On Power and Load Coupling in Cellular Networks for Energy Optimization
Abstract
We consider the problem of minimization of sum transmission energy in cellular networks where coupling occurs between cells due to mutual interference. The coupling relation is characterized by the signaltointerferenceandnoiseratio (SINR) coupling model. Both cell load and transmission power, where cell load measures the average level of resource usage in the cell, interact via the coupling model. The coupling is implicitly characterized with load and power as the variables of interest using two equivalent equations, namely, nonlinear load coupling equation (NLCE) and nonlinear power coupling equation (NPCE), respectively. By analyzing the NLCE and NPCE, we prove that operating at full load is optimal in minimizing sum energy, and provide an iterative power adjustment algorithm to obtain the corresponding optimal power solution with guaranteed convergence, where in each iteration a standard bisection search is employed. To obtain the algorithmic result, we use the properties of the socalled standard interference function; the proof is nonstandard because the NPCE cannot even be expressed as a closedform expression with power as the implicit variable of interest. We present numerical results illustrating the theoretical findings for a reallife and largescale cellular network, showing the advantage of our solution compared to the conventional solution of deploying uniform power for base stations.
I Introduction
Data traffic is projected to grow at a compound annual growth rate of from 2011 to 2016 [1], fueled mainly by multimedia mobile applications. This growth will lead to rapidly rising energy cost [2]. In recent years, information communication technology (ICT) has become the fifth largest industry in power consumption [3]. In cellular networks, in particular, base stations consume a significant fraction of the total endtoend energy [4], of which – of the power consumption is due to the power amplifiers [5, 6]. This observation has motivated green communication techniques for cellular networks [7, 8, 9, 10, 11, 12, 13, 14]. These technologies include adaptive approaches such as switching off power amplifiers to provide a tradeoff of energy efficiency and spectral efficiency [7, 8], selectively turning off base stations [9], as well as energy minimization approaches for relay systems [10], OFDMA systems [11, 12, 13], and SCFDMA systems [14]. Extensive survey of other savingenergy approaches are highlighted in [2, 15, 16].
In this paper, we focus on the important problem of minimizing the sum energy used for transmission in cellular networks. Besides reducing the energy cost for transmission, minimizing the transmission energy may lead to selection of power amplifiers with lower power rating, hence further reducing the overhead cost involved in turning on power amplifiers.
In a cellular network where base stations are coupled due to mutual interference, the problem of energy minimization is challenging, as each cell has to serve a target amount of data to its set of users, so as to maintain an appropriate level of service experience, subject to the presence of the coupling relation between cells. To tackle this energy minimization problem, we employ an analytical signaltointerferenceandnoiseratio (SINR) model that takes into account the load of each cell [17, 18, 19], where a load of a cell translates into the average level of usage of resource (e.g., resource units in OFDMA networks) in the cell. This loadcoupling equation system has been shown to give a good approximation for more complicated load models that capture the dynamic nature of arrivals and service periods of data flows in the network [20], especially at high data arrival rates. Further comparison of other approximation models concluded that the loadcoupled model is accurate yet tractable [21]. By using this tractable model, useful insights can then be developed for the design of practical cellular systems. In our recent works [22], we have used the load coupling equation to maximize sum utility that is an increasing function of the users’ rates.
Previous works [17, 18, 19, 20, 22] using the loadcoupling model all assume given and fixed transmission power. For transmission energy minimization, both power and load become variables and they interact in the coupling model, making the analysis more challenging. In fact, the coupling relation between cell powers cannot be expressed in closed form even for given cell loads. The key aspects motivating our theoretical and algorithmic investigations are as follows. First, is there an insightful characterization of the operating point in terms of load that minimizes the sum transmission energy? Second, given a system operating point in load, what are the properties of the coupling system in power? Third, even if power coupling cannot be expressed in closed form, is there some algorithm that converges to the power solution for given cell load?
