Ultra-Dense Networks: A New Look at the Proportional Fair Scheduler
Ming Ding David Lopez Perez Amir H. Jafari Guoqiang Mao Zihuai Lin









Ultra-Dense Networks: A New Look at 

the Proportional Fair Scheduler

    Ming Ding^, David López
Pérez^†, Amir
H. Jafari^∗, Guoqiang Mao^∦^,
Zihuai Lin^

^Data61, Australia, ^†Nokia
Bell Labs, Ireland

^∦School of Computing and Communication,
University of Technology Sydney, Australia

^∗Dept. of Electronic & Electrical
Engineering, University of Sheffield, UK

^The University of Sydney,
Australia

    December 30, 2023
============================================================================================================================================================================================================================================================================================================================================


In this paper, we theoretically study the proportional fair (PF) scheduler
in the context of ultra-dense networks (UDNs). Analytical results
are obtained for the coverage probability and the area spectral efficiency
(ASE) performance of dense small cell networks (SCNs) with the PF
scheduler employed at base stations (BSs). The key point of our analysis
is that the typical user is no longer a random user as assumed in
most studies in the literature. Instead, a user with the maximum PF
metric is chosen by its serving BS as the typical user. By comparing
the previous results of the round-robin (RR) scheduler with our new
results of the PF scheduler, we quantify the loss of the multi-user
diversity of the PF scheduler with the network densification, which
casts a new look at the role of the PF scheduler in UDNs. Our conclusion
is that the RR scheduler should be used in UDNs to simplify the radio
resource management (RRM)
[To appear in IEEE GLOBECOM2017. 1536-1276  2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. Please find the final version in IEEE from the link: http://ieeexplore.ieee.org/document/xxxxxxx/. Digital Object Identifier: 10.1109/xxxxxxxxx]

, which is justified by the diminishing multi-user diversity of the
PF scheduler in dense SCNs

.

For the case of sparse networks, we further derive an easy-to-compute
upper-bound performance.

 

In particular, for practical power models and 3GPP path loss functions,
our studies reveal that compared with the baseline scheme with all
BSs active, various IMC modes with different levels of power consumption
can double or even triple the EE performance in dense SCNs, e.g.,
when the BS density is 10^3 BSs/km^2.


 The results derived from our analysis shed valuable new light on
the deployment and the operation of future dense SCNs in 5G.


^Guoqiang Mao's research is supported by the Australian
Research Council (ARC) Discovery projects DP110100538 and DP120102030
and the Chinese National Science Foundation project 61428102.


 and it can largely improve the performance of the 5th-generation
(5G) networks


Stochastic geometry, Line-of-sight (LoS), Non-line-of-sight (NLoS),
Dense small cell networks (SCNs), Coverage probability, Area spectral
efficiency, Energy efficiency.






§ INTRODUCTION



Placeholder


Network densification is envisioned to be the key solution to meet
users' traffic demands in the 5th-generation (5G) networks <cit.>

, since it has the potential to linearly improve the network capacity
with the increasing number of deployed cells

.

Hence, dense small cell networks (SCNs) have attracted significant
attention as one of the most promising approaches to rapidly increase
network capacity and meet the ever-increasing data traffic demands.

 Indeed, the orthogonal deployment[The orthogonal deployment means that small cells and macrocells operate
on different frequency spectrum, i.e., Small Cell Scenario #2a <cit.>).] of dense small cell networks (SCNs) within the existing macrocell
networks <cit.> has been the workhorse for capacity enhancement
in the 4th-generation (4G), developed by the 3rd Generation Partnership
Project (3GPP), and this approach to capacity enhancement will continue
in 5G with the adoption of ultra-dense networks (UDNs) <cit.>.

This is due to its large spectrum reuse and its easy management; the
latter one arising from its low interaction with the macrocell tier,
e.g., no inter-tier interference.

 In this paper, we focus on the analysis of such orthogonal deployment
of UDNs.

Despite of its benefits, the SCN densification also opens up new research
questions. In particular, scheduling has been conceived as an effective
technique used at base stations (BSs) to efficiently use the available
spectrum and improve the overall system throughput. In most cases,
a proportional fair (PF) scheduler <cit.> serves as an
appealing technique that offers a good trade-off between maximizing
overall throughput and improving fairness among user equipments (UEs)
with diverse channel conditions. However, the gains of the PF scheduler
may be limited in UDNs, mostly because the number of UEs per active
BS is considerably reduced

, and partly because UEs may not experience very different channel
conditions on different subcarriers due to the dominance of LOS considering
the closeness of the UE and its serving BS

. This gives rise to the question of whether the PF scheduling
is as efficient for UDNs as it is for sparse networks, or whether
it can be substituted by other ones of lower complexity, such as a
round-robin (RR) one that semi-randomly selects UEs to serve. In this
paper, we answer this fundamental question by theoretical analyses.

This paper answers this fundamental question by comparing the performance
of the PF scheduler and that of the RR scheduler as the SCN densifies.


Network densification also brings questions about fundamental trade-offs.
The smaller the cell size, the closer the UE to its serving BS –
reducing the path loss, but the stronger the LOS – reducing
the multi-user diversity. Furthermore, decreasing the cell size will
not only diminish the multi-user diversity, but also will increase
the interference due to the dominance of LOS. As a result, it is necessary
to take a systematic view towards the tradeoffs of network densification.


To the best of our knowledge, there has been no prior work on the
theoretical study of the PF scheduler in the context of UDNs, the
density of which could be as large as tens of thousands of BSs per
km^2 <cit.>. Generally speaking, the
existing work on PF schedulers does not scale well with the network
densification. In more detail, the studies in the literature can be
classified into 3 categories: (i) analysis of a scenario with
merely one BS

, e.g., based on the Gaussian approximation (GA) of the instantaneous
data rate per user

 <cit.>, (ii) analysis of a
scenario with a limited number of BSs <cit.>,
which quickly becomes computationally infeasible for UDNs

, and does not consider the reduction of the number of UEs per BS
as the network densifies

, and (iii) system-level simulations for large-scale networks <cit.>,
which lacks analytical rigor.

