Robust Data Rate Estimation with Stochastic SINRModeling in Multi-Interference OFDMA Networks

Fei Liu, Janne Riihijarvi, Marina Petrova

Institute for Networked Systems, RWTH Aachen UniversityKackertstrasse 9, D-52072 Aachen, Germany

Email: fei@inets.rwth-aachen.de

Abstract—To meet the rapidly growing requirement for uni-versal coverage of high-speed mobile services, wireless cellularnetworks have been moving towards increasing density. Inter-cellcooperation in such densely deployed networks is becoming moresignificant than ever before. Among others, data rate estimationis a fundamental issue for inter-cell resource management andoptimization. In this paper, we focus on the data rate estimationproblem based on stochastic SINR models. We derive a closed-form solution of user SINR distribution in multi-interferencenetworks. We calculate upper and lower bounds of the SINRdistribution and extend them to a weighted sum SINR model toachieve more accurate estimation of data rate. The simulationresults reveal that our designed model can guarantee the accuracyof data rate estimation in diverse wireless network environmentssuch as urban and suburban scenarios. It decreases the errorof estimation and the ratio of high-error users even with verysmall signaling overhead fed back per user. Various factors, suchas the number of reported cells, low-SINR effect, propagationenvironments and inaccuracy of channel measurement, whichinfluence the estimation performance are analyzed and evaluatedas well. The weighted sum model is verified to have greatresistance to the influence of these factors and achieve accurateestimation.

Index Terms—Stochastic SINR modeling, multiple interfer-ence, data rate estimation, proportional fair.

I. INTRODUCTION

The population of mobile users, as well as the wirelessservice demand, is increasing dramatically [1]. In order tomeet the demand of enjoying mobile communication servicesanywhere and anytime, wireless cellular networks are trendingtowards increasing density, making inter-cell cooperation moresignificant than ever before. Inter-cell related operations, suchas user association, user handover, load balancing and inter-ference coordination, require a precise estimation of the long-term transmission performance to make decisions of systemactions dynamically [2], [3].

To estimate the long-term transmission performance, preciseanalysis of the received signal and the data rate model of theresource scheduler are prerequisites. In a multi-interferencenetwork, user devices can measure the reference signal re-ceived powers (RSRPs) of the serving base station (BS)and relatively strong inter-cell interference [4]. The statisticalresults of the measurements are reported to their base stationsand can be used for user data rate estimation and makingsystem decisions. The estimation accuracy is influenced bymultiple aspects, such as channel measurement precision, the

amount of feedback information, propagation environment, theanalytical models of the resource schedulers and so on.

In this paper, we consider using the proportional fair (PF)scheduler for resource scheduling which is a compromise-based scheduling algorithm and has been widely accepted [5].In the literature, there have been many related works on datarate estimation under PF scheduling. We can broadly classifythem into two groups according to their underlying models.The first model is designed for the single-cell scenario basedon the Gaussian approximation (GA) of the instantaneousdata rate per user [6]–[11]. The second model considersthe stochastic signal-to-interference-plus-noise ratio (SINR)distribution of each user in multi-cell networks and calculatesthroughput performance with the conditional probability distri-bution of the scheduled SINR under PF scheduling. There aretwo types of analytical SINR models. One can be referred toas interference as noise (IaN) in which the time-varying char-acteristics of interference signals are neglected [12]. They aremodeled as constant additive noise. This model simplifies theanalysis of instantaneous SINRs as only the average channelquality is necessary to be reported by each user device. Theother analytical SINR model is more meticulously designedand takes the available RSRP information into account. In [2]and [13], only one co-channel neighbor cell is considered andthe closed-form solution of SINR distribution is derived underthe Rayleigh fading channel. More general cases which havemultiple interferers are analyzed in [3], [14]–[16]. An impactof deriving the closed-form solution of SINR distribution is theuniqueness of interference RSRPs (IRSRPs). In [3] and [14],the IRSRPs are assumed mutually different. Besides this case,the same interference mean powers are also considered in [15]though the effect of noise is neglected. In order to overcomethese restrictions, the generalized integer gamma distributionlaw is used in [16] to derive a more universal stochastic SINRmodel [17]. However, this solution requires pretreatment of theinterference signals with the same IRSRPs and its calculationis highly complicated.

