Delay-Optimal Biased User Association in Heterogeneous Networks

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by  Fancheng Kong

1
Delay-Optimal Biased User Association in
Heterogeneous Networks
Fancheng Kong, Xinghua Sun, Member, IEEE, Victor C. M. Leung, Fellow, IEEE, and Hongbo Zhu
Abstract—In heterogeneous networks (HetNets), load balanc- macro BSs with a high transmission power. Macro BSs may
ing among different tiers can be effectively achieved by a biased thus easily become overloaded. To purposely push users to
userassociationschemewithwhicheachuserchoosestoassociate
micro BSs, a simple and efficient approach called the biased
with one base station (BS) based on the biased received power.
associationschemewasproposedin[3],withwhicheachuser
In contrast to previous studies where a BS always has packets
7 to transmit, we assume in this paper that incoming packets assigns a biased value to the measured received power from
1 intended for all the associated users form a queue in the BS. BSs of each tier, and associates with the BS that has the
0 In order to find the delay limit of the network to support real- largest mean biased received power. Ye et al. demonstrated
2 time service, we focus on the delay optimization problem by
in [4] that the user’s long-term rate can be greatly improved
properly tuning the biasing factor of each tier. By adopting
n by carefully tuning the biasing factor. With fixed locations of
a thinned Poisson point process (PPP) model to characterize
a
J the locations of BSs in the busy state, an explicit expression BSsandusers,notractableexpressionofaperformancemetric
of the average traffic intensity of each tier is obtained. On that such as the signal-to-interference-plus-noise ratio (SINR) or
4
basis,anoptimizationproblemisformulatedtominimizealower rate coverage can be derived, and the optimal biasing factor

] bound of the network mean queuing delay. By showing that the couldonlybefoundempirically.Stochasticgeometrywasthen
I optimizationproblemisconvex,theoptimalbiasingfactorofeach adopted in [5]–[10] to characterize the spatial distributions of
N tier can be obtained numerically. When the mean packet arrival
rate of each user is small, a closed-form solution is derived. BSsandusers,andtoquantifytheaverageperformancemetric
.
s The simulation results demonstrate that the network queuing of the network. For example, a Poisson point process (PPP)
c
performance can be significantly improved by properly tuning was adopted in [7], [8] to represent the irregular deployment
[
the biasing factor. It is further shown that the network mean of the BSs. The optimal biasing factor for BSs of each tier
1 queuing delay might be improved at the cost of a deterioration
was obtained therein to maximize the rate coverage. With a
v ofthenetworksignal-to-interferenceratio(SIR)coverage,which
4 indicatesaperformancetradeoffbetweenreal-timeandnon-real- similar PPP model, the biasing factor was optimized in [9],
0 time traffic in HetNets. [10] by maximizing the logarithm of the mean user rate.
9
Index Terms—heterogeneous network, biasing factor, average
0
traffic intensity, network mean queuing delay
0 A. Related Works and Motivations
.
1 Previous studies [4]–[10] assumed that the BSs always
0 I. INTRODUCTION have packets to transmit, which presents a worst case for
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With widespread use of portable devices such as smart the SINR and rate coverage. In practice, the BS load could
1
phonesandtablets,cellularnetworksarefacinganexponential vary significantly over a day. In particular, the amount of
:
v growth of mobile data traffic [1]. Meanwhile, real-time appli- user service requests can drop dramatically during non-peak
i
X cations such as video chat and online gaming become more traffic hours. The BSs are thus more likely to be idle during
andmorepopular,whichimposesstringentdelayrequirements such periods, but still consume energy [11]. On the other
r
a on the network. To deal with this ever-growing demand and hand, due to a high deployment density, one micro BS would
high service requirement, micro base stations (BSs) such as cover a small area. The void cell problem [12], [13] then
pico and femto BSs are deployed to undertake the traffic emerges, where some micro BSs don’t have any associated
pressure of macro BSs. The network architecture is thus users. Such BS thus solely acts as an interfering source. To
evolving to more dense and irregular heterogeneous networks improve energy efficiency and reduce the interference, the
(HetNets) [2]. techniquestoselectivelyswitchoffafractionofBSsaccording
Among all the techniques used in HetNets, load balancing to the traffic load have attracted extensive attention [14]–[18].
plays a key role to determine the network performance. For For example, the authors in [15] proposed a distributed on/off
example,ifthetraditionalmaximumdownlinkreceivedpower switching based algorithm in cellular networks to decide the
association scheme is adopted, users would tend to connect to minimumsetofactiveBSs.ByarguingthatacellularBScould
operateinnormalmode,sleepmode,orexpansionmode,Guo
F. Kong, X. Sun, and H. Zhu are with the Jiangsu Key Laboratory of etal.[16]proposedaschemethatdetermineswhichmodethe
Wireless Communications, College of Telecommunications and Information
BS should choose based on the load condition, such that the
Engineering, Nanjing University of Posts and Telecommunications, Nan-
jing210003,China(e-mail:[email protected];[email protected]; energy consumption is minimized. Dhillon et al. [17] adopted
[email protected]). a thinned-PPP model by assigning an active probability to
V. C. M. Leung is with the Department of Electrical and Computer
BSs of each tier and thus characterized the network signal-
Engineering, University of British Columbia, Vancouver, BC V6T 1Z4,
Canada(e-mail:[email protected]). to-interference ratio (SIR) coverage. With the same model,
2
Cao et al. [18] derived the optimal BS density of each tier associated BSs. The traffic intensity of one BS thus varies
to minimize the network energy consumption under a certain with the aggregate packet requests of all its associated users.
