Sarabjot Singh and Jeffrey G. Andrews This work has been supported by the Intel-Cisco Video Aware Wireless Networks (VAWN) Program and NSF grant CIF-1016649. A part of this paper is accepted for presentation at IEEE Globecom 2013 in Altlanta, USA [1]. The authors are with Dept. of Electrical and Computer Engineering at the University of Texas, Austin (email: sarabjot@utexas.edu and jandrews@ece.utexas.edu).

## I Introduction

The exponential growth in mobile traffic primarily driven by mobile video has led the drive to deploy low power base stations (BSs) both in licensed and unlicensed spectrum in order to complement the existing macro cellular architecture. Increased cell density increases the area spectral efficiency of the network [2] and is the key factor for the very large required capacity boost. Such a heterogeneous network (HetNet) consists of macro BSs coexisting with small cells formed by co-channel low power base stations like micro, pico and femto BSs, as well as unlicensed band WiFi access points (APs) [3, 4].

### I-a Motivation and Related Work

It has been established that without proactive offloading and resource partitioning only limited performance gains can be achieved from the deployment of small cells [8, 9, 10, 11, 12]. These techniques are strongly coupled and directly influence the rate of users, but the fundamentals of jointly optimizing offloading and resource partitioning are not well understood. For example, an excessively large association bias can cause the small cells to be overly congested with users of poor , which requires excessive muting by the macro cell to improve the rate of offloaded users. Earlier simulation based studies [11, 12] confirmed this insight and showed that excessive biasing and resource partitioning can actually degrade the overall rate distribution, whereas the choice of optimal parameters can yield about 2-3x gain in the rate coverage (fraction of user population receiving rate greater than a threshold). Although encouraging, a general tractable framework for characterizing the optimal operating regions for resource partitioning and offloading is still an open problem. The work in this paper is aimed to bridge this gap.

A “straightforward” approach of finding the optimal strategy is to search over all possible user-AP associations and time/frequency allocations for each network configuration. Besides being computationally daunting, this approach is unlikely to lead to insight into the role of key parameters on system performance. Another methodology is a probabilistic analytical approach, where the network configuration is assumed random and following a certain distribution. This has the advantage of leading to insights on the impact of various system parameters on the average performance through tractable expressions. Analytical approaches for biasing and interference coordination were studied in [13, 14, 15], but downlink rate (one of the key metrics) was not investigated. Optimal bias and almost blank subframes were prescribed in [13] based on average per user spectral efficiency. A related and mean throughput based analysis for resource partitioning was done in [14] and [16] respectively, but offloading was not captured. The choice of optimal range expansion biases in [15] was not based on rate distribution. In this paper, we use the metric of rate coverage, which captures the effect of both and load distribution across the network. Semi-analytical approaches in [17, 18] showed, through simulations, that there exists an optimal association bias for fifth percentile and median rate which is confirmed in this paper through our analysis. Also, to the best of our knowledge, none of the mentioned earlier works considered the impact of backhaul capacities on offloading, which is another contribution of the presented work.

### I-B Approach and Contributions

We propose a general and tractable framework to analyze joint resource partitioning and offloading in a two-tier cellular network in Section II. The proposed modeling can be extended to a multiple tier setting as discussed in Sec. III-D. Each tier of base stations is modeled as an independent Poisson point process (PPP), where each tier differs in transmit power, path loss exponent, and deployment density. The mobile user locations are modeled as an independent PPP and user association is assumed to be based on biased received power. On all channels, i.i.d. Rayleigh fading is assumed. Similar tractable frameworks were used for deriving distribution in HCNs in [19, 20, 21]. The empirical validation in [22] and theoretical validation in [23] for heavily shadowed cellular networks have strengthened the case of modeling macro cellular networks using a PPP. Due to the formation of random association/coverage areas in such network models, load distribution is difficult to characterize. An approximate load and rate distribution was derived for multiple radio access technology (RAT) HetNets in [24].

