How performance metrics depend on the

traffic demand in large cellular networks

B. B laszczyszyn (Inria/ENS) and M. K. Karray (Orange)Based on joint works [1, 2, 3] with M. Jovanovic (Orange)

Presented by M. K. Karray (http ://karraym.online.fr/)Simons Conference, UT Austin

May 18th, 2015

Outline

Introduction

Homogeneous network model [1, 2]

Homogeneous network performance [2]Typical cell modelCell-load equationsAverage user’s throughputMean cell model

Inhomogeneous networks [3]Scaling laws for homogeneous networksInhomogeneous networks with homogeneous QoS response

Numerical results [2, 3]

Conclusion

Introduction

◮ Performance metrics in cellular data networks

◮ cell loads, users number per cell, average user’s throughput

◮ They depend on

◮ traffic demand⇒ Dynamics of call arrivals and departures◮ base stations (BS) positioning

◮ irregularity⇒ Performance varies across cells

◮ inter-cell interference⇒ Performance in different cells areinterdependent

◮ In this work we propose

◮ an analytic approach accounting for the above three aspects inthe evaluation of the performance metrics in large irregularcellular networks

◮ validated by measurements performed in operational networks

Network geometry and propagation

◮ Base stations (BS) locations modelled by a point processΦ = {Xn}n∈Z on R

2

◮ assumed stationary, simple and ergodic◮ with intensity parameter λ > 0

◮ BS Xn emits a power Pn > 0 such that {Pn}n∈Z are marks ofΦ

◮ Propagation loss comprises

◮ a deterministic effect depending on the relative locationy −Xn of the receiver with respect to transmitter ; that is ameasurable mapping

l : R2 → R+

◮ and a random effect called shadowing

◮ The shadowing between BS Xn and all the locations y ∈ R2 is

modelled by a measurable stochastic process Sn (y −Xn)with values in R+

◮ the processes {Sn (·)}n∈Z

are marks of Φ

Network geometry and propagation

◮ The power received at location y from BS Xn is

PnSn (y −Xn)

l (y −Xn), y ∈ R

2, n ∈ Z

◮ Its inverse is denoted by LXn(y)

◮ The signal-to-interference-and-noise (SINR) power ratio in thedownlink for a user located at y served by BS X equals

SINR(y,Φ) =1/LX (y)

N +∑

Y ∈Φ\{X} ϕY /LY (y)

◮ where N ≥ 0 is the noise power◮ {ϕY }Y ∈Φ are additional (not necessarily independent) marks

in R+ of the point process Φ called interference factors

Service model◮ Each BS X ∈ Φ serves the locations where the received power

is the strongest among all the BS ; that is

V (X) ={

y ∈ R2 : LX (y) ≤ LY (y) for all Y ∈ Φ

}

called cell of X◮ A single user served by BS X and located at y ∈ V (X) gets a

bit-rateR (SINR (y,Φ))

called peak bit-rate

◮ Particular form of this (measurable) function R : R̄+ → R̄+

depends on the actual technology used to support the wirelesslink

◮ Each user in a cell gets an equal portion of time for his service.

◮ Thus when there are k users in a cell y1, y2, . . . , yk ∈ V (X),each one gets a bit-rate equal to his peak bit-rate divided by k

◮ i.e. the bit-rate of user located at yj equals1kR (SINR (yj ,Φ)), j ∈ {1, 2, . . . , k}

Traffic model

◮ There are γ arrivals per surface unit and per time unit

◮ Variable bit-rate (VBR) traffic : at their arrival, users requireto transmit some volume of data at a bit-rate decided by thenetwork

◮ Each user arrives at a location uniformly distributed andrequires to download a random volume of data of mean 1/µbits

◮ Arrival locations, inter-arrival durations as well as the datavolumes are assumed independent

◮ Users don’t move during their calls

◮ Traffic demand per surface unit

ρ =γ

µbit/s/km2

◮ The traffic demand in cell X ∈ Φ equals

ρ (X) = ρ |V (X)| bit/s

Cell performance metrics [1]

◮ Service in cell V (X) is stable when

ρ (X) < ρc (X) :=|V (X)|

∫

V (X) 1/R (SINR(y,Φ)) dy

called critical traffic : harmonic mean of the peak bit-rate. Incase of stability,

◮ User’s throughput

r (X) = max(ρc (X)− ρ (X) , 0)

◮ Number of users

N (X) =ρ (X)

r (X)

◮ Probability that BS is not idling equals min (θ (X) , 1) where

θ (X) :=ρ (X)

ρc (X)= ρ

∫

V (X)

1/R (SINR (y,Φ)) dy

called cell load

Typical cell

◮ Are there global metrics of the network allowing tocharacterize its macroscopic behaviour ?

