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Fundamentals of Cluster-Centric Content Placement in Cache-Enabled Device-to-Device Networks

Mehrnaz Afshang, Harpreet S. Dhillon, Peter Han Joo Chong

arXiv:1509.04747v4cs.ITmath.IT

TL;DR

Cluster-centric D2D caching seeks to improve performance across a whole cluster, while distributing content beyond a single cache device. The paper analyzes center-relative placement with a Poisson cluster model and finds performance improves when cached content or requesting devices are biased toward the cluster center.

Abstract

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This paper develops a comprehensive analytical framework with foundations in stochastic geometry to characterize the performance of cluster-centric content placement in a cache-enabled device-to-device (D2D) network. Different from device-centric content placement, cluster-centric placement focuses on placing content in each cluster such that the collective performance of all the devices in each cluster is optimized. Modeling the locations of the devices by a Poisson cluster process, we define and analyze the performance for three general cases: (i)$k$-Tx case: receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at the $k^{th}$ closest device to the cluster center, (ii) $\ell$-Rx case: receiver of interest is the $\ell^{th}$ closest device to the cluster center and its content of interest is available at a device chosen uniformly at random from the same cluster, and (iii) baseline case: the receiver of interest is chosen uniformly at random in a cluster and its content of interest is available at a device chosen independently and uniformly at random from the same cluster. Easy-to-use expressions for the key performance metrics, such as coverage probability and area spectral efficiency (ASE) of the whole network, are derived for all three cases. Our analysis concretely demonstrates significant improvement in the network performance when the device on which content is cached or device requesting content from cache is biased to lie closer to the cluster center compared to baseline case. Based on this insight, we develop and analyze a new generative model for cluster-centric D2D networks that allows to study the effect of intra-cluster interfering devices that are more likely to lie closer to the cluster center.

I. INTRODUCTION

The introduction motivates cluster-centric caching as collective optimization rather than device-specific placement, then frames a Poisson-cluster analysis of center-biased placement choices and network performance.

I. INTRODUCTION

  • Cluster-centric caching targets collective coverage and ASE, because device-centric placement can benefit one requester while degrading another seeking the same content.Source: §I · p. 1

A. Related Work and Motivation

Tractable model for cache-enabled D2D networks:

  • The framework proves coverage improves when each device’s content is cached at its cluster’s closest device, but studies distributed placement given storage, energy, and reuse concerns.Source: Tractable model for cache-enabled D2D… · p. 2

ASE and coverage probability analysis:

  • Coverage and ASE are derived for k-Tx, ℓ-Rx, and independent-uniform baseline cases, using center-relative receiver or serving-device choices and new distance characterizations.Source: ASE and coverage probability analysis · p. 2

New generative model for cluster-centric D2D networks:

  • A double-variance Thomas process models denser and sparser subclusters; sampling transmitters or receivers from the denser subcluster represents center-biased devices.Source: New generative model for… · p. 2

II. SYSTEM MODEL

  • Content is cached and requested within the same cluster; although the model is extendible, direct communication is restricted to cluster members.Source: §II · p. 3

A. System Setup and Key Assumptions

  • Devices are normally distributed around PPP cluster centers; Poisson cluster sizes yield a Thomas-process variant, and active-transmitter counts are Poisson with mean m̄a.Source: A. System Setup and Key Assumptions · p. 3

B. Channel Model

The channel model represents desired signal and interference at a receiver, then evaluates coverage and area spectral efficiency. The analysis frames cluster-centric placement around the cluster center and compares receiver and cache locations through three cases; concentrating all content at the nearest device maximizes cluster coverage, but limits reuse and may be infeasible.

B. Channel Model

  • The desired signal comes from the device caching the requested content, while interference is separated into independent intra-cluster and inter-cluster components.The model uses constant transmit power and Rayleigh fading; coverage is the probability that SIR exceeds the decoding threshold, and ASE measures average transmitted bits per unit time, bandwidth, and area.Source: B. Channel Model · pp. 3–4
  • Coverage probability is the chance that a link’s SIR exceeds its decoding threshold; ASE is the average transmitted bits per time, bandwidth, and area.ASE uses the density of simultaneously active transmitters.Source: B. Channel Model · p. 4