rate constraint. To simplify the analysis, frequency partitioning across tiers
Theaforementionedstudies[14]–[18]focusedonthetraffic is assumed in this paper. Although the queues of BSs from
load variance over a large time scale, i.e., peak and non-peak different tiers are not correlated, the queuing performance of
hours, and did not consider queuing in each BS. Most of one BS would affect the experienced interference of other co-
them aimed to improve the network energy efficiency. One channelBSsinthesametier.Todecouplethequeuingbehavior
BS, nevertheless, can vary between busy and idle states over of BSs in the same tier, we resort to the approximation of
a small time scale due to the dynamic packet arrivals of its replacingeachBS’sindividualtrafficintensitywiththeaverage
associated users, in which case the packet delay could be traffic intensity of its tier. The spatial distribution of BSs in
taken into account. In practice, with the proliferation of real- the busy state can thus be approximately characterized by a
time multimedia applications, the packet delay is becoming thinned-PPPmodel[17].TheSIRcoverageofeachtieristhen
animportantquality-of-service(QoS)metric.Forexample,an obtained,basedonwhichanexplicitexpressionoftheaverage
end-to-end latency over 250 ms for real-time multimedia is traffic intensity of each tier is further derived, and is shown to
generallyconsideredtobeunacceptable[19].Intheliterature, be an increasing function of the biasing factor of each tier.
there has been limited studies on the queuing analysis. The Inordertofindthedelaylimitthatthenetworkcanachieve,
queuingperformanceofasinglecellwasevaluatedin[20]and an optimization problem is formulated to minimize a lower
[21] in CDMA systems for the first time, based on which the bound of the network mean queuing delay by optimizing over
packetblockingprobabilityandthepacketqueuingdelaywere the biasing factor of each tier based on the derived average
characterized. These studies focused on only one independent traffic intensity. It is shown that the optimization problem is
queuebyassumingconstantinterferenceovertheentirecell.In convex, and the optimal biasing factor can be numerically
HetNets,nevertheless,BSsofvarioustypesaredeployedwith obtained. When the mean packet arrival rate of each user is
higherdensities,andthequeuingbehaviorofoneBSisclosely small, an explicit expression of the optimal biasing factor of
relatedwithothers,leadingtothecoupled-queueproblem[17]. each tier is obtained. With equal bandwidth allocation across
Specifically, whether a BS transmits or not will affect the tiers, it is further shown that each user should associate with
interference level experienced by other BSs that occupy the its nearest BS. A case study of a 2-tier HetNet shows that the
samespectrumresources,andwillconsequentlydeterminethe optimal biasing factor is sensitive to the bandwidth allocation
service rates of these co-channel BSs. The service rate in turn of each tier. To characterize the network capacity to support
affects the chance that one BS has packets to transmit or not. non-real-time services, the network SIR coverage is further
The analysis of coupled queues is a long-standing open derived. The contributions of this paper are summarized as
problem, and even solving a special case of two interacting follows.
queues is challenging [22]. Zhuang et al. [23] modeled mul- • By assuming queuing in each BS, an explicit expression
tiple interacting queues as a continuous-time Markov chain of the average traffic intensity of each tier is derived,
(CTMC) with fixed BS locations in the network. By minimiz- whichisshowntobeanincreasingfunctionofthebiasing
ing the average packet queuing delay, the optimal spectrum factor of each tier.
allocation pattern was obtained for each BS. Similarly, Cheng • Anoptimizationproblemofalowerboundofthenetwork
et al. [24] optimized the average queuing delay of network mean queuing delay is formulated, and is shown to be
subjected to the BS power constraints. They formulated a convex with respect to the biasing factor of each tier.
Markovdecisionprocess(MDP)problembasedontheinstan- When the mean packet arrival rate of each user is small,
taneouschannelstateinformationandqueuestateinformation, an explicit solution is obtained.
and proposed an adaptive user scheduling and power control • Simulation results of a 2-tier case demonstrate that the
policy. However, the state space of the Markov process may networkmeanqueuingdelaycanbesignificantlyreduced
becomehugeasthenetworkscalesup,andtheanalysiswould by properly tuning the biasing factor of each tier. In the
become intractable. Hence, this motivates us to deal with the meanwhile, a tradeoff is revealed between the network
coupledqueueproblemwiththetoolofstochasticgeometryto meanqueuingdelayandthenetworkSIRcoverage,which
account for the random BS deployment, and derive the mean indicatesthattheserviceprovidershouldstrikeabalance
performance metrics analytically such that some insights can between the performance of real-time and non-real-time
be gained for system design. services.
The rest of this paper is organized as follows. The system
model is presented in Section II. An optimization problem to
B. Our Approaches and Contributions
minimize a lower bound of the network mean queuing delay
In this paper, we consider a K-tier HetNet where users is formulated and studied in Section III. A case study of a 2-
and BSs of all tiers are randomly distributed, i.e., follow a tierHetNetispresentedinSectionIV.Conclusionsandfuture
PPP distribution. Similar to previous studies [5]–[10], it is works are given in Section V.
assumed that each user adopts a biased association scheme to
choose one BS with the maximum biased received power. In II. SYSTEMMODEL
contrast to previous studies [4]–[10], [14]–[18], we consider Consider a K-tier heterogeneous network where BSs in the
that the packet requests from the users form a queue in their kth tier form an independent PPP Φ with an intensity of
k
3
λk, k ∈ {1,...,K}. Users, on the other hand, form another III. QUEUINGDELAY
independent homogeneous PPP Φ with an intensity of λ
u u In this section, we will formulate an optimization problem
over the whole network. Frequency partitioning across tiers
of the network mean queuing delay, which is an important
is assumed in this paper. In particular, BSs of the same tier
performance metric of QoS. As the mean queuing delay of
share the spectrum with a bandwidth of W , k ∈ {1,...,K},
k a BS increases with a higher busy probability, we will first
and BSs of different tiers occupy non-overlapping frequency
characterize the average traffic intensity of each tier’s BSs,
bands.Therefore,foreachuserinthedownlink,theassociated ρ .