Placeholder


Compared with the existing work, the main contributions of this paper
are:


  * For the first time, we use stochastic geometry <cit.>
to derive the analytical results of the coverage probability and the
area spectral efficiency (ASE) performance for UDNs with the PF schedulers
used at BSs. The key point of our analysis is that the typical user
is no longer a random user as assumed in most studies of stochastic
geometry <cit.>.

Instead, a user with the maximum PF metric is chosen by its serving
BS as the typical user.


For the case of sparse networks, we further derive an easy-to-compute
upper-bound performance.


  * By comparing the previous results of the RR scheduler and our new
results of the PF scheduler, we quantify the loss of the multi-user
diversity of the PF scheduler with the network densification, which
leads to the conclusion that the RR scheduler should be used in UDNs
to simplify the radio resource management (RRM).


Placeholder


The rest of this paper is structured as follows. Section <ref>
describes the network scenario and the wireless system model considered
in this paper. Section <ref> presents our theoretical
results on the coverage probability and the ASE for UDNs while using
PF scheduling. The numerical results are discussed in Section <ref>,
with remarks casting a new look at the PF scheduler in the context
of UDNs. The conclusions are drawn in Section <ref>.





§ NETWORK SCENARIO AND SYSTEM MODEL



Placeholder


In this section, we present the network scenario, the wireless system
model and the PF scheduler considered in this paper.



 §.§ Network Scenario



Placeholder


For a certain time-frequency resource block, we consider a downlink
(DL) cellular network with BSs deployed on a plane according to a
homogeneous Poisson point process (HPPP) Φ with a density of
λ BSs/km^2. Active UEs are also Poisson distributed
in the considered DL network with a density of ρ UEs/km^2.
Here, we only consider active UEs in the network because non-active
UEs do not trigger data transmission

, and thus they are ignored in our analysis

.

Note that the total UE number in cellular networks should be much
higher than the number of the active UEs, but at a certain time slot
and on a certain frequency band, the active UEs with data traffic
demands are not too many.

 As shown in <cit.>, a typical density of the active
UEs in 5G is around 300UEs/km^2.

In practice, a BS will enter an idle mode if there is no UE connected
to it, which reduces the interference to neighboring UEs as well as
the energy consumption of the network. The set of active BSs should
be determined by a user association strategy (UAS). In this paper,
we assume a practical UAS as in <cit.>, where each
UE is connected to the BS having the maximum average received signal
strength, which will be formally presented in Subsection <ref>.

Note that in our previous work <cit.> and
some other existing work, e.g., <cit.>,
it was assumed that each UE should be associated with its closest
BS. Such assumption is not appropriate for the considered path loss
model in (<ref>), because in practice a UE should
connect to a BS offering the largest received signal strength. Such
BS does not necessarily have to be the nearest one to the UE, and
it could be a farther one with a strong LoS path.



Note that such BS idle mode operation is not trivial, which even changes
the capacity scaling law <cit.>. Since UEs are
randomly and uniformly distributed in the network, we assume that
the active BSs also follow an HPPP distribution Φ̃ <cit.>,
the density of which is denoted by λ̃ BSs/km^2.
Note that λ̃≤λ and λ̃≤ρ,
since one UE is served by at most one BS.

Also note that a larger ρ requires more active BSs with a larger
λ̃ to serve the active UEs.



From <cit.>, λ̃
is given by

 it was shown that the formula proposed in <cit.>
to calculate λ̃ is accurate for UDNs, which is given
by


    λ̃=λ[1-1/(1+ρ/qλ)^q],

where according to <cit.>, q depends on the path
loss model, which will be presented in Subsection <ref>.

According to <cit.>, 

Considering both active and inactive BSs,

the per-BS coverage area size X can be approximately characterized
by a Gamma distribution and the probability density function (PDF)
of X can be expressed as

    f_X(x)=(qλ)^qx^q-1exp(qλ x)/(q),

where (·) is the Gamma function <cit.>.
The UE number per BS is denoted by a random variable (RV) K, and
the probability mass function (PMF) of K can be calculated as

    f_K(k)    =    [K=k]
       (a)=   ∫_0^+∞(ρ x)^k/k!exp(-ρ x)f_X(x)dx
       (b)=   Γ(k+q)/Γ(k+1)Γ(q)(ρ/ρ+qλ)^k(qλ/ρ+qλ)^q,

where (a) is due to the HPPP distribution of UEs and
(b) is obtained from (<ref>).

Note that f_K(k) satisfies the normalization condition:
∑_k=0^+∞f_K(k)=1.

It can be seen from (<ref>) that K follows a Negative
Binomial distribution <cit.>, i.e., K∼NB(q,ρ/ρ+qλ).

As discussed in Subsection <ref>, we assume
that a BS with K=0 is not active

, which will be ignored in our analysis due to its muted transmission

. Thus, we focus on the active BSs and denote the UE number
per active BS by a positive RV K̃. Considering (<ref>)

and the fact that the only difference between K and K̃
lies in K̃≠0

, we can conclude that K̃ follows a truncated Negative Binomial
distribution

, i.e., K̃∼truncNB(q,ρ/ρ+qλ)

, the PMF of which is denoted by f_K̃(k̃),k̃∈{ 1,2,…,+∞}
and can be written as

    f_K̃(k̃)=[K̃=k̃]=f_K(k̃)/1-f_K(0).


where the denominator (1-f_K(0)) represents
the probability of a BS being active. Note that based on the definition
of λ̃ in Subsection <ref>,
we have that λ̃=(1-f_K(0))λ.


Note that f_K̃(k̃) satisfies the normalization
condition: ∑_k=1^+∞f_K̃(k̃)=1.



Example: As an example, we plot the results of f_K̃(k̃)
in Fig. <ref> with
the following parameter values[With a 3GPP path loss model incorporating both line-of-sight (LoS)
and non-LoS (NLoS) transmissions <cit.>, we have shown
in <cit.> that q=4.05 when the UE density equals
to ρ=300UEs/km^2. Note that our analytical
results in this paper can work with any value of q.]: UE density ρ=300UEs/km^2, BS density
λ∈{ 50,200,1000}BSs/km^2
and q=4.05.


 From this figure, we can draw the following observations:


  * The analytical results based on the truncated Negative Binomial distribution
match well with the simulation results. More specifically, the maximum
difference between the simulated PMF and the analytical PMF is shown
to be less than 0.5 percentile.