A. Coverage-Optimal Content Placement

  • The coverage-optimal single cache is the device closest to the cluster center, but concentrating content there limits simultaneous D2D links to one and may violate storage or energy constraints.The paper therefore considers distributing content across devices in a cluster.Source: §III · p. 4, A. Coverage-Optimal Content Placement · p. 4
  • Cluster-centric placement optimizes collective cluster performance by referencing the cluster center rather than an individual receiver.The analysis considers a receiver chosen uniformly with a transmitter at rank k, a receiver at rank ℓ with a uniformly chosen transmitter, and a baseline with both chosen uniformly.Source: A. Coverage-Optimal Content Placement · pp. 4–5

B. Coverage Probability Analysis

The coverage analysis characterizes serving and interfering-device distances, then uses conditional interference Laplace transforms to obtain tractable coverage expressions. Conditioning on distance to the cluster center handles shared-distance dependence, while the simpler expression for k-Tx approximates away ordered serving-device selection.

B. Coverage Probability Analysis

  • Conditioning on the receiver’s distance to the cluster center makes intra-cluster interferer distances conditionally i.i.d., enabling tractable interference Laplace transforms and coverage analysis.Without conditioning, the distances are correlated through their common cluster-center offset.Source: B. Coverage Probability Analysis · p. 5
  • For k-Tx, serving-device selection by rank couples serving and interfering distances; partitioning interferers into nearer and farther-from-center sets supports the exact interference analysis.The exact conditional Laplace transform is more complicated; the shorter paper version reported an approximation.Source: B. Coverage Probability Analysis · pp. 5–6
  • For ℓ-Rx and baseline, uniformly selecting the serving device removes dependence between serving and interfering-device distances, simplifying the intra-cluster interference transform.Conditioning on receiver distance to the cluster center remains necessary to handle the shared-offset correlation.Source: B. Coverage Probability Analysis · p. 6
  • The simpler ℓ-Rx and baseline transform approximates k-Tx by ignoring the effect of ordered serving-device selection on interferer distances; the paper reports it is numerically tight.It is exact for ℓ-Rx and baseline, but only an approximation for k-Tx.Source: B. Coverage Probability Analysis · p. 6
  • Assuming i.i.d. Rayleigh intra-cluster distances further simplifies the transform for k-Tx and baseline, but is not applicable to ℓ-Rx.This assumption ignores distance correlation induced by the common cluster-center offset.Source: B. Coverage Probability Analysis · p. 7

1) Coverage probability analysis of k-Tx case:

For k-Tx, the receiver is sampled uniformly within a cluster and its serving transmitter is the kth closest device to the center. Coverage expressions use the serving-distance distribution and interference transforms, with simpler alternatives that approximate the ordered-selection dependence.

1) Coverage probability analysis of k-Tx case:

  • The k-Tx coverage expression averages over the receiver and serving-distance distributions using the exact intra-cluster and inter-cluster interference transforms.The serving distance is conditioned on receiver distance to the center and the kth transmitter’s center distance.Source: 1) Coverage probability analysis of… · p. 7
  • The simpler k-Tx coverage expression uses the ℓ-Rx and baseline intra-cluster transform, approximating away the effect of ordered serving-device selection on interferer distances.A further simplification ignores intra-cluster distance correlations under Assumption 1; the paper says approximation tightness is validated numerically.Source: 1) Coverage probability analysis of… · pp. 7–8

2) Coverage probability analysis of ℓ-Rx case:

For ℓ-Rx, the receiver is the ℓth closest device to the cluster center and its serving transmitter is chosen uniformly within the cluster. The coverage expression averages over the receiver’s center-distance order statistic and serving distance.

2) Coverage probability analysis of ℓ-Rx case:

  • The ℓ-Rx coverage expression uses the ℓth receiver’s center-distance distribution and the serving-distance distribution induced by a uniformly chosen transmitter.Unlike k-Tx, uniform serving-device selection is used in this case.Source: 2) Coverage probability analysis of… · p. 8

3) Coverage probability analysis of the baseline case:

The baseline coverage analysis samples both the receiver and its serving transmitter uniformly at random within a cluster. It characterizes the serving-link distance using a conditional Rician distribution and the receiver’s distance from the cluster center using a Rayleigh distribution.

