SIR Coverage Analysis in Cellular Networks with Temporal Traffic: A Stochastic Geometry Approach
aa r X i v : . [ c s . I T ] J a n SIR Coverage Analysis in Cellular Networks withTemporal Traffic: A Stochastic Geometry Approach
Howard H. Yang,
Member, IEEE , and Tony Q. S. Quek,
Fellow, IEEE
Abstract —The bloom in mobile applications not just bring inenjoyment to daily life, but also imposes more complicated trafficsituation on wireless network. A complete understanding of theimpact from traffic profile is thus essential for network operatorsto respond adequately to the surge in data traffic. In this paper,based on stochastic geometry and queuing theory, we developa mathematical framework that captures the interplay betweenthe spatial location of base stations (BSs), which determines themagnitude of mutual interference, and their temporal traffic dy-namic. We derive a tractable expression for the SIR distribution,and verify its accuracy via simulations. Based on our analysis, wefind that i ) under the same configuration, when traffic conditionchanges from light to heavy, the corresponding SIR requirementcan differ by more than 10 dB for the network to maintaincoverage, ii ) the SIR coverage probability varies largely withtraffic fluctuation in the sub-medium load regime, whereas inscenario with very light traffic load, the SIR outage probabilityincreases linearly with the packet arrival rate, iii ) the mean delay,as well as coverage probability of cell edge user equipments (UEs)are vulnerable to the traffic fluctuation, thus confirms its appealfor traffic-aware communication technology. Index Terms —Poisson point process, cellular networks, ran-dom packet arrival, interacting queues, stochastic geometry,mean delay.
I. I
NTRODUCTION
The rapid evolution of mobile applications imposes morecomplicated traffic condition on wireless networks, wherenot only the data demand grows exponentially [1], but moreimportantly, the the content is largely changing from mobilevoice to multimedia [2]. To give an adequate response to thesurge in mobile data traffic, network operators need a completeunderstanding on the impact of temporal traffic. In this article,we aim to evaluate how the traffic statistic affects the wirelessnetworks, and to find those aspects that are most vulnerable.
A. Background and Related Work
Due to the broadcast nature of wireless channel, transmittersin space sharing a common spectrum will interact with eachother through the interference they cause. To characterizethe performance of such networks, stochastic geometry hasbeen recently introduced as a way to assess performance ofwireless links in large-scale networks [3]–[8]. The intrinsicelegance in modeling and analysis has popularized its ap-plication in evaluating performance among various wirelesssystems, including ad-hoc networks [3], cellular networks [9],or more advanced heterogeneous networks [10], even with
H. H. Yang and T. Q. S. Quek are with the Information Systems Technologyand Design Pillar, Singapore University of Technology and Design, Singapore(e-mail: howard [email protected], [email protected]). device-to-device (D2D) communication [11] and multiple-input multiple-output (MIMO) technology [12], [13]. How-ever, the main drawback of these models is that they heavilyrely on the full buffer assumption, i.e., every link alwayshas a packet to transmit, and do not allow one to representrandom traffic. While the additional dimension of randomnessin temporal domain increases the complexity in analysis, it isnevertheless a crucial factor in understanding network perfor-mance, especially for the next generation wireless system thatfaces more voliated traffic conditions [14], [15].The main difficulty with queuing in wireless network comesfrom the interdependency among the evolution of differentqueues, which is usually referred to as interacting queues [16].Because of interference, the queue status of one transmitter canaffect, and also be affected by, the queue status of its neigh-bors, hence making the analysis very difficult. Conventionally,the queuing interaction through wireless medium is studied us-ing simple collision models [16]–[19]. In such models, discretetime ALOHA protocol is usually employed, where each of the N terminals initiates a transmission attempt at every slot: Ifmore than two terminals transmit simultaneously, a collisionoccurs and all the terminals retransmit their packets in next slotwith the same probability [17]. Analytical results about systemstability can be obtained via exact form in scenarios with few(two or three) transmitters [16], or through approximationsin asymptotic regime with infinitely many transmitters [20].However, these models over simplify the wireless channeland lack the ability to tract the interference, which differsaccording to distance as well as channel gains, thus do notcapture the information-theoretic interactions precisely. Recentattempts to address this issue are made in [21]–[25], wherequeuing theory is combined with stochastic geometry to modelthe dynamic from both temporal and spatial domains. Theresults provide the necessary and sufficient conditions fornetwork to be stable [24], and different performance metrics,including success transmission probability [22], delay [21],and packet throughput [23] have been subsequently derived.While giving more refined analysis, these results either provideonly bounds that are not necessary tight [21], [24], or areonly appliable to networks with light traffic [23]. The mostrelated work is from [22], where the authors applied Geo/PH/1queuing model to account for the interference-based queuesamong UEs and analyzed the SIR performance under threedifferent transmission schemes. However, the requirement forfull channel inversion limits its generalization, and the re-stricted stable region constrains its application to relatively lowtraffic condition with small SIR detection threshold, and thusprevents one to take a complete treatment on traffic statistic. To this end, a mathematical framework that captures the spatial–temporal dynamic of the network, and adapts to scenarios withdifferent traffic conditions is of necessity to be explored. B. Approach and Summary of Contributions
In this paper, we model the BS deployment and UE loca-tions as independent Poisson point processes (PPPs), whereeach BS maintains an infinite capacity buffer to store theincoming packets. The queuing dynamic is modelled viaa descrete time system, where we consider the arrival ofpackets at each BS to be independent Bernoulli process.By combining stochastic geometry with queuing theory, weobtain a tractable expression for the SIR coverage probability.With the developed framework, we can explicitly characterizethe SIR variation due to change of traffic condition, and itsconsequential impact on system stability, as well as delaydistribution. Our main contributions are summarized below. • We develope a mathematical framework that capturesthe interplay between the spatial geometry of wirelesslinks and their temporal traffic dynamic. Our analysisis tractable, and takes into account all the key featuresof a cellular network, including traffic profile, small-scale fading and path loss, random network topology, andqueuing interaction. • Unlike [22], our result not only provides the standardSIR coverage probability, but also gives a more precisedescription about the fraction of UEs achieving SIR atdifferent levels. For instance, the SIR coverage probabil-ity of cell-edge UEs can be easily derived via our result. • We discuss the sufficient and necessary conditions for thenetwork to be stable, and provide an approximation forthe stable region. We also derive the mean delay distri-bution, by accounting for both queuing and transmissiondelay. • Using the developed analysis, we find that under thesame network configuration, there is more than 10 dBSIR difference between light and heavy traffic conditions.Moreover, in the very light traffic regime, the networkSIR outage probability is shown to increase linearly withpacket arrival rate. The result also reveals that the meandelay, as well as cell-edge UE rate, are vulnerable tothe variation of traffic condition, hence urging advancedsolution to adapt with traffic profile.The remainder of the paper is organized as follows. Weintroduce the system model in Section II. In Section III, wedetail the analysis of SIR distribution in cellular networks withtemporal traffic. We show the simulation and numerical resultsin Section IV, that confirm the accuracy of our analysis, andprovide insights about the impact of traffic profile on networkperformance. We conclude the paper in Section V.II. S
YSTEM M ODEL
In this section, we provide a general introduction to thenetwork topology, the traffic profile, as well as the propagationand failure retransmission model. The main notations usedthroughout the paper are summarized in Table I.