In this paper we simplify the analytical SINR model inmulti-interference networks and derive a closed-form solutionwith much lower computational complexity. Considering thelimited number of reported IRSRPs, two stochastic SINR mod-els are formulated and proved to be upper and lower bounds ofthe actual SINR distribution, respectively. With the analytical

results of SINR distribution, we estimate the long-term averageuser data rates under PF scheduling based on the methodsdeveloped in previous works [2], [3]. We further extend thetwo analytical models to a weighted sum SINR model in orderto improve the accuracy with limited reported interferenceinformation. The analytical SINR models are evaluated bysimulations, in which the realistic BS deployments in the urbanand suburban areas of Berlin, Germany are adopted. Theirrelative errors of data rate estimation are compared with theGA and IaN models. The various influence factors on theaccuracy of user data rate estimation are analyzed in detailand assessed in the simulation part. The results show that theweighted sum model has great resistance to the influence ofthese factors and achieves accurate estimation.

II. SYSTEM MODEL

We consider an orthogonal frequency division multipleaccess (OFDMA) downlink network containing multiple BSs.We denote the set of the BS indices by

B = {b |b = 1, . . . , B } . (5)

The index set of the user terminals is denoted by

U = {u |u = 1, . . . , U } . (6)

The traffic pattern is assumed to be full buffer, whichmeans that every user always has data to be transmitted. Thefrequency band is reused by all cells in the network and isdivided into K resource blocks (RBs). In each frame, BSsdistribute RBs to their connected users using the PF scheduler.All the RBs within a given frequency band are assumed toundergo the same independent identically distributed channelwhich is modeled by the Rayleigh distribution.

The instantaneous received power of the reference signal(RS) at user u from BS b is modeled as

Pu,b = pbLu,b∥hu,b∥2, (7)

where pb is the RS transmit power of BS b. Lu,b is thechannel gain of path loss and shadow fading, and hu,b isthe instantaneous Rayleigh fading gain of user u from BS bmodeled as a circularly symmetric complex Gaussian randomvariable. Its mean value is 0 and covariance is 1. The powergain of ∥hu,b∥2 is exponentially distributed with a unit meanvalue. Thus Pu,b is modeled as a random variable with the

TABLE IRSRP REPORT MAPPING

Reported index Measured value (dBm)

RSRQ 00 RSRP<-140RSRQ 01 -140≤RSRP<-139RSRQ 02 -139≤RSRP<-138

... ...RSRQ 96 -45≤RSRP<-44RSRQ 97 -44≤RSRP

exponential distribution and its mean value, i.e., the RSRP ofthe access point, is given as

pu,b = E [Pu,b] = pbLu,b. (8)

This is an average power of the symbols that carry cell-specificRSs over the entire bandwidth. A user reports this averagedvalue to its serving BS for path loss calculation and makingsystem decisions. The reporting range of RSRP is defined from-140 dBm to -44 dBm with 1 dB resolution. The mappedindices of real RSRP values are listed in Table I [18]. In thefollowing part of this paper, we use the RSRP in linear domainfor derivation.

The total instantaneous RS received power of user u is

Pu = Pu,b + P (Iu)u + σN ,

where Iu is the interfering BS set of user u, including |Iu|independent inter-cell interferers, P

(Iu)u is the sum of the

interference RS power from multiple interfering BSs in Iu,and σN is the noise power.

We denote the IRSRP as pu,i = E [Pu,i] , i ∈ Iu anddenote the mean value of the total received RS power aspu = E [Pu], which can be calculated according to the receivedsignal strength indicator (RSSI) as

pu =RSSI

NscK, (9)

where Nsc is the number of subcarriers per RB. The RSSI ispure wideband power measurement, including useful power,interference and noise over all used subcarriers [18]. Here weassume that all subcarriers within the frequency band are usedby the system.

The instantaneous SINR is expressed as

Φu =Pu,b

P(Iu)u + σN

=Pu,b∑

i∈IuPu,i + σN

. (10)

FΦu (ϕ) = 1−T∏

t=1

1

prtu,t

T∑t=1

rt∑r=1

(r − 1)!

(rt − 1)!