3) Coverage probability analysis of the baseline case:

  • The baseline case samples both the D2D receiver and transmitter of interest uniformly at random within a cluster.The passage identifies the coverage probability theorem for this case but does not provide its full expression.Source: 3) Coverage probability analysis of… · p. 8
  • The theorem states the baseline coverage probability, with a proof sketch provided by reference to the derivation for Theorem 1.The complete proof is referenced in [46].Source: 3) Coverage probability analysis of… · p. 8
  • Conditioned on the receiver’s distance from the cluster center, the serving distance follows a Rician distribution.The receiver’s distance from the cluster center follows a Rayleigh distribution because the receiver is sampled uniformly from the cluster.Source: 3) Coverage probability analysis of… · p. 8

C. Area Spectral Efficiency Analysis

The section defines ASE for the three placement cases using their coverage probabilities and the average number of simultaneously active transmitters. It frames the choice of active devices per cluster as a trade-off and reports that the ASE-maximizing number depends on the placement policy.

C. Area Spectral Efficiency Analysis

  • ASE for the three cases is specialized in terms of their coverage probabilities and the average number of simultaneously active transmitting devices.The proposition gives the ASE expressions; the average active-transmitter term is ¯maλc.Source: C. Area Spectral Efficiency Analysis · p. 8
  • Increasing simultaneous links increases spectrum reuse but also interference, while fewer links reduce intra-cluster interference at the cost of reuse.The section formulates the choice of ¯ma as maximizing ASE.Source: C. Area Spectral Efficiency Analysis · p. 8
  • The optimal number of simultaneously active links differs significantly among the three cases, showing that the ASE-maximizing choice depends on content placement policy.The authors report obtaining this result by numerically solving the ASE optimization problem.Source: C. Area Spectral Efficiency Analysis · p. 8

IV. NEW GENERATIVE MODEL FOR THE CLUSTER-CENTRIC D2D NETWORKS

The proposed double-variance Thomas model represents each cluster as denser and sparser device subclusters to study devices biased toward the cluster center. Its coverage analysis balances shorter serving links and reduced inter-cluster interference against increased intra-cluster interference.

IV. NEW GENERATIVE MODEL FOR THE CLUSTER-CENTRIC D2D NETWORKS

  • The double-variance model studies center-biased transmitters or receivers using denser and sparser subclusters while maintaining tractability.The authors motivate the model by noting that jointly handling intra-cluster distances would produce difficult multi-fold integrals.Source: §IV · p. 9

A. System Setup and Channel Model

  • Cluster centers follow a PPP, and each cluster contains normally distributed devices in subclusters with different scattering variances.The receiver is sampled uniformly from either subcluster, while the serving transmitter is sampled uniformly from the denser subcluster.Source: A. System Setup and Channel Model · p. 9

B. Coverage Probability

  • The model expresses intra-cluster and inter-cluster interference separately and characterizes their Laplace transforms for coverage analysis.The coverage theorem combines these interference transforms for a randomly chosen device in the double-variance model.Source: A. System Setup and Channel Model · p. 9, B. Coverage Probability · p. 9
  • For a randomly chosen receiver, the serving distance is conditionally Rician, while its distance from the cluster center follows a Rayleigh distribution.The receiver’s Gaussian scattering variance depends on whether it belongs to the denser or sparser subcluster.Source: B. Coverage Probability · p. 10
  • Center-biased transmitters or receivers shorten the serving link, while concentrating active transmitters in the denser subcluster reduces inter-cluster but increases intra-cluster interference.The first two effects improve coverage probability; increased intra-cluster interference degrades it.Source: B. Coverage Probability · p. 10

V. RESULTS AND DISCUSSION

The analysis matches simulations for the ℓ-Rx and baseline cases, while a k-Tx approximation is fairly tight. Biasing caching or requesting toward the cluster center improves coverage and ASE over the baseline in the reported comparisons, though distant caching can perform worse.

1) Validation of results:

  • The exact ℓ-Rx and baseline results match simulations, and the k-Tx approximation is fairly tight enough to serve as a proxy for its exact result.The exact k-Tx expression is not straightforward to compute numerically.Source: Figure 4 · p. 10, 1) Validation of results · p. 10

2) Performance comparison across three cases:

B. Applications of the Results to Total Hit Probability

The section applies the paper’s coverage results to total hit probability under uniform and cluster-centric content placement, then examines a double-variance cluster model. It concludes that cluster-center-biased placement can improve performance, while the new model captures denser concentrations of intra-cluster interferers.

