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Heterogeneous Cellular Networks with Spatio-Temporal Traffic: Delay Analysis and Scheduling
Yi Zhong, Tony Q. S. Quek, Xiaohu Ge
TL;DR
The paper addresses the limited analysis of multi-point delay in heterogeneous cellular networks with spatio-temporal traffic. It combines stochastic geometry and queueing theory, introduces delay outage, and evaluates scheduling and offloading. Results show traffic-dependent scheduling advantages and reduced macrocell traffic from cell-range expansion.
Problem
Multi-point-to-multi-point delay is insufficiently studied because it depends on all links and heterogeneous traffic and network factors.
Method
The paper combines stochastic geometry and queueing theory, modeling users by PPPs and individual packet arrivals by independent Bernoulli processes, then derives tractable delay bounds.
Results
Round-robin outperforms FIFO for heavy traffic but underperforms it for light traffic, while cell-range expansion greatly reduces macrocell traffic with a small picocell increase.
Takeaways & Limitations
Delay outage and the resulting bounds provide a framework for evaluating scheduling policies and supporting 5G deployment decisions where delay requirements are important.
Abstract
from arXiv · showhide
Emergence of new types of services has led to various traffic and diverse delay requirements in fifth generation (5G) wireless networks. Meeting diverse delay requirements is one of the most critical goals for the design of 5G wireless networks. Though the delay of point-to-point communications has been well investigated, the delay of multi-point to multi-point communications has not been thoroughly studied since it is a complicated function of all links in the network. In this work, we propose a novel tractable approach to analyze the delay in the heterogenous cellular networks with spatio-temporal random arrival of traffic. Specifically, we propose the notion of \emph{delay outage} and evaluated the effect of different scheduling policies on the delay performance. Our numerical analysis reveals that offloading policy based on cell range expansion greatly reduces the macrocell traffic while bringing a small amount of growth for the picocell traffic. Our results also show that the delay performance of round-robin scheduling outperforms first in first out scheduling for heavy traffic, and it is reversed for light traffic. In summary, this analytical framework provides an understanding and a rule-of-thumb for the practical deployment of 5G systems where delay requirement is increasingly becoming a key concern.
I. INTRODUCTION
The paper addresses the difficulty of analyzing delay for multi-point communications under spatio-temporal traffic and develops a tractable heterogeneous-network framework combining stochastic geometry with queueing theory. It evaluates delay outage, scheduling, and cell-range-expansion offloading for 5G network design.
- Motivation: Multi-point-to-multi-point delay remains insufficiently studied because it depends on all network links, traffic, access protocols, and propagation effects.
- Related works: Prior models often represent either user locations or packet arrivals, while joint models aggregate traffic by cell and cannot capture individual-user scheduling, offloading, throughput, or delay.
- Contributions: The framework combines stochastic geometry and queueing theory, modeling users with a PPP and packet arrivals at individual users with independent Bernoulli processes.
- Contributions: The paper introduces delay outage and evaluates random, FIFO, and round-robin scheduling policies for heterogeneous cellular networks.
- Contributions: Cell-range-expansion offloading greatly reduces macrocell traffic while causing a small increase in picocell traffic.
- Contributions: Round-robin scheduling outperforms FIFO under heavy traffic, whereas FIFO performs better under light traffic.
1) Random Scheduling:
The scheduling policies differ in how they select users or packets for service, and the framework assumes simple models that can be extended to more detailed wireless scenarios.
- FIFO Scheduling: FIFO serves packets in arrival order, treating all queues at a base station as one aggregate queue.
- Round-robin Scheduling: Round-robin schedules users sequentially, assigning each user one slot in a repeating cycle and retransmitting failed packets.
- Round-robin Scheduling: The analytical framework uses simplifying assumptions but can be extended to particular wireless scenarios and detailed technologies.
III. TRAFFIC STATISTIC
The paper models spatio-temporal traffic through user association, cell coverage, arrival rates, and delay requirements in a two-tier heterogeneous network. Numerical results show that cell range expansion substantially shifts traffic away from macrocells while only slightly increasing picocell traffic.
- Traffic characterization: The traffic model combines users’ spatial distribution with packet arrival rates and delay requirements for each user.The analysis begins by characterizing these traffic dimensions statistically.
- User association and coverage: The association probability of a typical user determines the tier-specific coverage-area approximation used for traffic statistics.Ergodicity equates the association probability with the average area fraction covered by a tier, yielding an effective PPP intensity λk/Pk.