Based on our proposed approach, the contributions of the paper can be divided into two categories:
Analysis. The rate complementary cumulative distribution function (CCDF) in a two-tier co-channel heterogeneous network is derived as a function of the cell range expansion/offloading and resource partitioning parameters in Section III. Rate coverage at a particular rate threshold is the rate CCDF value at that threshold. The derived rate distribution is then modified to incorporate a network setting where APs are equipped with limited capacity backhaul. Under certain plausible scenarios, the derived expressions are in closed form.
Design Guidelines. The theoretical results lead to joint resource partitioning and offloading insights for optimal and rate coverage in Section IV. In particular, we show the following:

• With no resource partitioning, optimal association bias for rate coverage is independent of the density of the small cells. In contrast, offloading is shown to be strictly suboptimal for in this case.

• With resource partitioning, optimal association bias decreases with increasing density of the small cells.

• In both of the above scenarios, the optimal fraction of users offloaded, however, increases with increasing density of small cells.

• With decrease in backhaul capacity/bandwidth the optimal association bias for the corresponding tier always decreases. However, in contrast to the trend in the “infinite”222Infinite bandwidth implies sufficiently large so as not to affect the effective end-to-end rate. backhaul scenario, the optimal association bias may increase with increasing small cell density.

The paper is concluded in Section V and future work is suggested.

## Ii Downlink System Model and Key Metrics

In this paper, the wireless network consists of a two-tier deployment of APs. The location of the APs of tier () is modeled as a two-dimensional homogeneous PPP of density (intensity) . Without any loss of generality, let the macro tier be tier and the small cells constitute tier . The locations of users (denoted by ) in the network are modeled as another independent homogeneous PPP with density . Every AP of tier transmits with the same transmit power over bandwidth . The downlink desired and interference signals from an AP of tier- are assumed to experience path loss with a path loss exponent . A user receives a power from an AP of tier at a distance , where is the random channel power gain. The random channel gains are assumed to be Rayleigh distributed with average unit power, i.e., . General fading distributions can be considered at some loss of tractability [25]. The noise is assumed additive with power . The notations used in this paper are summarized in Table I.

### Ii-a User Association

The analysis in this paper is done for a typical user located at the origin. This is allowed by Slivnyak’s theorem [26], which states that the properties observed by a typical333The term typical and random are interchangeably used in this paper. point of a PPP is same as those observed by a node at origin in the process . Let denote the distance of the typical user from the nearest AP of tier. It is assumed that each user uses biased received power association in which it associates to the nearest AP of tier if

 j =argmaxk∈{1,2}PkBkZ−αkk, (1)

where is the association bias for tier. Increasing association bias leads to the range expansion for the corresponding APs and therefore offloading of more users to the corresponding tier. For clarity, we define the normalized value of a parameter of a tier as its value divided by the value it takes for the serving tier. Thus,

 ^Pk≜PkPj,^Bk≜BkBj,% and ^αk≜αkαj,

are respectively the normalized transmit power, association bias, and path loss exponent of tier conditioned on the user being associated with tier . In this paper, association bias for tier 1 (macro tier) is assumed to be unity ( dB) and that of tier 2 is simply denoted by , where dB. In the given setup, a user can lie in the following three disjoint sets:

 u∈⎧⎪ ⎪⎨⎪ ⎪⎩U1 if j=1, P1Z−α11≥P2BZ−α22U¯B if j=2 and P2Z−α22>P1Z−α11UB if j=2 and P2Z−α22≤P1Z−α11

where clearly. The set is the set of macro cell users and the set is the set of unbiased small cell users. Thus, the set is independent of the association bias. The users offloaded from macro cells to small cells due to cell range expansion constitute and are referred to as the range expanded users. All the users associated with small cells are . We define a mapping from user set index to serving tier index. Thus, from (2), , .

The biased received power based association model described above leads to the formation of association/coverage areas in the Euclidean plane as described below.