(1

pu,t+

ϕ

pu,b

)−r T∏s=ts=1

(1

pu,s− 1

pu,t

)−rs

exp

(−ϕσN

pu,b

)(1)

F ∗Φu

(ϕ) = 1− exp (−ϕσN )∑i∈Iu

(p∗u,iϕ+ 1)−1 ∏

j∈Iu,j =i

p∗u,ip∗u,i − p∗u,j

(2)

User devices can measure and calculate the IRSRPs ofrelatively strong interferers and report them to its serving BSvia backhaul links. The maximum number of the reportedIRSRPs per user is controlled by a parameter maxReportCellsin the network [4]. We denote it as an integer NR. A usercan reports only NR IRSRPs to its serving BS. The reportedinterfering BS set of user u is denoted as I′u which includesthe indices of the largest NR BS in terms of IRSRPs. The setof other interferers is denoted as I′′u = Iu − I′u, which haverelatively lower IRSRPs than those in I′u.

To facilitate the calculation in following part, we normalizethe reported parameters of a user u and its received noisepower by the average useful power, pu,b, as follows.

pu =pupu,b

, pu,i =pu,ipu,b

,

σN =σN

pu,b, pu,b = 1.

III. STOCHASTIC SINR MODELS

From [16], the cumulative distribution function (CDF) ofthe user SINR, denoted as FΦu (ϕ), can be obtained by(1), where the power values are non-normalized. T is thenumber of tuples of interference. In each tuple there are rtinterference signals having the same mean power value pu,t.This formula overcomes the problem in [3], [14] and [15],where the IRSRPs must be mutually different. According tothe RSRP report mapping method, it is inevitable to confrontthe problem that the IRSRPs measured results of differentinterferers may be with the same RSRP index. Although (1)solved this problem, it has to firstly group the interferencesignal according to their IRSRPs. Moreover, the computationof the formula is obviously very complicated. Thus, we furtherderive a simplified solution based on [3] and [16] as follows.

A. Simplified Stochastic SINR Model

We firstly assume an infinitesimal variable ε and |Iu|variables 0 < εu,i < ε. We define the modified IRSRPs as

p∗u,i = pu,i + εu,i, (11)

where all p∗u,i can be mutually different by delicately settingthe values of εu,i. Then, according to (1) or the solution in [3],the modified CDF of the user SINR can be obtained as in (2).We use Theorem 1 in Appendix A to simplify the solution in(2) into

F ∗Φu

(ϕ) = 1− exp (−ϕσN )∏i∈Iu

(p∗u,iϕ+ 1

)−1. (12)

This formula has no limitation of the uniqueness of eachIRSRP. Thus we can obtain the simplified solution of userSINR distribution as

FΦu (ϕ) = limε→0

F ∗Φu

(ϕ)

= 1− exp (−ϕσN )∏i∈Iu

(pu,iϕ+ 1)−1 (13)

This closed-form solution has two advantages compared to thatin [16]: the pretreatment of the interference signals with thesame RSRP report index is avoidable, and the computation ofthe CDF is much simpler. Based on the obtained CDF solution,the possibility distribution function (PDF) of SINR can befurther derived as in (3).

B. Upper and Lower Bounds of the SINR Distribution

As we discussed in the system model, the number ofreported cells per user is controlled by the parameter NR.Thus, we consider the remaining part of the received undesiredpower, excluding the reported IRSRPs in I′u, as a whole anddenote it as

δu (I′u) = σN + p

(I′′u)u = pu − pu,b − p

(I′u)u , (14)

p(I′u)u =

∑i∈I′u

pu,i, p(I′′u)u =

∑i∈I′′u

pu,i.

Then, we design two analytical SINR models based on thereported information in the following part.

1) Stochastic SINR model 1 (S1): We consider the reportedIRSRPs separately. The other undesired power δu (I

′u) is

regarded as pure noise. We denote the user SINR based onthis assumption as Φ

(1)u . The CDF of Φ

(1)u can be calculated

according to (13) as

FΦ

(1)u

(ϕ, I′u)

= 1− exp[−ϕδu (I

′u)] ∏i∈I′u

(pu,iϕ+ 1)−1

. (15)

This analytical SINR model results in a higher value of theSINR CDF than the actual one, namely,

FΦ

(1)u

(ϕ, I′u) ≥ FΦu (ϕ) . (16)

fΦu (ϕ) = exp (−ϕσN )

[σN +

∑i∈Iu

pu,i(pu,iϕ+ 1)

] ∏j∈Iu

(pu,jϕ+ 1)−1

= [1− FΦu (ϕ)]

[σN +

∑i∈Iu

pu,i(pu,iϕ+ 1)

](3)

Ru = K

∫ ∞

0

r (ϕ)fΦu (ϕ)∏

v∈U/u

FΦv

(r−1

(r (ϕ)Rv/Ru

))dϕ (4)

We prove this as follows.

