Mattia Minelli, Maode Ma, Marceau Coupechoux, Jean-Marc Kelif, Marc Sigelle, and Philippe Godlewski M. Minelli is with the School of Electrical and Electronic Engineering, Nanyang Technological University, 50 Nanyang Avenue Singapore 639798 and with Télécom ParisTech and CNRS LTCI, 46, rue Barrault, Paris, France.
E-mail: mattia.minelli@gmail.com M. Maode is with the School of Electrical and Electronic Engineering, Nanyang Technological University
E-mail: EMDMa@ntu.edu.sg M. Coupechoux, M. Sigelle and P. Godlewski are with Télécom ParisTech and CNRS LTCI.
E-mails: {first name.name}@telecom-paristech.fr J.-M. Kelif is with Orange Labs, Issy-Les-Moulineaux, France.
Email: jeanmarc.kelif@orange.com
###### Abstract

Relay nodes-enhanced architectures are deemed a viable solution to enhance coverage and capacity of nowadays cellular networks. Besides a number of desirable features, these architectures reduce the average distance between users and network nodes, thus allowing for battery savings for users transmitting on the uplink. In this paper, we investigate the extent of these savings, by optimizing relay nodes deployment in terms of uplink energy consumption per transmitted bit, while taking into account a minimum uplink average user delay that has to be guaranteed. A novel performance evaluation framework for uplink relay networks is first proposed to study this energy-delay trade-off. A simulated annealing is then run to find an optimized relay placement solution under a delay constraint; exterior penalty functions are used in order to deal with a difficult energy landscape, in particular when the constraint is tight. Finally, results show that relay nodes deployment consistently improve users uplink energy efficiency, under a wide range of traffic conditions and that relays are particularly efficient in non-uniform traffic scenarios.

\justify

## I Introduction

Heterogeneous network deployments in modern cellular networks are regarded as a promising solution to meet the ever-increasing demand of wireless data and voice traffic. They consist in installing a number of low-power nodes, possibly of different types (e.g., femtocells, Relay Nodes (RNs), etc.), inside the coverage area of macro base stations (also called eNodes-B (eNBs) in this paper), resulting in a more dense network architecture. Indeed, a higher density of nodes entails a number of benefits, e.g., coverage and capacity are boosted [1] and power efficiency improves [2], due to the reduced distance between User Equipments (UE) and serving nodes [3]. This generates operational expenditures savings for operators, and lowers the environmental impact of their infrastructure. Results in [2, 4] et. al. show that the introduction of RNs and picocells can reduce downlink power consumption. However, this applies to the uplink as well (see e.g. [5, 6]), and the decrease in battery energy consumption for users can be consistent: [7, 8] indicate that uplink transmit power has a strong impact on users overall energy demand, especially for high transmit powers.

Hence, heterogeneous networks can be considered as an effective means to extend mobile users batteries duration, and several research works are dedicated to this topic. Authors of [5] and [9] propose a game theory-based framework for the maximization of femtocells uplink energy efficiency, by means of users transmit powers tuning [5] or radio resources assignment and power control parameters optimization [9]. In [10], closed-form expressions of the Signal to Interference Ratio (SIR) and of the outage probability are derived and used to measure the impact of femtocells coverage and users density on energy efficiency. Both [5] and [9] apply Quality of Service (QoS)-related constraints to the optimization problem, so as to avoid solutions with high energy efficiency but poor performance. Other works in the area of cellular networks, e.g. [11], address this issue by jointly optimizing power and user experienced QoS.

The close interaction between energy consumption and performance [11] makes the problem of energy efficiency maximization in relay-enhanced networks different from that addressed in [5, 9] et al. for the femtocells deployment case; this is because RNs communicate to their donor node through a wireless backhaul link [12], which represents a well-known performance bottleneck. On the contrary, small cells and femto cells benefit from a wired backhaul, where the delay issue is less crucial. Surprisingly, uplink RN networks energy consumption has received limited attention in the literature, to the best of our knowledge. Reference [13] proposes an optimization of the uplink power control in relay-based networks but ignores energy consumption. Authors of [6, 14] treat the maximization of uplink energy efficiency via either optimal assignment of subcarriers, users powers and bit allocations [6], or optimal radio resources allocation [14]. More than one order of magnitude can be achieved in user power consumption [14].

