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Optimizing Content Caching to Maximize the Density of Successful Receptions in Device-to-Device Networking

Derya Malak, Mazin Al-Shalash, Jeffrey G. Andrews

arXiv:1608.07857v1cs.ITcs.NImath.IT

TL;DR

The paper studies how to cache content in D2D networks to increase successful deliveries amid interference and noise. It models coverage and optimizes caching strategies, finding that a Zipf cache with exponent γr/3 closely matches the numerical optimum for γr = 0.5, while that approximation does not hold uniformly for γr = 2.

  • Problem

    The caching pmf that best maximizes DSR in D2D networks is not generally known, despite the need to place popular content on devices for nearby users.

    Source: §I · p. 2

  • Method

    Using a homogeneous PPP model with interference, noise, and Rayleigh fading, the paper derives coverage and optimizes caching for single-file and multi-file settings.

    Source: §II · p. 7, §III · pp. 8, 11, §VIII · p. 23

  • Results

    For γr = 0.5, Zipf caching with exponent γr/3 is close to the numerical optimum at SNR 1 and 10, but does not match the optimum consistently for γr = 2.

    Source: §VII · pp. 22–23

  • Takeaways & Limitations

    The γr/3 Zipf cache is a useful approximation in the reported γr = 0.5 cases, but its fit depends on the request exponent and operating condition.

    Source: §VII · pp. 22–23

    The model leaves transmit-diversity and intelligent-scheduling settings, as well as changing file popularities and simultaneous-transmission interference, for future work.

    Source: §VIII · p. 24

Abstract

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Device-to-device (D2D) communication is a promising approach to optimize the utilization of air interface resources in 5G networks, since it allows decentralized opportunistic short-range communication. For D2D to be useful, mobile nodes must possess content that other mobiles want. Thus, intelligent caching techniques are essential for D2D. In this paper we use results from stochastic geometry to derive the probability of successful content delivery in the presence of interference and noise. We employ a general transmission strategy where multiple files are cached at the users and different files can be transmitted simultaneously throughout the network. We then formulate an optimization problem, and find the caching distribution that maximizes the density of successful receptions (DSR) under a simple transmission strategy where a single file is transmitted at a time throughout the network. We model file requests by a Zipf distribution with exponent $\gamma_r$, which results in an optimal caching distribution that is also a Zipf distribution with exponent $\gamma_c$, which is related to $\gamma_r$ through a simple expression involving the path loss exponent. We solve the optimal content placement problem for more general demand profiles under Rayleigh, Ricean and Nakagami small-scale fading distributions. Our results suggest that it is required to flatten the request distribution to optimize the caching performance. We also develop strategies to optimize content caching for the more general case with multiple files, and bound the DSR for that scenario.

I. INTRODUCTION

The paper asks how to place cached content in D2D networks to maximize successful receptions, where existing placement results do not capture realistic simultaneous transmissions and interference. It develops a stochastic-geometry model and caching strategies for sequential and simultaneous file transmissions.

A. Related Work

  • Earlier D2D caching studies did not determine the caching pmf that maximizes DSR, and some used a model restricting each cluster to one pair without inter-cluster interference.The paper instead considers simultaneous D2D pairs without that restriction.Source: A. Related Work · p. 3

B. Contributions

  • Users are modeled as a homogeneous PPP, with transmitter and receiver fractions γ1 and γ2; DSR is the mean number of successful receptions per unit area.The analysis incorporates interference, noise, and Rayleigh fading, and counts a transmission as successful when SINR exceeds a threshold.Source: B. Contributions · p. 4, §II · p. 7
  • For sequential multi-file transmission, the paper optimizes caching using the request distribution and the transmitter and receiver fractions established in the single-file baseline.The single-file case optimizes γ1 and γ2; the multi-file model uses those fractions to optimize the caching pmf.Source: B. Contributions · p. 4
  • With Zipf requests, optimal caching is flatter than demand; under small-noise and approximated arbitrary-noise cases, γc is related to γr by γc = γr/(α/2+1), α > 2.For α = 4, the optimal distribution is approximated by Benford’s law; the stated small-fading extensions include Ricean and Nakagami distributions.Source: B. Contributions · pp. 4–5
  • For simultaneous transmissions of different files, popularity-based and global strategies yield lower coverage than the sequential model for the global strategy, and optimized caching skews toward popular content.The paper also develops DSR bounds for the multiple-file case, which simulations find accurate for particular γr values.Source: B. Contributions · pp. 5–6