FΦ

(1)u

(ϕ, I′u) = 1− exp (−ϕσN )

∏i∈I′u

(pu,iϕ+ 1)−1

∏i∈I′′u

exp (pu,iϕ)

≥ 1− exp (−ϕσN )

∏i∈I′u

(pu,iϕ+ 1)−1

∏i∈I′′u

(pu,iϕ+ 1)

= 1− exp (−ϕσN )∏i∈Iu

(pu,iϕ+ 1)−1

= FΦu (ϕ) .

Thus, this analytical SINR model can be adopted as the upperbound of the actual SINR CDF.

2) Stochastic SINR model 2 (S2): We can also assumeδu (I

′u) as the mean received power of an imaginary inter-

ference signal, which also has an exponential distribution ofthe delivered power. We denote the user SINR based on thisassumption as Φ

(2)u . According to (13), the CDF of Φ

(2)u can

be given as

FΦ

(2)u

(ϕ, I′u)

= 1−[δu (I

′u)ϕ+ 1

]−1 ∏i∈I′u

(pu,iϕ+ 1)−1

. (17)

This model results in a lower SINR CDF than the actualone, i.e.,

FΦ

(2)u

(ϕ, I′u) ≤ FΦu (ϕ) . (18)

We present the process of proof as

FΦu (ϕ) = 1− 1

exp (ϕσN )

∏i∈Iu

(pu,iϕ+ 1)−1

≥ 1− 1

σNϕ+ 1

∏i∈Iu

1

pu,iϕ+ 1

= 1−

∏i∈I′u

(pu,iϕ+ 1)−1

(σNϕ+ 1)∏

i∈I′′u

(pu,iϕ+ 1)

≥ 1−

∏i∈I′u

(pu,iϕ+ 1)−1

(σNϕ+

∑i∈I′′u

pu,iϕ+ 1

)= 1− 1(

σN + p(I′′u)u

)ϕ+ 1

∏i∈I′u

(pu,iϕ+ 1)−1

= FΦ

(2)u

(ϕ, I′u) .

Models S1 and S2 are used to estimate the expected datarates Ru according to (4) [3], [14]. r (ϕ) is the achievablebitrate of a RB and is computed as

r (ϕ) = NscSers (ϕ) /Ts, (19)

where Nsc is the number of subcarriers in a RB, Se is thenumber of effective OFDM symbols in a frame, and Ts is

the frame time duration. rs (ϕ) is the achievable bitrate ofeach symbol according to the Shannon capacity. The closed-form solution to the integral in (4) is unobtainable, we cannevertheless calculate the user data rate by numerical methods.

3) Weighted sum model of SINR (WS): In order to obtainmore accurate estimation of the user data rates with limitedreported information, we design a weighted sum SINR modelbased on the S1 and S2 models. The SINR expectations ofthe upper and lower bounds can be calculated with theirdistributions, respectively. We denote them as Φ

(1)

u and Φ(2)

u

and calculate them as

Φ(1)

u =

∫ ∞

0

ϕdFΦ

(1)u

(ϕ, I′u) , (20)

Φ(2)

u =

∫ ∞

0

ϕdFΦ

(2)u

(ϕ, I′u) . (21)

On the other hand, the reported reference signal receivingquality (RSRQ) indicates the ratio of the desired RSRP to theRSSI of a user [4]. It is defined as

RSRQ =K × RSRP

RSSI. (22)

According to the reported RSRQ, we can calculate the mea-sured average SINR as

Φu =

(1

Nsc × RSRQ− 1

)−1

. (23)

We consider to use a weighted method to combine the SINRdistributions of S1 and S2 and obtain an approximate resultof the real distribution. The WS model of SINR distributionis designed as the mixture

FΦ

(w)u

(ϕ, I′u) = w1FΦ(1)u

(ϕ, I′u) + w2FΦ(2)u

(ϕ, I′u) ,

where

w1 =

min

{max

{Φu−Φ

(2)u

Φ(1)u −Φ

(2)u

, 0

}, 1

}, Φ

(1)

u = Φ(2)

u ,

0.5, Φ(1)

u = Φ(2)

u ,

w2 = 1− w1.