A drawback of [6] is that the decisive impact of co-channel interference is neglected. Also, the paper assumes that users are fixed in number and position, and always have data to transmit. Hence, the influence of the traffic intensity is not considered, while a number of studies (see e.g. [15, 11]) show the importance of the traffic load for energy efficiency evaluation. Finally, [6, 14] lack of a thorough theoretical framework for the analysis of relay-enhanced cellular networks uplink energy efficiency, and they are based on the sole minimization of users energy consumption, without investigating the necessary tradeoff between this consumption and uplink performance. Hence, the need arises for a finer model to study this tradeoff.

The contributions of this paper are the following.

• We study the tradeoff between energy consumption and users experienced delay in uplink relays cellular networks. This is the first study of this type, to the best of our knowledge. We show that in many cases, relays can help both saving energy and reducing delays despite the constraint imposed by the wireless backhaul. We also highlight the interest of using relays in scenarios, where traffic is not uniform.

In order to find upper bounds on the achievable gains in terms of energy saving and/or average delay, we formulate a constrained optimization problem. Its objective is to place the relays and tune the network parameters so as to minimize the energy consumption per transmitted bit under delay constraints. To solve this problem, we rely on a Simulated Annealing (SA) algorithm enhanced with exterior search and penalty functions. We discuss the effectiveness of our approach for addressing our problem.

• We propose a framework for the analysis of energy efficiency in uplink relay networks. In order to study the delay constraint, we consider the dynamic nature of cellular traffic, by means of a model of UE arrivals and departures. This results in a hierarchical flow level analysis. The loads of eNBs and RNs are accounted in the estimation of interference and transmission delays. Our propagation model includes shadowing and fast fading, and UEs power control is considered. Overall, the proposed model is more comprehensive, compared to the existing literature on uplink energy efficiency in heterogeneous networks.

• We adopt a non-trivial scheduling scheme, which represents an approximation of the Proportional Fair (PF) scheduler, and derive the probability density function of the Signal to Interference plus Noise Ratio (SINR) of a scheduled UE. The choice of the scheduling algorithm has indeed a decisive impact on the delay performance. This is a novel contribution, to the best of our knowledge.

The paper is organized as follows. Section II introduces our system model, while Section III describes the framework used for evaluating energy consumption per transmitted bit and delay. Section IV is devoted to the optimization problem and our proposed algorithm, while Section V gives our numerical results. Finally, Section VI concludes our work.

## Ii System Model

We work on the uplink of a cellular network, where transmissions are performed on synchronized frames, and each network station (i.e. eNB or RN) is associated an uplink frame. Frames are partitioned into Radio Blocks (RBs) of the same bandwidth and time duration, which are labeled with an index . Each RB in the frame of station can be either granted to one of the UEs served by , or be left unused (if there is no UE to be served).

The network is composed of a set of stations, and divided into cells; let be the set of eNBs, and the set of RNs, so that . We focus on one given cell containing one donor eNB [16] and RNs, which are connected to the eNB by means of a wireless backhaul link (Fig. 2). The set of stations of cell is denoted . The surface of cell is defined as the region where UEs are served by one of the stations of , and its area is denoted . Similarly, surface , of area , is the region where UEs are served by station .

UEs select their serving station based on the highest product of received downlink pilot power times a station specific bias. They are then served over the access link. The technique of biasing the user association is referred to in literature as cell range expansion [17], and it is deemed beneficial for network performance as it allows for load balancing. We assume that eNBs have a bias of 1 and RNs have a common bias .

### Ii-a Frame Structure

Access link and backhaul link transmissions are dedicated two orthogonal sections of the frame (Fig. 2). RN transmissions on the backhaul link, which uses a quota of the RBs, do not interfere with UEs transmissions on the access link. This choice is widely adopted in the literature (see e.g. [13, 14]), as it avoids interference between RNs and UEs on the uplink. For the same reason, we assume that is the same for all network stations. The value of is supposed to be set by the operator and based on considerations on the overall network performance, while our analysis is focused on the performance related to one given network cell, given .

### Ii-B Traffic Model

We assume that UEs arrive in the network according to a spatial Poisson point process of intensity , transmit a flow of average size to their serving station, and leave the network. Flows transmitted from UEs to a RN on the access link are then forwarded by the RN to its donor eNB on the backhaul link.

The traffic density at a given location is denoted by , while the average traffic density in the network can be computed as , where is the overall network surface, of area 111We assume large enough so that interference in cell is accuratly computed.. We define , to account for the local variation of the traffic density with respect to . The ratio can be seen as the spatial Probability Density Function (PDF) of UEs arrivals. We also define for the sake of further developments.