k
BS acts as a desired signal transmitter, and other BSs of
the same tier are interfering sources. Consider a typical user
A. Average Traffic Intensity of Tier-k BSs, ρ
k
located at the origin. Denote the distance between this typical
For a randomly selected Tier-k BS , since it delivers a
user and a Tier-k BS as x , and the transmission power of a k,i
k
packet only when the SIR exceeds a certain threshold τ, its
Tier-k BS as P . The received power P for a typical user
k R
mean aggregate packet arrival rate can be obtained as
from this BS can then be written as
P =P g x−α, (1) γk,i =γNk,iPr[SIRk,i >τ] (5)
R k k k
whereN isthenumberofusersthatareassociatedtoBS
whereg isasmall-scalefadingcoefficient,whichisassumed k,i k,i
k
and Pr[SIR >τ] denotes the SIR coverage of BS , i.e.,
to follow an i.i.d exponential distribution of unit mean, i.e., k,i k,i
the probability that the SIR of a random user associated to
g ∼exp{1}, and α is the path-loss coefficient, which is
k
BS islargerthanthethresholdτ.Bycombining(3)-(5),the
assumed to be the same for all BSs in the network. Note k,i
average traffic intensity of Tier-k BSs can be obtained as
that shadowing, i.e., log-normal fading, can be modeled by
the randomness of the BSs’ and users’ locations [25]. (cid:20)γN Pr[SIR >τ](cid:21)
ρ =E[ρ ]=E k,i k,i
In this paper, we consider a biased association scheme k k,i µk
whereusersassociatewithoneBSaccordingtothemaximum
γ
mean biased received power [4]–[10]. In particular, for a = E[N ]E[Pr[SIR >τ]]
µ k,i k,i
typicaluserlocatedattheorigin,itmeasuresthemeanreceived k
γL
power from each tier’s BSs, and chooses a Tier-k BS if = E[N ]P[SIR >τ], (6)
W log (1+τ) k k
P B x−α ≥P B x−α ∀j ∈{1,...,K}, (2) k 2
k k k,min j j j,min where E[N ] denotes the average number of users associated
k
where B denotes the biasing factor of Tier j and x is with a Tier-k BS and P[SIR >τ] denotes the SIR coverage
j j,min k
the distance between the user and the nearest Tier-j BS. of all Tier-k BSs, i.e., the probability that the SIR of a typical
user associated with a Tier-k BS exceeds the threshold τ. As
Foreachuserinthenetwork,assumethatitspacketrequests
the average traffic intensity, ρ , is determined by the average
follow an independent Poisson process with a mean arrival k
number of associated users, E[N ], and the SIR coverage,
rate γ, and the packet length is exponentially distributed with k
P[SIR >τ], we will characterize these two components in
mean L. The incoming packets for all users form a queue k
the following.
in the associated BS, and the BS will transmit these packets
According to [6], the average number of users associated
in a first-in-first-serve (FIFS) fashion. To avoid users in poor
with a Tier-k BS, E[N ], has been obtained as
channelconditionsoccupyingtheBS,weconsiderafixedrate k
modulation and coding format. In particular, a BS will serve λ A
E[N ]= u k, (7)
a user only when its instantaneous SIR exceeds a threshold k λ
k
τ, and will drop its packet request otherwise. Note that due
whereA denotestheprobabilityforatypicalusertobeasso-
k
to a high BS deployment intensity, the background noise is
ciated with a Tier-k BS. Note that the association probability
dominated by the interference, and is therefore ignored in this
A has been derived in [6] as
k
paper. According to Shannon’s formula, the service rate for
each user that is associated to a Tier-k BS can be obtained as λ (P B )2/α 1
A = k k k = , (8)
k
W K K (cid:16) (cid:17)2/α
µk = Lklog2(1+τ). (3) (cid:80) λj(PjBj)2/α (cid:80) λ(cid:101)j B(cid:101)jP(cid:101)j
j=1 j=1
For a randomly selected Tier-k BS, BS , its traffic intensity,
k,i
ρ , can be obtained as where λ(cid:101)j = λj/λk, P(cid:101)j = Pj/Pk, and B(cid:101)j = Bj/Bk denote
k,i
the normalized intensity, the normalized transmission power,
γ
ρ = k,i, (4) and the normalized biasing factor of Tier j, respectively,
k,i µ
k conditioned on Tier k being a serving tier.
where γ is the mean aggregate packet arrival rate of all its RecallthatBSsofTierkformaPPPΦ withanintensityof
k,i k
associated users. Note that ρ can also be interpreted as the λ .Moreover,forarandomlyselectedBS wherei∈Φ ,the
k,i k k,i k
busyprobabilityortheutilizationofBS whenρ ≤1.Due traffic intensity ρ can be interpreted as its busy probability
k,i k,i k,i
to a varied association region, each BS has a different mean when ρ ≤ 1. The set of Tier-k BSs being busy, therefore,
k,i
aggregate packet arrival rate and thus has a different traffic forms a thinned point process Φ(cid:48) ⊆Φ by including BS ∈
k k k,i
intensity. Φ with the probability ρ [26]. Since the traffic intensity of
k k,i
4
one BS is different from each other, the thinned point process Intuitively, as the probability of a user being associated with
Φ(cid:48) is non-homogeneous. To simplify the analysis, it can be a Tier-k BS increases, more users from other tiers will be
k
approximately viewed as a homogeneous PPP with intensity offloaded to BSs of Tier k, i.e., E[N ] becomes larger, which
k
leads to an increment of the traffic intensity.
