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On the Accuracy of the Wyner Model in Cellular Networks

Jiaming Xu, Jun Zhang, Jeffery G. Andrews

arXiv:1009.5900v1cs.ITmath.IT

TL;DR

The Wyner model is widely used for its simplicity, but its validity as an approximation has not been seriously established. The paper examines accuracy under random user locations and concludes that it improves as the metric or system provides more averaging.

  • Problem

    The paper identifies a lack of serious validation of the Wyner model and asks when its simplifications provide a reasonable approximation.

    Source: §I · p. 1, A. Backgound and Related Work · p. 2

  • Method

    The paper analyzes cellular throughput under random user locations, including intracell TDMA and CDMA.

    Source: B. Average Throughput · p. 8

  • Results

    The more averaging there is, for example through spread spectrum, diversity, or mean-based metrics, the more accurate the Wyner model is.

    Source: §VIII · p. 18

  • Takeaways & Limitations

    The paper’s general conclusion is conditional: Wyner-model accuracy increases with averaging rather than holding uniformly across metrics.

    Source: §VIII · p. 18

Abstract

from arXiv · show

The Wyner model has been widely used to model and analyze cellular networks due to its simplicity and analytical tractability. Its key aspects include fixed user locations and the deterministic and homogeneous interference intensity. While clearly a significant simplification of a real cellular system, which has random user locations and interference levels that vary by several orders of magnitude over a cell, a common presumption by theorists is that the Wyner model nevertheless captures the essential aspects of cellular interactions. But is this true? To answer this question, we consider both uplink and downlink transmissions, and both outage-based and average-based metrics. For the uplink, for both metrics, we conclude that the Wyner model is in fact quite accurate for systems with a sufficient number of simultaneous users, e.g. CDMA. Conversely, it is broadly inaccurate otherwise. With multicell processing, intracell TDMA is shown to be suboptimal in terms of average throughput, in sharp contrast to predictions using the Wyner model. Turning to the downlink, the Wyner model is highly inaccurate for outage since it depends largely on the user locations. However, for average or sum throughput, the Wyner model serves as an acceptable simplification in certain special cases if the interference parameter is set appropriately.

I. INTRODUCTION

The paper asks when the simplified Wyner model remains a reasonable approximation of cellular networks with random user locations. Its findings make accuracy conditional on the metric, access scheme, processing, and degree of averaging.

I. INTRODUCTION

  • The linear Wyner model simplifies cellular networks by considering only two adjacent interferers, fixing user locations, and representing each neighbor’s interference with one constant α.The paper contrasts these assumptions with real networks, where users are randomly located and interference varies across a cell.Source: §I · pp. 1–2

A. Backgound and Related Work

  • Prior work used the model for capacity and throughput analyses, including multicell processing, fading, MIMO, and precoding, but did not establish when it approximates more realistic scenarios.The paper notes that grid-based multicell models are analytically complex, helping explain the appeal of the simpler Wyner model.Source: A. Backgound and Related Work · p. 2
  • Despite extensive use of the Wyner model, the paper identifies a lack of serious validation and asks how much its simplifications cost and when α is appropriate.It offers analysis and simulations as a mathematics-based starting point; “accurate” means capturing major trends within a small constant, while “inaccurate” means key trends may be missed and errors may be large.Source: A. Backgound and Related Work · p. 2

B. Contributions

  • With single-cell processing, the Wyner model is broadly inaccurate for few simultaneous users, but a tuned α captures key outage and throughput trends when users are numerous.The paper identifies TDMA and other orthogonal access protocols as cases that evade the model, while CDMA and other non-orthogonal uplink protocols fit its averaging assumptions.Source: B. Contributions · p. 3
  • With multicell processing, CDMA outperforms intracell TDMA, contradicting Wyner’s original prediction that TDMA is optimal and capacity-achieving.For equal power allocation, a properly selected α also makes the model quite accurate for CDMA sum throughput.Source: B. Contributions · p. 3
  • For downlink outage, the model misses significant within-cell variations; for channel-inverted average throughput, tuned α works reasonably for CDMA but not TDMA.With equal transmit power per user, the model is much less accurate for both TDMA and CDMA.Source: B. Contributions · p. 3
  • The paper’s unifying conclusion is that more averaging, such as spread spectrum, diversity, or mean-based metrics, corresponds to greater Wyner-model accuracy.The model can handle average or sum throughput in certain scenarios, but generally not location-dependent outage; CDMA uplink is an exception.Source: B. Contributions · p. 3