The typical number of bits carried by a RB is assumed to be much smaller than . Hence, access link buffer queue of all stations can be modeled as an M/G/1/PS queue (Fig. 3), and its load is denoted with . It is the sum of load contributions over , so that . Only stable scenarios, i.e., are considered in this work. We define the vectors and , for the sake of further developments. Let introduce a binary random variable , which is equal to one when RB in station frame is granted to a UE, and equal to zero otherwise. We will assume that the probability for any station to receive data on RB depends only on , and is equal to .

Backhaul link buffer queue is also modeled as a multi-class M/G/1/PS (Fig. 3). Each flow in the backhaul queue belongs to a class , according to the transmitting RN. The model shown in Fig. 3 thus allows for a hierarchical flow level analysis.

### Ii-D SINR model

Consider a UE located in and a RB . We define the instantaneous SINR experienced by station for the considered UE as:

 γk(s,τ)=Pk(s,τ)∑j∈K,j≠kΘj(τ)Pk(fj(τ),τ)+N, (1)

where yields the position of the UE scheduled by station on RB , and is the thermal noise power. In the following, we drop the dependency of all variables on , for the sake of simplicity. Let be the PDF of . Following [20, 21], we approximate by a lognormal distribution, i.e., (see App. A for the derivation of and .

### Ii-E Scheduling Policy

Let assume that stations have perfect Channel State Information (CSI) about all served UEs and consider a given frame of station . We label the UEs served by during the considered frame with an index , while their locations are denoted by respectively. Every station implements a Maximum Quantile Scheduling [22, 23] (MQS) among served UEs. The MQS scheduler of station assigns RB to the UE that maximizes its instantaneous SINR , with respect to over a window of RBs (more details can be found in App. B). This policy has the property of being fair in RBs allocation between UEs, while maintaining a good throughput performance [23]. Moreover, [23] shows that the performance degradation introduced by an imperfect estimation of the SINR distribution can be lower than that incurred in practical implementations of other popular scheduling algorithms. The MQS scheduling differs from the well-known PF scheduler [24], because the latter allocates each radio resource to the user maximizing its scaled SINR on , where represents the average SINR experienced by for user over the last RBs. However, for large values of its behavior approximates that of a PF scheduler (see e.g. [22]).

Now, consider the distribution of conditioned to , i.e., the PDF of the SINR of a scheduled UE. Contrary to , this distribution is not lognormal.

###### Theorem 1.

The distribution can be expressed as:

 πγ,k(si,z)=pγ,k(si,z)W∑n=1Qn(i,z)Tk(W,n), (2)

where

 Qn(i,z) = (W−1n−1)(∫z0pγ,k(si,t)dt)W−n× (1−∫z0pγ,k(si,t)dt)n−1, Tk(W,n) = W2(1−ρk)(W−ρk(W−n))2.

See App. B. ∎

### Ii-F UEs Physical Data Rate

We define the function , yielding the physical data rate achieved, when the receiver experiences SINR (or equivalently the throughput of a user, if it were alone in the serving area of its serving station). We assume that is non decreasing in .

## Iii Performance Evaluation

In this section, we define and derive our performance parameters in terms of delay and energy consumption per transmitted bit. We first obtain the expressions of access and backhaul link loads, and then we use them to get flow transmission delays.

Let focus on station , and recall that is the contribution of to the load of station . We have the following Lemma:

###### Lemma 1.

The local contribution to the load of the access link buffer queue of station is expressed by:

 ϱk(s)=11−βEτ[¯ωϕ(s)C(γk(s,τ))∣∣∣fk(τ)=s]. (3)
###### Proof.

The contribution to the load is expressed as the ratio of the traffic density in to the uplink rate achieved by a scheduled UE in . The term takes into account the quota of the frame RBs dedicated to the backhaul link. ∎

###### Theorem 2.

The load of the access link of station can be expressed as:

 ρk=¯ω1−β∫Akϕ(s)∫∞0πγ,k(s,z)C(z)dzds. (4)
###### Proof.

The load is the sum of the contributions over : . Now, the term in (3) can be expressed according the law of total probability, obtaining (4). ∎

Note first that if all UEs transmit at power , then and equation (4) reduces to: . can thus be seen as the effective area covered by , in terms of traffic density.

Note then that is a function of , as the load depends on the interference from other stations. We define the operator , which yields the value of corresponding to a given .

###### Lemma 2.

The operator maps to a finite -dimensional interval.