λ(cid:48) =ρ λ . (9)
k k k
To this end, an explicit expression of ρ has been derived
k
It will be demonstrated in Section IV that the average traffic in (12). When the mean packet arrival rate of each user γ is
intensity achieves a good approximation. For a typical user small, the average traffic intensity ρ of Tier-k BSs can be
k
that is associated with a Tier-k BS, the interference all comes approximately written as
from busy BSs of the same tier. According to (1), the SIR of
(cid:104) (cid:105)1
this typical user can then be written as −1+ 1+4γλ A2(λ R )−1LZ 2
u k k k
ρ =
P g x−α k 2A Z
SIR = k xk,0 k,0 , (10) k
k (cid:80) P g x−α
j∈Φ(cid:48)k\BSk,0 k k,j k,j (≈a)−1+1+2γλuA2k(λkRk)−1LZ
2A Z
where x and x denote the distance from the typical user k
k,0 k,j
γλ LA
to the associated BS BSk,0 and the jth interfering Tier-k = u k, (14)
BS, respectively; gk,0 and gk,j denote the small-scale fading λkRk
coefficient of BSk,0 and the jth interfering Tier-k BS, re- where (a) follows from the fact that
spectively.In(10),BS andΦ(cid:48)\BS denotetheassociated
Tier-k BS of this typicka,0l user aknd thke,0set of interfering Tier- (cid:20)1+ 4γλuA2kLZ(cid:21)12 ≈1+ 2γλuA2kLZ. (15)
k BSs, respectively. Note that as frequency partitioning is λkRk λkRk
assumedacrosstiers,thereisnointer-tierinterference,andthe
interfering sources consist of all the busy Tier-k BSs besides
the associated BS . Following a similar approach in [7], we
k,0
have the following lemma that presents the SIR coverage of a B. Queuing Delay Optimization
Tier-k BS.
In this section, we will further minimize a lower bound of
Lemma 1. TheSIR coverageof aTier-k BScan bewritten as
the network mean queuing delay based on the average traffic
1 intensity by optimally tuning the biasing factors of all tiers.
P[SIR >τ]= , (11)
k A ρ Z(τ,α,1)+1 As each BS can be modeled as a M/D/1 queuing system, the
k k
´
mean queuing delay D of Tier k BSs can be obtained as
whereZ(τ,α,1)=τα2 (∞1/τ)α2 1+duuα2 ,andtheprobabilityAk k (cid:20) L (cid:21)
for a typical user to be associated with a Tier-k BS is given
D =E . (16)
in (8). k Rk(1−ρk,i)
According to Lemma 1, the outage probability of Tier Since (16) is difficult to characterize, we resort to its lower
k, P[SIRk ≤τ], can be written as P[SIRk ≤τ] = bound using the convexity of 1/(1−ρk,i), i.e., we have
AkρkZ(τ,α,1) . If Tier-k BSs are always busy, i.e., ρ = 1, L L
tAhkeρokZut(aτg,αe,1p)+ro1bability P[SIRk ≤τ] reduces to [6, Eq.k(17)]. Dk ≥D¯k = Rk(1−E[ρk,i]) = Rk(1−ρk). (17)
Bycombining(6),(7),and(11),theaveragetrafficintensity
Bycombining(3),(12)and(17),thelowerboundofthemean
ρ of Tier-k BSs can be derived as
k queuing delay of the whole network D¯ can then be written as
−λ R +(cid:104)(λ R )2+4γλ λ A2R LZ(cid:105)21 (18), which is shown on the top of next page.
k k k k u k k k
ρ = (12) It can be observed from (18) that the lower bound of
k 2AkλkRkZ the mean queuing delay D¯ is critically determined by the
where association probability A . To minimize D¯, we have the
k
following optimization problem
R =W log (1+τ), (13)
k k 2
D¯∗ = minimize D¯, (19a)
andZ denotesZ(τ,α,1)forsimplicity.Itisindicatedin(12)
{Ak}∀k∈{1,...,K}
that ρ is critically determined by the mean packet arrival
k
K
rate of each user γ and the association probability A . It (cid:88)
k s.t. A =1, (19b)
is clear that ρ increases as γ increases. On the other hand, k
k
k=1
the following lemma presents the monotonicity of the average
trafficintensityρ ofTier-kBSswithrespecttotheassociation
k ρ <1, k ∈{1,...,K}. (19c)
k
probability A .
k
Notethatinsteadofdirectlyoptimizingoverthebiasingfactor
Lemma 2.Theaveragetrafficintensityρ ofTier-k BSsisan
k of each tier, we optimize over the association probabilities
increasing function of its association probability, A .
k {A } in (19) to obtain the optimal solution {A∗} . The
k ∀k k ∀k
Proof: See Appendix B. optimal normalized biasing factor of Tier k conditioned on
5
D¯ =(cid:88)K λk ·D¯ =(cid:88)K 2Akλ2kLZ (18)
K k K (cid:18) (cid:104) (cid:105)1(cid:19)
k=1 (cid:80) λ k=1 (cid:80) λ 2A λ ZR +λ R − (λ R )2+4γλ λ A2R LZ 2
j j k k k k k k k u k k k
j=1 j=1
(cid:32) (cid:33)−2 (cid:32) (cid:33)−3 (cid:32) (cid:33)−2 −12
K−1 K−1 K−1
RK 1− (cid:80) Aj −RK2 1− (cid:80) Aj RK2 1− (cid:80) Aj +4γλuλ−K1RKLZ
∂D¯ j=1 j=1 j=1
=2λ Z
∂Ak K  (cid:32) K−1 (cid:33)−1  (cid:32) K−1 (cid:33)−2 122
2ZRK +RK 1− (cid:80) Aj −RK2 1− (cid:80) Aj +4γλuλ−K1RKLZ
 j=1 j=1 
R A−2−R2A−3(cid:0)R2A−2+4γλ λ−1R LZ(cid:1)−21
−2λ Z k k k k k k u k k =0, k ∈{1,...,K−1}. (25)
k (cid:104)2ZR +R A−1−(cid:0)R2A−2+4γλ λ−1R LZ(cid:1)21(cid:105)2
k k k k k u k k
Tier i, {B(cid:101)k∗}∀k, can then be readily obtained as whichleadsto (cid:80)K Ak <1accordingto(23b).Inthiscase,the
B(cid:101)∗ = Pi(λiA∗k)α2 , k ∈{1,...,K}, (20) lower bound ofk=th1e network mean queuing delay will always
k Pk(λkA∗i)α2 beunbounded.If(23)doesnothold,thefeasibleregionofthe
optimization problem (19) can be further written as
accordingto(8).Ontheotherhand,theconstraint(19b)comes
(cid:40) (cid:12)
farnodmthtehecofnacsttrathinatt(e1a9cch) uensesrursehsotuhladt athsseolcoiwateerwbiotuhnodnoefBthSe, A= (A1,...,AK−1),(cid:12)(cid:12)(cid:12)0<Aj <min(cid:110)1,γλ Lλ−jRλjR Z(cid:111),
u j j
network’s mean queuing delay is bounded, which leads to the
following lemma.