Toward these ends, our contributions are as follows. We show that if full load is feasible, i.e., the users’ data requirements can be satisfied, then operating at full load is optimal in minimizing sum transmission energy (Section IVC, Theorem 1). If full load is not feasible, however, then no feasible solution exists (Section IVC, Corollary 1). Thus, full load is necessary and sufficient to achieve the minimum transmission energy. Moreover, the optimal power allocation for all base stations is unique (Section VB, Theorem 2), and can be numerically computed based on an iterative algorithm that can be implemented iteratively at each base station (Section VD, Algorithm 1). To prove the algorithmic result, we make use of the properties of the socalled standard interference function [23]; the proof is however nonstandard, because the function of interest does not have a closedform expression, and hence we use an implicit method to verify its properties. We also characterize the load region over all possible power allocation given some minimum target data requirements (Section VC, Theorems 3–4). Finally, we obtain numerical results to illustrate the optimality of the fullload solution on a cellular network based on a reallife scenario [24]. Compared with the conventional solution where the uniform power is used for base stations, we show the significant advantage of the poweroptimal solution in terms of meeting user demand target and reducing the energy consumption.
The rest of the paper is organized as follows. Section II gives the system model of the loadcoupled network. Section III formulates the energy minimization problem. Section IV characterizes the optimality of full load, while Section V derives properties of the powercoupling system and an iterative power allocation algorithm that achieves the power solution. Numerical results are given in Section VI. Section VII concludes the paper.
Notations: We denote a column vector by a bold lower case letter, say , a matrix by a bold capital letter, say , and its th element by its lower case . We denote a positive matrix as if for all . Similarly, we denote a nonnegative matrix as if for all . Similar conventions apply to vectors. Finally, and denote the allzeros and allones vectors of suitable lengths.
Ii System Model
Iia Preliminaries
We consider a cellular network consisting of base stations that interfere with each other due to resource reuse. We focus on the downlink communication scenario where base station transmits with power per resource unit (in time and frequency). We refer to cell interchangeably with base station . For notational convenience, we collect all power as vector .
We assume a given association of the users to the base stations. In this association, each base station serves one unique group of users, denoted by set , where User is served in cell at rate that has to be at least a rate demand of nats. Thus, relates to a qualityofservice (QoS) constraint. We collect all the rates as vector and the corresponding minimum demands as . Thus, a rate vector meets the QoS constraints if .
IiB Load Coupling
We first consider the load coupling model for the cellular network. We denote by the load in the network, where . In LTE systems, the load can be interpreted as the fraction of the timefrequency resources that are scheduled to deliver data. We model the SINR of user in cell as [17, 18, 19, 20]
(1) 
where represents the noise power and is the channel power gain from base station to user ; note that represents the channel gain from the interfering base stations. The function depends on for , but not on ; the dependence on the entire vector is maintained in (1) for notational convenience. The SINR model (1) gives a good approximation of more complicated cellular network load models [20]. Intuitively, can be interpreted as the likelihood of receiving interference from cell on all the resource units. Thus, the combined term is interpreted as the average interference taken over time and frequency for all transmissions.
Given the SINR, we can transmit reliably at the maximum rate nat/s per resource block, where is the bandwidth of a resource unit and is the natural logarithm. To deliver a rate of nat for user , the th base station thus requires resource units. We assume that resource units are available. Thus, we get the load for cell as , i.e.,
(2) 
for . Without loss of generality, we normalize by in (2) and so we set . Let . In vector form, we obtain the nonlinear load coupling equation (NLCE)
(3) 
for , where we have made the dependence of the load on the rate and power explicit.
In the NLCE, the load appears in both sides of the equation and cannot be readily solved as a fixedpoint solution in closed form. Intuitively, this is because the load for base station affects the load of another base station , which would then in turn affect the load . This difficulty in obtaining the in the NLCE remains despite that the function (and similarly function ) depends on for but not on .
