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Spatiotemporal Stochastic Modeling of IoT Enabled Cellular Networks: Scalability and Stability Analysis

Mohammad Gharbieh, Hesham ElSawy, Ahmed Bader, Mohamed-Slim Alouini

arXiv:1609.05384v3cs.ITmath.IT

TL;DR

Massive, spatially distributed IoT devices create uplink scalability challenges because their queue-dependent transmissions interfere. The paper combines stochastic geometry and queueing models to assess stability and compare transmission strategies. Its results show stability regions contract with higher device intensity and SINR threshold, while strategy performance depends on operating conditions.

  • Problem

    IoT uplink models need to capture per-device traffic, spatially distributed devices, and mutual interference, which standalone queueing or stochastic-geometry models do not jointly represent.

    Source: §I · p. 2

  • Method

    The paper combines stochastic geometry with a two-dimensional Geo/PH/1 DTMC to model spatially interacting IoT queues and their transmission-protocol states.

    Source: §I · p. 3, A. Methodology, Contribution, and… · p. 3

  • Results

    Stability regions shrink as device intensity and SINR threshold rise; power ramping extends stability at high SINR, baseline performs better at low SINR, and backoff matches baseline for stable operation.

    Source: §V · pp. 26–27

  • Takeaways & Limitations

    Maintaining stability as IoT intensity rises calls for lower SINR thresholds or traffic, or network densification or more channels; the more effective transmission strategy varies with SINR and intensity.

    Source: §V · p. 27

    The analysis ignores spatial correlation among interfering devices’ transmit powers for tractability, an approximation the paper reports as having weak, negligible effect on success probability.

    Source: B. Stochastic Geometry Analysis for… · p. 11

Abstract

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The Internet of Things (IoT) is large-scale by nature, which is manifested by the massive number of connected devices as well as their vast spatial existence. Cellular networks, which provide ubiquitous, reliable, and efficient wireless access, are natural candidates to provide the first-mile access for the data tsunami to be generated by the IoT. However, cellular networks may have scalability problems to provide uplink connectivity to massive numbers of connected things. To characterize the scalability of cellular uplink in the context of IoT networks, this paper develops a traffic-aware spatiotemporal mathematical model for IoT devices supported by cellular uplink connectivity. The developed model is based on stochastic geometry and queueing theory to account for the traffic requirement per IoT device, the different transmission strategies, and the mutual interference between the IoT devices. To this end, the developed model is utilized to characterize the extent to which cellular networks can accommodate IoT traffic as well as to assess and compare three different transmission strategies that incorporate a combination of transmission persistency, backoff, and power-ramping. The analysis and the results clearly illustrate the scalability problem imposed by IoT on cellular network and offer insights into effective scenarios for each transmission strategy.

I. INTRODUCTION

IoT uplink scalability is challenged by spatially widespread, sporadic devices whose queue-dependent transmissions interfere. The paper combines spatial and queueing models to characterize stability and compare access strategies.

I. INTRODUCTION

  • Cellular uplink must support massive IoT traffic, but prior queueing models –[11] simplify interactions as collisions while stochastic-geometry models – assume saturated traffic.The paper motivates a joint model that accounts for per-device traffic, spatial distribution, and mutual interference.Source: §I · p. 2

A. Methodology, Contribution, and Organization

  • A two-dimensional Geo/PH/1 DTMC integrated with stochastic geometry models interference-coupled queues and protocol states, supporting stability conditions and a stability-region frontier for maximum spatial traffic intensity.The framework uses the frontier to identify parameters under which packets can be served in finite time.Source: A. Methodology, Contribution, and… · pp. 3–4

A. Spatial & Physical Layer Parameters

  • Base stations and IoT devices are modeled as independent homogeneous PPPs, with nearest-base-station association, power-law path loss, Rayleigh fading, and full path-loss inversion.The model assumes dense base stations so devices can invert path loss toward the nearest base station without a binding maximum transmit-power constraint.Source: A. Spatial & Physical Layer Parameters · pp. 4–5

B. MAC Layer Parameters

  • Devices generate geometric arrivals into infinite FCFS buffers and transmit without base-station scheduling; baseline persistence, power ramping, and deterministic/probabilistic backoff define the compared protocols.Power ramping raises the power-control threshold after failures, while backoff defers attempts; the study compares success probability, delay, and queue size.Source: A. Methodology, Contribution, and… · p. 4, B. MAC Layer Parameters · pp. 5–6

III. OUTLINE OF THE ANALYTICAL FRAMEWORK

The analytical framework combines matrix-analytic queueing methods with stochastic geometry to model service and interference. Its queueing analysis represents packet service with phase-type transitions, derives stability conditions, and treats unstable queues through their protocol-phase distribution.