K−1
j ∈{1...,K−1}; max(cid:110)0,1− λKRK (cid:111)< (cid:88) A
Lemma 3. For the lower bound D¯ , when the mean packet γλ L−λ R Z j
k u K K
arrival rate γ > (Z+1)λkRk, it is bounded if the association (cid:41) j=1
λuL
probability
<1 , (24)
λ R
0<A < k k ; (21)
k γλ L−λ R Z
u k k where A is eliminated according to the constraint (19b)
K
otherwise, it will become unbounded. When γ < (Z+1)λkRk, without loss of generality. It is shown in Appendix D that
λuL
it is always bounded. the objective function (19a) is convex within the feasible
region A. Nevertheless, there may not exist a solution in
Proof: See Appendix C. A through solving (25), which is shown at the top of this
page, by setting the partial derivative of D¯ with respect to the
According to Lemma 3, constraint (19c) can be further associationprobabilityA tozero.Thefollowinglemmarules
k
written as outthispossibilityandguaranteesthattheoptimalassociation
(cid:40) 0<A0k<<Aγλu<Lλ−k1λR,kkRkZ, γγ >< ((ZZ++λ11u))λλLkkRRkk , (22) psorolubtaiobniliotifes(2{5A)∗kw}i∀thkincaAn.always be obtained by finding the
k λuL
Lemma4.(25)hasauniquesolutionwithinthefeasibleregion
where k ∈ {1,...,K}. First note that (22) does not have a
A, which is the optimal association probabilities {A∗} .
feasible region if and only if k ∀k
(cid:26)(Z+1)λ R (cid:27) Proof: See Appendix E.
γ >max k k (23a)
∀k λuL Sofarwehavedemonstratedhowtofindtheoptimalassoci-
and ation probability of each tier A∗ by solving (25) numerically.
k
K Recall that it is indicated in Lemma 3 that when the mean
(cid:88) λkRk <1, (23b) packet arrival rate of each user γ < min(cid:110)(Z+1)λkRk(cid:111), we
γλuL−λkRkZ ∀k λuL
k=1 have the average traffic intensity ρ < 1 for all tiers, and
k
according to (22). Intuitively, when the mean packet arrival the lower bound of the mean queuing delay D¯ is always
k
rate of each user γ is too large, (22) can be written as boundedforeachtier.Inthiscase,theaveragetrafficintensity
0 < Ak < γλuLλ−kλRkkRkZ for each Tier k, k ∈ {1,...,K}, ρk issimplywrittenas(14),andanexplicitoptimalassociation
6
Algorithm 1 Procedure to optimize the association proba-
TABLEI
bility when the mean packet arrival rate of each user γ <
(cid:110) (cid:111) SIMULATIONPARAMETERS
min (Z+1)λkRk
∀k λuL Parameter Value
1: Input: λk, Wk for each tier, and other system parameters User Density λ 10−2 m−2
u
λu, L, γ, τ. Tier-1 BS Density λ 10−4 m−2
1
2: Initialize: a set of index C ={1,...,K} where optimal Tier-2 BS Density λ 5∗10−4 m−2
2
association probability of Tier k is not determined.
Tier-1 BS Transmission Power P 46dBm
1
3: Calculate the solution set {A∗} by (26).
k ∀k∈C Tier-2 BS Transmission Power P2 35dBm
4: for ∀k ∈C, construct a set S ={m|A∗ <0,∀m∈C}.
m Path Loss Coefficient α 4
5: if S =∅, return {A∗} .
k ∀k∈C Mean Packet Length L 0.1Mb
6: else, for ∀m∈S, let λ =0 and A∗ =0, delete m from
m m
C.
7: end if
8: go to Step 3.
probability A∗ for each tier can be obtained, which is shown
k
in the following lemma.
Lemma 5. When the mean packet arrival rate of each user
(cid:110) (cid:111)
γ <min (Z+1)λkRk ,theoptimalassociationprobabilityof
∀k λuL
Tier k A∗ to minimize the lower bound of the network mean
k
queuing delay D¯ can be written as
K
(cid:80)
λ log (1+τ) λ (W −W )
k 2 j k j
A∗ = λk + j=1 . (26)
k K K
(cid:80) (cid:80)
λ γλ L λ
j u j
j=1 j=1
The detailed derivation can be found in Appendix F. Fig.1. Averagetrafficintensityofeachtierρkversusthenormalizedbiasing
factorB(cid:101)2 withvariousvaluesofthemeanpacketarrivalrateofeachuserγ.
Intuitively, if the bandwidth of Tier k is larger than that of W1=10MHz,W2=6MHz,andτ =1.