  * For the active BSs, f_K̃(k̃) shows a
more dominant peak at k̃=1 as λ, and in turn λ̃,
increases. This is because the ratio of ρ to λ̃
gradually decreases toward 1 as λ increases, approaching
the limit of one UE per active BS in ultra-dense SCNs. In particular,
when λ=1000BSs/km^2, more than 80%
of the active BSs will serve only one UE. Intuitively speaking, each
of those BSs should dynamically engage all subframes for DL or UL
transmissions based on the specific traffic demand of its served UE.
In such way, the subframe resources can be fully utilized in dynamic
TDD. Note that such operation is not available in static TDD due to
its fixed TDD configuration.


Furthermore, the cumulative mass function (CMF) of K̃ can
be written as

    F_K̃(k̃)=∑_t=1^k̃f_K̃(t).





 §.§ Wireless System Model



Placeholder


Following <cit.>, we adopt a general path loss
model, where the path loss ζ(r) is a multi-piece
function of r written as



    ζ(w)=ζ_1(w)=[  ζ_1^L(w),; ζ_1^NL(w), ]   [     with probability Pr_1^L(w); with probability (1-Pr_1^L(w)) ],    when 0≤ w≤ d_1
    ζ_2(w)=[  ζ_2^L(w),; ζ_2^NL(w), ]   [     with probability Pr_2^L(w); with probability (1-Pr_2^L(w)) ],    when d_1<w≤ d_2
    ⋮   ⋮
    ζ_N(w)=[  ζ_N^L(w),; ζ_N^NL(w), ]   [     with probability Pr_N^L(w); with probability (1-Pr_N^L(w)) ],    when w>d_N-1.




    ζ(r)=ζ_1(r),    when 0≤ r≤ d_1
    ζ_2(r),    when d_1<r≤ d_2
    ⋮   ⋮
    ζ_N(r),    when r>d_N-1,

where each piece ζ_n(r),n∈{ 1,2,…,N}
is modeled as

    ζ_n(r)=[    ζ_n^L(r)=A_n^Lr^-α_n^L,; ζ_n^NL(r)=A_n^NLr^-α_n^NL, ]   [    LoS Prob.: Pr_n^L(r); NLoS Prob.: 1-Pr_n^L(r) ],

where

In practice, A_n^L, A_n^NL,
α_n^L and α_n^NL are constants
obtainable from field tests <cit.>.




  * ζ_n^L(r) and ζ_n^NL(r),n∈{ 1,2,…,N}
are the n-th piece path loss functions for the LoS transmission
and the NLoS transmission, respectively,

  * A_n^L and A_n^NL are the path losses
at a reference distance r=1 for the LoS and the NLoS cases, respectively,

  * α_n^L and α_n^NL are the path
loss exponents for the LoS and the NLoS cases, respectively.

Moreover, Pr_n^L(r) is the n-th
piece LoS probability function that a transmitter and a receiver separated
by a distance r has a LoS path, which is assumed to be a
monotonically decreasing function with regard to r

. Such assumption has been confirmed by existing measurement studies

 <cit.>. 

For convenience, {ζ_n^L(r)}
and {ζ_n^NL(r)} are
further stacked into piece-wise functions written as

    ζ^Path(r)=ζ_1^Path(r),    when 0≤ r≤ d_1
    ζ_2^Path(r),   when d_1<r≤ d_2
    ⋮   ⋮
    ζ_N^Path(r),    when r>d_N-1,

where the string variable Path takes the value of “L” and “NL”
for the LoS and the NLoS cases, respectively.


For convenience, {Pr_n^L(r)}
is stacked into a piece-wise function as

    Pr^L(r)=Pr_1^L(r),    when 0≤ r≤ d_1
    Pr_2^L(r),   when d_1<r≤ d_2
    ⋮   ⋮
    Pr_N^L(r),    when r>d_N-1.



As a special case to show our analytical results, we consider the
two-piece path loss and the exponential LoS probability functions
defined by the 3GPP <cit.>. Specifically, we have N=2,
ζ_1^L(w)=ζ_2^L(w)=A^Lw^-α^L,
ζ_1^NL(w)=ζ_2^NL(w)=A^NLw^-α^NL,
Pr_1^L(w)=1-5exp(-R_1/w),
and Pr_2^L(w)=5exp(-w/R_2),
where R_1=156 m, R_2=30 m, and d_1=R_1/ln10=67.75 m <cit.>.
For clarity, this case is referred to as the 3GPP Case hereafter.

As discussed in Subsection <ref>, we assume
that each UE is connected to the BS having the maximum average received
signal strength, which is equivalent to the BS with the largest ζ(r).

Note that the generality and the practicality of the adopted path
loss model (<ref>) have been well established in <cit.>.
In more detail, this model is consistent with the ones adopted in
the 3GPP <cit.>, <cit.>, and includes
those models considered in <cit.> and <cit.>
as its special cases. In <cit.>, only NLoS transmissions
were considered, i.e., assuming Pr_n^L(r)=0,∀ n∈{ 1,2,…,N}
in (<ref>). In <cit.>, only
one piece of path loss function with one LoS path loss exponent and
one NLoS path loss exponent was considered, i.e., assuming N=1
in (<ref>).


Another important note is on the practical usage of the proposed path
loss model, which is summarized in the following:


  * As have been addressed in <cit.>, the standard
single-slope path loss function does not accurately capture the dependence
of the path loss exponent on the link distance in many future networks,
such as the multi-ray transmission environments, the dense/clustered
small cells, the millimeter wave communications with blockage, etc.
Therefore, a dual or more slope path loss function such as (<ref>)
should be considered.

  * The realistic LoS probability functions usually take complicated mathematical
forms, e.g., a sum of exponential functions in the 3GPP standards <cit.>,
which will be addressed in more detail in Section . Therefore, to
achieve both the analytical tractability and the accurate results,
it is desirable to approximate the complicated LoS probability function
as a few pieces of elementary functions such as the linear function.
Such piece-wise LoS probability function is well captured by (<ref>)
and can be studied in our framework.

  * With the justification of both (<ref>) and (<ref>),
we can conclude that the proposed path loss model in(<ref>)
is versatile to cope with various practical network scenarios.