III. OUTLINE OF THE ANALYTICAL FRAMEWORK

  • The framework combines the matrix-analytic method for each Geo/PH/1 queue with stochastic geometry for interference-based interactions among queues.The analysis accounts for packet traffic and transmission phases alongside spatial interference.Source: §III · p. 7

A. Queueing Analysis

  • The PH-type service model uses an absorbing Markov chain: S captures transitions among transient phases, while s gives successful-transmission absorption probabilities.Restarting protocol phases after successful transmissions sets the initialization vector β to {1, 0}.Source: A. Queueing Analysis · pp. 7–9
  • The QBD transition blocks A0, A1, and A2 encode queue-level increases, no change, and decreases, respectively, while B, C, and E encode boundary transitions.The respective level changes correspond to an arrival during service, a completion paired with an arrival or continued service, and a completion without an arrival.Source: A. Queueing Analysis · p. 8
  • The queue is stable if and only if its stationary departure rate exceeds its stationary arrival rate, expressed as πA2e > πA0e.For stable systems, the steady-state distribution can then be obtained; when the condition fails, the queue becomes unstable.Source: A. Queueing Analysis · pp. 8–9
  • For unstable queues, the analysis abstracts away queue size and solves for the marginal distribution Π over protocol phases in a finite ergodic chain.The phase distribution is obtained from ΠP = Π with Πe = 1.Source: A. Queueing Analysis · p. 9, Figure 3 · p. 9

B. Stochastic Geometry Analysis for Service Probability

The service-probability analysis links transmission success to SINR and aggregate interference, while approximations address spatial and temporal correlations. Because queue states determine interfering activity and success probabilities determine service, an iterative solution resolves their dependence.

B. Stochastic Geometry Analysis for Service Probability

  • Success probabilities are determined by whether SINR exceeds threshold θ, with interference modeled through the active-device point processes and their channel, fading, and power-control effects.For power ramping, the success probability depends on the power-control phase and its corresponding interfering-device process.Source: B. Stochastic Geometry Analysis for… · pp. 9–10
  • The model ignores spatial correlation among interfering-device transmit powers; the authors report that this correlation has a negligible effect on success probability in their model.The passage describes this approximation as verified by numerical results and simulations.Source: B. Stochastic Geometry Analysis for… · p. 11
  • Approximation 2 is stated to be accurate only for full channel-inversion power control, with accuracy not assured for constant-power uplink or downlink transmissions.This bounds the stated scope of the temporal-correlation approximation.Source: B. Stochastic Geometry Analysis for… · p. 11
  • Under full channel inversion, temporal interference correlation is treated as negligible, and Fig. 4 compares conditional with marginal success probabilities.The text reports negligible correlation effects even in an aggressive case with all devices active, two pseudo codes, and 50% common interferers across slots.Source: B. Stochastic Geometry Analysis for… · pp. 11–12, Figure 4 · p. 11

C. Causality problem and the iterative solution

D. Performance Metrics

The framework derives steady-state transmission success and delay measures from the queue and protocol-phase distributions. For stable systems, it also provides average queue size and waiting time, and combines waiting with transmission delay.

D. Performance Metrics

  • Steady-state transmission success probability is computed using the phase-dependent SINR and the protocol-phase distribution Π.The text defines the idle-state probability and the marginal distribution over transmission phases as components of the steady-state solution.Source: D. Performance Metrics · p. 13
  • Expected transmission delay is estimated using an absorbing Markov chain for the protocol phases a packet encounters until successful transmission.The delay measure accounts for transmission phases, distinct from queueing delay.Source: D. Performance Metrics · p. 13
  • For stable systems, the framework gives average queue size and average queueing delay, while average queueing-plus-transmission delay is E[Wq] + D.Queueing delay counts time slots spent waiting before service starts.Source: D. Performance Metrics · pp. 13–14

IV. PERFORMANCE ANALYSIS

The performance-analysis section is organized by transmission scheme, examining stability conditions before applying the analytical framework to each scheme.