TABLE IN
OTATION S UMMARY
Notation Definition Φ b ; λ b PPP modeling the location of base stations(BSs); BS deployment density Φ u ; λ u PPP modeling the location of UEs; UE deploy-ment density P mt ; α BS transmit power; path loss exponent S ε ε -stable region of wireless network, under whichthe fraction of unstable queues is less than εξ ; ξ c ,ε Packet arrival rate; critical arrival rate for ε -stability γ x ,t ; θ Received SIR of typical UE at time slot t ; SIRdecoding threshold µ Φ x ,t Conditional SIR coverage probability at timeslot t A. Network Topology and Traffic Model
We consider the downlink of a cellular network, as depictedin Fig. 1, that consists of randomly deployed BSs whose spatiallocations follow independent Poisson point process (PPP) Φ b with spatial densities λ b . The location of UEs is modelled asanother independent PPP Φ u with spatial density λ u , whereeach UE associates with its closest BS for transmission. Weassume the UE density is high enough that every BS has atleast one UE associates with it. In this network, all BSs andUEs are assumed to be equipped with single antenna, and eachBS transmits with constant power P mt . In light of its spectralefficiency, we employ universal frequency reuse throughoutthe network, i.e., every BS transmits in the same spectrum.We use a discrete time queuing system to model the randomtraffic profile. In particular, the time axis is segmented into asequence of equal time intervals, referred to as time slots.We further assume all queuing activities, i.e., arrivals anddepartures, take place around the slot boundaries. Specifically,at the m th time slot, a potential packet departure may occurin the interval ( m − , m ) , and a potential packet arrival canhappen in the interval ( m, m + ) . In other words, departuresoccur at the moment immediately before the slot boundarieswhile arrivals occur at the moment immediately after the slotboundaries. For a generic UE located in the cell of x i ∈ Φ b ,we model its packet arrival as a Bernoulli process with rate ξ ∈ [0 , , which represents the probability of a new arrivaloccurs in a slot. We further assume that each BS accumulatesall the incoming packets in an infinite-size buffer for furthertransmission purpose.In order to investigate the evolution of queuing dynamic,we limit the mobility of transceivers by considering a staticnetwork, i.e., the locations of the BSs and UEs are generatedonce at (0 − , , and remained unchanged in all the following PPPs serve as a good model for the planned deployment of macro cellBSs, as verified by both empirical evidence [8], [26] and theoretical analysis[27]. The one UE per cell set up is mainly for the sake of notational simplicity.By adopting a similar approach as [23], the analysis in this paper can be easilyextended to consider more realistic scenario where multiple UEs in each cellshare the same wireless channel.
Fig. 1. A snapshot of the Poisson cellular network with temperal traffic. Thecoverage area of different BSs form the Voronoi cells, whereas each BS maybe active or inactive, depending on its buffer status. time slots. B. Propagation Channel and Failure Retransmission
In this network, we adopt a block-fading propagation model,where the channels between any pair of antennas are assumedindependent and identically distributed (i.i.d.) and quasi-static,i.e., the channel is constant during one transmission slot,and varies independently from slot to slot. We consider allpropagation channels are narrowband and affected by twoattenuation components, namely small-scale Rayleigh fadingwith unit mean power, and large-scale path loss that followspower law. Affected by the random channel fading and aggregatedinterference, the process of packet departure does not possessa constant rate and can lead to failure packet deliverty.Retransmission is thus necessary to guarantee the packetcan be correctly received. By enabling retransmission, thetransmission model at each BS becomes: During each timeslot, every node with a non-empty buffer sends out a packetfrom the head of its queue. If the received SIR exceedsa predefined threshold, the transmission is successful andthe packet can be removed from the queue; otherwise, thetransmission fails and the packet remains in the buffer. Weassume the feedback of each transmission, either success orfail, can be instantaneously awared by the BSs such that theyare able to schedule transmission at next time slot. Moreover,for the BSs with empty buffer, they mute the transmissions toreduce power consumption and inter-cell interference.
C. Signal-to-interference ratio (SIR)
By applying Slivnyak’s theorem [5] to the stationary PPPof BS, it is sufficient to focus on the SIR of a typical UE at Note that most of the pratical networks can be approximately regarded asstatic, since the locations of any end device cannot change drastically in arelatively short period [24]. The analysis in this paper is not necessary constrained to simple propaga-tion model, it can be further extended to incorporate more realistic setups thatinclude multi-slope path loss [28], [29] and complicated fading environment[30], [31].