II. SYSTEM MODEL AND DEFINITION

The system model contrasts a deterministic Wyner-style interference assumption with randomly located users whose interference depends on their positions. It specifies a one-dimensional cellular network and path-loss-based channels, omitting fading and shadowing.

A. The System Model

  • The model places N cells along a line, each covering a segment of length 2R, with base stations at cell centers and K active users per cell.Intercell interference comes from two neighboring cells; users are independently and uniformly distributed within each cell.Source: A. The System Model · p. 4
  • Unlike the Wyner model’s fixed user locations and deterministic, homogeneous interference parameter α, this model makes intercell interference depend on random user locations.The channel gain from a base station to its home user is normalized to 1.Source: A. The System Model · p. 4
  • Channel strength is determined by path loss, with received power decaying as r^-β for β ≥ 2; fading and lognormal shadowing are neglected.In the uplink, users are assumed to use perfect channel inversion so each signal reaches its home base station with received power P.Source: A. The System Model · p. 4

B. Terminology

The section defines the cellular access schemes, processing settings, and performance metrics, then finds that the Wyner model captures uplink CDMA outage well when enough users are active but not intracell TDMA outage.

B. Terminology

  • The section compares intracell TDMA, CDMA, and OFDMA, and defines outage probability, average throughput, and sum throughput as its performance metrics.TDMA allows one user per cell to transmit at a time; CDMA allows simultaneous transmissions across the band; OFDMA assigns users distinct subcarriers.Source: B. Terminology · pp. 5–6
  • OFDMA lets users transmit simultaneously while assigning each user a distinct subset of subcarriers.Source: B. Terminology · p. 5
  • Single-cell processing treats intercell interference as Gaussian noise on uplink, while multicell processing jointly decodes received signals; downlink transmission is likewise restricted or shared by processing mode.Under SCP, base stations transmit only information for home users; under MCP, each base-station signal contains information for all users.Source: B. Terminology · pp. 5–6
  • Outage probability is the chance that received SIR falls below a threshold; average throughput averages user throughput over possible locations, while sum throughput is the maximum total user rate..Source: B. Terminology · p. 6

III. UPLINK WITH SCP

  • The uplink SCP analysis compares intracell TDMA and CDMA using outage probability and average throughput.For the outage analysis, it assumes an interference-limited case and ignores Gaussian noise.Source: §III · p. 6, A. Outage Probability · p. 6

A. Outage Probability

  • For intracell TDMA, random user locations make intercell interference and SIR random, so the Wyner model is inaccurate for outage probability.The Wyner model instead yields no notion of outage in the described 1-D setting.Source: A. Outage Probability · p. 7
  • At sufficiently large K, the uplink SIR becomes deterministic and homogeneous, matching the Wyner model when α equals average intercell interference intensity.The stated simulation finds the Gaussian approximation accurate and outage probability rises sharply from 0 to 1 when K ≥5.Source: A. Outage Probability · p. 8
  • With more than 5 active users per cell, the Wyner model characterizes CDMA outage probability very accurately.The analysis also states that sufficiently large K makes the SIR deterministic and homogeneous when α is set to the average intercell interference intensity.Source: A. Outage Probability · p. 8

B. Average Throughput

For uplink single-cell processing, random user locations improve average throughput, and intracell TDMA outperforms CDMA; the Wyner model misses these findings for TDMA but approximates CDMA at higher K.