II. SYSTEM MODEL

  • In the randomized model, each user transmits with probability γ1 or receives with probability γ2, draws a cached file or request accordingly, and receivers associate with the closest transmitter.For multiple files, different file types can transmit simultaneously.Source: §II · p. 6, Figure 2 · p. 7

III. DSR FOR A SINGLE FILE

For a single file, the paper expresses density of successful receptions (DSR) as receiver density times coverage probability, then optimizes the transmitter and receiver fractions. The optimum uses fewer transmitters than receivers, with the balance and DSR responding to network conditions.

III. DSR FOR A SINGLE FILE

  • The single-file DSR is receiver density λγ2 multiplied by the coverage probability pcov(T, λγ1, α).Receivers associate with their nearest transmitter, while other transmitters interfere.Source: §III · p. 8
  • The optimal fractions satisfy γ1 < a/2 < γ2 ≤ a ≤ 1, so receivers outnumber transmitters.Here a is the total fraction of transmitting and receiving users.Source: §III · p. 8
  • As SNR increases with T = SNR/2, the maximum DSR and optimal transmitter fraction γopt decrease.For increasing T, higher γ1 compensates for lower coverage; at low T, a higher receiver fraction is preferred.Source: §III · pp. 10–11, Figure 3 · p. 10
  • As user density λ increases, DSR rises while γopt falls, which the authors attribute to increased interference from more transmitters.The single-file model serves as the baseline for the multiple-file caching problem.Source: §III · p. 11

IV. OPTIMIZING THE DSR OF THE SEQUENTIAL SERVING MODEL WITH MULTIPLE FILES

The sequential model transmits one file type at a time and optimizes its caching distribution to maximize DSR. The analysis finds flatter caching than demand for Zipf requests and extends the optimization to general demand profiles and several fading distributions.

A. Sequential Serving-based Model

  • The sequential strategy schedules one file type at a time, with transmitters drawing files from the caching pmf and receivers drawing requests from the demand pmf.The optimal transmitter fraction and caching pmf are coupled and are solved in sequence.Source: A. Sequential Serving-based Model · pp. 11–12
  • For Zipf requests, the optimal caching distribution is also Zipf, with exponent γc = γr/(α/2+1) for α > 2, making caching flatter than demand.For small non-zero noise this is the stated Rayleigh result; under arbitrary noise it is an approximation based on a separability assumption.Source: B. Rayleigh Fading DSR Results · p. 13
  • For arbitrary request pmfs, the approximate optimum applies under Rayleigh, Nakagami, or Ricean fading and requires flattening demand; uniform requests yield uniform caching.The arbitrary-noise result uses the separability approximation for coverage.Source: B. Rayleigh Fading DSR Results · pp. 13–14
  • For arbitrary noise with α = 4, the paper gives an approximate caching distribution described as a variety of Benford’s law.The passage identifies it as a special bounded case of Zipf’s law.Source: B. Rayleigh Fading DSR Results · pp. 14–15

V. A LOWER AND UPPER BOUND ON THE DSR AND DIFFERENT CACHING STRATEGIES

Because the multiple-file sequential DSR is intractable for general caching distributions, the paper derives upper and lower bounds and proposes two caching strategies. The strategies either protect the least popular file or optimize DSR across all files without prioritization.