BS 1 BS 2UE 1 UE 2BS 1BS 2 UE 1UE 2
Transmission linkInterference link
Fig. 2. Example of interacting queues with two BSs sharing the wirelesschannel. Due to impact from mutual interference, the service rates of BS 1and BS 2 are different, and dynamically change with their queuing status. the origin, with its tagged BS located at x . Given the UE isreceiving data at time slot t , the received SIR can be writtenas γ x ,t = P mt h x k x k − α P x ∈ Φ b \ x P mt ζ x,t h x k x k − α , (1)where h x ∼ exp(1) denotes the small scale Rayleigh fadingfrom BS x to the origin, k·k is the Euclidean distance, α standsfor the path loss exponent, and ζ x,t ∈ { , } is an indicatorshowing whether a node located at x ∈ Φ b is transmitting attime slot t ( ζ x,t = 1 ) or not ( ζ x,t = 0 ).It is important to note that since the spectrum is sharedamong BSs, the queuing status of each BS is coupled withother transmitters and hence results in interacting queue. Assuch, the BS active state, ζ x,t , is both spatial and temperaldependent, since the location affects the pathloss and furtherthe aggregated interference, and the time changes the queuelength at each node. To better illustrate this concept, Fig. 2gives a simple example of queuing interaction between twoBSs. Note that compared to UE 1, UE 2 has an advantage lo-cation and hence enjoys better path loss and fewer interference.Consequently, BS 2 can quickly empty its queue, and the dis-parity between their communication conditions results in BS 2activates less frequently than BS 1. Moreover, depending onwhether packets appear at both BSs or not, the correspondingactive durations are also different: If both transmitters havepackets to send, the mutual interference will reduce the servicerate and prolong the active duration of each BS individually.On the other hand, when one transmitter becomes silent, theother one can benefit from the reduced cross-talk and speedup its queue flushing process, hence also decreases the activeperiod. Extending this concept to a large-scale network, wefind that as the realization of PPP is irregular, there are alwayssome BSs experience poor transmission environment, e.g.,their UEs are located at the cell edge, and some others havinggood communication condition, e.g., their UEs are around thecell centers. In this regard, even the packet arrival rate is thesame for all transmitters, the queuing status and active state F Φ ( u ) = 12 − π Z ∞ ω Im u − jω " δ ∞ X k =1 (cid:18) jωk (cid:19) (cid:18) F Φ ( ξ ) + Z ξ ξ k t k F Φ ( dt ) (cid:19) Z ( k, δ, θ ) − dω (4)can varies largely from BS to BS, and the characterization ofSIR in such network is very challenging.III. A NALYSIS
We now present the main technical results of the paper.In particular, we first detail the definition of conditional SIRcoverage probability, and the analysis on its distribution. Thenwe discuss the conditions for the queuing network to be stable.After that, we give moments as well as a computationallyefficient approximation for the conditional SIR coverage prob-ability. Finally, we derive the distribution of mean delay, whichinvolves both queuing and transmission delay.
A. Conditional SIR Coverage Probability
Since both the received signal strength and the interfer-ence at a given UE are governed by a number of stochas-tic processes, e.g., random spatial distribution of transmit-ting/receiving nodes, random packet arrivals, and queuingdynamics, the SIR in (1) is a random variable and can only becharacterized via distribution. In this regard, conditioning onthe realization of the point process Φ , Φ b ∪ Φ u , we definethe conditional SIR coverage probability as follows [24], [32]. Definition 1:
Given the typical UE is receiving data attime slot t ,, its conditional SIR coverage probability is definedas µ Φ x ,t = P ( γ x ,t ≥ θ | Φ)= E exp − θ k x k α X x ∈ Φ b \ x ζ x,t h x k x k α (cid:12)(cid:12)(cid:12) Φ . (2)Note that the conditional SIR coverage probability µ Φ x ,t isstill a random variable (as we condition on the realizationof Φ ), which contains all the information about the UE SIR(and therefore achievable rate) distribution across the network.Moreover, the interaction of queues is also captured by (2) viathe accumulated interference.In order to analyze the distribution of µ Φ x ,t , we need toaddress two issues: i ) due to random packet arrival and re-transmission of failed deliveries, the active state, i.e., ζ x,t = 1 ,at each transmitter varies over time, and ii ) there may existcommon interferencing BSs seen by the same UE from onetime slot to another, which introduces temporal correlationfor the SIR coverage probability [33]–[35]. The dynamicallychanging active state of BSs, together with the temperal corre-lation, involve memory to the queues and highly complicatesthe analysis. Fortunately, when the number of transmittersasymptotically approaches infinity, a mean field property startsto emerge in the evolution of queues, i.e., the interactionbetween queues become “weak” and “global”, and the impactfrom aforementioned temperal and spatial correlation tends to be negligible on the employed system model [19]. Motivatedby this fact, we make the following assumption.
Assumption 1:
The temporal interference correlation has anegligible effect on the transmission SIR coverage probability.Hence, we assume the typical UE sees almost independentinterference at each time slot.
It is now safe to assume that all UEs experience i.i.d. steadystate queue distributions, and each BS activates independently,whereas the active probability depends on the specific servicerate, or equivalently, the SIR coverage probability. To faciliateanalysis in the following, we introduce a simple result fromqueuing theory as a preliminary, which describes the activeprobability of a transmitter under fixed arrival and departurerates.
Lemma 1:
Given the arrival rate being ξ , the service rate µ , the active probability at a generic BS is η a = (cid:26)
1, if µ ≤ ξ , ξ/µ , if µ > ξ . (3) Proof:
The result is a standard conclusion fromGeo/Geo/1 queue, which can be found in [36].After all the above preparason, we are now ready to deriveour main result of this paper, i.e., distribution of the conditionalSIR coverage probability.