 Finally, we assume that each BS/UE is equipped with an isotropic
antenna, and that the multi-path fading between a BS and a UE is modeled
as independently identical distributed (i.i.d.) Rayleigh fading <cit.>.
In order to make the 3GPP Case even more practical, we will further
consider distance-dependent Rician fading <cit.>

and correlated shadow fading <cit.>

 in our simulations to show their minor impact on our conclusions.
More specifically, we adopt the practical Rician fading defined in
the 3GPP <cit.>, where the K factor in dB
unit (the ratio between the power in the direct path and the power
in the other scattered paths) is modeled as K[dB]=13-0.03r,
where r is the distance in meter. 

We consider a practical correlated shadow fading defined in 3GPP <cit.>,
where the shadow fading in dB is modeled as zero-mean Gaussian random
variables, e.g., with a standard deviation of 10dB <cit.>.
The correlation coefficient between the shadow fading values associated
with two different BSs is denoted by τ, e.g., τ=0.5 in <cit.>.





 §.§ The PF Scheduler



Placeholder


The original operation of the PF scheduler is as follows <cit.>,


  * First, the average throughput of each UE is tracked by an exponential
moving average at the BS.

  * Second, each UE frequently feeds back its channel state information
(CSI) to its serving BS, so that such BS can calculate the ratio of
the instantaneous achievable rate to the average throughput for each
user, which is defined as a PF metric for UE selection.

  * Finally, the UE with the maximum PF metric will be selected for DL
transmission, which is formulated as

    u^*=u∈{ 1,2,…,k̃}max{R̃_u/R̅_u} ,

where u, u^*, R̃_u and R̅_u denote the
UE index, the selected UE index, the instantaneous achievable rate
of UE u and the average throughput of UE u, respectively. Note
that the distribution of k̃ has been discussed in (<ref>).

according to our modeling of the UE number per active BS in (<ref>),
we have that u∈{1,2,…,k̃}.




Placeholder


From a network performance analysis point of view, it is very difficult,
if not impossible, to analyze the original PF scheduler given by (<ref>).
This is because that the objective of a performance analysis is usually
to derive the average user throughput R̅_u or aggregate
inter-cell interference, but in this case it is part of the PF metric,
i.e., R̃_u/R̅_u, and it should be known
and plugged into the UE selection criterion of (<ref>)
before the performance analysis of R̅_u is carried out.
A widely adopted approach to tackle this dilemma is to use alternative
measures of CSI in a PF metric, instead of R̃_u and R̅_u <cit.>.

Fortunately, there is another way without approximation, but requiring
modification of the definition of the PF metric.


Here, we follow the framework developed in <cit.>, where
the authors proposed to use the ratio of the instantaneous signal-to-noise
ratio (SNR) to the average SNR as a PF metric instead of the original
one. More specifically, the UE selection criterion of the PF scheduler
proposed in <cit.> is given by

    u^*=u∈{ 1,2,…,k̃}max{Z̃_u/Z̅_u} ,

where Z̃_u and Z̅_u denote the instantaneous
SNR of UE u and the average SNR of UE u, respectively. Although
this criterion of (<ref>) is not exactly the same as that
of (<ref>), it captures the important characteristics
of the PF scheduler: (i) allowing preference to UEs with relatively
good instantaneous channels with respect to their average ones since
R̃_u is a strictly monotonically increasing function
of Z̃_u, and (ii) allocating the same portion
of resource to each UE in the long term to enforce fairness, because
the chance of Z̃_u≥Z̅_u is almost the same for
all UEs. Since the accuracy and the practicality of (<ref>)
have been well established in <cit.>, we will focus on
studying the PF scheduler characterized by (<ref>)

in the following sections

.



§ MAIN RESULTS



Placeholder


In this section, we study the coverage probability and the ASE performance

of SCNs in terms of the coverage probability and the ASE by considering
the performance

of a typical UE located at the origin o.



 §.§ The Coverage Probability



Placeholder




Considering the path loss model in (<ref>)
and the PF scheduler model in (<ref>), we can derive p^cov(λ,γ)
as

    p^cov(λ,γ)=∑_n=1^N(T_n^L+T_n^NL),

where T_n^L=∫_d_n-1^d_n𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} f_R,n^L(r)dr,
T_n^NL=∫_d_n-1^d_n𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}(r)dr,
and d_0 and d_N are defined as 0 and +∞, respectively.
Moreover, f_R,n^L(r) and f_R,n^NL(r)
(d_n-1<r≤ d_n), are represented by

    f_R,n^L(r)=exp(-∫_0^r_1(1-Pr^L(u))2π uλ du)exp(-∫_0^rPr^L(u)2π uλ du)Pr_n^L(r)2π rλ,

and

    f_R,n^NL(r)=exp(-∫_0^r_2Pr^L(u)2π uλ du)exp(-∫_0^r(1-Pr^L(u))2π uλ du)(1-Pr_n^L(r))2π rλ,

where r_1=r_1{ζ^NL(r_1)=ζ_n^L(r)}
and r_2=r_2{ζ^L(r_2)=ζ_n^NL(r)}.


are given implicitly by the following equations as

    r_1=r_1{ζ^NL(r_1)=ζ_n^L(r)} ,

and

    r_2=r_2{ζ^L(r_2)=ζ_n^NL(r)} .





See Appendix A.





Considering the truncated Negative
Binomial distribution of the UE number per active BS, K̃,
characterized in (<ref>), we can derive 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]},
which will be used in Theorem <ref> as

    𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]} =∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;  t ])(-δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))]f_K̃(k̃),

where K̃^max is a large enough integer that makes
F_K̃(K̃^max) in (<ref>)
close to one with a gap of a small value ϵ so that the expectation
value in (<ref>) can be accurately evaluated
over k̃, f_K̃(k̃) is obtained
from (<ref>), δ_n^L(r)
is expressed by

    δ_n^L(r)=exp(-γ P_N/Pζ_n^L(r)),

and ℒ_I_agg^L(s) is
the Laplace transform of I_agg for LoS signal transmission
evaluated at s, which can be further written as

    ℒ_I_agg^L(s)=exp(-2πλ̃∫_r^+∞Pr^L(u)u/1+(sPζ^L(u))^-1du)exp(-2πλ̃∫_r_1^+∞[1-Pr^L(u)]u/1+(sPζ^NL(u))^-1du).