◮ Consider spatial averages of the cell characteristics over anincreasing network window A

◮ By the ergodic theorem of point processes (discrete version),these averages converge to Palm-expectations of therespective characteristics of the “typical cell” V (0)

◮ For example, for traffic demand

lim|A|→∞

1

Φ(A)

∑

X∈A

ρ(X) = E0[ρ(0)]

and for cell load

lim|A|→∞

1

Φ(A)

∑

X∈A

θ(X) = E0[θ(0)]

◮ Analoguous convergence holds for other cell characteristics :critical traffic, user’s throughput, number of users

Typical cell characteristics

◮ Technical condition : Assume that location 0 belongs to aunique cell a.s.

◮ Then by the inverse formula of Palm calculus, typical celltraffic demand

E0[ρ(0)] =

ρ

λ

and cell load

E0[θ(0)] =

ρ

λE

[

1

R (SINR (0,Φ))

]

◮ Right-hand side : Expectation of the inverse of the peakbit-rate of the typical user with respect to the stationary

distribution of Φ

◮ By the ergodic theorem of point processes (continuous version)

E

[

1

R (SINR (0,Φ))

]

= lim|A|→∞

1

|A|

∫

A

1

R (SINR (y,Φ))dy

Cell-load equations◮ The above results hold true

◮ whatever is the point process Φ of BS locations provided it issimple stationary and ergodic (not necessarily Poisson)

◮ whatever are the marks {ϕY }Y ∈Φ pondering the interference

◮ In real networks a BS transmits only when it serves at leastone user, thus we take ϕY equal to the probability that Y isnot idling

ϕY = min (θ (Y ) , 1)

Then

SINR(y,Φ) =1/LX (y)

N +∑

Y ∈Φ\{X} min (θ (Y ) , 1) /LY (y)

◮ Recalling the expression of the cell load

θ (X) = ρ

∫

V (X)1/R (SINR (y,Φ)) dy

we see that cell loads θ(X) are related to each other by asystem of cell-load equations

Average user’s throughput

◮ Define the average user’s throughput in the network as theratio of mean volume of data request to mean service duration

r0 := lim|A|→∞

1/µ

mean service time in A ∩ S

where S is the union of stable cells

◮ By Little’s law and ergodic theorem, it is shown in [2] that

r0 =ρ

λ

P(0 ∈ S)N0

where N0 := E0[N(0)1 {N(0) < ∞}]

◮ N0 and P(0 ∈ S) do not have explicit analytic expressions !

Mean cell model◮ Virtual cell defined as a queue having the same traffic demand

and load as the typical cell ; that is

ρ̄ := E0[ρ(0)] =

ρ

λ

θ̄ := E0[θ(0)] =

ρ

λE

[

1

R (SINR (0,Φ))

]

◮ Remaining characteristics are related to the above two via therelations of cell performance metrics

◮ critical traffic demand

ρc (X) =ρ (X)

θ (X)→ ρ̄c :=

ρ̄

θ̄

◮ user’s throughput

r (X) = max(ρc (X)− ρ (X) , 0) → r̄ := max (ρ̄c − ρ̄, 0)

◮ number of users

N (X) =ρ (X)

r (X)→ N̄ :=

ρ̄

r̄

Mean cell load equation

◮ Assume that all BS emit at the same power

◮ In the mean cell model, we consider the following (single)equation in the mean-cell load θ̄

θ̄ =ρ

λE

[

1/R

(

1/LX∗ (0)

N +min(

θ̄, 1)∑

Y ∈Φ\{X∗} 1/LY (0)

)]

where X∗ is the location of the BS whose cell covers theorigin.

◮ We solve the above equation with θ̄ as unknown◮ We will see in the numerical section that the solution of this

equation gives a good estimate of the empirical average of theloads {θ (X)}X∈Φ obtained by solving the system of cell-loadequations for the typical cell model

Scaling laws for homogeneous networks

◮ Consider a homogeneous network model with a deterministicpropagation loss of the form

l (x) = (K |x|)β , x ∈ R2

where K > 0 and β > 2 are two given parameters

◮ For α > 0 consider a network obtained from this original oneby scaling

◮ the base station locations Φ′ = {X ′ = αX}X∈Φ,◮ the traffic demand intensity ρ′ = ρ/α2,◮ distance coefficient K ′ = K/α◮ and shadowing processes S′

n (y) = Sn

(

yα

)

,

while preserving the original powers P ′n = Pn

◮ For the rescaled network consider the cells V ′(X ′) and theircharacteristics ρ′(X ′), ρ′c(X

′), r′(X ′), N ′(X ′), θ′(X ′)