Theorem 1:
The cumulative distribution function of theconditional SIR coverage probability is given by the fixed-point equation (4) on top of this page, which can be iterativesolved as follows F Φ ( u ) = lim n →∞ F Φ n ( u ) (5) where F Φ n ( u ) is given by F Φ n ( x ) = 12 − π Z ∞ ω Im ( x − jω h δ ∞ X k =1 (cid:18) jωk (cid:19) η ( k ) n − × Z ( k, δ, θ ) i − (cid:27) dω (6) whereas δ = 2 /α , Im {·} denotes the imaginary part of acomplex number, and Z ( k, δ, θ ) has the form as Z ( k, δ, θ ) = ( − k +1 θ k k − δ F ( k, k − δ ; k − δ + 1 , − θ ) , (7) with F ( a, b ; c, d ) being the hypergeometry function [37], and η ( k ) n − is given as η ( k ) n − = F Φ n − ( ξ ) + Z ξ ξ k t k F Φ n − ( dt ) . (8) In particular, when n = 1 , we have η ( k )0 = ξ k , ∀ k ∈ N . The mean field effect appears because of the cellular infrastruture, whereinterference is bounded away from the tagged transmitter. In Poisson bipolarnetworks, where the interaction is “strong” and “local”, such approximationmay not hold.
Packet arrival rate, ξ A v e r age a c t i v e p r obab ili t y , η ( ξ ) θ = 0 dB θ = 5 dB θ = 10 dB Fig. 3. Average active probability versus packet arrival rate, under differentSIR detection thresholds.
Proof:
See Appendix A.The expression in (4) not only quantifies how all the keyfeatures of a cellular network, i.e., deployment strategy, inter-ference, and traffic profile, affect the distribution of SIR, butalso illustrates how the interacting queues are affecting the SIRcoverage via a fixed-point functional equation. The function F Φ ( u ) can be interpreted from two aspects: If we regard atypical cell as a queuing system, equation (4) describes thedistribution of the random service rate; if we look at (4) in theview of network performance, then the CCDF, i.e., − F Φ ( u ) ,gives the level of certainty that at least u fraction of UEsin the network can attain SIR threshold θ . Several numericalresults based on (4) will be shown in Section IV to providemore practical insights. In the following, we discuss the stableregion, moments of the conditional SIR coverage probability,and approximation for the CDF. Remark 1:
We introduce an auxiliary function as η ( ξ ) = F Φ ( ξ ) + Z ξ ξt F Φ ( dt ) (9) which can be regarded as “average” active probability. Notethat as ξ grows from 0 to 1, while the value of η ( ξ ) alsovariates accordingly, the trend is very different. As depictedin Fig. 3, we can see that, with failed packet retransmissions,the average active probability goes up on a fast-and-then-slowbasis with respect to the packet arrival rate ξ , and this effectis especially significant in networks with high SIR detectionthreshold.B. Stable Region The primal consideration in queuing systems is about sta-bility, i.e., the critical conditions under which all the queuescan remain finite length and do not explode. For an isolatedsystem, even with random arrival and departure process, thestable region can be explicitly determined to be when averageservice rate is larger than the average arrival rate (Loynestheorem [38]). However, such condition cannot be directlygeneralized to large-scale queuing networks, as the strict sta-bility, i.e., all queues are finite-length, is not achievable (except for the trivial case of ξ = 0 ). Recall our discussion in SectionII, the random locations of both BS and UE always result insome UEs located at poor coverage area, e.g., the cell edge,and having unbounded queue length. In this regard, insteadof requiring all the queues to be stable, a more meaningfulalternative will be to maintain the fraction of unstable queuesto be below certain level. To this end, we introduce the ε -stable region [24], which gives the conditions for a networkto be operated with less than ε portion of saturated queues.Following [24], a formal definition is in the sequel. Definition 2:
For any ε ∈ [0 , , the ε - stable region S ε and the critical arrival rate ξ c are defined respectively as S ε = ( ξ ∈ R + : P ( lim T →∞ T T X t =1 µ Φ x o ≤ ξ ) ≤ ε ) (10) and ξ c = sup S ε . (11) The network is ε -stable if and only if ξ ≤ ξ c . The critical arrival rate gives an explicit stable boundary,beyond which there are more queues transfering from finitesize to infinite length and the requirement of less than ε portion of unstable queues cannot be guaranteed. While anexact expression for the critical arrival rate is not available,we can nevertheless obtain explicit conditions for the networkto be ε -stable. Theorem 2:
The sufficient condition for the network to be ε -stable is ξ ≤ ξ Sc ε = sup ( ξ ∈ R + : 12 − π Z ∞ ω Im ( x − jω h δ ∞ X k =1 (cid:18) jωk (cid:19) × Z ( k, δ, θ ) i − (cid:27) dω ≤ ε (cid:27) , (12) and the necessary condition for the network to be ε -stable is ξ ≤ ξ Nc ε = sup ( ξ ∈ R + : 12 − π Z ∞ ω Im ( x − jω h δ ∞ X k =1 (cid:18) jωk (cid:19) × ξ k Z ( k, δ, θ ) i − (cid:27) dω ≤ ε (cid:27) . (13) Proof:
For the sufficient condition, we consider a domi-nant system, where all the nodes keep transmitting irrespect oftheir buffer status (for a node with empty buffer, it transmits“dummy” packets). As such, the interference in dominantsystem is larger than the actual ones, and the result can beattained via having ξ = 1 in (4).For the necessary condition, we consider a favorable system,where each node transmits without retransmission, i.e., eventhere is transmission failure, the transmitter simplily ignoresthe failure and withdraw the failed packets. In this scenario,the interference is smaller than that of the actual one, whereaswe can obtain the result as making η k = ξ k .From the above results, we note that the actual value ofthe critical arrival rate falls in the interval of [ ξ Sc , ξ Nc ] . Usingthe ergodic property of PPP, an approximation for the ε -stableregion is given in the sequel. Corollary 1:
The ε -stable region of the network is approx-imated as follows ξ ≤ ξ Ac ,ε = sup (cid:8) ξ ∈ R + : F Φ ( ξ ) ≤ ε (cid:9) . (14) Proof:
Note that due to stationary and ergodicity, theensemble average obtained by averaging over the point processequals the spatial averages obtained by averaging an arbitraryrealization of PPP over a large region, i.e., P x lim T →∞ T T X t =1 µ Φ x ,t ≤ ξ ! = F Φ ( ξ ) , (15)and the result follows according to the definition. Remark 2:
With the notion of stability, the function η ( ξ ) defined in (9) can be interpreted as: On average, there are F Φ ( ξ ) portion of UEs in the network with infinity queue sizeand keep transmitting, while the rest − F Φ ( ξ ) portion ofUEs maintain stable queues, and active independently withprobability R ξ ξ/tF Φ ( dt ) .C. Moments Based on the CDF of the conditional SIR coverage proba-bility, in this part we give the corresponding moments, whichcan faciliate the assessment of network performance.