###### Proof.

For all stations , the corresponding is increasing with respect to every coordinate, as an increase in the load of any station produces an increase in the interference received at . We define , which represents an upper bound for the load of . Then, . Hence,

 F(ρ)∈∏k∈K[0,ρmaxk],ρmaxk∈R+,∀k∈K. (5)

###### Theorem 3.

There exists at least one such that .

###### Proof.

This follows from Lemma 2 and (5): if is a continuous mapping from an -dimensional closed interval to itself, then the Brouwer’s fixed point theorem guarantees the existence of at least one fixed point. ∎

It is not feasible in general to draw any conclusion about the uniqueness of the fixed point. However, following the approach of [25], we start from a single cell without interference () and iterate function , so as to model a scenario of increasing traffic.

We now focus on the backhaul link queue of the eNB of cell . Let be the index of the eNB and the load of the queue. The probability for a given RN served by to be selected for backhaul transmission is equal to , where . Hence, can be written as:

 ρBL,j=¯ωβnRN∑k=1ΦkRBL,j,k, (6)

where the inverse of is the average of the inverse of the rate on the backhaul link between RN and eNB , which depends on propagation conditions, and is equal to: , where denotes the SINR on the backhaul of RN served by eNB , during RB . A detailed derivation of is presented in App. C.

### Iii-C Flow Average Transmission Delay

The flow average transmission delay is an effective parameter to measure network uplink performance. It is the sum of the average access delay and the average backhaul delay. In this section, we derive its formulation. Consider a user located in , served by RN with donor eNB . We denote by the average access delay of a flow in , by the average delay of a flow served by and by the average delay on the backhaul link.

###### Lemma 3.

The average access delay in is expressed by:

 Dk(s)=ξ1−ρk∫∞0πγ,k(s,z)(1−β)C(z)dz. (7)
###### Proof.

The delay to transmit a bit of information is the inverse of the UE rate, multiplied by the number of UEs served by the same station during a frame (because each UE is scheduled on a fraction of the RBs with MQS). The average transmission delay for the whole flow is hence given by: . Considering that , the law of total probability can be used to average over : , obtaining expression (7). ∎

###### Corollary 1.

The average access delay for a UE served by is:

 Dk=∫Akϕ(s)ΦkDk(s)ds. (8)
###### Proof.

The statement can be verified by means of the Little’s law: . Now, we can substitute (4) to the numerator and conclude the proof. ∎

###### Lemma 4.

The average backhaul link delay is expressed by:

 DBL,j,k=ξ(1−ρBL,j)βRBL,j,k,k∈KR,j∈KB. (9)
###### Proof.

Similarly to what performed for Lemma 3, we proceed by expressing as: and apply the law of total probability, obtaining:

 DBL,j,k = ξβRBL,j,k∞∑v=1P(xBL,j(τ)=v∣∣xBL,j(τ)>0)× Eτ[xBL,j(τ)∣∣xBL,j(τ)=v],

where is the number of flows in the backhaul queue of during the considered frame. Conventionally, we set . From this equation, we obtain (9). ∎

###### Proposition 1.

The average transmission delay for an uplink flow transmitted in cell is equal to:

 ¯Dc=∑k∈KcΦkAk(Dk+DBL,j,k). (10)

### Iii-D Energy Consumption per Transmitted Bit

Let be the energy consumed by a UE located in for transmitting one bit. This metric is sometimes called Energy Consumption Rating in the literature [26, 27]. We have:

 ϵ(s) = Eτ[T(s)C(γk(s,τ))∣∣∣fk(τ)=s] (11) = T(s)∫∞0πγ,k(s,z)C(γk(s,z))dz,s∈Ak

which can be also expressed as a function of : . The average uplink energy consumption per bit associated to UEs in cell is given by:

 Π = 1Ac∫Acϕ(s)ϵ(s)ds (12) = 1−βAc∑k∈Kc(1−ρk)∫AkT(s)Dk(s)ϕ(s)ds.

## Iv Optimization

In this Section, we discuss the optimization of UEs uplink energy efficiency. The proposed algorithm aims at minimizing the UE average energy consumption per transmitted bit in (12), by optimizing , and the RN placement in the cell of interest, for a given , and traffic spatial profile . Our optimization is constrained to respect a maximum tolerable value of in (10), so as to take into account the users experienced quality of service.

RN placement is usually optimized with respect to the downlink performance of the network because of its larger traffic volume. On the contrary, we consider here the uplink performance because we aim at finding upper bounds on the possible gains that an operator can achieve in terms of energy and/or average delay on this link. Note however that with the growing traffic related to multimedia content sharing, social networks and other peer-to-peer applications, uplink and downlink traffics tend to be more balanced333A joint uplink and downlink RN placement optimization is left for further work..