Scaling laws for homogeneous networks

◮ Proposition : Assume ϕ′X′ = ϕX , X ∈ Φ. Then for any

X ′ ∈ Φ′, we have V ′ (αXn) = αV (Xn) while ρ′(X ′) = ρ(X),ρ′c(X

′) = ρc(X), r′(X ′) = r(X), N ′(X ′) = N(X),θ′(X ′) = θ(X)

◮ Corollary : Assume ϕX = min (θ (X) , 1),ϕ′X = min (θ′ (X) , 1). Then the load equations are the same

for the two networks Φ and Φ′. Therefore θ′ (X ′) = θ (X),X ∈ Φ and by above Proposition, ρ′c(X

′) = ρc(X),r′(X ′) = r(X), N ′(X ′) = N(X), θ′(X ′) = θ(X)

◮ Corollary : Assume ϕ′X′ = ϕX , X ∈ Φ (possibly satisfying the

load equations). Then E′0 [ρ′ (0)] = E

0 [ρ (0)] andE

′0 [θ′ (0)] = E0 [θ (0)]. Consequently, the mean cells

characteristics associated to Φ and Φ′ are identical

Inhomogeneous networks with homogeneous QoSresponse

◮ A country is composed of urban, suburban and rural areas

◮ The parameters K and β of the deterministic part ofpropagation loss depend on the type of the zone.

◮ Assume that for each zone i

Ki/√λi = const

◮ Then the scaling laws say that

◮ locally, for each homogeneous area of this inhomogeneousnetwork, one will observe the same relation between the meanperformance metrics and the (per-cell) traffic demand

◮ In other words, one relation is enough to capture the keydependence between performance and traffic demand fordifferent areas of this network

Numerical setting

◮ 3G network at carrier frequency f0 = 2.1GHz with frequencybandwidth W = 5MHz

◮ Distance-loss function l(r) = (Kr)β, with K = 7117km−1,β = 3.8 (COST Walfisch-Ikegami model [4])

◮ Log-normal shadowing with standard deviation 9.6dB and themean spatial correlation distance 100m

◮ Transmision power is P = 60dBm, with fraction ǫ = 0.1 forpilot channel, noise power N = −96dBm

◮ 3D antenna pattern specified in [5, Table A.2.1.1-2]

◮ R (SINR) = 0.3 ×WE

[

log2

(

1 + |H|2 SINR)]

where H

Rayleigh fading satisfying E[|H|2] = 1

◮ Poisson process of BS with intensity λ = 1.27km−2 (averagecell radius 0.5km) within a disc of radius 5km

European city

◮ Typical cell and mean cell models predict similar values of theaverage load

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000

Cell load Proportion of stable cells

Traffic demand per cell [kbps]

Typical cell loadMean cell load

Measured cell loadProportion of stable cells

Large region in an European country

◮ comprizing urban, suburban and rural areas

0

0.05

0.1

0.15

0.2

0.25

0.3

0 100 200 300 400 500

Cell load

Traffic demand per cell [kbit/s]

Mean cellMeasurements

Conclusion

◮ Two approaches based on stochastic geometry in conjunctionwith queueing and information theory are developed

◮ In order to evaluate performance metrics in large irregularcellular networks

◮ Typical cell approach : spatial averages◮ Mean cell approach : simpler, approximate but fully analytic

◮ We validate the proposed approach by showing that it allowsto predict the performance of a real network

◮ Further work

◮ Spatial distribution of the performance metrics [6]◮ For multi-tier networks, calculate the characteristics of each

tier [7]

Bibliography

[1] M. K. Karray and M. Jovanovic, “A queueing theoretic approach to the dimensioning of wireless cellular networks serving variablebit-rate calls,” IEEE Trans. Veh. Technol., vol. 62, no. 6, July 2013.

[2] B. B laszczyszyn, M. Jovanovic, and M. K. Karray, “How user throughput depends on the traffic demand in large cellular networks,”in Proc. of WiOpt/SpaSWiN, 2014.

[3] B. B laszczyszyn and M. K. Karray, “What frequency bandwidth to run cellular network in a given country ? - a downlinkdimensioning problem,” in Proc. of WiOpt/SpaSWiN, 2015.

[4] COST 231, Evolution of land mobile radio (including personal) communications, Final report, Information, Technologies andSciences, European Commission, 1999.

[5] 3GPP, “TR 36.814-V900 Further advancements for E-UTRA - Physical Layer Aspects,” in 3GPP Ftp Server, 2010.

[6] B. B laszczyszyn, M. Jovanovic, and M. K. Karray, “QoS and network performance estimation in heterogeneous cellular networksvalidated by real-field measurements,” in Proc. of PM2HW2N, 2014.

[7] ——, “Performance laws of large heterogeneous cellular networks,” in Proc. of WiOpt/SpaSWiN, 2015.