Theorem 3:
The m -th moment of the conditional SIRcoverage probability is given by M m = 11 + δ P mk =1 (cid:0) mk (cid:1) η ( k ) Z ( k, δ, θ ) , (16) where η ( k ) is given as η ( k ) = F Φ ( ξ ) + Z ξ (cid:18) ξt (cid:19) k F Φ ( dt ) (17) with F Φ ( u ) giving in (4) . In particular, when m = 1 , we havethe standard SIR coverage probability given as P ( γ x > θ ) = 11 + δη Z (1 , δ, θ ) . (18) Proof:
See Appendix B.Because of its important role in network performance as-sessment, we further provide bounds as well as an approxima-tion for the SIR coverage probability to attain better insights.
Corollary 2:
The SIR coverage probability can be respec-tively bounded by the following
11 + δ Z (1 , δ, θ ) < P ( γ x > θ ) <
11 + δξ Z (1 , δ, θ ) (19) and when ξ ≪ , the SIR coverage probability can beapproximated as follows P ( γ x > θ ) ≈ − ξδ Z (1 , δ, θ ) ( α =4) = 1 − ξδ √ θ arctan( √ θ ) . (20) Proof:
The upper and lower bounds can be obtainedvia similar approach as in the proof of Theorem 2, i.e., byconsidering a favorable system without retransmission and adominant system that keeps transmitting, respectively.For the approximation, we notice that F Φ ( ξ ) → as ξ → .Hence, we assume all the queues are stable in the regime with very light traffic load, and approximate the active probabilityusing the mean, i.e., ξ/ P ( γ x > θ ) [23]. The result thenfollows from solving the following equation P ( γ x > θ ) = 11 + δξ P ( γ x >θ ) Z (1 , δ, θ ) . (21) Remark 3:
Note that the gap between the upper andlower bounds decreases with the increment of the packetarrival rate ξ , which is consistent with the conclusion in [21].Furthermore, in the light traffic regime, the outage probability, − P ( γ x > θ ) , increases linearly with respect to the packetarrival rate ξ , which demonstrates the significant impact oftraffic profile on the system performance.D. Approximation Motivated by the fact that F Φ ( u ) is supported on the interval [0 , , we content ourselves in this part by approximatingthe function F Φ ( u ) via a Beta distribution to reduce thecomputational complexity [32]. A formal operation is statedin the sequel. Corollary 3:
The probability distribution function (pdf) of F Φ n ( x ) in Theorem 1 can be tightly approximated via thefollowing f X n ( x ) = x µn ( βn +1) − − µn (1 − x ) β n − B ( µ n β n / (1 − µ n ) , β n ) (22) where B ( a, b ) denotes the Beta function, µ n and β n arerespectively given as µ n = M ( n )1 , (23) β n = ( µ n − M ( n )2 )(1 − µ n ) M ( n )2 − µ n (24) where M ( n ) m can be written as M ( n ) m = 11 + δ P mk =1 (cid:0) mk (cid:1) ˆ η ( k ) n − Z ( k, δ, θ ) , (25) with ˆ η ( k ) n − being ˆ η ( k ) n − = Z ξ f X n − ( t ) dt + Z ξ ξ k t k f X n − ( t ) dt. (26) When n = 1 , we have ˆ η ( k )0 = ξ k , ∀ k ∈ N .Proof: At each step of the iteration in Theorem 1, thefunction F Φ n ( u ) is supported on [0 , . As such, by respec-tively matching the mean and variance to a Beta distribution B ( a n , b n ) , it yields a n a n + b n = M ( n )1 , (27) a n b n ( a n + b n ) ( a n + b n + 1) = M ( n )2 − h M ( n )1 i (28)and the result follows by solving the above system equations.The accuracy of Corollary 3 will be verified in Fig. 6. D =2 D =4 D =3 ... ... Packet arrivalPacket departure
Fig. 4. Illustration of delay in queuing system. The number of required slots todeliver each packet varies due to different queuing and retransmission results.
E. Mean Delay and Analysis
In the context of wireless networks, delay is an importantfactor that determines the Quality of Service (QoS) [14],[21]. As shown in Fig. 4, the random arrival of packets andinterference-limited channel will inevitably incur waiting andretransmission in the packet delivery process, thus inducemultiple slots for one successful delivery. Besides, since thenumber of required time slots differs among each packet,we can only quantify the delay in average manner. In itsaccordance, the following definition formalizes the notion ofmean day.
