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A Stochastic Geometry Analysis of Inter-cell Interference Coordination and Intra-cell Diversity

Xinchen Zhang, Martin Haenggi

arXiv:1403.0012v4cs.ITmath.IT

TL;DR

PPP-based analyses have often lacked careful treatment of ICIC and ICD under temporal or spectral interference correlation. This paper derives coverage expressions accounting for that correlation and finds different ICIC and ICD gains across reliability and spectral-efficiency regimes.

Abstract

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Inter-cell interference coordination (ICIC) and intra-cell diversity (ICD) play important roles in improving cellular downlink coverage. Modeling cellular base stations (BSs) as a homogeneous Poisson point process (PPP), this paper provides explicit finite-integral expressions for the coverage probability with ICIC and ICD, taking into account the temporal/spectral correlation of the signal and interference. In addition, we show that in the high-reliability regime, where the user outage probability goes to zero, ICIC and ICD affect the network coverage in drastically different ways: ICD can provide order gain while ICIC only offers linear gain. In the high-spectral efficiency regime where the SIR threshold goes to infinity, the order difference in the coverage probability does not exist, however the linear difference makes ICIC a better scheme than ICD for realistic path loss exponents. Consequently, depending on the SIR requirements, different combinations of ICIC and ICD optimize the coverage probability.

I. INTRODUCTION

The paper addresses limited PPP-based analysis of ICIC and ICD under temporal or spectral interference correlation, presenting a correlated cellular model and comparing the schemes’ coverage effects. It situates its user-centric coordination approach among prior ICIC and diversity analyses.

A. Motivation and Main Contributions

  • PPP-based cellular analyses have often lacked careful treatment of ICIC and ICD, partly because temporal or spectral correlation is difficult to handle.The baseline PPP model predicts coverage below 60% at 0 dB SINR even when noise is neglected.Source: A. Motivation and Main Contributions · p. 1
  • ICIC and ICD are analyzed in a homogeneous PPP model that accounts for temporal or spectral signal and interference correlation.The typical user associates with the BS providing the strongest signal averaged over small-scale fading, but not shadowing; fading is Rayleigh and shadowing may be almost arbitrary.Source: A. Motivation and Main Contributions · p. 1, A. System Model · p. 2

B. ICIC, ICD and Related Works

C. Paper Organization

  • The paper’s roadmap covers separate ICIC and ICD analyses, their combination, and model validation with load trade-offs.The introduction also contrasts diversity under interference correlation in cellular networks with the no-diversity result for ad hoc retransmission in.Source: C. Paper Organization · p. 2, B. ICIC, ICD and Related Works · p. 2

A. System Model

  • BS locations and shadowing remain static across time and frequency, while small-scale fading is iid across resource blocks.Spatial shadowing correlation due to common obstacles is omitted for analytical tractability.Source: A. System Model · p. 2

B. The Path Loss Process with Shadowing (PLPS)

The paper maps the shadowed cellular network into an ordered path loss process, then uses that representation to define coverage and compare the system load induced by coordination and diversity.

B. The Path Loss Process with Shadowing (PLPS)

  • The path loss process with shadowing (PLPS) is an ordered process that captures shadowing and BS spatial distribution and determines BS association.The PLPS is a one-dimensional PPP with intensity measure Λ((0, r]) = λπr^δE[S^δ], where δ = 2/α.Source: B. The Path Loss Process with… · p. 3
  • The shadowing moment constraint is necessary: without it, aggregate received power is unbounded almost surely.The stated constraint is on the δ-th moment of the shadowing variable Sx.Source: B. The Path Loss Process with… · p. 3

C. The Coverage Probability and Effective Load Model

  • Coverage with K-BS coordination and M-RB selection combining means the typical user’s SIR exceeds θ on at least one of the M resource blocks.The model assumes coordinated non-serving BSs are muted, while the activity of BSs outside the coordinating set is represented by iid Bernoulli variables with transmitting probability 1/κ.Source: C. The Coverage Probability and… · p. 3

D. System Load and Comparability

  • The ICIC-ICD system load is proportional to κM, combining the effective ICIC load κ with the M-fold load from selection combining.The baseline comparison treats ICIC and ICD as increasing load by factors κ and M, respectively.Source: D. System Load and Comparability · p. 3

III. INTERCELL INTERFERENCE COORDINATION (ICIC)

This section derives a finite-integral expression for coverage under K-BS coordination and examines how coordination size and effective load change coverage. It also gives an efficient numerical alternative for evaluating part of the expression.