- Traffic statistics: The PGF and PMF of Nk,ξ,β quantify the number of tier-k users whose arrival rates and delay requirements fall below specified thresholds.The count is modeled conditionally on cell area and the spatial PPP of users.
- Offloading effects: B2 = 10 greatly reduces users served by each macrocell, while users served by each picocell increase only slightly under offloading.With equal bias factors, macrocells serve more users because they provide larger coverage.
- Arrival-rate effects: Increasing the picocell bias substantially decreases each macrocell’s total arrival rate while slowly increasing each picocell’s arrival rate.The result indicates effective macrocell traffic offloading with a small per-picocell traffic cost; path-loss conditions also affect macrocell traffic.
IV. SUCCESS PROBABILITY AND DELAY
The paper analyzes success probability and delay while accounting for heterogeneous link conditions, queue states, and scheduling. It characterizes network-wide delay through the distribution of conditional mean delay and bounds the coupled SIR process using modified systems.
- Delay metric: Delay is measured in time slots and consists of queueing delay plus service time.Queueing delay covers waiting before service, while service time covers transmission of the packet.
- Delay characterization: Because static link success probabilities differ, the network-wide metric is the statistical CDF of users’ mean delays.Ergodicity allows this distribution to be obtained from conditional mean delay at a typical transmission across PPP realizations.
- Conditional analysis: The analysis conditions on the BS spatial realization and derives the conditional mean delay for a typical user associated with a tier-k BS.The serving-link distance distribution is conditioned on the user’s tier association.
- Coupled network dynamics: Queue states and SIR are coupled because interfering BS activity depends on queue status while SIR affects queue evolution.This coupling makes direct delay analysis difficult in the heterogeneous network.
- SIR bounds: The paper bounds SIR using a dominant system with dummy transmissions and a modified system that drops unscheduled or failed interfering packets.These constructions respectively increase and decrease interference relative to the original system.
A. Statistic of Success Probability
The paper derives the statistical cdf of users’ success probabilities in heterogeneous cellular networks by conditioning on base-station configurations and modeling interfering-base-station activity. The cdf depends on inter-tier density, bias, and transmit-power ratios, while simulation results show how picocell bias and path loss affect success probabilities.
- Increasing the picocell bias raises the proportion of picocell users with low success probability, whereas larger path loss exponents increase success probabilities in both tiers.Biased users tend to lie near picocell edges, while stronger attenuation suppresses inter-cell interference.
- The statistical cdf of all users’ success probabilities equals the cdf for a typical user conditioned on the base-station point process.This result assumes each interfering base station is independently active with probability q.
- The cdf equals the proportion of users whose success probabilities are below a threshold u.
- In K-tier networks, the success-probability cdf depends only on ratios of base-station densities, bias factors, and transmit powers across tiers.
- For a single-tier network, the success-probability cdf is independent of base-station density, bias factor, and transmit power.
B. Statistic of Delay
The paper introduces the statistical analysis of delay under different scheduling policies. It evaluates delay distributions by relating users’ service processes to scheduling, activity, and traffic conditions.
- Delay statistics are analyzed separately for different scheduling policies in the heterogeneous cellular network.
- Random scheduling selects one user uniformly from each active base station’s covered users in every time slot.The analysis also accounts for base stations being idle with probability 1 − p.
1) Random Scheduling:
Under random scheduling, the typical user’s queue is modeled through independent slot transmissions and geometric service times. The analysis then obtains network-wide delay distributions and bounds using alternative interference activity systems.
- Random Scheduling: Random scheduling yields a Geo/G/1 queue for the typical user, with Bernoulli packet arrivals of rate ξ0 and geometric service times with success probability µ.The conditional mean delay follows from this equivalent discrete-time queue.
- Random Scheduling: The statistical cdf of users’ mean delays equals the cdf of the typical user’s conditional mean delay.
- Random Scheduling: Theorem 1 bounds the statistical cdfs of success probability and mean delay for random scheduling.
- Random Scheduling: The lower and upper bounds use dominant and modified systems with different interfering-base-station activity probabilities.The dominant system uses q = p, while the modified system uses q = (ξmax + ξmin)/2 p as given in the passage.
2) FIFO Scheduling:
FIFO scheduling aggregates packets at a base station into one queue and models its arrivals and service using a Geo/G/1 system. The resulting network-wide delay and success-probability distributions are bounded analytically.
- FIFO Scheduling: The aggregated FIFO queue is modeled as a Geo/G/1 system with Bernoulli arrivals at rate Σ_i=0^{N−1} ξi.The approximation assumes that simultaneous arrivals of more than two packets are very unlikely.