V. A LOWER AND UPPER BOUND ON THE DSR AND DIFFERENT CACHING STRATEGIES

  • The paper derives lower and upper bounds to characterize sequential multi-file DSR when the caching distribution lacks a simple form.It compares these bounds with alternative caching strategies.Source: §V · p. 15, A. Bounds on DSRS · p. 15

1) Upper Bound (UB):

  • The upper bound uses concavity of coverage probability in the caching probability and Jensen’s inequality.The derivation also uses the assumption that file 1 is the most popular.Source: 1) Upper Bound (UB) · p. 15

2) Lower Bound (LB):

  • For Zipf requests, the optimal caching distribution is also Zipf, and a uniform distribution performs worse because it is not skewed toward popular files.The result supplies a lower bound for the sequential multi-file DSR.Source: 2) Lower Bound (LB) · pp. 15–16

B. Caching Strategies for the Sequential Serving Model with Multiple Files

VI. SIMULTANEOUS TRANSMISSIONS OF DIFFERENT FILES WITH ARBITRARY NOISE

The section analyzes simultaneous transmission of multiple files using popularity-based and global models, with coverage depending on the requested-file state. It compares these strategies with sequential serving and examines how request and caching skew relate to DSR.

Multiple-file coverage analysis

  • For a receiver in state j, candidate transmitter density is λ_j = λ_t p_j, while transmitters without its requested files form the remaining interferer process.The receiver gets content from the closest transmitter candidate; interference includes both transmitters with requested files and those without them.Source: §VI · p. 17

Coverage probability

  • Coverage probability increases monotonically with p_j, the fraction of transmitters that can serve a receiver in state j.The section gives a special-case analysis for path-loss exponent α = 4 and notes that coverage approaches 1 as the threshold T grows without bound.Source: §VI · p. 18

A Discussion on the Various Transmission Models

  • Sequential serving mitigates interference and provides higher coverage than popularity-based and global transmission, but its DSR is reduced by weighting the file-transmit pmf.The popularity-based and global models depend on global file popularity distributions, while sequential serving activates only a subset of transmitters and candidate receivers.Source: B. Global DSR · p. 19

Numerical results

  • The optimal caching pmf need not be Zipf, but the paper optimizes Zipf exponents for simulations because the general formulations are intractable.The nonlinear coverage probability in transmitter density makes the unrestricted optimization formulations intractable.Source: B. Global DSR · p. 20
  • As the request distribution becomes more skewed, DSR increases in the popularity-based and global models, and rises with caching skew γ_c.The sequential model’s optimal Zipf caching exponent γ_c increases with request exponent γ_r; the discussion states the optimal caching distribution should favor popular files.Source: B. Global DSR · pp. 20–21

VII. NUMERICAL RESULTS AND DISCUSSION

Numerical comparisons show that the quality of Zipf and Benford caching approximations depends on the request exponent and noise level. The conclusions extend the caching results to general demand profiles and fading models, while identifying synchronization and broader transmission settings as open issues.

VII. NUMERICAL RESULTS AND DISCUSSION

  • As γr increases, the range of file counts where Benford and Zipf caching are comparable narrows; for γr > 0.3, Zipf is not a practical approximation to Benford.As noise decreases, optimal caching converges to Zipf; at high SNR, Zipf and Benford behave similarly for relatively high γr compared with the general-noise case.Source: §VII · p. 21
  • For γr = 0.5 and SNR = 1, Zipf caching with exponent γr/3 is close to the numerical optimum and lower bound, while Benford caching has similar characteristics.The no-noise case has the highest DSR among the compared SNR or T values.Source: Figure 11 · p. 22, §VII · p. 22
  • At γr = 2 and SNR = 1, Zipf caching with exponent γr/3 does not match the numerical optimum, while Benford caching performs similarly to that Zipf strategy..Source: Figure 13 · p. 22, §VII · p. 23
  • At γr = 0.5 and SNR = 10, the upper and lower bounds are closer, and Zipf caching with exponent γr/3 gives a density close to the numerical solution and lower bound.Benford caching performs worse than Zipf caching and even worse than the lower bound in this comparison.Source: Figure 12 · p. 22, §VII · pp. 22–23
  • At γr = 2 and SNR = 10, Zipf and Benford caching both perform similarly to the numerical solution.Source: Figure 14 · p. 22, §VII · p. 23

VIII. CONCLUSIONS

  • For sequential transmission, the optimal caching pmf can be modeled by a Zipf law whose exponent γc is related to γr through an expression involving the path-loss exponent.Source: §VIII · p. 24
  • For more general demand profiles under Rayleigh, Ricean, and Nakagami fading, the results suggest flattening the request distribution to optimize caching performance.Source: §VIII · p. 24
  • The analysis leaves optimal caching with transmit diversity, intelligent scheduling, changing file popularities, and interference from simultaneous transmissions for future work.Diversity may reduce outage, but practical use is a critical issue because physical-layer synchronization may be required; assuming full synchronization provides an upper bound.Source: §VII · p. 23, §VIII · p. 24

APPENDIX

The proofs derive parameter conditions for the DSR optimization and use coverage bounds and optimality conditions to obtain the optimum.

