1});

4. 4.

   Event\({\text{S}}\) The instantaneous data rate from the nearest BT to the reference IT is larger than or equal to \(R_{0}\) and the distance between them is shorter than \(R_{\text{d}}\).

\(\varvec{\mathbb{P}}\left[ {\text{C}} \right] = p_{f}^{t}\) is evident. The probability that a transmitter transmits \(f\) is \(p_{f}^{t} u_{f}^{t}\) and the probability that a transmitter is busy is \(p_{\text{t,b}}^{t}\). Thus, the probability that a BT transmits \(f\) is \(\varvec{\mathbb{P}}\left[ {\text{T}} \right] = {{p_{f}^{t} u_{f}^{t} } \mathord{\left/ {\vphantom {{p_{f}^{t} u_{f}^{t} } {p_{\text{t,b}}^{t} }}} \right. \kern-0pt} {p_{\text{t,b}}^{t} }}\). Let \(\mathcal{\mathcal{F}}_{\text{i,b}}^{t}\) be a set containing all the contents \(f \in \mathcal{\mathcal{F}}\) that satisfy the following two conditions: 1) The reference IT doesn’t cache \(f\); 2) The nearest BT of the reference IT transmits \(f\). Then we have \({\mathbb{E}}\left[ {\left| {\mathcal{\mathcal{F}}_{\text{i,b}}^{t} } \right|} \right] = \sum\nolimits_{f = 1}^{F} {\frac{{p_{f}^{t} u_{f}^{t} }}{{p_{\text{t,b}}^{t} }}\left( {1 - p_{f}^{t} } \right)}\). Each content in \(\mathcal{\mathcal{F}}_{\text{i,b}}^{t}\) is chosen to push with equal probability \(\varvec{\mathbb{P}}\left[ {\text{P}} \right] = \frac{1}{{{\mathbb{E}}\left[ {\left| {\mathcal{\mathcal{F}}_{\text{i,b}}^{t} } \right|} \right]}} = \frac{{p_{\text{t,b}}^{t} }}{{\sum\nolimits_{f = 1}^{F} {p_{f}^{t} u_{f}^{t} \left( {1 - p_{f}^{t} } \right)} }}\) according to the RCP mechanism. We denote \(\varvec{\mathbb{P}}\left[ {\text{S}} \right]\) by \(p_{\text{t,r}}^{t}\). With slight modification to the proof of Theorem 2 in [29], we can easily obtain

where \(R_{\text{p}}^{t}\) is the instantaneous pushing data rate, and \(f_{{r^{t} }} \left( r \right)\) is the PDF of the distance from the nearest BT to the reference IT.

In order to obtain \(p_{f}^{t + 1}\) for a given content \(f\), we list all events for an IT at the \(t\)-th time period in Table 2. The probabilities in the second column are derived based on the fact that the event \({\text{C}}\), \({\text{T}}\) and \({\text{S}}\) are mutually independent events.

For an IT, according to the law of total probability, the probability that it stores \(f\) at the \(\left( {t + 1} \right)\)-th time period is

Case 3

The probability of a user being a receiver is \(\rho\). For a receiver, if it caches \(f\) at the \(t\)-th time period, it also caches \(f\) at the \(\left( {t + 1} \right)\)-th time period. If the receiver doesn’t cache \(f\) but requests \(f\), it certainly stores \(f\) at the next time period according to the file request protocol. Otherwise, the receiver will not store \(f\) at the next time period. Thus, the probability that a receiver caches \(f\) at the \(\left( {t + 1} \right)\)-th time period is \(p_{f}^{t} + \left( {1 - p_{f}^{t} } \right)q_{f}\).

Applying the law of total probability again based on the above derivation, (5) is obtained.

1.3 Proof of Proposition 4

According to the file request protocol and the assumptions in Sect. 5.2, the content request delay comes from three cases: (1) Receiving files via D2D transmission successfully; (2) failing in receiving files via D2D transmission and then accessing to BSs; (3) finding no transmitter with the requested files within \(R_{\text{d}}\) and then connecting to BSs. In a word, delay is evaluated for the receivers that 1) request \(f\) and 2) don’t cache \(f\) (\(f\) represents any file). We refer to these receivers as \(f\)-outward receivers, and the probability that a receiver is an \(f\)-outward receiver is \(q_{f} \left( {1 - p_{f} } \right)\). We present the probability and the value of delay for an \(f\)-outward receiver in each case as follows.