The weighted sum model is a straightforward linear combina-tion of the SINR CDFs obtained by S1 and S2. Therefore, it iseasy to implement in realistic systems. w1 and w2 are weightfactors of S1 and S2 in the range of 0 to 1, respectively.

IV. INFLUENCE FACTORS OF ESTIMATION ERROR

The accuracy of data rate estimation is influenced bymultiple factors as we mentioned. In this section, we analyzethe major ones in detail and verify our analysis by means ofsimulation in the next Section.

A. The Maximum Number of the Reported Cell

Recall that the number of the maximum reported neighborcells per user is controlled by the system parameter NR,namely maxReportCells which is an integer in the range of1 to 8 [4]. Higher NR allows more IRSRPs to be reported by

-10 -5 0 5 10 15 20-20

-15

-10

-5

0

5

10

15

20

+1.0dB

+0.8dB

+0.6dB

+0.4dB

+0.2dB

0dB

-0.2dB

-0.4dB

-0.6dB

-0.8dB

Rela

tiv

e e

rro

r o

f sp

ectr

al

eff

icie

ncy

(%

)

SINR (dB)

-1.0dB

Fig. 1. The relative error of theoretical spectral efficiency with differentSINR values and offsets.

every user. This can increase the power proportion of knowableinterference in the undesired signal, which is defined as

ηu(I′u) =

p(I′u)u(

p(Iu)u + σN

) = 1− δu (I′u)(

p(Iu)u + σN

) . (24)

Obviously, the analytical results of user SINR are moreaccurate with fewer unknown parts included in the undesiredsignal. However, more IRSRPs fed back can increase the sig-naling overhead per user. On the contrary, decrease of NR mayresult in larger error of the analytical SINR distribution as wellas the data rate estimation. This is because that more IRSRPsare added to δu (I

′u) and not considered independently.

B. The Increase of Error Due to the Low-SINR Effect

The inaccurate SINR result can undoubtedly lead to anerror in the estimated data rate. Due to the natural featuresof the relationship between SINR and link capacity, the errormagnitude of data rate estimation is related to the SINR levelof a user. We calculate the relative error (RE) of theoreticallyachievable spectral efficiency (SE) with various SINR offsetsaccording to the Shannon capacity as follows,

γ (ϕdB, θdB) =

log2(1 + 10

(ϕdB+θdB)10

)log2

(1 + 10

ϕdB10

) − 1

× 100%,

where ϕdB is the SINR and θdB is the offset of SINR in dB.According to the equation, we plot γ (ϕdB, θϕdB) in Fig. 1.The same SINR offset at high and low SINR levels resultsin different errors of the achievable SE. A user with a lowerSINR is more sensitive to the deviation of its SINR. For theusers with high SINRs, the same proportion of SINR errorleads to smaller RE of the calculated capacity. For instance, auser with the SINR of 20 dB can tolerate ±1.0 dB errors ofSINR to guarantee a RE between ±5%. However, with a -10dB SINR, only ±0.2 dB and smaller deviations are allowable.

C. The Impact of the Propagation Environment

The path loss exponent varies in different communicationenvironments. It is larger in urban areas than in suburbanand rural areas, and can influence the accuracy of data rateestimation via two aspects. We denote the path loss exponentas α and rewrite the normalized power of an interferer as

pu,i =pu,ipu,b

= 10α logdu,bdu,i

−Xu,i +Xu,b,

where du,i and du,b are the distances of user u apart frominterferer i and access point b, respectively. Xu,b and Xu,i areloss factors of shadow fading in dB. A higher α results in alarger difference between the closest serving BS and furtherinterferers. Thus the high exponent enhances user SINR. Thislaw is also verified in [19] in detail. According to Section IV-B,higher SINRs are favorable for reducing the estimation errorscaused by inaccurate channel measurements.

The same influence of path loss exponent also applies to therelationship among interferers. A higher α enlarges the powerdifferences among interferers at different locations, i.e.,

pu,ipu,j

= 10α logdu,jdu,i

−Xu,i +Xu,j .