IV. PERFORMANCE ANALYSIS

  • For each transmission scheme, the analysis first considers necessary and sufficient stability conditions, then applies the framework described earlier.Source: §IV · p. 14

A. Baseline Scheme

The baseline scheme models a device with a single transmission phase and uses its success probability in a Geo/Geo/1 queue. Stability bounds identify supported device intensity, while queue metrics require the steady state to be recomputed when arrival rates change.

A. Baseline Scheme

  • A single transmission phase reduces the baseline Geo/PH/1 model to a Geo/Geo/1 queue governed by success probability p.The phase is represented by the transient state S = p̄.Source: A. Baseline Scheme · p. 14
  • Stochastic geometry characterizes the baseline transmission success probability, with the η = 4 case simplifying its expression to an arctan function.The general expression uses a Gaussian hypergeometric function and relies on an approximate Voronoi-cell-area PDF.Source: A. Baseline Scheme · p. 15
  • The activity factor x◦ and success probability p are interdependent, so an iterative solution is needed to characterize the baseline network.The paper gives necessary and sufficient stability conditions before applying the iterative solution.Source: A. Baseline Scheme · pp. 15–16
  • The stability bounds expose the tradeoff between IoT spatial intensity and traffic requirements, identifying intensities that guarantee stability or exceed the network’s supported limit.The sufficient condition guarantees stable operation; the upper bound identifies the maximum supported device intensity for given traffic requirements and SINR threshold.Source: A. Baseline Scheme · pp. 15–16
  • When arrival rate changes, equation (19) cannot be applied directly; first recalculate x◦ and p, then use it for a stable network.The average queue length and queueing delay are obtained from equation (19) for stable networks.Source: A. Baseline Scheme · pp. 16–17

B. Power-Ramping Scheme

The power-ramping scheme models transmission phases in which devices raise their power-control threshold after failures. Its phase-specific success probabilities and queue-state probabilities are interdependent, and the paper characterizes them iteratively alongside stability and queue-performance measures.

B. Power-Ramping Scheme

  • Power-ramping devices increase their power-control threshold after a transmission failure, making success probability depend on the transmission phase.The scheme uses a Geo/PH/1 queueing model for its multiple phases.Source: B. Power-Ramping Scheme · p. 17
  • The marginal phase distribution Π is required to calculate the phase-specific transmission success probabilities p_m.The success probability is characterized for a device operating at threshold ρ_m.Source: B. Power-Ramping Scheme · p. 18
  • The interdependence among Π, the transient matrix S, and phase-specific success probabilities requires an iterative solution for the steady state.Algorithm 3 obtains the marginal steady-state probability vector Π for the power-ramping scheme.Source: B. Power-Ramping Scheme · pp. 18–19
  • The scheme provides necessary and sufficient network-stability conditions before the iterative solution is used to find phase probabilities.The conditions are evaluated under traffic and interference bounds described for the power-ramping system.Source: B. Power-Ramping Scheme · p. 18
  • For a stable power-ramping network, the steady-state distribution supports calculations of transmission success, retransmissions, average queue length, and average delay.The delay calculation uses the transient matrix S and the steady-state quantities obtained through Algorithm 3.Source: B. Power-Ramping Scheme · pp. 19–20

C. Backoff Scheme

In the backoff scheme, devices defer transmission after failures using deterministic and probabilistic backoff states. The design parameters N and q trade transmission success against transmission activity, and are selected with queue stability and waiting time in view.

C. Backoff Scheme

  • Backoff devices defer transmission after failures, using deterministic and probabilistic backoff states.The scheme’s steady-state probabilities are computed with the backoff-specific transient matrix.Source: C. Backoff Scheme · pp. 20–21
  • For a generic IoT device using the backoff transmission scheme, marginal steady-state distribution Π is obtained via Algorithm 4 and is required to obtain transmission success probability.Π1 is the probability of being in transmission state; Πn for n > 1 is the probability of being in the backoff phase.Source: C. Backoff Scheme · pp. 20–21
  • Larger N or smaller q gives more conservative spectrum access and higher transmission success probability, while reducing the probability of being in transmission.N sets deterministic backoff-slot size, and q parameterizes probabilistic backoff.Source: C. Backoff Scheme · p. 22
  • The design selects N and q to minimize queue waiting time over combinations that yield stable performance.If no combination yields a stable network, the objective switches to minimizing the average number of retransmissions.Source: C. Backoff Scheme · p. 22

V. NUMERICAL RESULTS & SIMULATIONS

The results show that scalability depends on device intensity, SINR threshold, and transmission strategy: stability regions shrink as intensity and threshold rise, while scheme performance varies by load and operating regime. Power-ramping helps in some low-to-moderate-load or high-threshold settings, whereas backoff is useful under congestion.