Definition 3:
Let A x ( T ) be the number of packets arrivedat a typical transmitter x within period [0 , T ] , and D i,x bethe number of time slots between the arrival of the i -th packetand its successful delivery. The mean delay is defined as D Φ x , lim T →∞ P A x ( T ) i =1 D i,x A x ( T ) . (29)Note that D i,x in (29) represents the number of time slotsrequired to successfully deliver the i -th packet, and its value isaffected by: ( i ) queueing delay, caused by other accumulatedunsent packets, and ( ii ) transmission, or equivalently local,delay, due to link failure and retransmission [39]–[41]. Byaveraging over all time slots, (29) provides information onthe average number of slots to successfully deliver a packet.Furthermore, using results from queuing theory and stochasticgeometry, we can characterize its distribution with the follow-ing expression. Theorem 4:
The CDF of mean delay at a typical UE isgiven as P (cid:0) D Φ x ≤ T (cid:1) = 1 − F Φ (cid:18) − ξT + ξ (cid:19) ≈ lim n →∞ Z ξ + − ξT x µn ( βn +1) − − µn (1 − x ) β n − B ( µ n β n / (1 − µ n ) , β n ) dx (30) where F Φ ( u ) is given by (4) , and µ n and β n are respectivelygiven in (23) and (24) . Proof: Without loss of generality, we focus on the typ-ical UE. Given the service rate, i.e., the conditional successprobability, µ Φ x , we have the conditional mean delay being[21] D Φ x = − ξµ Φ x − ξ , if µ Φ x > ξ ,+ ∞ , if µ Φ x ≤ ξ . (31)The distribution of mean delay can then be computed as P (cid:0) D Φ x ≤ T (cid:1) ( a ) = P (cid:0) D Φ x ≤ T | µ Φ x > ξ (cid:1) P (cid:0) µ Φ x > ξ (cid:1) + P (cid:0) D Φ x ≤ T | µ Φ x ≤ ξ (cid:1) P (cid:0) µ Φ x ≤ ξ (cid:1) ( b ) = P (cid:0) D Φ x ≤ T, µ Φ x > ξ (cid:1) = P (cid:18) − ξT + ξ ≤ µ Φ x (cid:19) (32)where ( a ) is by the law of total probability, and ( b ) follows bynoticing P ( D Φ x ≤ T | µ Φ x ≤ ξ ) = 0 . We then obtain the resultby using (4) and Corollary 3 to the above.The accuracy of Theorem 4 will be verified in Fig. 10.Moreover, two observations immediately follows from (30). Observation 1:
When T = 1 , we have P (cid:0) D Φ x ≤ T (cid:1) = 0 ,which states that the UEs who are able to success withoutretransmission form a probabilistic null set in large queuingnetworks. Observation 2: As T ≫ , the CCDF of mean delaycan be approximated as, P ( D Φ x > T ) ≈ F Φ ( ξ ) + 1 /T ,which implies the distribution is heavy tail. In addition, when T → ∞ , we have the delay outage, i.e., − P ( D Φ x ≤ T ) converges to F Φ ( ξ ) , which represents the fraction of UEs thatare experiencing unstable queues, and it is in line with ourdiscussion in Section-III-B. IV. N
UMERICAL R ESULTS AND S IMULATIONS
In this section, we validate the accuracy of our analysisthrough simulations, and explore the impact of traffic con-dition on network performance from several aspects. Duringeach simulation run, the BSs and UEs are realized over a100 km area via independent PPPs. Packets arrive at eachnode according to independent Bernoulli process. We averageover 10,000 realizations and collect the statistic from eachcell to finally calculate the SIR coverage probability. Unlessdifferently specified, we use the following parameters for pathloss exponent, BS density, and packet arrival rate, respectively: α = 3 . , λ b = 10 − BS / km , and ξ = 0 . packet / slot.In Fig. 5, we compare the simulated CDF of conditional SIRcoverage probability to the analysis proposed in Theorem 1,for various values of SIR detection threshold θ and packetarrival rate ξ . First, the results show a close match for all valuesof θ and ξ , which validate the our mathematical framework.Next, note that from Fig. 5(a) we can quantify the level ofconfidence about how much fraction of UEs are able to achievethe targeted SIR at different thresholds, it is also coined as thereliability of the network [32]. For instance, when ξ = 0 . ,the probability that 80% of UEs in the network can attain-5 dB SIR is 0.85, showing that with high possibility themajority of UEs can have their packets correctly decoded Conditional SIR coverage probability, µ Φ x C u m u l a t i v e D i s t r i bu t i on F un c t i on , F Φ ( u ) SimulationsAnalysis θ = 0 dB (a) Conditonal SIR coverage probability, µ Φ x C u m u l a t i v e d i s t r i bu t i on f un c t i on , F Φ ( u ) SimulationsAnalysis ξ = 0.10 (b)Fig. 5. Simulation versus analysis: CDF of conditional success probability. In Fig. (a), we fix the packet arrival rate to be ξ = 0 . , and varies the SIRthreshold as θ = − , − , , , , dB. In Fig. (b), we fix the SIR threshold to be θ = 0 dB, and varies the arrival rate as ξ = 0 . , . , . , . , . . without retransmission. Such probability can drop to 0.08if the SIR detection threshold increases to 5 dB, indicatingthat in large-scale queuing networks, it is very unlikely tomaintain the majority of UEs at high SIR. On the other hand,from Fig. 5(b) we observe that the conditional SIR coverageprobability monotonically decreases with the growth of packetarrival rate, due to the fact that more and more BSs areactivated because of their non-empty buffers and thus imposeadditional inter-cell interference. It is also worthwhile to notethat the increment of traffic load defects the network SIRcoverage in a non-linear manner, whereas the degradation ofSIR coverage is more severe as traffic load goes from light( ξ = 0 . ) to medium ( ξ = 0 . ), and the decreasing trendslows down as the network load further increases to heavytraffic regime ( ξ = 0 . ). The reason comes from the