A. Integral Form of Coverage Probability

  • Theorem 1 gives a finite-integral coverage probability expression for the typical user under K-BS coordination.The integral can be evaluated numerically.Source: A. Integral Form of Coverage Probability · p. 5
  • The incomplete-beta representation of C1 is at least 30 times faster in Matlab than evaluating the hypergeometric function.The alternative is proposed because the hypergeometric function can be cumbersome when embedded in an integral.Source: A. Integral Form of Coverage Probability · p. 5
  • Increasing K raises coverage for all θ, but the marginal coordination gain decreases as K grows.The paper attributes the smaller marginal gain to attenuation of interference from distant BSs.Source: A. Integral Form of Coverage Probability · p. 5, Figure 2 · p. 5
  • Larger effective load κ yields greater coordination gain in SIR by muting strong interferers and thinning interferers outside the coordination cluster.The SIR gain comes with increased load and bandwidth loss because fewer BSs actively serve users.Source: A. Integral Form of Coverage Probability · p. 5

B. ICIC in the High-Reliability Regime

The section studies ICIC as the SIR threshold approaches zero, when the typical user is covered almost surely. In this high-reliability regime, coordination changes outage probability linearly, with its usefulness increasing at larger path loss exponents.

B. ICIC in the High-Reliability Regime

  • In the high-reliability regime (θ →0), the typical user is covered almost surely, and the section examines ICIC’s asymptotic behavior.The authors seek a simpler estimate that makes ICIC’s benefit easier to interpret.Source: B. ICIC in the High-Reliability Regime · p. 5
  • For pure ICIC, the number of coordinating BSs affects outage probability only linearly in the high-reliability regime.The coefficient difference between coordination levels indicates ICIC’s usefulness.Source: B. ICIC in the High-Reliability Regime · p. 6
  • ICIC is more useful at larger path loss exponents α; the same trend holds for other κ values.Fig. 3 plots the asymptotic coverage coefficient aK against α for K=1–5, assuming κ=1.Source: B. ICIC in the High-Reliability Regime · p. 6, Figure 3 · p. 6

C. ICIC in High Spectral Efficiency Regime

The section characterizes ICIC at high spectral efficiency and then presents coverage analysis for intra-cell diversity (ICD) without ICIC. At high thresholds, ICIC’s coverage decay depends on path loss exponent rather than coordination parameters, while selection combining improves coverage with diminishing marginal gains as more resource blocks are assigned.

C. ICIC in High Spectral Efficiency Regime

  • At high spectral efficiency, coverage decays faster for smaller α, and its decay is not affected by the choice of K or κ.This regime corresponds to θ →∞, where coverage tends to zero as spectral efficiency increases.Source: C. ICIC in High Spectral Efficiency… · p. 6

IV. INTRA-CELL DIVERSITY (ICD)

  • Selection combining transmits to the user on M resource blocks simultaneously and covers the user when the maximum SIR across them exceeds θ.This ICD analysis is presented as a baseline without ICIC.Source: §IV · p. 6

A. General Coverage Expression

  • Theorem 2 gives the joint success probability over M resource blocks without ICIC, and Corollary 1 gives the resulting selection-combining coverage probability.The coverage expression follows by applying inclusion-exclusion to the joint success probabilities.Source: A. General Coverage Expression · pp. 6–7
  • With M-RB selection combining and no ICIC, coverage increases as more resource blocks are assigned, but the marginal gain diminishes with M.Fig. 4 compares M=1–5 at α=4.Source: A. General Coverage Expression · p. 7, Figure 4 · p. 7

B. ICD in the High-Reliability Regime

For ICD, selection combining can produce a diversity gain in the high-reliability regime, unlike retransmission under the comparison in. Its effect changes both the outage coefficient and exponent, whereas ICIC changes only the coefficient; the asymptotic approximation is most accurate at small thresholds.

B. ICD in the High-Reliability Regime

  • Numerically evaluating the M-th derivative of the reciprocal confluent hypergeometric function may not be straightforward.The authors suggest Faà di Bruno’s formula or Bell polynomials as alternatives.Source: B. ICD in the High-Reliability Regime · p. 7
  • Selection combining provides diversity gain, unlike the retransmission result in, under the paper’s association assumptions.The paper identifies desired-signal/interference correlation as critical to time/spectral diversity.Source: B. ICD in the High-Reliability Regime · p. 7
  • ICIC changes the outage coefficient but leaves its θ →0 order unchanged, whereas ICD changes both the coefficient and exponent.The paper attributes the difference to selection-combining diversity under its strongest-average-signal association, which correlates desired signal and interference; assumes a fixed desired-link distance independent of interferers.Source: B. ICD in the High-Reliability Regime · p. 7
  • Fig. 5’s asymptotic approximation matches the exact outage expression reasonably well at small θ, over a wider range for smaller M.The approximation is aMθ^M; the authors note its potential use for small-coding-rate applications such as spread spectrum/UWB communication and node discovery.Source: B. ICD in the High-Reliability Regime · pp. 7–8, Figure 5 · p. 8

C. ICD in High Spectral Efficiency Regime

The paper extends its coverage analysis to high spectral efficiency and to joint ICIC–ICD operation. In the high-spectral-efficiency limit, ICIC and ICD have no order difference, but ICIC is often more effective for realistic path-loss exponents.