Thus the power of the reported closer and stronger interferersoccupy higher proportion of the total undesired power, i.e.,higher ηu(I

′u). As we explained in Section IV-A, this can

improve the accuracy of the analytical results of SINR as wellas the data rate estimation.

According to the above two reasons, the error of data rateestimation is smaller in urban areas, where the path lossexponent is higher, than that in suburb.

D. The Accuracy of Measured RSRPs

The RSRPs are statistical results calculated according tothe RS measurement results. Since an RS exists only for onesymbol at a time, the measurement is made on all the RBswhich contain cell-specific RSs. The accuracy of the statisticalRSRP results can be improved with more samples of RSpowers taken into account for calculation. In other words, themeasurement of RSRP over larger number of RBs can decreasethe deviation of the result and improve the accuracy of datarate estimation.

V. SIMULATION RESULTS AND ANALYSIS

We consider OFDMA-based downlink networks for the sim-ulations with the configuration parameters listed in Table II.The BS deployments in 13x12 km2 rectangular areas in theurban area of Berlin and a nearby suburban area are adoptedfor the simulation scenarios, as shown in Fig. 2 and Fig. 3 [20].The minimum inter-cell distance of cochannel BSs is assumedto be 200 meters. User terminals are uniformly randomlydistributed in each scenario. To avoid edge effects, only theuser data rates in the central 7 cells are computed and theBSs in other cells perform as pure interference sources. Thenumber of RBs used for the statistic results of RSRPs is setto 1000.

TABLE IISIMULATION PARAMETERS

Parameter Value

BS Tx power 46 dBmBS Tx antenna gain 18 dBi

Carrier frequency 1800 MHzUrban path loss model 138.47 + 38.22 log (d) dB

Suburban path loss model 130.41 + 33.77 log (d) dBStandard deviation of shadowing 8 dB

Noise power density -174.5 dBm/HzNoise figure 7 dBBandwidth 10 MHz

Number of RBs (K) 50Number of subcarries per RB (Nsc) 12Number of effective symbols (Se) 10 per frame

Frame duration (Ts) 1 msMinimum distance to BS 20 m

Activated user density 20 per cellMaximum reported cell number (NR) 1 to 8

13.3 13.32 13.34 13.36 13.38 13.4 13.42 13.44 13.46 13.48 13.5

52.46

52.47

52.48

52.49

52.5

52.51

52.52

52.53

52.54

52.55

Latitude

Longit

ude 1 2

34 567

BS in computed cell

BS as Interferer

Fig. 2. The BS deployments in the Urban scenario. (Map source andcopyright Google Maps, Google Inc.)

Fig. 4 presents the SINR CDFs of three randomly selectedusers in the central no. 4 cell, which are obtained by differentstochastic SINR models as well as the simulation. The resultsobtained by the S1 and S2 models result in upper and lowerbounds of the simulation results, respectively. This is inaccordance with our analysis. As the parameter NR decreases,the gaps between our analytical results and the actual onesincrease, i.e., the errors of the analysis become larger. This isbecause that fewer IRSRPs reported from the user devices leadto higher error of the SINR distribution. The results based onthe IaN model are also presented in the figure. In comparisonwith the S1 model, it results in much higher values and largererrors. In fact, it is a special case of model S1 while NR = 0,which considers no independent interference signals.

The relative errors of the estimated user data rates withdifferent models are shown in Fig. 5. The parameter NR isset to 2 so that 2 IRSRPs are allowed to be reported per

13.04 13.06 13.08 13.1 13.12 13.14 13.16 13.18 13.2 13.22

52.5

52.51

52.52

52.53

52.54

52.55

52.56

52.57

52.58

52.59

Latitude

Lo

ngit

ud

e 123 4

56

7

BS in computed cell

BS as Interferer

Fig. 3. The BS deployments in the Suburban scenario. (Map source andcopyright Google Maps, Google Inc.)

-10 -5 0 5 10 15 20 250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

S1 S2

NR = 1

NR = 2

NR = 3

IaN

Simulation

Cu

mu

lati

ve d

istr

ibu

tio

n f

un

cti

on

SINR (dB)

User 18

(Cell edge)

User 7

User 4

(Cell center)

Fig. 4. The CDFs of user SINR with different analytical models.

user. The SINR CDF obtained by the S1 model is an upperbound of the actual one. So its probability distribution at lowSINR is larger, resulting in lower values of user SINRs andaccordingly lower data rates. The case of the S2 model isopposite, namely, higher estimated data rates. Even with only2 IRSRPs available for the data rate estimation, the WS modelachieves extremely low REs of the estimation which are nomore than 5%. The users with low SINRs have higher REsthan those with better channel qualities, which is consistentwith our analysis in Section IV-B. However, this shortcomingis overcome effectively by the WS model.