compositeeffect of temporal traffic. In light traffic condition, as packetarrival rate goes up, the increased traffic load not only wakesup more BSs, but also brings in more accumulated packets atthe buffer. Together with the reduced service rate, the activeduration of transmitters is extended, which in turns defectthe SIR across the network. In the heavy traffic regime, asmost of the queues are already saturated, the additional activecells cannot largely change the interference, and thus the SIRcoverage probability descent is leveled off. Such explanationalso goes to the observation in Remark 1.Taking a closer look between Fig. 5(a) and Fig. 5(b),we notice that under the same cellular configuration, theconditional SIR coverage probability with θ = 0 dB is largerthan that with θ = 5 dB, even their respective packet arrivalrates are ξ = 0 . and ξ = 0 . . This observation indicates thatpackets with smaller detection threshold, i.e., short packets,are more preferable to the network. The reason comes fromthe fact that higher SIR threshold not merely increases thepossibility of failure in decoding process, but triggers moreretransmissions thus prolong the active period of interferers,which further reduces the coverage probability.Fig. 6 compares the CDF of conditional SIR coverage Conditional SIR coverage probability, µ Φ x C u m u l a t i v e d i s t r i bu t i on f un c t i on , F Φ ( u ) Approximation: Corollary 3Analysis: Theorem 1 α = 3.5 α = 4 α = 3 Fig. 6. CDF of conditional SIR coverage probability (Theorem 1) and theapproximation (Corollary 3). Parameters are set as: SIR detection threshold θ = 0 dB, and packet arrival rate ξ = 0 . . probability to its approximation in Corollary 3. We can seethat the approximation matches well with the analysis underdifferent path loss exponents, thus confirms its accuracy. Wefurther observe that smaller path loss exponent reduces theSIR coverage, which is due to the fact that while smaller α enhances the received signal power, it nevertheless also resultsin higher inter-cell interference, and the latter subpass thepower gain and hence defacts the SIR.Fig. 7 plots the maximum arrival rates per sufficient andnecessary conditions as functions of ε . The figure shows alarge difference between the necessary and sufficient con-ditions, while the approximation locates in between, henceconfirms the necessity for better SIR characterization in trafficnetwork. It also reveals that a slight change of arrival rate canlargely impact the stable region, i.e., the network stability isvulnerable to traffic condition.Fig. 8 depicts the standard SIR coverage probability as a ǫ M a x i m u m a rr i v a l r a t e , ξ Necessary conditionApproximationSufficient condition
Fig. 7. Comparison of necessary and sufficient conditions. -15 -10 -5 0 5 10 15
SIR threshold, θ [dB] S I R c o v e r age p r obab ili t y , P ( γ x > θ ) Upper boundsLower boundsSimulationsAnalysis ξ = 0.05 ξ = 0.50 Fig. 8. Success probability versus SIR threshold θ . function of SIR detection threshold θ , under different packetarrival rates. We first note the close match between the analysisand the simulation, which validates the accuracy of Theo-rem 3. More importantly, the figure confirms that the trafficprofile plays a crucial role in the SIR coverage probability.For instance, to maintain a 90% coverage probability, therequired SIR detection threshold can differ more than 10dB in light traffic ( ξ = 0 . ) and heavy traffic ( ξ = 0 . )regimes, and this gap will be larger as the packet arrivalrate keeps increasing. Fig. 8 also reveals that the upper andlower bounds are not tight, whereas even in heavy trafficregime the difference between the two bounds is around 5 dB,and this gap increases dramatically in light traffic condition.Moreover, the bounds fail to capture the trend of coverageprobability variates according to the detection threshold θ . Aswhat we have discussed in Remark 1, both ξ and θ significantlyaffect the active probability, hence, simply approximating thetransmitting BSs to a thinned point process cannot providedesire results and hence validate the importance of the traffic-aware analysis.In Fig. 9 we plot the cell-edge coverage probability, i.e., Packet arrival rate, ξ [Packets/slot] C e ll - dege c o v e r age p r obab ili t y , - F Φ ( . ) Fig. 9. 95%-likely rate versus packet arrival rate ξ , for various SIR thresholds θ = − , − , , , dB. − F Φ (0 . , as a function of packet arrival rate ξ . Thisquantity represents the performance of the “5% UEs”, i.e., theUEs in the bottom 5th percentile in terms of performance, andis particularly interested to operators [15], [32]. We can seethat the cell-edge coverage probability is vulnerable to trafficfluctuation. Even in the case with very small SIR detectionthreshold, e.g., θ = − dB, the cell-edge coverage probabilitycan drop by half as the packet arrival rate changes from lowto high, and this defection is more severe in scenarios withhigh SIR thresholds. As such, it is critical to employ moreadvanced technology to boost up the SIR performance at thecell edge [13].Fig. 10 compares the CDF of mean delay given in The-orem 4 to the values obtained from simulations. The figureshows that simulation results agree well with the analyticalvalues. Moreover, Fig. 10 confirms the observation in SectionIII by showing a heavy tail behavior in the distribution ofmean delay. Fig. 10 also reveals that the mean delay is verysensitive to the variation of traffic load, e.g., the delay outageis round 0.01 when ξ = 0 . , but this value climbs to around0.25 when the packet arrival rate doubles, i.e., ξ = 0 . . Itis becuase higher the traffic load, on one hand incurs moreretransmissions via increased interference, on the other, alsoprolongs waiting time of each packet since the additionalincomings quickly occupy all available buffers. This compositeeffect significantly affects the mean delay.V. C ONCLUSION