### Iv-a Problem Statement

We assume that RNs can be placed on a discrete and finite grid of candidate sites inside cell 444This assumption is consistent with real deployment scenarios, where normally only a certain number of locations inside the cell are available for RN installation [28]., while the positions of RNs outside cell are assumed to be already set by the network operator, and not modifiable during the optimization. Position of a given RN is denoted with . Moreover, we define . Target received powers and can vary between a minimum and a maximum value, denoted with and respectively. Similarly, we assume (see Section II). Now, the configuration of the network is defined as the set of positions of the RNs in cell , plus the adopted and :

 x:{z(1),⋯,z(nRN),¯P,B}, z(i)∈Ac,∀i∈{1⋯nRN}, ¯Pmin≤¯P≤¯Pmax,Bmin≤B≤Bmax. (13)

We name configuration space the set of all configurations, and denote it with .

Our problem is to minimize energy consumption (12) under delay constraint (10):

 minxΠ(x) (14)
 s.t. ¯Dc(x)≤D{\tiny max},x∈Ω,D{\tiny max}∈R+, (15)

where constraint (15) restricts the domain of feasible solutions to the subspace:

 ~Ω={x∈Ω s.t. Dc(x)≤D{\tiny max}} (16)

Recall that the computation of the station loads via the fixed point iteration of Section III-A is required for the evaluation of the delay.

Now, the problem (14) is in general non-convex, and the cardinality of , especially for high , makes it intractable via exhaustive search. Hence, we rely on stochastic optimization algorithms, and propose a customized version of the well-known Simulated Annealing algorithm to solve (14). In the following, we first briefly recall the generic SA, and then introduce our version.

#### Iv-A1 Generic SA Algorithm

The SA is a metaheuristic aimed at solving large non-convex problems, which has been first proposed by Metropolis [29] and then applied on a wide range of optimization problems (see, e.g., [30, 31]). The literature shows that the SA is an effective algorithm, if its parameters are appropriately tuned (see, e.g., [32]). Let denote the energy555Not to be confused with the energy in J involved in the energy consumption (12). associated with configuration , and consider the problem of minimizing over the configuration space . The SA explores only a subset of the configuration space, where usually , and is able to find the optimal configuration by means of an appropriate selection of the analyzed configurations. At temperature , the algorithm proceeds by assigning to each configuration an exponential probability, given by:

 PT(x)=exp(−F(x)T)ZT,∀x∈Ω, (17)

where is a normalizing constant. Hence, the solution is the that maximizes . According to the Metropolis-Hastings variant [33] of the SA, the set of configurations to be analyzed is determined according to the following procedure:

• At step a configuration is arbitrarily selected and designated as current solution for step : .

• At any step : assume is the current solution for step . The SA will first draw a new candidate solution for step , which complies with the given proposal law (defined by the algorithm designer). Then, assuming that is symmetric, will be accepted as current solution for step with probability:

 PTm(x∗(m)←x′(m))= min(1, exp(−F(x′(m))−F(x∗(m−1))Tm)),

where is a parameter called temperature, which decreases to zero slowly enough as , and is such that . If is not accepted, then . In most practical implementations of the SA, the temperature is updated following a law of the kind , where is close to 1.

#### Iv-A2 Proposed Exterior Search Approach for SA

We seek for a minimizer of over , where the cardinality of the feasible configurations space depends on the value of the constraint: . In particular, is an increasing function of , because the number of feasible configurations increases as the constraint loosens.

When is small, we can reasonably expect to have . This may reduce the effectiveness of the SA, as the energy surface that the algorithm explores could be fragmented into many isolated regions, some of which are unreachable for the algorithm. Moreover, we can expect the optimal solution to tightly respect the constraint, i.e., to lie close to the border of [34]. Thus, the standard SA may not appropriately cover the region where the solution is, as many of its configurations are non feasible.

This problem can be solved by means of an exterior search approach [35, 36]. It consists in extending the search to configurations outside , while adding a penalty to the energy of those configurations that violate the constraint. This method presents analogies with the Lagrangian relaxation method (see the classic works [37, 38]), and it is regarded as a powerful instrument to deal with constrained optimization problems (see, e.g., [39, 40]). This idea is illustrated in Fig. 4, where we see on a fictitious example how exterior search can reduce the path to the optimal configuration.