For the convergence of the results, we calculate the CDFsof the relative error amplitude (REA) of the estimated userdata rate, which are presented in Fig. 6. REA is the absolutevalue of relative error. Low NR leads to few reported IRSRPsand more interference power included in the combined termδu (I

′u). This makes it more similar to a constant value like

-15 -10 -5 0 5 10 15 20 25

0

10

20

30

40

50

60

70

80

S1

S2

WS

Rela

tiv

e e

rro

r o

f est

imate

d d

ata

rate

(%

)

SINR (dB)

Fig. 5. The relative errors of the estimated user data rates with the designedmodels. (NR = 2, urban scenario)

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

S1 S2 WS

NR = 1

NR = 2

NR = 3

IaN GA

Cu

mu

lati

ve d

istr

ibu

tio

n f

un

cti

on

Relative error amplitude of estimated data rate (%)

Fig. 6. CDFs of relative error amplitude of the estimated user data rate.(Urban scenario)

additive noise [16]. Thus the REA of model S1 is smallerthan that of S2. As NR decreases, the REAs of both S1 andS2 become lager. However, the result of the WS model is stillsatisfactory with the REA no more than 10% while the resultof S2 deteriorates much. The results of the estimation basedon the IaN and GA models are better only than S2 when NR

is low.To verify our analysis in Section IV-C, we compare the error

of the data rate estimation in the urban and suburban areasas in Fig. 7. The REA of all the three models in the urbanscenario is lower than that in the suburb due to a higher pathloss exponent. To confirm the affect of path loss exponent,the distributions of user average SINR and ηu(I

′u) in (24)

are shown in Fig. 8 and Fig. 9, respectively. In the urbanarea, users have higher SINRs than those in suburb, and thepower proportion of reported interferers per user is alwayshigher with various NR. Both reasons contribute to better datarate estimation results in the urban scenario as we analyzed.

0 5 10 15 20 25 300.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu

mu

lati

ve d

istr

ibu

tio

n f

un

cti

on

Relative error amplitude of estimated data rate (%)

S1 S2 WS

Urban

Suburban

Fig. 7. Comparison of the REA CDFs in the urban and suburban scenarios.(NR = 2)

-15 -10 -5 0 5 10 15 20 250.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Cu

mu

lati

ve d

istr

ibu

tio

n f

un

cti

on

SINR (dB)

Urban

Suburban

Fig. 8. Comparison of the SINR CDFs in the urban and suburban scenarios.

However, the WS model makes up the accuracy loss causedby the propagation environment in suburb, achieving a veryclose performance of estimation as in the urban area, as shownin Fig. 7.

In practical systems, small error of the estimation is tol-erable. Thus we mainly focus on the users who have REAslarger than 5% and denote the population ratio of these usersas R0.05. We consider the impact of the maximum number ofreported cells on the estimation of user data rate. Fig. 10 showsthis ratio R0.05 with various NR. The R0.05 of the WS modelis the smallest among all the models and always lower than2% in both of the urban and suburban scenarios. When NR

is small, the accuracy of the S1 and S2 results is influencedlargely, especially the latter one. The result of S2 is better thanthose of IaN and GA only if enough IRSRPs are reported, i.e.,NR ≥ 3 in the urban scenario and NR ≥ 6 in suburb.

We also consider the error caused by the inaccurate mea-surements of RSRPs. We assume that each reported RSRPis the mean value of multiple received RS power samples

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

U SU

NR = 1

NR = 2

NR = 3

Cu

mu

lati

ve d

istr

ibu

tio

n f

un

cti

on

The reported power proportion of useless signals

Fig. 9. Comparison of the CDFs of reported interference power proportion,ηu(I′u), in the urban and suburban scenarios.