We collect the QoS constraints as . Without loss of generality, we assume is strictly positive, as those users with zero rate can be excluded from further consideration. Hence the power vector satisfies so as to serve all the users. Consequently, the load must be strictly positive, i.e., .
Iii Energy Minimization Problem
Our objective is to minimize the sum transmission energy given by . We note that the product measures the transmission energy used by base station , because the load reflects the normalized amount of resource units used (in time and frequency) while the power is the amount of energy used per resource unit.
The energy minimization problem is given by Problem .
(4a)  
(6a)  
As was mentioned earlier, the power vector and rate vector vector are strictly positive to satisfy the nontrivial QoS constraint. The load vector is in fact determined by the NLCE constraint (6a) and thus may be treated as an implicit variable. The second constraint (6a) is imposed so that the rate satisfies the QoS constraint.
We denote an optimal solution to Problem as and the corresponding load as as determined by the NLCE. A key challenge of Problem is that a positive solution pair is considered feasible only if there exists a load such that (6a) holds. Whether this existence holds is not obvious due to the nonlinearity of the NLCE. As such, the convexity of the optimization problem cannot be readily established, and hence standard convex optimization techniques do not apply readily.
Iv Optimality of Full Load
In Section IVA and Section IVB, we consider fundamental properties of rate and load, respectively, such that there exists a power satisfying the NLCE. To study the fundamental properties, we consider the existence of a load satisfying . The additional constraint that is taken into account in Section IVC, in which we prove the key result that full load, i.e., , is a necessary and sufficient condition for the solution in Problem to be optimal.
Iva Satisfiability of Rate
We first establish conditions on rate vector such that a load exists and satisfies the NLCE, possibly with . We denote the spectral radius of matrix as , defined as the absolute value of the largest eigenvalue of .
Lemma 1
For any power , there exists a unique load satisfying the NLCE if and only if
(7) 
where the th element of is given by
(8) 
which is a function of (but not ).
Proof:
Follows directly from [22, Theorem 1]. \qed
Due to Lemma 1, we say that the rate vector is satisfiable if . If is not satisfiable, then there does not exist any power that results in a load satisfying constraint (6a). We note that even if is satisfiable, it is still possible that the load does not satisfy and hence violates its upper bound. Thus, satisfiability is a necessary condition for a feasible solution to exist in Problem , but it may not be sufficient.
Henceforth, we assume that a rate is satisfiable; otherwise no feasible solution exists in Problem . Given , we can then numerically obtain by the iterative algorithm for load (IAL) [22, Lemma 1], as follows. Specifically, starting from an arbitrary initial load , define the output of the th algorithm iteration as
(9) 
for , where is the total number of iterations. Then converges to the fixedpoint solution of the NLCE as goes to infinity. The IAL is derived using [23] by showing that is a socalled standard interference function, to be defined in Section VA.
IvB Implementability of Load
Although Lemma 1 states that any given power vector always corresponds to a load vector that satisfies the NLCE, the reverse is not true. To obtain some intuition why this inverse mapping may fail, let us consider the special case of base stations with channel gain for all , rate , and noise variance . We randomly choose the power using a uniform distribution over which is plotted in Fig. 1(a). The corresponding load obtained using the IAL is shown in Fig. 1(b). We see that indeed there is a load region that does not appear to correspond to any power .
Given that is satisfiable, we say that a load is implementable if there exists power such that the NLCE is satisfied.
The following toy scenario shows that full load may not always be implementable. In practice, this may occur during peak times in cellular hot spots, such as train stations, when mobile data cannot be sustained at high speeds even if all timespectrum resources are used no matter how power is set (cf. Lemma 1).
We assume cells with one user per cell, each with rate nat, and . The channel gains from a base station to the user it serves and the user it does not serve is set as and , respectively. Note that the rate is satisfiable since we get which satisfies (7). By symmetry, the power allocated for all cells must be the same with and must thus satisfy (2), i.e., . For any , the left hand side is upper bounded by nat, which is less than nat. Hence, regardless of the power allocation, (2) cannot hold and so full load is not implementable in this case. and hence
IvC Main Result: Full Load is Optimal
Our first main result is given by Theorem 1, which states that full load, if implementable, is optimal to minimize the sum energy in Problem .