We thus adopt an exterior search approach, and reformulate (14) as:

 minxF(x),x∈Ω, (18)
 where F(x)=Π(x)+1Dc(x)>DmaxV(x,Dmax),

and where is a function accounting for constraint (15) and named penalty function, which plays a crucial role in the optimization [36].

Penalty functions are classified into static, dynamic, and adaptive. A static penalty function does not change during the optimization. It can be represented, e.g., by a fixed constant added to the energy of those configurations , or by a function proportional to the Euclidean distance of the considered configuration to the feasible region [41]. On the contrary, dynamic penalty functions can be adjusted in accordance to the progress of the optimization. A common approach is to gradually increase the penalty with the number of explored configurations [42], so as to guarantee the convergence of the optimization towards a feasible solution. Finally, an adaptive penalty function (see, e.g., [40, 43]) considers further aspects of the search, such as steering the algorithm towards regions of the energy surface which are deemed promising for the search.

We propose the following exterior penalty function, which is dynamic and adaptive at the same time:

 V(x,Dmax)=α(m−1)Π(x)Dmax(Dc(x)−Dmax), (19)

where is a constant and is the SA step. The benefits of using function (19) are manifold. First, the penalty is proportional to the violation of constraint , favoring the exploration of configurations which are out of but close to it, while penalizing more those which are far. At the same time the penalty is independent of the adopted value of , depending rather on the percentage of exceeding delay, with respect to . This represents an important feature when dealing with constraints of a different order of magnitude, compared to . Moreover, the penalty is proportional to , so as to favor the exploration of configurations outside when they appear to be promising, i.e., their energy is sensibly lower to that of the current solution (hence the adaptive nature of (19)). Finally, acting on we can modify the percentage of acceptance for configurations outside . We use here the works of Geman and Robini [44, 45]: Choosing is sufficient to guarantee convergence of the SA algorithm towards a feasible minimizer. App. D explains why the convergence property of the SA is maintained with such a penalty function.

Surprisingly, stochastic constrained optimization based on exterior penalties has been seldom employed in wireless networks optimization. There are however three recent interesting references using this method [46, 47, 48]. In this context, two important problems arise:

1. The generic choice of the penalty term itself. To our knowledge, the most interesting analysis appears in [48], using the so-called logarithmic barrier. Our approach is somewhat similar: since our penalty term is the product of the objective function and the constraint , its gradient is a linear combination of the gradients of both previous expressions. Thus, in a deterministic, continuous framework, during a phase when the penalty is active () it adapts the minimizing direction search to both the objective function and the constraint opposite gradients (see Fig. 4). We expect the stochastic, discrete SA to behave in a somewhat similar way.

2. The selection of the Lagrangian multipliers and their evolution during the algorithm. Apart from the usual choice of constant parameters, an interesting development can be found in [46]: at each step of the SA algorithm, each penalty weight itself either increases if the associated constraint is violated, or decreases (in a geometric way) if this constraint has been satisfied during several previous steps. In [47] all penalty weights increase regularly.

To the best of our knowledge, our approach is the first one to use stochastic constrained optimization for station placement in wireless networks. The originality of our work also lies in the multiplicative penalty function and the theoretical choice of the penalty coefficient .

#### Iv-A3 Initial Temperature Choice

Following [49], we address the problem of finding a good value for the initial temperature in each considered optimization problem via dichotomic search during a series of preliminary runs of the algorithm.

## V Results

### V-a Considered Scenario and SA Parameters

The proposed theoretical framework is here applied on a test scenario drawn according to [12, case 1] for what concerns propagation, shadowing and stations transmit power. System bandwidth is  MHz. The network is composed by one central eNB surrounded by 6 eNBs regularly distributed around it on a circle of ray  m. We optimize RN placements in the central cell only, while RNs in the surrounding cells are assumed to be regularly distributed on a circle of radius 160 m around their donor eNB. All stations have omnidirectional antennas. The same realization of shadowing (drawn according to [12]) is used for all simulations, so as to ensure that results be comparable. If not otherwise mentioned, we adopt a uniform traffic density spatial distribution, i.e., is constant , in order to draw general conclusions about energy efficiency. In Section V-B5 we show the performance under a non-uniform traffic distribution. We set .

Function is approximated by means of the Modulation and Coding Scheme (MCS) indicated in [50], so as to take into account the effect of an upper-bounded capacity function, which is the typical case in real deployments. Also, fast fading on the backhaul link is not considered. This choice is justified by the assumption that RNs do not move, and optimization of their positions is performed on a long-term basis (see, e.g. [51]). We have assumed and  dB, with a step of 1 dB. The grid of candidate RN sites has a step of  m.