- FIFO Scheduling: The statistical cdf of users’ mean delays equals the cdf of the typical user’s conditional mean delay under FIFO scheduling.
- FIFO Scheduling: Theorem 2 provides bounds on the statistical cdfs of success probability and mean delay for FIFO scheduling.
- FIFO Scheduling: The FIFO upper bound is obtained by choosing the minimum active probability among all base-station tiers.
3) Round-robin Scheduling:
The round-robin analysis models the new queueing system as a Geo/G/1 queue and derives bounds for conditional and network-wide mean-delay distributions. It also indicates that the framework can extend to FDMA with thinned interference.
- Round-robin delay model: The new queueing system is modeled as a Geo/G/1 queue, with mean delay measured in scheduled time slots needed to transmit a packet.The quantity DRR denotes the mean delay for the typical user.
- Round-robin delay model: Round-robin conditional delay accounts for the initial average scheduling position and additional scheduling periods after unsuccessful transmission.A successful first scheduled transmission requires N/2 slots on average; later attempts add multiples of N slots.
- Delay distribution: The statistical cdf of network-wide mean delay equals the cdf of typical-user mean delay conditioned on the base-station configuration.This result is stated for independently active interfering base stations with probability q.
- Delay distribution: Theorem 3 provides bounds for the statistical cdfs of success probability and mean delay for all users in the network.The bounds are developed specifically for round-robin scheduling.
- Extension to FDMA: FDMA can be analyzed using the same approach, with interference modeled through a thinned set of interfering base stations.Orthogonal sub-channels serve users simultaneously, simplifying user scheduling while thinning interference.
C. Delay Outage
Delay outage measures the proportion of users whose mean-delay requirements are unmet, and the paper bounds and numerically evaluates this metric under traffic, offloading, scheduling, and network-density variations.
- Definition and analysis: A user is in delay outage when its practical mean delay exceeds its mean-delay requirement βi.The delay outage probability ηDO is the proportion of outage users in the wireless network.
- Definition and analysis: The paper uses prior results to bound the delay outage probability and evaluates its statistical behavior numerically.The numerical discussion covers mean-delay distributions and delay outage.
- Scheduling: Round-robin outperforms FIFO under heavy traffic, whereas FIFO outperforms round-robin under light traffic; random scheduling performs worst.Round-robin can waste slots on empty queues in light traffic, while FIFO can be blocked by poor-link users in heavy traffic.
- Offloading: Increasing the picocell bias factor initially decreases delay outage by offloading macrocell traffic, but excessive bias can increase round-robin outage.With large B2, excessive offloading leaves macrocell resources underused when all users are served each round-robin period.
- Traffic intensity: Increasing active probability p generally decreases delay outage in light traffic, while the analytical bounds become looser for large p.Higher p increases packet-scheduling opportunities without much additional interference under light traffic.
- Network density: Delay outage first decreases and then increases with user density, while deploying more BS tiers helps only at high user density.At low density, adding users increases the share of singly occupied cells; at high density, additional tiers reduce outage.
- Overall findings: The framework analyzes spatio-temporal random traffic, offloading, and scheduling through upper and lower delay bounds.The bounds are reported as especially tight for heavy traffic and small active probability.
- Overall findings: Cell range expansion greatly reduces macrocell traffic while causing only a small increase in picocell traffic.The result suggests potential for better utilization of idle small-cell resources.
APPENDIX A PROOF OF LEMMA 5
The proof evaluates the conditional success-probability distribution through the logarithm of the conditional success probability, its moment generating function, and the Gil-Pelaez theorem.
- Conditional success probability: The proof conditions success probability on the base-station configuration, serving tier, and serving-distance variable.The conditional expressions distinguish the serving base station's tier and distance.
- Distribution construction: The moment generating function of Y is derived using independence across BS tiers and the probability generating functional of the PPP.These steps connect the tiered spatial model to the distribution calculation.
- Distribution construction: The cdf of Y is obtained with the Gil-Pelaez theorem and then mapped to the cdf of conditional success probability through logarithms.The resulting conditional cdf is subsequently averaged over serving conditions using total probability.
APPENDIX B PROOF OF LEMMA 7
The appendix proof obtains the target cdf by applying total probability, substituting earlier results, and invoking a preceding lemma.
- Proof steps: The proof begins by applying the total probability formula.
- Proof steps: Earlier expressions are substituted to obtain the target cdf.The proof explicitly plugs in equations (32) and (33).
- Proof steps: A result from Lemma 5 is used before the proof concludes.The final statement records that the required results follow.