1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

90

100

Use

r ra

tio

of

RE

A>

0.0

5 (

%)

Maximum number of reported IRSRPs

U SU

S1

S2

WS

IaN

GA

Fig. 10. The user ratio of REA>0.05 with various NR.

over Km RBs. Based on the Rayleigh fading channel model,the mean value of the power samples follows the Erlangdistribution. In order to improve the accuracy of the estimatedresults, higher Km is necessary to reduce the variance of thestatistical results. Fig. 11 presents the results of R0.05 of theWS model with various Km. The WS model is more sensitiveto the change of Km than IaN and GA because its analysis isbased on the multiple independent IRSRPs. The improvementof the precision of the IRSRPs has very slight benefit for theestimation accuracy of the GA model because it considers nodetailed independent interferers. Different from other models,the performance of the IaN model enhances with a smallerKm. From Fig. 6 we can find that the REA of the IaN modeldistributes largely above 5%. Thus the statistics with fewerRBs result in higher variance of the estimated data rate sothat more users have the chance to obtain the REA lower than5%. But this is meaningless for improvement of estimationaccuracy because R0.05 is too large with IaN.

1 10 100 10000

10

20

30

40

50

60

70

80

90

100

Use

r ra

tio

of

RE

A>

0.0

5 (

%)

Number of RBs for RSRP calculation

NR = 1

NR = 2

NR = 3

IaN

GA

Fig. 11. The user ratio of REA>0.05 with various Km. (Urban scenario)

VI. CONCLUSION

In this paper, we have focused on the problem of data rateestimation with stochastic SINR models in multi-interferencenetworks. We derived the closed-form solution of the SINRdistribution in which the pretreatment of the same IRSRPsis avoidable. Based on the simplified model, we developedthe upper and lower bounds of the SINR distribution with theconsideration of limited number of the reported interferers. Wethen extended these models to a weighted sum SINR modelwith the purpose of more accurate estimation of user data rate.The simulation results show that it has significant superioritycompared with the IaN and GA models in both urban andsuburban scenarios. It can largely decrease the error of datarate estimation and enhance the ratio of the low-RE users evenwith very small signaling overhead. We analyzed the factorswhich can affect the estimation accuracy and assessed themby simulations. The impact of the low SINR on the estimationis overcome greatly by the weighted sum model. Inaccuratechannel measurements can increase the error of estimationseriously only if very small amount of RBs are used for theRSRP calculation. We are currently extending the present worktowards more heterogeneous networking scenarios, in whichvarious types of BSs are deployed with different densities.User mobility is expected to have an impact on the accuracyof dynamic estimation as well. Thus another key objective isto extend our analysis and evaluation towards networks withdifferent degrees of mobility, ranging from pedestrian usageto high mobility in vehicular networks.

APPENDIX APROOF OF THEOREM 1

Theorem 1. If

x ∈ R+, ai ∈ R+, i ∈ {1, 2, . . . , N}∀i = j, ai = aj ,

M+1∑i=1

x

x+ ai

M+1∏j=1,j =i

aiai − aj

=

M∑i=1

x

x+ ai

aiai − aM+1

M∏j=1,j =i

aiai − aj

+x

x+ aM+1

M∏j=1

aM+1

aM+1 − aj

(A2)=

M∑i=1

x

x+ ai

aiai − aM+1

M∏j=1,j =i

aiai − aj

+x

x+ aM+1

M∑i=1

aM+1

aM+1 − ai

M∏j=1,j =i

aiai − aj

(A3)

=M∑i=1

( x

x+ ai

aiai − aM+1

+x

x+ aM+1

aM+1

aM+1 − ai

) M∏j=1,j =i

aiai − aj

=

x

x+ aM+1

M∑i=1

x

(x+ ai)

M∏j=1,j =i

aiai − aj

(A2)=

x

(x+ aM+1)

M∏j=1

x

x+ aj=

M+1∏j=1

x

x+ aj.

thenN∏j=1

x

x+ aj=

N∑i=1

x

x+ ai

N∏j=1,j =i

aiai − aj

. (A1)

Proof of Theorem 1 with induction:• When N = 1,

x

x+ a1=

x

x+ a1;

• Assuming (A1) true when N = M , i.e.,

M∏j=1

x

x+ aj=

M∑i=1

x

x+ ai

M∏j=1,j =i

aiai − aj

, (A2)

the result holds for N = M + 1 as shown in (A3).

ACKNOWLEDGEMENTS

We thank the financial support from the German ResearchFoundation (DFG) through UMIC Research Centre.

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