Tierj,i.e.,W >W ,theservicerateofTierk willbelarger,
k j
indicatingabetterqueuingperformance.Therefore,theTier-k
A∗ < 0. To minimize the lower bound of the network mean
BSs will undertake more traffic from other tiers by having a m
queuingdelay,theassociationprobabilityofTiermshouldbe
largerassociationprobability.Withequalbandwidthallocation
close to zero. Intuitively, if the bandwidth of Tier m is much
amongalltiers,i.e.,W =W ,i,k ∈{1,...,K},theoptimal
i j smallerthanthatofothertiers,thenfewusersshouldassociate
association probability of a Tier-k BS can be further written
with Tier-m BSs due to the low service rate. In this case, the
as
λ association probability A could then be simply set as A =
A∗ = k (27) m m
k K 0, i.e., Tier-m BSs are turned off. The procedure to obtain the
(cid:80) λj optimal association probability when γ < min(cid:110)(Z+1)λkRk(cid:111)
j=1 ∀k λuL
is summarized in Algorithm 1.
according to (26). The corresponding optimal normalized
biasing factor B(cid:101)∗ of Tier k, conditioned on Tier i, is thus
k IV. CASESTUDY
given by
1 In this section we will demonstrate the analytical results in
B(cid:101)k∗= , (28) SectionIIIbysimulationsofa2-tierHetNet.Thebasestations
P(cid:101)k
and the users are drawn from PPPs with high intensities,
where P(cid:101)k is the normalized transmission power of Tier k and the background noise is ignored in the simulations. This
conditioned on Tier i. It is indicated in (28) that in this setting,forexample,cancorrespondtoadenseheterogeneous
case, each user chooses the nearest BS. The traffic load is network that consists of macro cellular BSs and micro Wi-Fi
thus evenly distributed among all BSs, which leads to similar accesspoints,eachofwhichusesanon-overlappingfrequency
queuing performance with the same service rate of each tier’s band. Each point of the simulation results is obtained by
BSs. averaging all the BSs on a time scale of 105 s. The system
Notefrom(26)thatifthereexistsonetier,sayTierm,such parameters used in simulations are summarized in Table I.
(cid:80)K Fig. 1 illustrates how the average traffic intensity of each
thatlog (1+τ) λ (R −R )<−γλ L,andthenwehave
2 j m j u tier, i.e., ρ and ρ , varies with the normalized biasing factor
j=1 1 2
7
Fig.2. NetworkmeanqueuingdelayD anditslowerboundD¯ versusthe
normalizedbiasingfactorB(cid:101)2 withvariousvaluesofthemeanpacketarrival Fig.3. MinimumlowerboundofthenetworkmeanqueuingdelayD¯∗versus
rateofeachuserγ.W1=10MHz,W2=6MHz,andτ =1. themeanpacketarrivalrateofeachuserγ.W1 =10MHz,W2 =6MHz,
andτ =1.
B(cid:101)2 withvariousvaluesofthemeanpacketarrivalrateofeach
user γ. It can be observed from Fig. 1 that the average traffic to that of D. Therefore, by properly tuning the normalized
intensityofTier1,ρ1,decreasesasthenormalizedbiasingfac- biasingfactorB(cid:101)2accordingto(20)and(25),themeanqueuing
torB(cid:101)2 increases,whilethatofTier2,ρ2,increases.Intuitively, delay performance can be improved significantly. With the
since the association probability of a Tier-2 BS, A , increases meanpacketarrivalrateofeachuserγ =1.9,forinstance,the
2
asthenormalizedbiasingfactorB(cid:101)2 increasesaccordingto(8), optimalnormalizedbiasingfactorisobtainedasB(cid:102)2∗ =1.7dB,
more users that associate with Tier-1 BSs would be offloaded and the corresponding network mean queuing delay D can be
to Tier-2 BSs, which leads to an increment of ρ according reduced to be 48 ms.
2
to Lemma 2. Moreover, due to a larger deployment intensity Recall that it is indicated in Lemma 5 that when the
(cid:110) (cid:111)
oftheTier-2BSs,theusersthatoriginallyassociatewithonly mean packet arrival rate satisfies γ < min (Z+1)λkRk , the
∀k λuL
one Tier-1 BS can be offloaded to several neighboring Tier-2 minimum lower bound of the network mean queuing delay
BSs.Hence,thedeclinerateofρ1 islargerthantheincreasing D¯∗ can be obtained by combining (18) and (26). Fig. 3
rateofρ2.ItcanbeclearlyseenfromFig.1thatthesimulation further compares the minimum lower bound of the network
results match with the analysis well with a wide range of the mean queuing delay D¯∗ obtained by combining (18) and (25)
normalized biasing factor, indicating that replacing each BS’s with that by combining (18) and (26), respectively. It can be
traffic intensity by the average traffic intensity in (9) achieves observed from Fig. 3 that the gap between the two curves
a good approximation. diminishes as the mean packet arrival rate of each user γ
(cid:110) (cid:111)
Fig. 2 further demonstrates how the network mean queuing decreases. When γ <min (Z+1)λkRk , the minimum lower
delay D, as well as its lower bound D¯, vary with the ∀k λuL
bound of the network mean queuing delay D¯∗ obtained by
normalized biasing factor B(cid:101)2. For the sake of comparison,
combining (18) and (26) is quite close to that obtained by
the y-axis on the left hand side of Fig. 2 denotes the network
combining (18) and (25). When the mean packet arrival rate
meanqueuingdelayD whileontherighthandsideitdenotes (cid:110) (cid:111)
the lower bound D¯. To obtain the network mean queuing of each user γ is large, i.e., γ ≥m∀ikn (Z+λ1u)LλkRk , there is a
delay in simulations, BSs that have an unbounded queuing large gap between the curves in Fig. 3. Therefore, the optimal
delay are not taken account of. It can be observed from association probabilities {A∗k}∀k should be instead obtained
Fig. 2 that the trend of the network mean queuing delay D by numerically solving (25). As Lemma 4 guarantees, (25)
resembles that of its lower bound D¯. Both D and D¯ are has a unique solution within the feasible region A, which is
very sensitive to the normalized biasing factor B(cid:101)2. If B(cid:101)2 is the optimal association probability {A∗k}∀k.