C. ICD in High Spectral Efficiency Regime

A. The General Coverage Expression

  • The general coverage analysis extends a Laplace-transform result to fading variables H ≥ 0 with E[H^δ] < ∞.For integer-shape gamma fading, Lemma 7 supplies a result used to derive Theorem 3.Source: A. The General Coverage Expression · pp. 8–9
  • Theorem 3 gives the joint coverage probability over M resource blocks under K-cell coordination for all M ∈ N and K > 1.Corollary 2 gives the coverage probability for K-BS coordination with M-RB selection combining.Source: A. The General Coverage Expression · p. 9

B. The High-Reliability Regime

The high-reliability analysis finds that pure ICD maximizes coverage, while ICIC’s impact on outage is linear. The section also validates the effective-load model against simulations, including under shadowing.

B. The High-Reliability Regime

  • Proposition 5 gives quantitative evidence that pure ICD maximizes coverage probability in the high-reliability regime.The plotted outage probabilities for K=1–5 and M=1,2 are reported as consistent with Proposition 5.Source: B. The High-Reliability Regime · pp. 9–10

A. The Effective Load Model

  • The effective-load simulation estimates κ from the ratio of base stations to scheduled users, and finds that its estimates are approximated by an affine function of K.The fitted function depends on shadowing variance.Source: A. The Effective Load Model · p. 10
  • With λu=10λ, the deviation from a PPP is small, and at 10 dB shadowing simulated and predicted coverage are almost visually indistinguishable.The latter comparison supports the effective-load model for non-asymptotic ICIC analysis.Source: A. The Effective Load Model · p. 10

B. ICIC-ICD Trade-off

ICIC and ICD perform differently across SIR regimes, while a hybrid scheme can balance their coverage benefits. Accounting for system load changes the preferred combination of coordination and diversity.

B. ICIC-ICD Trade-off

  • The combined schemes’ load is proportional to κM, the ICIC–ICD load factor used to compare their resource costs.ICIC contributes effective load κ, while ICD contributes a factor of M.Source: B. ICIC-ICD Trade-off · p. 10
  • At intermediate θ, a hybrid ICIC–ICD scheme could provide the highest coverage probability, with the crossing point depending on system parameters.Fig. 9 compares three combined schemes with different but similar load factors using analysis and simulation.Source: B. ICIC-ICD Trade-off · pp. 10–11, Figure 9 · p. 11

C. ICIC-ICD-Load Trade-off

  • As θ increases, increasing K is beneficial; without an outage constraint, keeping K and M small is more desirable, especially at small θ.With an outage constraint, a slight increase in M can significantly reduce optimal K, whose coordination cost is usually higher due to signaling overhead.Source: C. ICIC-ICD-Load Trade-off · pp. 11–12

VII. CONCLUSIONS

  • ICD provides diversity gain as outage vanishes, whereas ICIC affects outage linearly; at high spectral efficiency, ICIC is more effective because its linear gain is typically larger.Both schemes generate extra system load.Source: B. ICIC-ICD Trade-off · p. 10, §VII · p. 12
  • The paper validates its effective-load model with simulations and uses the analysis to explore the trade-off between combined ICIC–ICD schemes and system load.The model is described as very accurate in the non-asymptotic regime.Source: §VII · p. 12

APPENDIX A PROOF OF PROPOSITION 1

The proof establishes the K = 1 case using a hypergeometric-function series expansion, then simplifies the relevant limit with repeated applications of L’Hospital’s rule. Combining (22) and (23) completes the proof.

  • The proof establishes (11) for K = 1 using the hypergeometric-function series expansion.Source: Appendix A · p. 12
  • A change of variable y = θx rewrites the integral used in the argument.Source: Appendix A · p. 12
  • Repeated applications of L’Hospital’s rule simplify the right-hand side limit.Source: Appendix A · p. 12
  • Combining (22) and (23) completes the proof.Source: Appendix A · p. 12

APPENDIX B PROOF OF PROPOSITION 3

The proof uses Taylor expansion and derivative identities to obtain Proposition 3’s asymptotic expansion. In particular, the outage probability is O(θ^M) as θ approaches zero, with its leading coefficient determined.

  • Expanding a derivative by the Leibniz rule and reusing Faà di Bruno’s formula yields the proposition’s concise expression.Source: Appendix B · pp. 13–14
  • Lemma 9 is proved by induction, treating the cases n = 1, 0 < An < M, and An = M.Source: Appendix B · pp. 13–14
  • The proof expands 1/C1(x, m) at x = 0, using Faà di Bruno’s formula to obtain its derivatives.Source: Appendix B · p. 14
  • P∩o_1,M = O(θ^M) as θ → 0, and Lemma 9 helps determine the coefficient of θ^M.Source: Appendix B · p. 14
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