###### Definition 1.

Association Region: The region of the Euclidean plane in which all users are served by an AP is called its association region. Mathematically, the association region of an AP of tier located at is

 Cxj={y∈R2:∥y−x∥≤(PjBjPkBk)1/αj∥y−X∗k(y)∥^αk∀k}, (3)

where .

The random tessellation formed by the collection of association regions is a general case of the multiplicatively weighted Voronoi [27, Chapter 3], which results by using the presented model with equal path loss exponents.

### Ii-B Resource Partitioning

A resource partitioning approach is considered in which the macro cell shuts its transmission on certain fraction of time/frequency resources and the small cell schedules the range expanded users on the corresponding resources, which protects them from macro cell interference.

###### Definition 2.

: The resource partitioning fraction is the fraction of resources on which the macro cell is inactive, where .

Thus, with resource partitioning fraction of the resources at macro cell are allocated to users in and those at small cell are allocated to users in . The fraction of the resources in which the macro cell shuts down the transmission, the small cells schedule the range expanded users, i.e., . Let denote the inverse of the effective fraction of resources available for users in . Then, for and for . The operation of range expansion and resource partitioning in a two-tier setup is further elucidated in Fig. 1. In these plots, the power ratio is assumed to be dB and dB.

As a result of resource partitioning (), the of a typical user , when it belongs to , is

 (4)

where denotes the indicator of the event , is the channel power gain from the tagged AP (AP serving the typical user) at a distance , denotes the interference from the tier. The interference power from tier is

 Iy,k=Pk∑x∈Φk∖slHxx−αk. (5)

In this paper, all APs of a tier are assumed to be active, when the corresponding tier is active. However, if each AP of tier is independently active with a probability , the submission in (5) can then be treated as that over a thinned PPP of density .

Let denote the set of users associated with the tagged AP. If the tagged AP belongs to macro tier, then denotes the total number of users (or load henceforth) sharing the available fraction of the resources. Otherwise, if the tagged AP belongs to tier 2, then the load is of which users share the fraction of the resources and users share the rest ; (one is subtracted to account for double counting of the typical user). The available resources at an AP are assumed to be shared equally among the associated users. This results in each user having a rate proportional to its link’s spectral efficiency. Round-robin scheduling is an approach which results in such equipartition of resources. Further, user queues are assumed saturated implying that each AP always has data to transmit to its associated mobile users. Thus, the rate of a typical user is

 R=∑l∈{1,¯B,B}11(u∈Ul)γlNlWlog(1+SINR). (6)

The above rate allocation model assumes infinite backhaul bandwidth for all APs, which may be particularly questionable for small cells. Discussion about limited backhaul bandwidth is deferred to Sec. III-C.

### Ii-C Rate and SINR Coverage

The rate and coverage can be formally defined as follows.

###### Definition 3.

Rate/ Coverage: The rate coverage for a rate threshold is

 R(ρ)≜P(R>ρ), (7)

and coverage for a threshold is

 S(τ)≜P(SINR>τ). (8)

The coverage can be equivalently interpreted as (i) the probability that a randomly chosen user can achieve a target threshold, (ii) the average fraction of users in the network who at any time achieve the corresponding threshold, or (iii) the average fraction of the network area that is receiving rate/ greater than the rate/ threshold.

## Iii Rate Distribution

This section derives the load distribution and distribution, which are subsequently used for deriving the rate distribution (coverage) and is the main technical section of the paper.

### Iii-a SINR Distribution

For completely characterizing the and rate distribution, the average fraction of users belonging to the respective three disjoint sets (, , and ) is needed. Using the ergodicity of the PPP, these fractions are equal to the association probability of a typical user to these sets, which are derived in the following lemma.