not carefully tuned, the delay performance could be greatly Fig.4furtherillustrateshowtheoptimalnormalizedbiasing
degraded. For example, when γ = 1.9, the network mean factor, B(cid:102)∗, and the corresponding minimum lower bound of
2
queuing delay D is as high as 135 ms with the normalized thenetworkmeanqueuingdelay,D¯∗,varywiththebandwidth
biasingfactorB(cid:101)2 =−10dB,whichisnotacceptabletomany ratioofTier2,W2/W,withvariousvaluesofthemeanpacket
delay-sensitive applications. Moreover, due to a similar trend arrival rate of each user γ. Note that the total bandwidth
between the network mean queuing delay D and its lower W =W +W isfixedhere.ItcanbeobservedfromFig.4(a)
1 2
bound D¯, the optimal normalized biasing factor of D¯ is close that for a given γ, the optimal normalized biasing factor B(cid:102)∗
2
8
(a)
(a)
(b)
(b)
Fig.4. OptimalnormalizedbiasingfactorB(cid:102)∗andtheminimumlowerbound
of the network mean queuing delay D¯∗ ver2sus the bandwidth ratio of Tier
2 W2/W with various values of the mean packet arrival rate of each user
γ. W=12MHz and τ = 1. (a) Optimal normalized biasing factor B(cid:102)∗. (b)
MinimumlowerboundofthenetworkmeanqueuingdelayD¯∗. 2
increasesasW /W increases.Intuitively,asthebandwidthof
2
Tier2,W ,increases,Tier-2BSscanprovideahigherservice
2
rate to the associated users. The optimal B(cid:101)∗ should thus
2
become larger so as to encourage more users to be associated
with Tier-2 BSs. Moreover, it can be observed from Fig. 4(a)
thatasW /W increases,theoptimalnormalizedbiasingfactor
2
B(cid:101)2∗ becomes insensitive to the mean packet arrival rate of (c)
each user γ. The minimum lower bound of the network mean
queuing delay D¯∗, on the other hand, decreases as W /W Fig. 5. The network SIR coverage and the network mean queuing delay
2 performancewithvariousbandwidthratiosofTier2W2/W.γ=1.8,W =
increases, as Fig. 4(b) demonstrates. 12MHz, and τ = 1. (a) Network SIR coverage P[SIR>τ]. (b) Network
While minimizing the network mean queuing delay is meanqueuingdelayD.(c)Lowerboundofthenetworkmeanqueuingdelay
D¯.
desirableforreal-timetraffic,theSIRcoverageisanimportant
9
performance metric to support non-real-time traffic for ser-
vice providers. According to (11), the network SIR coverage
P[SIR>τ] can be written as
K
(cid:88)
P[SIR>τ]= A ·P[SIR >τ]
k k
k=1
K
=(cid:88) Ak . (29)
A ρ Z(τ,α,1)+1
k k
k=1
Fig. 5(a) demonstrates how the network SIR coverage
P[SIR>τ] varies with the normalized biasing factor B(cid:101)2
with various values of the bandwidth ratio W /W. It can
2
be observed from Fig. 5(a) that there exists an optimal nor-
malized biasing factor with which the network SIR coverage
is maximized. Intuitively, when B(cid:101)2 is too large, a large
fraction of users that are originally associated with Tier-1
BSs are offloaded to Tier-2 BSs. As these users are close (a)
to the interfering Tier-1 BSs and have long distances to
their associated Tier-2 BSs, they have very poor channel
conditions,whichleadstoalowSIRcoverageofthenetwork.
Similarly,whenB(cid:101)2istoosmall,thenetworkSIRcoveragealso
deteriorates. In addition, it can be seen from Fig. 5(a) that
the optimal normalized biasing to maximize P[SIR>τ] is
insensitive to the bandwidth allocation. In the meanwhile, the
optimalnormalizedbiasingfactorB(cid:102)∗ tominimizeD¯ increases
2
as W /W increases, as illustrated in Fig. 5(c) indicating a
2
tradeoff between the network mean queuing delay and the
network SIR coverage. For example, if W /W = 1/2, the
2
optimal normalized biasing factor is obtained as B(cid:102)∗ =24dB,
2
with which the network SIR coverage greatly deteriorates. In
thiscase,theserviceprovidersshouldproperlytunethebiasing
factor in HetNets such that a desired point on the tradeoff
curve can be achieved to balance the performances of real-
time traffic and non-real-time traffic.
(b)
As the SIR threshold τ critically determines the network
mean queuing delay and the network SIR coverage, Fig. 6
further demonstrates the impact of the SIR threshold τ on
these two performance metrics. It can be observed from Fig.
6 that for a given normalized biasing factor B(cid:101)2, the network
SIR coverage P[SIR>τ] decreases as the SIR threshold τ
increases. In the meanwhile, both the network mean queuing
delay D and its lower bound D¯ decrease as τ increases.
Intuitively, with a higher SIR threshold τ, the mean aggregate
packetarrivalrateofeachBSbecomeslowerwhiletheservice
rate becomes higher, leading to a better queuing performance.