B. Applications of the Results to Total Hit Probability

  • Total hit probability requires both that the requested content is available in the cluster and that the receiver is in coverage of its serving device.The section assumes Zipf popularity but states that the analysis extends to other popularity distributions.Source: B. Applications of the Results to… · p. 11

1) Uniform content placement:

  • Under uniform content placement, the total hit probability uses baseline coverage, where the file of interest is available at a uniformly selected device.Source: 1) Uniform content placement · p. 11

2) Cluster-centric content placement:

  • Cluster-centric placement assigns the most popular file to the nearest transmitter to the cluster center, the next most popular to the second-nearest, and so on.The analysis considers a receiver chosen uniformly at random and confines the two placement strategies to the k-Tx and baseline cases.Source: B. Applications of the Results to… · p. 11, 2) Cluster-centric content placement · p. 12
  • Total hit probability is significantly higher with cluster-centric placement, and increasing Zipf parameter γ improves it while leaving uniform-placement results unchanged.More popular files are requested more often and cached closer to the center, where lower-k coverage is higher.Source: 2) Cluster-centric content placement · p. 12

C. Performance of the New Generative Cluster-Centric Model

  • When the same number of transmitters is active, coverage is higher with more active transmitters in the sparser subcluster because they cause less intra-cluster interference.Coverage is lowest when both devices of interest are in the sparser subcluster and highest when both are in the denser subcluster.Source: C. Performance of the New Generative… · p. 12

VI. CONCLUSION

  • The paper derives coverage and ASE for k-Tx and ℓ-Rx cases, compares them with independent uniform selection, and reports gains when either device of interest is closer to the cluster center.It also proposes a double-variance Thomas cluster process to model more intra-cluster interferers near the center.Source: §VI · pp. 12–13
  • The authors identify extensions in popularity, social interaction, cache constraints, in-band transmissions, broader cluster processes, and measurement of real-world cluster statistics.Source: §VI · p. 13

APPENDIX

The appendix supports the main analysis with distance-ordering, conditional distance-distribution, and intra-cluster interference arguments.

A. Proof of Lemma 1

  • The serving distance from a uniformly displaced receiver is stochastically ordered by the transmitter’s distance from the cluster center, supporting the closest-transmitter comparison.The proof uses the Rician distance distribution and monotonicity of the Marcum Q-function.Source: A. Proof of Lemma 1 · p. 13

B. Proof of Lemma 2

  • For the distance between a receiver and an intra-cluster device, the proof conditions on their radii and uses a uniform angle; device radii follow a Rayleigh distribution.Distances below or above the serving-device radius are represented by truncated Rayleigh distributions.Source: B. Proof of Lemma 2 · p. 13

C. Proof of Lemma 3

  • The k-Tx interference derivation partitions active intra-cluster devices by whether they lie closer to the center than the serving device and averages over their number.The number closer to the center is binomial under the stated count conditions, with the active-device total conditioned to be below the cluster’s transmitter count.Source: C. Proof of Lemma 3 · pp. 13–14

D. Proof of Lemma 4

These appendix proofs derive an intra-cluster interference transform and the serving-distance distribution used in the coverage analysis.

D. Proof of Lemma 4

  • The intra-cluster interference Laplace transform averages over Poisson-distributed device counts and uses exponential fading and the conditional Rician distance distribution.The proof also gives an approximation under the assumption that the mean active-transmitter count is much smaller than the cluster transmitter count.Source: D. Proof of Lemma 4 · p. 14

E. Proof of Corollary 3

  • The serving-distance distribution is obtained by conditioning on the transmitter’s radius from the cluster center and converting the Gaussian displacement to polar coordinates.The proof states that conditioning on the transmitter’s radius suffices and follows the approach of Theorem 1.Source: E. Proof of Corollary 3 · p. 14

F. Proof of Lemma 7

The appendix derives interference transforms for the double-variance cluster model, including the special case in which one subcluster has no active transmitters.

F. Proof of Lemma 7

  • The Lemma 7 derivation combines interference from two independent device sets using their independent fading and the Poisson probability generating functional.It then changes from Cartesian to polar coordinates.Source: F. Proof of Lemma 7 · p. 15

G. Proof of Lemma 8

  • The Lemma 8 transform uses conditional independence of the two subcluster sets given cluster centers; Lemma 5 is its special case when the second set has zero mean active count.Source: G. Proof of Lemma 8 · p. 15
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