The fixed point of is found by iteratively computing stations loads. At iteration , probabilities are computed according to , obtained at iteration , and then fed into (4) to get , starting from . Iterations are stopped when either , or . In the latter case, the analyzed configuration is labeled as non-valid. Similarly, fixed point iterations are used to compute the loads on the backhaul link, after is determined. We were not able to analytically prove the convergence of the fixed point iteration (by showing for example that is a contraction mapping). We have however numerically observed that the iteration always converges in less than 10 iterations.

SA algorithm is implemented as described in Section IV. Once again, we emphasize the need to optimize network parameters and in particular RN locations in order to obtain upper bounds on the achievable performance. The algorithm yields a solution after 45 temperature steps. For each tested temperature value, the energy of a certain number of network configurations is calculated. This number varies according to , e.g., for , 400 network configurations are tested at each temperature step. For every optimization, the SA is repeated 4 times, and the best solution among the 4 obtained solutions is elected as a final result. The configuration corresponding to the final result is denoted with .

### V-B Numerical Results

We denote with the configuration with minimum energy when no RNs are deployed in the whole network, and with its corresponding energy. Unless otherwise mentioned, results are expressed in terms of the energy consumption ratio , so as to show the energy gain (or loss) resulting from RN deployment, and they are plotted against the normalized constraint , where is the value of corresponding to , i.e., the average delay when no RNs are deployed. Hence, a point in the region corresponds to an energy consumption per bit gain with respect to the network with no RNs, whereas a point in the region corresponds to an average uplink transmission delay gain. All the curves that we obtained exhibit a hyperbolic-like shape, which follows from the nature of our constrained optimization. When is large, , the delay doesn’t play any role in the optimization and so the energy consumption gain reaches its maximum. Instead, if the constraint is tight we have , and it is unlikely that any configuration that performs well in terms of energy efficiency lies in .

#### V-B1 Energy - Delay Trade-off

Fig. 6 shows the trade-off between energy consumption and delay, for a varying number of RNs. Note how RN deployment can bring consistent UEs energy savings. Let consider the region where . We observe that the energy consumption gain increases with . This is an expected result, as RNs reduce the average distance between UEs and their serving station. The corresponding increase in uplink interference is mitigated by a lower users transmit power. On the contrary, a lower number of RNs performs better when . This behavior is related to the cardinality of the valid configurations set, which tends to decrease with , narrowing the space for network optimization and reducing the achievable energy consumption gains. In this case adding RNs is detrimental for energy efficiency, as more UEs served by RNs imply a higher backhaul delay, that further reduces .

#### V-B2 Effect of Offered Traffic

We analyze the effect of on energy consumption in Fig. 6, where and are scaled with the constants and resp., obtained in a network with no RN and a traffic density  bits/s/m. This choice has been made in order to keep the same scaling constants for all curves. As we can see, the deployment of 1 RN is sufficient to reduce the average energy consumption of UEs to less than a half, compared to the eNBs-only network. Now, let focus on the line (same delay as in the eNB-only case). We note that adding a single RN can help the terminals to be more energy efficient without touching the quality of service and even if the load increases to  bits/s/m. Nonetheless, beyond approximately  bits/s/m, delay and interference negatively affect user performance.

#### V-B3 Energy Efficiency of RNs vs Small Cells

Fig. 8 compares the results obtained with our system model, with those obtained using small cells instead of RNs (i.e., ), highlighting the difference of our work with respect to those dedicated to devices with wired (or ideal) backhaul. As expected, RNs allow for a smaller energy consumption gain compared to small cells, for a fixed . This is due to the transmission delay on the backhaul link, to the increased delay on the access link (due to ) and to the constraints on RN placement related to backhaul path-loss and shadowing. Performance of RNs is more impaired when the traffic density or increase. However, RN deployment still yields consistent uplink energy consumption gains.

#### V-B4 Effectiveness of Exterior Search with Penalty Function

Fig. 8 compares the results of the optimization using penalty function and exterior search, with those obtained by means of an interior search. Both the interior and the exterior search have been carried on with the same number of iterations, for an unbiased comparison. The exterior search proves to be more effective when the delay constraint is tight, as is expected to be fragmented and . No meaningful gain in terms of search effectiveness is observed when the constraint is loose, as in this case the two approaches tend to coincide.

#### V-B5 Effect of Offered Traffic Spatial Distribution

Fig. 9 compares the results obtained using a uniform with those obtained when is the sum of a uniform traffic profile and a bi-dimensional gaussian function, centered at  m, with the same average traffic density. We can see that RNs allow for larger energy consumption gains when traffic is non-uniform. This is due to the flexibility of the RN solution that allows relays to be placed close to the hot-spot center.