Lemma 2
For Problem , the optimal solution is such that the load vector satisfies .
Proof:
Note that the load satisfies for all cell to satisfy nontrivial rate demands. Assume that at optimality, we have where there exists at least one cell with load and power . With all other power and load fixed, , we reduce the power to . Using (2), the corresponding load strictly increases to . We choose such that . With this new powerload pair for cell , we claim that (see proofs below): (i) the objective function is reduced, and (ii) the corresponding rate vector is such that , i.e., the NLCE constraint is satisfied since . The two claims together imply that with is not optimal, independent of the actual cell . By contradiction, for all , i.e, .
We now prove the first claim. Denote the energy used in cell , as a function of its power , as where does not depend on nor . Then
(10) 
It can be verified by calculus that the numerator of each summand is strictly increasing for . Since the numerator equals zero at , the numerator is strictly positive for . Clearly the denominator is strictly positive for . Thus, . Hence, when the power for cell is decreased, the energy decreases. Thus, the objective function also decreases.
To prove the second claim, we first note that for cell , we have constrained the new powerload pair to satisfy (2). Thus, the new rate for cell , denoted by , is the same as the optimal rate corresponding to the powerload pair . Next, we observe that the product is strictly smaller as compared to , according to the first claim. Thus, for user in cell , strictly increases. It follows that the NLCE for cell is satisfied with the same load but with a larger rate as compared to the optimal rate . In summary, we thus have . \qed
Theorem 1
Suppose full load, i.e., , is implementable. Then the optimal solution for Problem is as follows: , and is such that . The optimal power vector is thus given implicitly by the NLCE as
(11) 
Proof:
From Theorem 1, serving the minimum required rate is optimal. This observation is intuitively reasonable as less resources are used and hence less energy is consumed. Interestingly, Theorem 1 states that having full load is optimal. This second observation is not as intuitive, since it is not immediately clear the effect of using higher load on both the sum energy and interference. This is because using a high load may lead to more interference to neighbouring cells, which may then require other cells to use more energy to serve their users’ rates. Mathematically, the reason can be attributed to the proof of Lemma 2, which shows that, as the power decreases, the energy as well as the interference for each cell decreases, while concurrently the load increases. Thus, by using full load, the energy is minimized.
Next, Corollary 1 provides a converse type of result to Theorem 1. The result follows from a theorem with a generalized statement, which we defer to Section VB because the proof requires the use of algorithmic notions for finding power given load.
Corollary 1
If full load is not implementable, then there is no other load with that is implementable. Thus, there is no feasible solution for Problem .
Remark 1
V Optimal Power Solution
Although full load is optimal for Problem , it is still not clear if the optimal power is unique and how to numerically compute in (11). Our second main result, Theorem 2, answers both questions, but in a more general setting. Namely, we provide theoretical and algorithmic results for finding power given arbitrary load that is implementable (not necessarily all ones) and arbitrary rate that is satisfiable (not necessarily equal to ), so as to satisfy the NLCE.
Va Standard Interference Function
Before we state the main result of the section, we recap the standard interference function and the iterative algorithm introduced in [23]. The algorithm shall be used to obtain the optimal power , and is also a key step in the proof of the implementability of load.
Consider a function . We denote the input as as we shall focus on using power as the input. We say is a standard interference function if it satisfies the following properties for all input power [23].

Positivity: ;

Monotonicity: If , then .

Scalability: For all , .
Next, we consider both the synchronous and asynchronous versions of the iterative algorithm for power (IAP), similar to the two versions of iterative algorithm in [23]. IAP generates a sequence of power vectors via multiple iterations. In each iteration, the power vector produced amounts to evaluating function with the previous iterate as the input. As power is a vector, when the calculation of one power element is performed, there is a choice of whether or not to use this updated power value in the function evaluation for the remaining power elements. These two choices lead to the synchronous and asynchronous IAPs. We consider a specific form of asynchronous IAPs which will turn out to be useful for our proof of Theorem 3 later.