In a similar way, 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}
is computed by

    𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]} =∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;  t ])(-δ_n^NL(r))^tℒ_I_agg^NL(tγ/Pζ_n^NL(r))]f_K̃(k̃),

where δ_n^NL(r) is expressed by

    δ_n^NL(r)=exp(-γ P_N/Pζ_n^NL(r)),

and ℒ_I_agg^NL(s) is
the Laplace transform of I_agg for NLoS signal transmission
evaluated at s, which can be further written as

    ℒ_I_agg^NL(s)=exp(-2πλ̃∫_r_2^+∞Pr^L(u)u/1+(sPζ^L(u))^-1du)exp(-2πλ̃∫_r^+∞[1-Pr^L(u)]u/1+(sPζ^NL(u))^-1du).



See Appendix B.




First, we investigate the coverage probability that the typical UE's
signal-to-interference-plus-noise ratio (SINR) is above a designated
threshold γ:

    p^cov(λ,γ)=Pr[SINR>γ],

where the typical UE's SINR is computed by

    SINR=Pζ(r)y(k̃)/I_agg+P_N,

where y(k̃) is the channel gain on condition
of the UE number k̃

 in the typical UE's serving BS

, P and P_N are the BS transmission power and the
additive white Gaussian noise (AWGN) power at each UE, respectively,
and I_agg is the cumulative interference given by

    I_agg=∑_i: b_i∈Φ̃∖ b_oPβ_ig_i,

where b_o is the BS serving the typical UE, and b_i, β_i
and g_i are the i-th interfering BS, the path loss from b_i
to the typical UE and the multi-path fading channel gain associated
with b_i (exponentially distributed), respectively. Note that

when all BSs are assumed to be active, the set of all BSs Φ
should be used in the expression of I_agg <cit.>.
Here,

in (<ref>), only the BSs in Φ̃∖ b_o
inject effective interference into the network, where Φ̃
denotes the set of the active BSs. 

In other words, the BSs in idle modes are not taken into account in
the analysis of I_agg.


It is very important to note that the distribution of y(k̃)
should be derived according to (<ref>). More specifically,
based on the variable definition in (<ref>), we can reformulate
(<ref>) as

    u^*=u∈{ 1,2,…,k̃}max{Pζ(r)h_u/P_N/Pζ(r)×1/P_N} =u∈{ 1,2,…,k̃}max{ h_u} ,

where h_u is an i.i.d. RV with a unit-mean exponential
distribution due to our consideration of Rayleigh fading mentioned
in Subsection <ref>. Thus, y(k̃)
can be modeled as the maximum RV of k̃ i.i.d. exponential
RVs. The complementary cumulative distribution function (CCDF) of
y(k̃) is <cit.>

    F̅_Y(k̃)(y)=Pr[Y(k̃)>y]=1-(1-exp(-y))^k̃.

It is easy to see that Pr[Y(k̃)>y]
increases as k̃ grows, which in turn improves the typical
UE's channel gain. Note that for the RR scheduler, the typical UE
is randomly selected in the BS. Consequently, we have that k̃=1
in (<ref>)

, which leads to the degeneration of y(k̃) to
h_u

 and the analytical results for RR have been derived in <cit.>.


Based on the path loss model in (<ref>) and the
PF scheduler model in (<ref>), we present our result of
p^cov(λ,γ) in Theorem <ref>.

 shown on the next page




Digging into Theorem <ref> and considering the truncated
Negative Binomial distribution of the UE number per active BS, K̃
, we present our results on 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}
and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}
in Theorem <ref>.

Plugging Theorem <ref> into Theorem <ref>,
yields our theoretical results on p^cov(λ,γ).
From Theorems <ref> and <ref>,
we can draw an important and intuitive conclusion in Lemma <ref>.

The p^cov(λ,γ)
of the PF scheduler converges to that of the RR scheduler as .





See Appendix C.




Placeholder




𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}
and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}
can be respectively upper bounded by

    𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}≤∑_k̃=1^K̃^max{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃} f_K̃(k̃),

and

    𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}≤∑_k̃=1^K̃^max{ 1-[1-δ_n^NL(r)ℒ_I_agg^NL(γ/Pζ_n^NL(r))]^k̃} f_K̃(k̃).



See Appendix D.




Although we have obtained the closed-form expressions of p^cov(λ,γ)
for the PF scheduler in Theorems <ref> and <ref>,
it is important to note that Theorem <ref>
is computationally intensive for the case of sparse networks, where
the maximum UE number per active BS K̃^max could
be very large, leading to complex computations for ℒ_I_agg^L(tγ/Pζ_n^L(r))
and ℒ_I_agg^NL(tγ/Pζ_n^NL(r)),t∈{ 0,1,…,K̃^max}
in (<ref>) and (<ref>),
respectively. For example, when the UE density is ρ=300UEs/km^2
and the BS density is λ=10BSs/km^2,

the average UE number per active BS is around 30UEs/BS,
but

K̃^max should be as large as 102 to make F_K̃(K̃^max)
sufficiently close to one with a gap smaller than ϵ=0.001

 with a gap of 0.001

. As a result, we need to calculate the integrals in (<ref>)
and (<ref>) at least 102
times for every possible value of r.

 The problem becomes even worse when the network is more sparse than
λ=10BSs/km^2.

 The good news is that Theorem <ref>
is very efficient for 5G UDNs, where K̃^max is
expected to be less than 10 <cit.>. In the next
subsection, we derive alternative and more efficient expressions for
sparse networks.

Based on the considered UAS, it is straightforward to derive

    Pr[Pζ_n^L(r)g/I_agg+P_N>γ]     =    𝔼_[I_agg]{Pr[g>γ(I_agg+P_N)/Pζ_n^L(r)]}
        =    𝔼_[I_agg]{F̅_G(γ(I_agg+P_N)/Pζ_n^L(r))}
        =    𝔼_[I_agg]{𝔼_[k̃]{ 1-[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}}
        =    𝔼_[k̃]{ 1-𝔼_[I_agg]{[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}}
        =    𝔼_[k̃]{ 1-∑_t=0^k̃([ k̃;  t ])(-1)^texp(tγ P_N/Pζ_n^L(r))𝔼_[I_agg]{exp(tγ I_agg/Pζ_n^L(r))}}
        =    𝔼_[k̃]{ 1-∑_t=0^k̃([ k̃;  t ])(-1)^t(δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))}
        =    ∑_k̃=1^K̃^max[1-∑_t=0^k̃([ k̃;  t ])(-1)^t(δ_n^L(r))^tℒ_I_agg^L(tγ/Pζ_n^L(r))]f_K̃(k̃)







 §.§ A Low-Complexity Upper-Bound of p^cov(λ,γ)



Placeholder


We present upper bounds of 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}
and 𝔼_[k̃]{Pr[Pζ_n^NL(r)y(k̃)/I_agg+P_N>γ]}
in Theorem <ref>.