###### Lemma 1.

(Association probabilities) The association probability, defined as , is given below for each set

 A1=2πλ1∫∞0zexp(−π2∑k=1λk(^Pk^Bk)2/αkz2/^αk)dz, (9)
 A¯B=2πλ2∫∞0zexp(−π2∑k=1λk(^Pk)2/αkz2/^αk)dz, (10)
 AB=2πλ2∫∞0z{exp(−π2∑k=1λk(^Pk^Bk)2/αkz2/^αk)−exp(−π2∑k=1λk(^Pk)2/αkz2/^αk)}dz. (11)

If path loss exponents are same, i.e., , the association probabilities simplify to:

 A1 =λ1∑2k=1λk(^Pk^Bk)2/α,A¯B=λ2∑2k=1λk(^Pk)2/α,
 AB=λ2∑2k=1λk(^Pk^Bk)2/α−λ2∑2k=1λk(^Pk)2/α. (12)
###### Proof.

See Appendix A. ∎

Equation (1) corroborates the intuition that increasing association bias leads to decrease in the mean population of macro cell users implied by the decreasing . On the other hand, the mean population of range expanded users increases implied by the increasing . Further, is the probability of a typical user associating with the tier 2.

The conditional coverage, when a typical user is

###### Lemma 2.

( Coverage) For a typical user in the setup of Sec. II, the coverage is

 S(τ)=A1S1(τ)+A¯BS¯B(τ)+ABSB(τ), (13)

where the conditional coverage are given by (14)-(16),

, and .

###### Proof:

See Appendix C. ∎

The result in Lemma 2 is for the most general case and involves a single numerical integration along with a lookup table for . The expressions can be further simplified as in the following corollary.

###### Corollary 1.

With noise ignored, , assuming equal path loss exponents , the coverage of a typical user is

 S(τ)=λ1∑2k=1λk(Pk/P1)2/αQ(τ,α,Bk)+λ2∑2k=1λk(Pk/P2)2/αQ(τ,α,1)+λ2λ2Q(τ,α,1)+λ1{P1/(P2B2)}2/α−λ2λ2Q(τ,α,1)+λ1(P1/P2)2/α. (17)

As evident from the above Lemma and Corollary, coverage is independent of the resource partitioning fraction because of the independence of on the amount of resources allocated to a user in our model. Further, the distribution of the small cell users, , is independent of association bias, as is independent of bias. Further insights about coverage are deferred until the next section. In general, we show that coverage with and without resource partitioning show considerably different behavior, which is also reflected in the rate coverage trends.

### Iii-B Main Result

Similar to the conditional coverage, conditional rate coverage, when a typical user is The following theorem gives the rate distribution over the entire network.

###### Theorem 1.

(Rate Coverage) For a typical user in the setup of Sec. II, the rate coverage is

 R(ρ)=A1R1(ρ)+A¯BR¯B(ρ)+ABRB(ρ), (18)

where the conditional rate coverage are given by (19)-(21), , , and .

###### Proof:

Using (4) and (6), the probability that the rate requirement of a random user is met is

 P(R>ρ) =∑l∈{1,¯B,B}P(u∈Ul)P(WγlNllog(1+SINR)>ρ|u∈Ul) (22) =∑l∈{1,¯B,B}AlENl[Sl(t(^ρNlγl))], (23)

where and . In general, the load and are correlated, as APs with larger association regions have higher load and larger user to AP distance (and hence lower ). However for tractability of the analysis, this dependence is ignored, as in [24], resulting in , where . Using Lemma 2, the rate coverage expression is then obtained. ∎

The probability mass function of the load depends on the association area, which needs to be characterized.

###### Remark 1.

(Mean Association Area) Association area of an AP is the area of the corresponding association region. Using the ergodicity of the PPP, the mean of the association area of a typical AP of tier is .

The association region of a tier 2 AP can be further partitioned into two regions. The non-shaded region in Fig. 1 surrounding a small cell at can be characterized as

 Cx¯B≜{y∈R2:∥y−x∥≤(P2/P1)1/α2∥y−X∗k(y)∥^α1,∀k}. (24)

As per (2), all the users lying in are the small cell users (belonging to ) and recalling (3) all users lying in are the offloaded users that belong to . In Fig. 1, is the shaded region surrounding a tier 2 AP.