We assume iterations are performed in each case. For the synchronous IAP, the entire power vector is updated in each iteration. In contrast, for the asynchronous IAP, there are inner iterations for each (outer) iteration, and in each inner iteration, only one power element is updated.

Synchronous IAP: Assume an arbitrary initial power given by . The output for iteration , is given by
(12) Clearly, any power element of is solely determined by .

Asynchronous IAP: In each iteration , we perform inner iterations. Assume an arbitrary initial power given by . The output of the th inner iteration, is given by
(13) where represents the power vector containing the most current elements after outer iterations and inner iterations (i.e., during the th iteration). After (outer) iterations are fully completed, each with inner iterations, we obtain as the final power vector solution.
Lemma 3 demonstrates the use of the IAP algorithms to obtain the unique fixedpoint point solution.
Lemma 3
Suppose a fixedpoint solution exists for . If is a standard interference function, then starting from any initial power vector, both the synchronous and asynchronous IAP algorithms converge to the fixedpoint solution , which is unique.
Proof:
We omit the proof which is found in [23, Theorems 2,4]. \qed
VB Main Result: Existence and Computation of Power Solution
Before proving the main result, we present and prove some properties on how the elements of the power vector relate to each other in NLCE. The properties will then be used to establish that the results in [23] with the notion of standard interference function can be applied.
Let be the vector of length that contains all elements in vector except for element . For example, if , then . Lemma 4 shows that given and , the dependency of on (such that the NLCE holds) qualifies as a function, even if the function is not in closed form.
Lemma 4
Let satisfy the NLCE, where the vectors are strictly positive. Then there exists function satisfying for all . Writing ’s and ’s in vector form, we get .
Proof:
We fix and drop these notations in the function for simplicity. To prove the existence of the function , we need to show that given , there exists a unique for . First, we write the NLCE in (2) as
(14) 
where
(15)  
(16) 
are both independent of . We fix and . It follows that and so . Observe that is a strictly decreasing function of . Since as , and as , there exists a unique such that , and thus satisfies (14). Hence there exists a function of the form , for any . \qed
Remark 3
The function does not submit to a closedform solution. For example, consider expressing in terms of in (14) where the number of summands is . Because each of them is nonlinear in , the dependency of on is not explicit.
Remark 4
Although cannot be expressed in closed form, we can numerically obtain the output of the function given the input . Equivalently, this means that we want to obtain the value of such that (14) holds. This is computed, for example, by a bisection search on , making use of the property that is a strictly decreasing function. Specifically, we first choose an arbitrary but small power such that and an arbitrary but large power such that . Next we use the new power and evaluate if is greater or smaller than one, then replace or by , respectively. By performing this procedure iteratively, we have guaranteed convergence to the desired that satisfies . This forms the basis for the proposed algorithm later in Section VI.
We observe that is to some extent similar to in the NLCE (3). From Remark 3, however, the function cannot be readily written as a closedform expression. Thus, proving properties related to is more challenging, as compared to the case of for which a closedform solution is available. Nevertheless, Lemma 5 states that qualifies as a standard interference function as defined in [23].
Lemma 5
Given load and rate , is a standard interference function in .
Proof:
Henceforth we assume that load and rate are given. For notational convenience, we drop the dependence of these entities in the notation of . We consider an arbitrary and refer to as defined in (14), (15) and (16), respectively, throughout the proof. For this proof, it is useful to denote the function explicitly as to ease the discussion. We prove each of the three properties required for standard interference function below.
Positivity: From the proof of Lemma 4, there exists a unique that satisfies (14), i.e., . This holds for all , thus .