## Vi Conclusion

We have proposed a comprehensive framework for the optimization and performance evaluation of uplink energy consumption per bit in relays-enhanced cellular networks. This framework considers the arrival and departure of users, and the loads of network stations. Moreover, shadowing and fast fading are both taken into account in the propagation model. A realistic radio resource scheduling scheme is assumed, and its impact on users performance is derived. A customized optimization algorithm based on exterior search with penalty functions is proposed for the optimization, which is carried on under quality of service constraints. Results show a meaningful boosting of users energy efficiency given by the deployment of RNs, even if the average flow delay is imposed to be the same as in the network without relays. Proposed exterior search approach is shown to be more effective than the traditional interior search.

## Appendix A Derivation of μγ,k and σγ,k

We use here a simple and classical approach consisting in introducing a auxiliary Random Variable (RV). (1) can indeed be rewritten as: , where is a realization of a lognormal RV , and is the denominator. Note that does not depend on . Several authors have shown that the interference term can be well approximated by a lognormal distribution even when the number of interferers is variable (see e.g. [20, 21]). In [20], it is also shown that the product of an exponential RV () and a lognormal RV () as well as the sum of two lognormal RVs can be again approximated by a lognormal RV. Parameters are then obtained by moment matching. The derivation of , which is less standard, is now detailed. The mean of is equal to:

 M1,Ik = Eτ⎡⎣∑j≠kΘjPk(fj)⎤⎦ = ∑j≠kρjEτ[Pk(fj)] (1)= ∑j≠kρj∫Ajϱj(s)ρjT(s)Gk(s)ds = ∑j≠kρjYj,k,

where . (1) is obtained by weighting the power received from location with the local load in . Note that are input parameters of the considered deployment scenario (propagation and transmit power assumptions). Now depends on the SINR distribution in and hence on the first moment of . In order to avoid an additional complexity to our model, we make the approximation . This is justified by the fact that in a urban environment, it is unlikely that UEs transmit at their maximum power [13] so that every user of is received with the same average power. The approximation comes then from the expressions (3), (4) of and . In the same way, we derive the variance of : , where . Again, we approximate . Finally, and are found by matching and with the mean and variance of the approximating lognormal.

## Appendix B Proof of Theorem 1: Derivation of πγ,k(s,z) under MQS scheduling

The MQS scheduler orders the values of SINR of each UE in ascending order. The ranking of UE located in on RB is the ranking of in the ordered vector . The lower the ranking, the better the SINR on (wrt the SINR on previous blocks). Station assigns RB to the UE with the lowest . We assume large enough so that the probability for two or more UEs to have the same ranking is negligible. We have: 666as in our block Rayleigh fading environment each realization of is independent of the others.. Moreover, the probability for any UE served by to be scheduled is (due to the fairness in RBs allocation). Let denote the Cumulative Distribution Function (CDF) of . We have:

 P(ri(τ)=n|γk(si,τ)=z)= (20) (W−1n−1)Πγ,k(si,z)W−n(1−Πγ,k(si,z))n−1.

We denote the right hand side, which does not depend on . Now, we derive , given . Applying the Bayes theorem on we obtain:

 πγ,k(s,z) = P(fk(τ)=si|γk(si,τ)=z)pγ,k(si,z)P(fk(τ)=si) (21) = P(fk(τ)=si|γk(si,τ)=z)pγ,k(si,z)U.

We then work on applying the law of total probability, conditioning it with respect to the possible rankings :

 (22) W∑n=1P(ri(τ)=n|γk(si,τ)=z)×

The probability of being scheduled depends on the instantaneous SINR only through the ranking: once we know , knowing does not add any additional information regarding the probability of being scheduled by . Hence,

 (23)

and (21) becomes

 πγ,k(s,z)=pγ,k(si,z)UW∑n=1Qn(i,z)(W−nW)U−1. (24)

So far, we have found given . We now define , and use the law of total probability, summing up given for all the possible values of , and obtaining its general expression:

 P(γk(si,τ)=z|fk(τ)=si,U>0)= (25) pγ,k(si,z)W∑n=1Qn(i,z)(1−ρk)∞∑U=1ρU−1kUΨU−1,

where

 (1−ρk)∞∑U=1ρU−1kUΨU−1=W2(1−ρk)(W−ρk(W−n))2. (26)

We name and get t