B. Average Throughput

  • Under the Wyner model, uplink TDMA and CDMA average throughput are the same when CDMA uses G = K and optimal multiuser detection is assumed.The comparison assumes perfect synchronization; intracell interference is suppressed effectively while intercell interference remains and is treated as Gaussian noise.Source: B. Average Throughput · p. 8
  • The Wyner model is quite accurate for CDMA average throughput when K > 20, but inaccurate for TDMA.Source: B. Average Throughput · p. 9
  • Across the reported K and β ranges, intracell TDMA outperforms CDMA, and random user locations increase throughput, which the Wyner model fails to capture.For β = 4, throughput increases by 100%.Source: B. Average Throughput · p. 9

IV. UPLINK WITH MCP

With multicell processing and random user locations, CDMA has higher average throughput than intracell TDMA, contrary to the Wyner-model prediction that TDMA is optimal.

IV. UPLINK WITH MCP

  • The section compares average throughput for intracell TDMA and CDMA with multicell processing.The throughput expressions average over i.i.d. random user locations.Source: §IV · p. 9
  • The average throughput of CDMA exceeds that of TDMA, with equality only when user locations are fixed and their distances from home base stations are the same.Source: §IV · p. 10
  • The Wyner model predicts intracell TDMA is optimal and achieves capacity, but random user locations make TDMA suboptimal and render that conclusion inaccurate.Source: §IV · p. 10

V. DOWNLINK WITH SCP

For downlink single-cell processing, the Wyner model misses location-dependent outage behavior, while appropriately tuning its interference parameter can provide a lower bound on average throughput.

V. DOWNLINK WITH SCP

  • The analysis considers downlink outage and average throughput for TDMA and CDMA under perfect channel inversion and equal transmit power per user.Source: §V · p. 10

A. Perfect Channel Inversion (PCI)

  • Under the 1-D Wyner downlink model, SIR is 1 and there is no notion of outage, unlike the random-location setting.In the random-location model, outage probability can be 0 or 1 depending on both the SIR threshold and user locations.Source: A. Perfect Channel Inversion (PCI) · pp. 11–12
  • Downlink outage probability depends on user location, a key fact the Wyner model fails to capture.With random locations, SIR is random and outage changes with the SIR threshold; the lower bound is tight in Fig. 6.Source: A. Perfect Channel Inversion (PCI) · p. 11
  • With random user locations, appropriately choosing α in the Wyner model provides a lower bound on average throughput.This result is presented for the average throughput comparison under PCI.Source: A. Perfect Channel Inversion (PCI) · p. 13

B. Equal Transmit Power Per User (ETP)

With equal transmit power per user, the Wyner model can characterize average throughput in CDMA with perfect channel inversion when α is tuned, but is inaccurate in other cases, especially ETP. Under ETP, a user's SIR depends on that user's location, not neighboring users' locations, so the analysis focuses on average throughput rather than user-specific outage.

B. Equal Transmit Power Per User (ETP)

  • Under ETP, a user's SIR depends on that user's location and not on neighboring users' locations, so there is no user-specific outage and the focus is average throughput.The section assumes equal transmit power per user and continuously transmitting base stations.Source: B. Equal Transmit Power Per User (ETP) · p. 13
  • The Wyner model can provide a lower bound to average throughput when its interference parameter α is tuned.The passage states that this is established by a theorem and refers to the Wyner model expression in (7).Source: B. Equal Transmit Power Per User (ETP) · p. 13
  • With perfect channel inversion, tuning α lets the Wyner model reasonably characterize average throughput in CDMA, but it is inaccurate in other cases, especially ETP.Perfect inversion partly counters random user distributions, and interference averaging makes intercell interference independent of specific neighboring-user locations.Source: B. Equal Transmit Power Per User (ETP) · p. 13

VI. DOWNLINK WITH MCP

With multicell processing, equal-power CDMA and intracell TDMA differ under random user locations: CDMA throughput matches the Wyner model only under restrictive conditions, while TDMA is suboptimal relative to CDMA. In TDMA, even as the user count grows, the random channel matrix does not converge to a deterministic limit that would permit tuning α to match the lower bound.