The proposed upper bounds in Theorem <ref>
require to calculate the integrals in (<ref>)
and (<ref>) only
once for every possible value of r, which makes the analysis of
sparse networks very efficient.

Based on the considered UAS, it is straightforward to derive

    Pr[Pζ_n^L(r)g/I_agg+P_N>γ]     =    𝔼_[I_agg]{Pr[g>γ(I_agg+P_N)/Pζ_n^L(r)]}
        =    𝔼_[I_agg]{F̅_G(γ(I_agg+P_N)/Pζ_n^L(r))}
        =    𝔼_[I_agg]{𝔼_[k̃]{ 1-[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}}
        =    𝔼_[k̃]{ 1-𝔼_[I_agg]{[1-exp(γ(I_agg+P_N)/Pζ_n^L(r))]^k̃}}
       ≤   𝔼_[k̃]{ 1-[1-𝔼_[I_agg]{exp(γ(I_agg+P_N)/Pζ_n^L(r))}]^k̃}
        =    𝔼_[k̃]{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃}
        =    ∑_k̃=1^K̃^max{ 1-[1-δ_n^L(r)ℒ_I_agg^L(γ/Pζ_n^L(r))]^k̃} f_K̃(k̃)



 Consequently, plugging Theorem <ref>
into Theorem <ref>, yields our theoretical results
on an upper bound of p^cov(λ,γ),
which is particularly useful for sparse networks.



 §.§ The Area Spectral Efficiency



Placeholder


We also investigate the area spectral efficiency (ASE) performance
in bps/Hz/km^2, which is defined as <cit.>

    A^ASE(λ,γ_0)=λ̃∫_γ_0^+∞log_2(1+γ)f_Γ(λ,γ)dγ,

where γ_0 is the minimum working SINR in a practical SCN,
and f_Γ(λ,γ) is the PDF of
the SINR γ observed at the typical UE for a particular value
of λ. Based on the definition of p^cov(λ,γ)
in (<ref>) and the partial integration theorem <cit.>,
(<ref>) can be reformulated as

Based on , which is the complementary cumulative distribution function
(CCDF) of SINR, f_Γ(λ,γ)
can be computed as

    f_Γ(λ,γ)=∂(1-p^cov(λ,γ))/∂γ.



    A^ASE(λ,γ_0)    =    λ̃/ln2∫_γ_0^+∞p^cov(λ,γ)/1+γdγ
          +λ̃log_2(1+γ_0)p^cov(λ,γ_0).


Regarding A^ASE(λ,γ_0), it is
important to note that:


  * Unlike <cit.>, in this work,
λ̃ is used in the expression of A^ASE(λ,γ_0)
because only the active BSs make an effective contribution to the
ASE.

  * The ASE defined in this paper is different from that in <cit.>,
where a constant rate based on γ_0 is assumed for the typical
UE, no matter what the actual SINR value is. The definition of the
ASE in (<ref>) can better capture the dependence of the
transmission rate on SINR, but it is less tractable to analyze, as
it requires one more fold of numerical integral compared with <cit.>.

  * Previously in Subsection <ref>,
we have obtained a conclusion from Theorem <ref>:
p^cov(λ,γ) with the BS IMC should
be better than that with all BSs being active in dense SCNs due to
λ̃≤λ. Here from (<ref>), we may
arrive at an opposite conclusion for A^ASE(λ,γ_0)
because it scales linearly with the active BS density λ̃,
which caps at ρ in dense SCNs with the BS IMC. The takeaway
message should not be that the IMC generates an inferior ASE in dense
SCNs. Instead, since there is a finite number of active UEs in the
network, some BSs are put to sleep and thus the spatial spectrum reuse
in practice is fundamentally limited by ρ. The key advantage
of the BS IMC is that the per-UE performance should increase with
the network densification as discussed in Subsection <ref>.



Regarding the computational complexity, in order to obtain p^cov(λ,γ)
presented in Theorem <ref>, for a general case, three
folds of integrals are respectively required for the calculation of
{ f_R,n^Path(r)}, {ℒ_I_agg(γ/Pζ_n^Path(r))},
and { T_n^Path}, where the string variable Path
takes the value of “L” (for the LoS case) or “NL” (for the
NLoS case). Note that an additional fold of integral is needed for
the calculation of A^ASE(λ,γ_0)
in (<ref>), making it a 4-fold integral computation.


Placeholder





§ SIMULATION AND DISCUSSION



Placeholder


In this section, we investigate network performance and use numerical
results to validate the accuracy of our analysis. According to Tables
A.1-3, A.1-4 and A.1-7 of <cit.>

 and <cit.>

, we adopt the following parameters for the 3GPP Case: α^L=2.09,
α^NL=3.75, A^L=10^-10.38, A^NL=10^-14.54

An important footnote: Note that for the purpose of numerical calculation,
all the distances should be converted to km because A^L
and A^NL are defined for distances in km.

, 

BW=10 MHz,

P=24 dBm, P_N=-95 dBm (including a noise figure
of 9 dB at each UE). Besides, the UE density ρ is set to 300UEs/km^2,
which leads to q=4.05 in (<ref>) and (<ref>) <cit.>.