Case 1

The probability that an \(f\)-outward receiver receives \(f\) via D2D transmission successfully is \(p_{{{\text{r,d}},f}}\) according to (13). For a BT, the average number of different files it transmits is \(N_{\text{t,b}} \triangleq \frac{1}{{p_{\text{t,b}} }}\sum\limits_{f = 1}^{F} {p_{f} u_{f} }\). The average proportion of time in a frame allocated to one file is \({1 \mathord{\left/ {\vphantom {1 {N_{\text{t,b}} }}} \right. \kern-0pt} {N_{\text{t,b}} }}\). Thus, the average delay for an \(f\)-outward receiver in Case 1 is \({\mathbb{E}}\left[ {{{N_{\text{t,b}} S} \mathord{\left/ {\vphantom {{N_{\text{t,b}} S} {\hat{R}_{f} }}} \right. \kern-0pt} {\hat{R}_{f} }}} \right]\), in which \(\hat{R}_{f}\) is the instantaneous data rate of an \(f\)-outward receiver receiving \(f\) via D2D transmission successfully and the expectation is taken with respect to \(\hat{R}_{f}\). Nevertheless, the accurate derivation involves evaluating the expectation of the reciprocal of \(\hat{R}_{f}\), which entails lengthy intractable integral expression. To keep tractability, we approximate \({\mathbb{E}}\left[ {{{N_{\text{t,b}} S} \mathord{\left/ {\vphantom {{N_{\text{t,b}} S} {\hat{R}_{f} }}} \right. \kern-0pt} {\hat{R}_{f} }}} \right]\) by \({{N_{\text{t,b}} S} \mathord{\left/ {\vphantom {{N_{\text{t,b}} S} {{\mathbb{E}}\left[ {\hat{R}_{f} } \right]}}} \right. \kern-0pt} {{\mathbb{E}}\left[ {\hat{R}_{f} } \right]}}\). Denoting \({\mathbb{E}}\left[ {\hat{R}_{f} } \right]\) by \(\bar{R}_{f}\), the average delay for an \(f\)-outward receiver in Case 1 is approximated by \({{N_{\text{t,b}} S} \mathord{\left/ {\vphantom {{N_{\text{t,b}} S} {\bar{R}_{f} }}} \right. \kern-0pt} {\bar{R}_{f} }}\).

According to the definition of \(\bar{R}_{f}\), we have

where the expectation in the second line is taken over both the PPP and the fading distribution, and \(f_{{r_{f} }} \left( {r|r_{f} \le R_{\text{d}} } \right)\) is the conditional PDF of \(r_{f}\). Given \(f_{{r_{f} }} \left( r \right) = 2\pi \lambda_{{{\text{t}},f}} re^{{ - \pi \lambda_{{{\text{t}},f}} r^{2} }}\) (\(r \ge 0\)), \(f_{{r_{f} }} \left( {r|r_{f} \le R_{\text{d}} } \right)\) is simply given by

According to the definition of the conditional expectation, we have

Plugging (17) and (18) into (16), we have

Case 2

The probability that an \(f\)-outward receiver finds at least one transmitter with \(f\) within \(R_{\text{d}}\) (termed event “Hit”) is \(p_{{{\text{in}},f}} \triangleq 1 - e^{{ - \pi \lambda_{{{\text{t}},f}} R_{\text{d}}^{2} }}\) [18]. Under the condition that event “Hit” occurs, successful D2D transmission and failed D2D transmission are complementary events. Thus, the probability that an \(f\)-outward receiver fails to receive \(f\) via D2D transmission is \(p_{{{\text{in}},f}} - p_{{{\text{r,d}},f}}\). In this case, additional delay \(D_{0}\) is entailed to detect failed D2D transmission. The delay of cellular transmission is simply \({S \mathord{\left/ {\vphantom {S {R_{0} }}} \right. \kern-0pt} {R_{0} }}\), and the total delay for an \(f\)-outward receiver in Case 2 is \({S \mathord{\left/ {\vphantom {S {R_{0} }}} \right. \kern-0pt} {R_{0} }} + D_{0}\).

Case 3

Based on the previous analysis, the probability that an \(f\)-outward receiver finds no transmitter caching \(f\) within \(R_{\text{d}}\) is \(p_{{{\text{out}},f}}\). In this case, the request for \(f\) has to be handled by BSs and the delay is \({S \mathord{\left/ {\vphantom {S {R_{0} }}} \right. \kern-0pt} {R_{0} }}\).

Given the above results, the approximated average delay for an \(f\)-outward receiver is \(p_{{{\text{r,d}},f}} \frac{{N_{\text{t,b}} S}}{{\bar{R}_{f} }} + \left( {p_{{{\text{in}},f}} - p_{{{\text{r,d}},f}} } \right)\left( {\frac{S}{{R_{0} }} + D_{0} } \right) + p_{{{\text{out}},f}} \frac{S}{{R_{0} }}\). Averaging over all the files in \(\mathcal{\mathcal{F}}\), the average delay for a receiver is approximated by (7).

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