In this paper, we introduced an analytical toolset to evaluatethe impact of temporal traffic on the performance of cellularnetworks. We used a general model that accounts for keyfeatures from both spatial and temporal domain, including thechannel fading, path loss, network topology, traffic profile,and queuing evolution. By exploiting queuing theory andstochastic geometry, we obtain the SIR distribution through afixed-point functional equation, and validated its accuracy bysimulation. Our results confirmed that temporal traffic profilecan largely affect the network SIR performance. In particular, Delay, D x Φ [Time slots] CD F o f m ean de l a y , P ( D x Φ < T ) SimulationsAnalysis ξ = 0.3 ξ = 0.4 ξ = 0.2 Fig. 10. Simulation versus analysis: Mean delay distribution. it showed that under the same configuration, when trafficcondition changes from light to heavy, the correspondingSIR requirement can differ by more than 10 dB for thenetwork to maintain coverage. Moreover, the SIR coverageprobability varies largely with traffic fluctuation in the sub-medium load regime, whereas in scenario with very lighttraffic load, the SIR outage probability increases linearly withthe packet arrival rate. In addition, the mean delay, as wellas coverage probability of cell-edge UEs are vulnerable to thetraffic fluctuation, thus confirms its appeal for traffic-awarecommunication technologies.The derivation of SIR distribution as a tractable form ofsystem parameters opens various areas to gain further designinsights. On one hand, the framework can be extended to adoptmore sophisticated point process, e.g., the Poisson cluster pointprocess [42], or determinantal point processes [43]. On theother hand, the analysis can be applied to investigate thedesign of different wireless technologies. For example, thetraffic scheduling problem in large-scale wireless networks[21], or the resource allocation problem in Dynamic TDDsystem [44]. Analyzing impact of temporal traffic on theperformance of Massive-MIMO system is also a concretedirection to investigate in the future.A
PPENDIX
A. Proof of Theorem 1
To faciliate the presentation, we denote F n as the σ -algebrathat contains all the information about queuing status of everynode x ∈ Φ b up to time t = n . We further introduce Y Φ x,n and q x,n to denote Y Φ x,n = ln P ( γ Φ x,n > θ | Φ) and q x,n = P ( ζ x,n =1) , respectively.As such, the σ -algebra {F n } ∞ n =0 forms a filtration, where F n − ⊂ F n . At slot t = 0 , every transmitter has one packetarrival with probability ξ , and actives with probability ξ , i.e., q x, = ξ , ∀ x ∈ Φ b . We can thus compute the momentgeneration function of Y Φ x, at a generic BS x ∈ Φ b , via the following M Y Φ x, ( s ) = E (cid:2) exp (cid:0) sY Φ x, (cid:1)(cid:3) = E Y z ∈ Φ \ x (cid:18) ξ θ k x k α / k z k α + 1 − ξ (cid:19) s ( a ) = exp − λ b Z ∞ B c (0 , k x k ) (cid:20) − (cid:18) ξ θ k x k α / k x k α +1 − ξ (cid:19) s (cid:21) dx ! ( b ) = Z ∞ πλ b r exp (cid:0) − λ b πr (cid:1) × exp − πλ b r ∞ X k =1 (cid:18) sk (cid:19) ( − k +1 Z ∞ (cid:18) ξ v α/ /θ (cid:19) k vdv ! dr ( c ) = h δ ∞ X k =1 (cid:18) jωk (cid:19) ξ k Z ( k, δ, θ ) i − (33)where ( a ) follows from the probability generating functional ofPPP, ( b ) comes from polar coordinate transform and decondi-tioning k x k with its pdf, which follows a Rayleigh distributionas f R ( r ) = 2 πλ b re − λ b πr , and ( c ) is by algebraic operation.By the Gil-Pelaez theorem [45], we have the CDF of µ Φ x, given as F Φ0 ( u ) = P ( P ( γ x, > θ | Φ) < u ) = P (cid:0) Y Φ x, < ln u (cid:1) = 12 − π Z ∞ ω Im n u − jω M Y Φ x, ( jω ) o dω. (34)Next, consider all the queues have evolved to time t = n ,and we have obtained the CDF of µ Φ x,n − as P ( µ Φ x,n − θ | Φ) s ]= E (cid:2) E (cid:2) P ( γ x,n > θ | Φ) s (cid:12)(cid:12) F n − (cid:3)(cid:3) = E E Y z ∈ Φ \ x (cid:18) q z,n θ k x k α / k z k α +1 − q z,n (cid:19) s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − = Z ∞ πλ b r exp (cid:0) − λ b πr (cid:1) exp − πλ b r ∞ X k =1 (cid:18) sk (cid:19) ( − k +1 × E "Z ∞ (cid:18) q x,n v α/ /θ (cid:19) k vdv (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F n − dr = h δ ∞ X k =1 (cid:18) sk (cid:19) E (cid:2) q kx,n |F n − (cid:3) Z ( k, δ, θ ) i − . (35)Leveraging Lemma 1, we have E (cid:2) q kx,n |F n − (cid:3) = E " χ { µ Φ x,n ≤ ξ } + (cid:18) ξµ Φ x,n (cid:19) k χ { µ Φ x,n >ξ } |F n − = F Φ n − ( ξ ) + Z ξ ξ k t k F Φ n − ( dt ) (36)where χ {·} is the indicator function. By substituting (36) backinto (35), and using the the Gil-Pelaez theorem for another time, we have the CDF of µ Φ x,n given as F Φ x,n ( u ) = P ( P ( γ x,n > θ | Φ) < u ) = P (cid:0) Y Φ x,n < ln u (cid:1) = 12 − π Z ∞ ω Im n u − jω M Y Φ x,n ( jω ) o dω. (37)Noet that F Φ n ( u ) appears on the left hand side of (37),and F Φ n − ( u ) appears on the right hand side. As F Φ n ( u ) ≤ F Φ0 ( u ) , ∀ u ∈ [0 , , n ∈ N , by the Dominant ConvergenceTheorem [46], we have F Φ n ( u ) → F Φ ( u ) , as n → ∞ , and theresult follows. B. Proof of Theorem 3
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