###### Remark 2.

(Association Area Distribution) A linear scaling based approximation for the distribution of association areas proposed in [24], which matched the first moment, is generalized in this paper to the setting of resource partitioning as below

 C1=C(λ1A1), (25) C¯B=C(λ2A¯B), and CB =C(λ2AB), (26)

where is the area of a typical cell of a Poisson Voronoi (PV) of density (a scale parameter).

Using the area distribution proposed in [28] for PV , the following lemma characterizes the probability mass function (PMF) of the load seen by a typical user.

###### Lemma 3.

(Load PMF) The PMF of the load at tagged AP of a typical user is

 pl(n)≜P(Nl=n)=3.53.5(n−1)!Γ(n+3.5)Γ(3.5)(λuAlλJ(l))n−1(3.5+λuAlλJ(l))−(n+3.5)n≥1, (27)

where is the gamma function.

###### Proof:

See Appendix B. ∎

The rate distribution expression for the most general setting requires a single numerical integral after use of lookup tables for and . The summation over in Theorem 1 can be accurately approximated as a finite summation to a sufficiently large value, (say), since both the terms and decay rapidly for large .

The rate coverage expression can be further simplified if the load at each AP is assumed to equal its mean.

###### Corollary 2.

(Mean Load Approximation) Rate coverage with the mean load approximation is given by

 ¯R(ρ)=A1¯R1(ρ)+A¯B¯R¯B(ρ)+AB¯RB(ρ), (28)

where the conditional rate coverage are given by (29)-(31) and

###### Proof:

Lemma 3 gives the first moment of load as . Further, using the result that [29], along with an approximation , the simplified rate coverage expression is obtained. ∎

The mean load approximation above simplifies the rate coverage expression by eliminating the summation over . The numerical integral can also be eliminated by ignoring noise and assuming equal path loss exponents (as is done in Sec IV-B). As can be observed from Theorem 1 and Corollary 2, the rate coverage for range expanded users increases with increase in resource partitioning fraction , as users in can be scheduled on a larger fraction of (macro) interference free resources. On the other hand, the rate coverage for the macro users and small cell (non-range expanded) users decreases with the corresponding increase. Further insights on the effect of biasing are delegated to the next section.

### Iii-C Rate Coverage with Limited Backhaul Capacities

Analysis in the previous sections assumed infinite backhaul capacities and thus the air interface was the only bottleneck affecting downlink rate. However, with limited backhaul capacities for BSs of tier , the rate is given by

 (32)

where is the rate of the user with infinite backhaul bandwidth. The above rate allocation assumes that the available backhaul bandwidth for a BS of tier , , is shared equally among the associated users/load . This allocation model is similar to the fair round robin scheduling and results in the peak rate of a typical user (associated with an AP of tier ) being capped at . The analysis can be extended to incorporate a generic peak rate dependency on backhaul bandwidth and load at the AP (which may result from a different backhaul allocation strategy)444Exact analysis of wired backhaul allocation among the competing TCP flows could be an area of future investigation.. The following lemma gives the rate distribution in this setting.

###### Lemma 4.

(Rate Coverage with Limited Backhaul) The rate coverage in the setting of Sec. II and with rate model of (32) is

 R′(ρ)=P(R′>ρ)=A1R′1(ρ)+A¯BR′¯B(ρ)+ABR′B(ρ), (33)

where

 R′1(ρ) (34) R′¯B(ρ) =⌈O2/ρ−2⌉∑m=0pB(m)⌈O2/ρ−m−1⌉∑n=1p¯B(n)S¯B(t(γ¯Bn^ρ)), R′¯B(ρ)

and is given by Lemma 2.

###### Proof:

Since the maximum rate of a user is . Thus, for this user to have positive rate coverage, i.e., , a necessary condition is . When this necessary condition is satisfied, the rate coverage is equivalent to

 Rl′ =P(R′>ρ) (35)

Using for , and independence of and the conditional rate coverage are obtained. ∎

It is evident from the above Lemma that rate coverage decreases with decreasing backhaul bandwidth. Therefore, decreasing will lead to decrease in the rate of the user when it is associated to small cell and thus decreasing the optimal offloading bias (this is further explored in subsequent sections). As the backhaul bandwidth increases to infinity, Lemma 4 leads to Theorem 1, or, .