A. Proof of Lemma 1

B. Proof of Lemma 2

  • For α = 4, the proof rewrites the coverage expression using a change of variables and the Gaussian tail probability.Source: B. Proof of Lemma 2 · pp. 24–25
  • The proof of Lemma 2 uses coverage bounds and the optimality condition to obtain the maximum DSR and the optimal value of γ1.Source: B. Proof of Lemma 2 · p. 25

C. Proof of Lemma 3

These proof excerpts use concavity and derivative conditions for optimization, then apply a small-noise approximation and Lagrange multipliers to the caching problem.

C. Proof of Lemma 3

  • The proof of Lemma 3 identifies a concave objective in γ1 and uses its derivative to obtain a polynomial equation.Source: C. Proof of Lemma 3 · p. 25

D. Proof of Lemma 4

  • For Lemma 4, the proof uses exp(−x) = 1 − x + o(x) as x → 0, then applies the resulting optimality condition to coverage as σ2 → 0.Source: D. Proof of Lemma 4 · p. 26

E. Proof of Lemma 5

  • The proof of Lemma 5 sets up a Lagrangian for the caching probabilities and multiplier, with its multiplier derivative recovering the constraint.Source: E. Proof of Lemma 5 · p. 26

F. Proof of Lemma 6

The proof approximates coverage as separable in caching intensity and threshold, drawing on the near-linear dependence of β(T, α)^(α/2) on T.

F. Proof of Lemma 6

  • The proof investigates the general solution using Lagrange multipliers and obtains M equations by differentiating with respect to the M caching probabilities.Source: F. Proof of Lemma 6 · p. 27
  • The near-linearity is used to motivate the separable approximation, with β(T, α)^(α/2)/T treated as invariant to T.Source: F. Proof of Lemma 6 · pp. 27–28
  • The analysis treats coverage as separable, pcov(T, λtpc(j), α) = f(λtpc(j), α)g(T), based on the observed near-linear dependence of β(T, α)^(α/2) on T.The observed slope is mainly determined by α and changes only slightly with μ.Source: F. Proof of Lemma 6 · p. 28

G. Proof of Lemma 7

The proofs derive caching solutions for maximizing DSR, including a Zipf caching exponent related to the request exponent and path-loss exponent. For least-popular-file optimization, the solution has a threshold structure in the reception fractions.

G. Proof of Lemma 7

  • The Lemma 7 proof uses Lagrange multipliers to derive differences between file caching probabilities as functions of network parameters, γ_r, and file indices.Source: G. Proof of Lemma 7 · p. 28

H. Proof of Lemma 8

I. Proof of Lemma 9

The all-files DSR optimization uses water-filling when the transmission fraction is below one. A uniform-fraction formulation provides an upper bound on that optimization.

I. Proof of Lemma 9

  • For ξ = 1, the optimal solution is ξ_i = p_c(i), giving ρ_i = 1 for every file.Source: §I · p. 29
  • When ξ < 1, the all-files maximum-DSR solution uses water-filling, with ρ_i ordered by p_r(i)p_c(i).Source: §I · p. 29
  • The uniform choice ρ_i = ξ for every file is used to upper-bound the all-files DSR formulation.Source: §I · p. 30

J. Proof of Theorem 2

The proof derives coverage probability for a typical randomly located user using interference Laplace transforms. It also gives a Rayleigh-fading result for path-loss exponent 4, with coverage depending on receiver state through its active-user density.

J. Proof of Theorem 2

  • The interference Laplace-transform derivation uses the PPP probability generating functional and independent fading gains.Source: J. Proof of Theorem 2 · p. 30
  • The typical-user coverage probability is derived from the interference Laplace transforms and the stated changes of variables.Source: J. Proof of Theorem 2 · p. 31

K. Proof of Corollary 3

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