Parameters     Description     Valued_1     The cut-off point in the linear LoS probability function     300mα^L     The path loss exponent for LoS     2.09α^NL     The path loss exponent for NLoS     3.75A^L     The constant value in the path loss function for LoS     10^-14.54A^NL     The constant value in the path loss function for NLoS     10^-10.38P     The BS transmission power     24dBmP_N     The noise power     -95dBm





 §.§ The Coverage Probability Performance of the 3GPP Case



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In Fig. <ref>, we plot
the results of p^cov(λ,γ) with
the PF scheduler for the 3GPP Case when ρ=300 UEs/km^2
and γ=0 dB. Note that our analytical results on
the exact performance are obtained from Theorems <ref>
and <ref>. In contrast, our analytical
results on the upper-bound performance are obtained from Theorems <ref>
and <ref>. As a benchmark, we also
provide simulation results, and display the analytical results of
the RR scheduler reported in <cit.>. Moreover, we
show the ratio of the simulated p^cov(λ,γ)
of the PF scheduler to that of the RR scheduler in Fig. <ref>.

From these two figures, we can observe that:


  * As can be seen from Fig. <ref>,
our analytical results well match the simulation results, which validates
the accuracy of Theorems <ref> and <ref>.
However, as discussed in Subsection <ref>,
the evaluation of our analytical results is only efficient for dense
and ultra-dense networks. Thus, in Fig. <ref>,
we are only able to show the results of p^cov(λ,γ)
for λ≥100 BSs/km^2. When λ<100 BSs/km^2,
the proposed upper-bound results in Theorems <ref>
and <ref> successfully capture the
qualitative performance trend of the PF scheduler, with a maximum
error of 0.04 in terms of p^cov(λ,γ)
for sparse networks.



  * As predicted in Lemma <ref>, although
the PF scheduler shows a better performance than the RR one for all
BS densities, such performance gain diminishes as the network evolves
into an UDN due to the loss of multi-user diversity. As can be seen
from Fig. <ref>,
the performance gain of the PF scheduler continuously decreases from
around 175% (ratio=2.75) when λ=1 BSs/km^2
toward zero (ratio=1) in UDNs, e.g., λ=10^4 BSs/km^2.

  * The detailed explanation of the performance behavior in Fig. <ref>
is provided as follows:


  * When λ∈[10^0,10^1] BSs/km^2,
the network is noise-limited, and thus the coverage probabilities
of both RR and PF increase with the BS density λ as the network
is lightened up with more BSs.

  * When λ∈[10^1,10^2] BSs/km^2,
p^cov(λ,γ) of the PF scheduler
shows an interesting flat trail. This is because (i) the signal
power is enhanced by LoS transmissions, as shown by the p^cov(λ,γ)
of the RR scheduler in that BS density region; while (ii) the
multi-user diversity decreases in that BS density region as exhibited
in Fig. <ref>;
and (iii) the above two factors

the LoS signal power increase and the multi-user diversity decrease

 roughly cancel each other out.

  * When λ∈[10^2,10^3] BSs/km^2,
the coverage probabilities of both RR and PF decrease with λ,
as the network is pushed into the interference-limited region, and
this performance degradation is due to the transition of a large number
of interfering paths from NLoS to LoS, which accelerates the growth
of the aggregate inter-cell interference <cit.>.

  * When λ>10^3 BSs/km^2, the coverage probabilities
of both RR and PF continuously increase <cit.>.

Such performance improvement is referred to as the Coverage
Probability Takeoff in <cit.>.

 Such performance behavior can be attributed to the BS idle mode operations,
i.e., (i) the signal power continues increasing with the network
densification, and (ii) the interference power is controlled
because not all BSs are turned on and emit interference.

, since only the BSs with active UE(s) eject interference.






 §.§ The Performance Impact of Rician Fading



Placeholder


In this subsection, we investigate the performance for the 3GPP Case
with distance-dependent Rician fading

 and correlated shadow fading

, which has been introduced in Subsection <ref>.
Due to its complex modeling, we conduct simulations to investigate
this enhanced 3GPP Case, and the results are plotted in Fig. <ref>.




As one can observe from this figure,

All the conclusions in Subsections <ref>
are qualitatively valid for Fig. <ref>,
which shows the usefulness of our analysis and a less urgency to consider
Rician fading in this performance analysis.

that (i) our theoretical analysis is useful to predict the
performance trend for the PF scheduler, and that (ii) it is
not urgent to consider distance-dependent Rician fading in this performance
analysis.

 Note that comparing Fig. <ref>
with Fig. <ref>, we
can see that Rician fading closes the performance gap between PF and
RR more quickly than Rayleigh fading, due to its less variance in
channel fluctuation for the PF scheduler to exploit. 

Only some quantitative deviations exists, which shows that (i)
our theoretical analysis is useful to predict the performance trend
for dense and ultra-dense SCNs of the PF scheduler, and that (ii)
it is not urgent to consider distance-dependent Rician fading in this
performance analysis.


and correlated shadow fading





 §.§ The ASE Performance of the 3GPP Case



Placeholder


In Fig. <ref>, we
plot the results of A^ASE(λ,γ_0)
when ρ=300 UEs/km^2 and γ_0=0 dB,
with q^*=4.18 plugged into Proposition <ref>.

 From Fig. <ref>,
we can draw the following conclusions:


  * For 3GPP Case 1, the ASE suffers from a slow growth or even a slight
decrease when λ∈[20,200] BSs/km^2
because of the interference transition from NLoS to LoS <cit.>.
Such performance degradation has also been confirmed in Fig. <ref>.

  * After such BS density region of interference transition, for both
path loss models with ρ=300 UEs/km^2 and the BS
IMC, the ASEs monotonically grow as λ increases in dense
SCNs, but with noticeable performance gaps compared with those with
ρ=+∞ UEs/km^2.

  * As discussed in Section <ref>, the takeaway message
should not be that the IMC generates an inferior ASE in dense SCNs.
Instead, since there is a finite number of the active UEs in the network,
some BSs are put to sleep and thus the spatial spectrum reuse in practice
is fundamentally limited by ρ. The key advantage of the BS IMC
is that the per-UE performance should increase with the network densification
as exhibited in Fig. <ref>.



The intuition is that the randomness in channel
fluctuation associated with Rayleigh fading is much stronger than
that associated with Rician fading due to the large K factor in
ultra-dense SCNs <cit.>, and the
resulting stronger channel fluctuation implies a larger probability
that the SINR could drop below the minimum working SINR γ_0
defined in (<ref>).