In addition, it is illustrated in Fig. 6(a) that the optimal nor-
malized biasing factor to maximize the network SIR coverage
P[SIR>τ] is insensitive to the SIR threshold τ, while the
optimalnormalizedbiasingfactorB(cid:102)∗ tominimizeD¯ increases
2
asτ decreases,asFig.6(c)demonstrates.Intuitively,although
the service rates of both macro and micro BSs become lower
with a smaller τ, macro BSs are more likely to become
(c)
overloaded as their deployment density is much lower than
that of micro BSs. The optimal normalized biasing factor B(cid:102)∗ Fig. 6. The network SIR coverage and the network queuing delay
2
shouldthusbecomelargertoundertaketheloadpressurefrom performance with various values of the SIR threshold τ. W1 = 8MHz,
W2 = 4MHz, and γ = 3.8, (a) Network SIR coverage P[SIR>τ]. (b)
macro BSs. By comparing Fig. 6(a) with Fig. 6(b) and Fig. Network mean queuing delay D. (c) Lower bound of the network mean
6(c), it can be found that with a smaller SIR threshold τ, the queuingdelayD¯.
10
deterioration of the network mean queuing delay D becomes λ2R2Z(cid:104)(cid:0)λ2R2 +4γλ λ R A2LZ(cid:1)12 −λ R (cid:105)
k k k k u k k k k k
muchmoresevereifthenormalizedbiasingfactorisoptimally = >0.
tuned to maximize the network SIR coverage P[SIR>τ], ∆12
(31)
indicatingamoresignificanttradeoffbetweenthenetworkSIR
coverage and the network mean queuing delay. By combining (30) and (31), we have dρk > 0, which
dAk
indicates that ρ monotonically increases as A increases.
k k
V. CONCLUSIONANDFUTUREWORK
In this paper we have studied how to optimally tune the
biasing factor of each tier in HetNets in order to minimize a APPENDIXB
lower bound of the network mean queuing delay. It is shown PROOFOFLEMMA3
that the network queuing performance can be significantly
improved when the biasing factor of each tier is optimally
Proof: It has been shown in Lemma 2 that the average
tuned. The characterization of the optimal biasing factor pro-
traffic intensity ρ monotonically increases as the association
k
vides guidance for real-time service provisioning in HetNets.
probability A increases. With A <1, we then have
k k
The case study of a 2-tier HetNet further illustrates that the
(cid:104) (cid:105)1
network mean queuing delay and the network SIR coverage −λ R + (λ R )2+4γλ λ R A2LZ 2
k k k k u k k k
might not be optimized simultaneously by tuning the biasing ρ =
k 2A λ R Z
factor,indicatingaperformancetradeoffbetweenreal-timeand k k k
non-real-time services.
(cid:104) (cid:105)1
It is worth mentioning that it is assumed in this paper that −λ R + (λ R )2+4γλ λ R LZ 2
k k k k u k k
oneBSwillserveauserwithaconstantrateifitsSIRexceeds < . (32)
2λ R Z
a threshold. In practice, nevertheless, the service rate could k k
depend on the channel conditions. In this case, as the biasing In the following, we divide the discussion into two parts:
factor of one tier decreases, the mean service rate of this tier 1) If −λkRk+[(λkRk)2+4γλuλkRkLZ]12 < 1, i.e., γ <
increases as the users located at the edge of the cells are 2λkRkZ
offloaded. The queuing performance of this tier can thus be (Z+1)λkRk, we have
λuL
improved due to a lower mean aggregate packet arrival rate
ρ <1 (33)
and a higher mean service rate. Therefore, there would exist k
anoptimalbiasingfactorforeachtiersuchthatthetrafficload according to (32). In this case, D¯ will always be bounded if
k
is balanced across tiers and the network mean queuing delay γ < (Z+1)λkRk.
λuL
is minimized. On the other hand, if the biasing factor of one 2) If γ > (Z+1)λkRk, D¯ will be bounded if and only if
tier is too large, the SIR coverage of this tier degrades, which λuL k
would drag down the network SIR coverage. Therefore, the (cid:104) (cid:105)1
−λ R + (λ R )2+4γλ λ A2R LZ 2
network mean queuing delay may be optimized at the cost of k k k k u k k k
<1. (34)
the network SIR coverage. The tradeoff between the network 2A λ R Z
k k k
SIRcoverageandthenetworkmeanqueuingdelayinthiscase
Accordingly, we have
is an interesting issue that needs further study.
λ R
Inaddition,itisassumedthatorthogonalspectrumresources A < k k . (35)
k γλ L−λ R Z
are allocated to different tiers. In practice, nevertheless, uni- u k k
versal frequency reuse may be adopted so that all the other
BSs may act as interfering sources for one BS. Therefore, the
averagetrafficintensitiesofdifferenttierswouldbecorrelated.
The characterization of the queuing performance under such
APPENDIXC
circumstances deserves much attention in future study.
PROOFOFCONVEXITYOF(19)
APPENDIXA
Proof: According to (17), the second-order derivative of
PROOFOFLEMMA2
D¯ with respect to A can be written as
k k
Proof: According to (12), the first-order derivative of the
average traffic intensity ρk with respect to Ak is given by d2D¯k = 2L ·(cid:18)dρk (cid:19)2+ L · d2ρk .
dρk = 4γLλuλ2kRk2A2kZ2∆−12−λkRkZ(cid:16)−λkRk+∆12(cid:17), dAk2 Rk(1−ρk)3 dAk Rk(1−ρk)2 dAk(326)
dAk 2(AkλkRkZ)2 Substituting (30) into (36) yields
(30) d2D¯ L (cid:34) (cid:18)dρ (cid:19)2 d2ρ (cid:35)
k > · 2 k + k
where ∆=λ2kRk2 +4γλuλkRkA2kLZ. The numerator on the dAk2 Rk(1−ρk)2 dAk dAk2
right hand side of (30) can be further written as (cid:32)
L
(cid:16) (cid:17) = · 4γλ Lλ2R2Z2A3 +2∆
4γLλuλ2kRk2A2kZ2∆−12 −λkRkZ −λkRk+∆12 Rk(1−ρk)2A4kZ2∆ u k k k