### Iii-D Extension to Multi-tier Downlink

The analysis in the previous sections discussed a two-tier setup, which can be generalized to a -tier () setting. In this setting, location of the BSs of tier are assumed according to a PPP of density . Further, is assumed to be the association bias corresponding to tier , where dB and dB . Similar to (2), a user associated with tier can be classified into two disjoint sets:

 u∈{U¯Bj if PjZ−αjj>PkBkZ−αkk ∀k≠jUBj if u∉U¯Bj and PjBjZ−αjj>PkBkZ−αkk ∀k≠j. (36)

With resource partitioning, an AP of tier schedules the offloaded users, , in fraction of the resources, which are protected from the macro-tier interference and the non-range expanded users are scheduled on fraction of the resources. Thus, the of a user associated with tier is

 SINR=11(u∈UBj)PjHyy−αj∑Kk=2Iy,k+σ2+11(u∈U¯Bj)PjHyy−αj∑Kk=1Iy,k+σ2. (37)

By using similar techniques as in a two-tier setting, the coverage for this setting is given in (38)-(39). The rate is given by

 R={11(u∈UBj)ηNBj+11(u∈U¯Bj)1−ηN¯Bj}Wlog(1+SINR). (40)

The rate coverage for this setting can be derived by using (38)-(39) and a generalization of Lemma 3.

### Iii-E Validation of Analysis

We verify the developed analysis, in particular Theorem 1, Corollary 2, and Lemma 4, in this section. The rate distribution is validated by sweeping over a range of rate thresholds. The rate distribution obtained through simulation and that from Theorem 1 and Corollary 2 for two values for the pair of bias and resource partitioning fraction () is shown in Fig. (a)a. The respective densities used are BS/km, BS/km, and users/km with , . The assumed transmit powers are dBm and dBm. Thermal noise power is assumed to be dBm. The rate distribution for the case with limited backhaul obtained through simulation and that from Lemma 4 is shown in Fig. (b)b. The rate distribution is shown for two different backhaul bandwidths for a bias of dB and without resource partitioning. Both the plots show that the analytical results, Theorem 1 and Lemma 4, give quite accurate (close to simulation) rate distribution. Furthermore, the mean load approximation based Corollary 2 is also not that far off from the exact curves in Fig. (a)a. This gives further confidence that the rate distribution obtained with mean load approximation in Corollary 2 can be used for further insights (as is done in the following sections).

## Iv Insights on optimal SINR and Rate Coverage

As it was mentioned earlier the extent of resource partitioning and offloading needs to be carefully chosen for optimal performance. Although a simplified setting is considered in the following results for analytical insights, it is shown that these insights extend to more general settings through numerical results.

### Iv-a SINR Coverage: Trends and Discussion

Although rate coverage is the main metric of interest, insights obtained from coverage should be useful in explaining key trends in rate coverage. As stated before, the coverage with and without resource partitioning exhibits different behavior in conjunction with offloading. The following lemma presents some key trends for coverage in both settings.

###### Corollary 3.

Ignoring thermal noise (), assuming equal path loss exponents and equal to four ( 555 typically varies from to depending on the propagation environment.), the coverage without resource partitioning is

 Sw(τ)=1√τtan−1(√τ)+1+a√p(√τtan−1(√τ/b)+√b) +1√τtan−1(√τ)+1+1a√p(√τtan−1(√bτ)+√1/b), (41)

where , , and . The coverage with resource partitioning for the corresponding setting is

 S(τ)=1√τtan−1(√τ)+1+a√p(√τtan−1(√τ/b)+√b) +1√τtan−1(√τ)+1+1a√p(√τtan−1(√τ)+1) +1√τtan−1(√τ)+1+1a√pb−1√τtan−1(