The Impact of 3GPP Case 2 on the ASE Performance

In this subsection, we investigate the ASE performance for 3GPP Case 2,
which has been introduced in Subsection <ref>.
The parameters in the LoS probability function Pr^L(r)
of 3GPP Case 2 are set to R_1=156 m and R_2=30 m <cit.>.
Here, we directly apply the numerical integration in Theorem <ref>
to evaluate the ASE performance for 3GPP Case 2.

First, we study the value of q^* for the approximate λ̃
in 3GPP Case 2. The simulated results on the active BS density in
case of various values of ρ are very similar to those shown
in Fig. <ref>. For
brevity, in Fig. <ref>,
we only plot the average error of the active BS density for ρ=300UEs/km^2.

Similar conclusions can be drawn for Fig. <ref>
as discussed in Subsection <ref>,
except for the small numerical difference of q^*=4.05 for ρ=300UEs/km^2
in 3GPP Case 2.

In Fig. <ref>,
we show the results of A^ASE(λ,γ_0)
for 3GPP Case 2.

 As one can observe, all the conclusions in Subsection <ref>
are qualitatively valid for Fig. <ref>.
Only some quantitative deviations exists.


Finally, we investigate the ASE performance, which is calculated from
the results of p^cov(λ,γ) using
(<ref>). Since the analytical results of p^cov(λ,γ)
obtained from Theorems <ref> and <ref>
have been validated in Fig. <ref>,
we only plot the analytical results of A^ASE(λ,γ_0)
for the PF and RR schedulers in Fig. <ref>

with the 3GPP Case when ρ=300 UEs/km^2 and γ_0=0 dB

. Note that we only investigate dense and ultra-dense SCNs with λ≥100 BSs/km^2
in Fig. <ref> because
they are the focus of 5G and Theorem <ref>
is computationally efficient to evaluate them.

as explained at the end of Subsection <ref>,
Theorem <ref> is computationally efficient
for the BS density λ≥100 BSs/km^2, which allows
us to obtain the ASE performance.


As can be seen from this figure, the ASE performance of the PF scheduler
quickly converges to that of the RR scheduler in UDNs, e.g., when
λ=10^3 BSs/km^2, the ASE of the PF scheduler
is around 590.1 bps/Hz/km^2, which is merely 4.52%
higher than that of the RR scheduler around 564.6 bps/Hz/km^2.
Such gain becomes practically zero as λ further increases.



§ CONCLUSION



Placeholder


In this paper, we studied the network performance of the PF scheduler.
Analytical results that are computationally efficient have been derived
for dense and ultra-dense SCNs. For sparse networks, easy-to-compute
analytical results of an upper-bound performance have been obtained
to achieve a balance between accuracy and efficiency. Considering
the negligible gain of the more complex channel-dependent PF scheduling
in UDNs,

 which is justified by the diminishing multi-user diversity of the
PF scheduler in dense SCNs,

 it is recommended to adopt simpler scheduling mechanisms such as
the RR scheduler to simplify the RRM, and thus reduce network complexity
for UDNs. As our future work, a non-full-buffer traffic model <cit.>
will be studied.

Acknowledgments

Guoqiang Mao's research is supported by Australian Research Council
(ARC) Discovery projects DP110100538 and DP120102030 and Chinese National
Science Foundation project 61428102.





§ APPENDIX A: PROOF OF THEOREM <REF>




Placeholder




Due to the page limit, we only provide the proof sketch of Theorem <ref>
as follows. In (<ref>), T_n^L and
T_n^NL are the components of the coverage probability
for the case when the signal comes from the n-th piece LoS
path and for the case when the signal comes from the n-th
piece NLoS path, respectively. The calculation of T_n^L
is based on (<ref>), in which f_R,n^L(r)
characterizes the geometrical density function of the typical UE with
no other LoS BS and no NLoS BS providing a better link
to the typical UE than its serving BS (a BS with the n-th
piece LoS path). Besides, 𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}
gives the expected coverage probability over all possible values of
k̃ on condition of r. The logic of the calculation of
T_n^NL is similar to that of T_n^L.



§ APPENDIX B: PROOF OF THEOREM <REF>




Placeholder




Due to the page limit, here we only provides the key of the proof
for Theorem <ref>. The derivation of
(<ref>) is as follows,


𝔼_[k̃]{Pr[Pζ_n^L(r)y(k̃)/I_agg+P_N>γ]}


    =    𝔼_[k̃]{Pr[y(k̃)>γ(I_agg+P_N)/Pζ_n^L(r)]}
       (a)=   𝔼_[k̃,I_agg]{F̅_Y(k̃)(γ(I_agg+P_N)/Pζ_n^L(r))}
       (b)=   ∑_k̃=1^K̃^max𝔼_[I_agg]{1-∑_t=0^k̃([ k̃;  t ])(-δ_n^L(r))^texp(-sI_agg)}f_K̃(k̃),

where the step (a) of (<ref>) comes
from (<ref>), and in the step (b) of (<ref>)
s=tγ/Pζ_n^L(r) and 𝔼_[I_agg]{exp(-sI_agg)} =ℒ_I_agg^L(s)
should be further plugged into (<ref>)
to obtain (<ref>). The calculation of
ℒ_I_agg^L(s) can be
referred to <cit.>. The derivation of (<ref>)
is very similar to (<ref>), which is
omitted for brevity.



§ APPENDIX C: PROOF OF LEMMA <REF>




Placeholder






The key of the proof for Lemma <ref>
lies in (<ref>) and (<ref>)
of Theorem <ref>. When λ→+∞,
we have that K̃^max→1 and f_K̃(1)=1

, because the resulting UDN is approach the limit of one UE per active
BS

. Hence, Theorem <ref> will degenerate
to the results for the RR scheduler addressed in <cit.>.





§ APPENDIX D: PROOF OF THEOREM <REF>




Placeholder






The key of the proof for Theorem <ref>
lies in using Jensen's inequality as follows <cit.>

which is because the function [1-(1-exp(-x))^k̃]
defined in (<ref>) is a concave function with regard
to exp(-x) when k̃≥1 and x∈[0,1].
Plugging (<ref>) into the step (a) of (<ref>)
and with some mathematical manipulations, yields the proof.


    𝔼_[I_agg]{ 1-(1-exp(-x))^k̃}≤1-(1-𝔼_[I_agg]{exp(-x)})^k̃.




IEEEtran