VI. DOWNLINK WITH MCP

  • For CDMA, the Wyner model matches random-location throughput under equal power allocation, or when channel-gain dependencies are ignored and intercell-interference variance is set to zero.The channel gains are generally location-dependent and highly correlated, interference variance is not zero, and unequal power allocation is often used.Source: §VI · pp. 15–16
  • With an appropriately chosen α, the Wyner model characterizes the lower bound on random-location CDMA sum throughput quite accurately in simulations.The comparison is between the Wyner-model per-cell sum throughput and a lower bound for random user locations.Source: §VI · p. 16
  • With equal power allocation, CDMA has larger per-cell sum throughput than intracell TDMA; equality holds when user locations are fixed and home-base-station distances match.The paper contrasts this with the Wyner-model result that originally makes intracell TDMA capacity-achieving.Source: §VI · p. 16
  • In intracell TDMA, the random channel matrix does not converge to a deterministic limit even as K grows, preventing α from being tuned to match the sum-throughput lower bound.The paper identifies minimax uplink-downlink duality as the key tool in its sum-throughput analysis.Source: §VI · pp. 14, 16

VII. EXTENSION TO OFDMA

The accuracy of the Wyner model for OFDMA depends on how many users contribute interference over a codeword: with many users it resembles CDMA, while systems with few contributors resemble TDMA. The paper notes that OFDMA implementations can fall between these cases, depending on system design.

VII. EXTENSION TO OFDMA

  • OFDMA lies between TDMA and CDMA, with the specific system design determining which it more closely resembles.The number of users M contributing interference over a codeword depends on resource allocation, frequency hopping, and ARQ design.Source: §VII · pp. 16–17
  • OFDMA users receive distinct time-frequency resource blocks, eliminating intracell interference, while universal frequency reuse leaves intercell interference.A codeword's interference may come from transmissions by M users as resource-block subcarriers change across OFDM symbols.Source: §VII · p. 17
  • When M is much greater than 1, OFDMA resembles CDMA and the Wyner model accurately characterizes the SIR distribution; simulations suggest uplink M may need to exceed 5.As M tends to infinity, the SIR tends to 1, matching the Wyner model.Source: §VII · p. 17
  • In most LTE implementations, M is expected to be closer to 1 than to K, so the TDMA-model conclusions are expected to hold quite accurately.The passage attributes this to fixed resource blocks and protocols designed to allow orthogonality among users within a cell and potentially between neighboring cells.Source: §VII · p. 17

VIII. CONCLUSION AND FUTURE WORK

The paper compares a one-dimensional Wyner model with a one-dimensional grid model with random user locations. Its results suggest that the model is more accurate with more averaging, while the study leaves random channel effects and many system variations unexamined.

VIII. CONCLUSION AND FUTURE WORK

  • The study does not consider random channel effects such as fading or shadowing, or many other power-control, multiple-access, and multicell-processing permutations.The authors suggest that the two-dimensional Wyner model would presumably be even less accurate against two-dimensional grid or more randomly spatial counterparts because of additional randomness.Source: §VIII · p. 18
  • The authors suggest that more averaging, such as from spread spectrum, diversity, or mean-based metrics, corresponds to greater Wyner-model accuracy.The study compares a one-dimensional model with random user locations and summarizes its main results in Table I.Source: §VIII · pp. 17–18

APPENDIX

The appendix supports its results with an approximation argument and proof steps using Jensen’s inequality, covariance decomposition, and the law of large numbers.

APPENDIX

  • The approximation argues that the numerator contributes more than the denominator, then uses the uniform distribution of L_n^(k).The passage presents this reasoning in the appendix’s approximation of μ and σ².Source: APPENDIX · p. 18

B. Proof of Theorem 1

  • Jensen’s inequality and convexity of log(1 + ρx^-1) support the bound in the proof of Theorem 1.The proof also invokes the definition of α².Source: B. Proof of Theorem 1 · p. 18

C. Proof of Theorem 2

  • For CDMA, covariance matrix Λ_N is rewritten as a sum of K covariance matrices Λ_k; Jensen’s inequality and the concavity of log(det Λ) for covariance matrices then support the following step.Source: C. Proof of Theorem 2 · p. 19

K HHT

  • The proof of Lemma 1 applies the law of large numbers to entries involving 1 + U_n^(k) + V^(k).The lemma’s derivation is extended to the other entries in the displayed matrix.Source: D. Proof of Lemma 1 · p. 19, K HHT · p. 19
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