V. NUMERICAL RESULTS & SIMULATIONS

  • Transmission success probability curves closely match between analysis and simulation, and their discontinuities distinguish stable from unstable operation.Increasing the SINR threshold reduces success probability and can increase interference by slowing queue flushing.Source: Figure 5 · p. 23, §V · p. 23
  • At low and moderate device densities, power-ramping lowers average queue length and queueing delay; at high intensity it provides similar stability with a higher average queue length.The authors attribute the low-to-moderate-density result to quicker buffer flushing before instability.Source: Figure 6 · p. 23, Figure 7 · p. 24, §V · pp. 24–25
  • Power-ramping performs better for low and moderate device intensities in service delay, while backoff performs best at high intensity when the network is unstable.For stable networks at higher traffic intensities, baseline and backoff perform better than power-ramping.Source: Figure 8 · p. 24, §V · p. 25
  • Power-ramping becomes unstable at x0 = 0.71 for α̃ = 4 and x0 = 0.45 for α̃ = 8, with the larger discontinuity at α̃ = 4.The paper relates the discontinuity to the transition from stable operation, where x0 > 0, to unstable operation, where x0 = 0.Source: §V · p. 24
  • For stable operation, baseline and backoff perform equivalently because the optimal backoff parameters reduce the backoff scheme to baseline; under unstable congestion, backoff outperforms both alternatives.The reported optimal parameters are N = 0 and q = 1 in stable operation.Source: §V · pp. 25–26, Table I · p. 26
  • Stability regions shrink as device intensity and SINR threshold rise; power-ramping extends stability at high SINR, while the baseline is stronger at low SINR.At moderate SINR, the schemes perform approximately similarly, with a slight power-ramping advantage.Source: Figure 9 · p. 26, §V · pp. 26–27
  • Stable operation cannot be maintained as IoT device intensity scales up; lower SINR thresholds or traffic, denser networks, and more channels are reported as ways to retain stability.At high device intensity, backoff is necessary under unstable operation; it relaxes interference and increases transmission success probability.Source: §V · p. 27

VI. CONCLUSIONS

The paper combines stochastic geometry and queueing theory to model IoT cellular uplinks and compare persistency, backoff, and power-ramping. Its results characterize stability limits and show that effective transmission strategies depend on SINR threshold, device intensity, and whether the network is stable.

VI. CONCLUSIONS

  • The model represents base stations and IoT devices as independent Poisson point processes and uses a two-dimensional DTMC to track queue and transmission-protocol states.Transmission-success probabilities from stochastic geometry populate the DTMC, whose steady state feeds back into the spatial analysis through an iterative solution.Source: §VI · pp. 27–28
  • The results map stability boundaries and compare three strategies: power-ramping is effective at high SINR, persistent fixed-power transmission at low SINR, and deferring transmission is unnecessary when the network is stable.The paper also reports different measures for stable and unstable networks, including queue size, packet delay, and service time for stable networks, and service time for unstable networks.Source: §VI · pp. 27–28

APPENDIX

The appendix derives expressions for joint transmission failure and aggregate interference, using spatial point-process modeling and stated approximations.

A. Proof of Proposition 1

  • The proof derives a joint transmission-failure probability across two time slots, accounting for shared interfering devices and intra-cell and inter-cell interference.The derivation describes common and non-common interferers and averages over the numbers of devices in the test cell.Source: A. Proof of Proposition 1 · pp. 28–29

B. Proof of Lemma 1

  • The interference derivation uses nearest-base-station association and power control, then approximates inter-cell interference by a PPP with independent transmit powers while ignoring power correlations.It conditions intra-cell interference on the number of neighbors and uses an approximate PPP Voronoi-cell-area distribution.Source: B. Proof of Lemma 1 · pp. 29–30

C. Proof of Lemma 4

Lemma 4 obtains (23) using a procedure similar to the proof of Lemma 1; the approximation ignores correlations between device transmission powers in the same and adjacent Voronoi cells.

C. Proof of Lemma 4

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