Monotonicity: From (14), we observe that strictly increases as decreases, or as any element of increases. Hence, to satisfy , strictly increases if any element of increases. We note that an equivalent representation of is . It follows that is increasing in any of the arguments.
Scalability: Let and , where . Observe that
(17) 
since both equal one according to (14). It is easy to check that . That is, multiplying all the terms in the triplet by a positive constant still allows (14) to be satisfied. Thus we get from (17)
(18) 
With the second argument in fixed, we note that the output of the function strictly decreases with and strictly increases with . By the equality in (18), it follows that because . Taking into account of the definition of and , we have proved . \qed
Using Lemma 4 and Lemma 5, we are ready to provide the main result, stating that NLCE can be expressed in an alternative form with the power taken as the subject of interest. The proof is nonstandard, because the relations among the power elements do not submit to a closed form (Remark 3). Hence, it has been necessary to first establish that the relation between one power element and the others qualifies as a function (Lemma 4). Next, we have used an implicit method to prove that is indeed a standard interference function (Lemma 5).
Theorem 2
Given load and rate , the power that satisfies the NLCE can be represented equivalently in the form of a nonlinear power coupling equation (NPCE) given by
(19) 
where is a standard interference function. Given that a solution exists, then is unique and can be obtained numerically by the IAP.
Proof:
Remark 5
So far we have assumed that there is no maximum power constraint imposed for any element of power . If such power constraints are imposed, then a socalled standard constrained interference function defined in [23] can be used instead to perform the IAP, in which the output of each iteration is set to the maximum power constraint value, if that returned from is higher. This type of iteration converges to a unique fixed point [23, Corollary 1].
VC Characterization on Implementability of Load
Theorem 3 provides a monotonicity result for load implementability. We recall that a load vector is said to be implementable if there exists power such that the NLCE holds.
Theorem 3
Consider two load vectors with and . If is implementable, then is implementable. Moreover, the respective corresponding powers and satisfy .
Proof:
Suppose is implementable, i.e., there exists power such that the NPCE (or equivalently the NLCE) holds. From Theorem 3, is a standard interference function. We shall prove that is also implementable, i.e., exists.
Before we consider the general case of , we first focus on the special case that strict inequality holds only for the first element (with reindexing if necessary), i.e., and with . We now use the asynchronous IAP (13) with load , and we set the initial power as . Our objective is to show that the power converges to that satisfies the NPCE with .
Consider the asynchronous IAP (13) with outer iteration and inner iteration :

For : Consider the NLCE for cell with the original load and power :
(20) In the first iteration, and are updated by the actual load of interest and the iterated power , respectively, with other load and power unchanged. Since , we must have .

For : We shall show that the iterated power satisfies . The NLCE for cell with the original load and power can be written as:
Upon updating cell 2, we have updated to the newly iterated , respectively. Since as mentioned earlier, .

For : For subsequent iterations, it can be shown similarly that for . This completes the first outer iteration.
At this point, we get . It can be similarly shown that for .
For large number of iterations , the decreasing sequence must converge since (i.e., it is bounded from below) for any due to the positivity of the standard interference function. Thus, the power solution exists, i.e., is implementable.
At convergence, we have . So far we have assumed that only one element of the load is strictly increased. In general, if more than one load element is increased, repeating the argument sequentially for every such element proves that power exists and is decreased. Thus, in general is implementable for , where . \qed
From Theorem 3, we also obtain the equivalent result that is not implementable if is not implementable.
The next theoretical characterization is on the implementable load region over all nonnegative power vectors for any given satisfiable rate , i.e., . Theorem 4 states that the boundary of this region is open. The norm in Theorem 4 can be any norm, e.g., the 2norm or the maximum norm .
Theorem 4
Suppose load is implementable with power and rate . Then there exists , such that any load vector with is implementable. Moreover, the implementable load region is open.
Proof:
Let with , and let the corresponding load satisfying the NLCE with rate be . Note that exists, because the existence of load does not depend on power (cf. Lemma 1). By applying the IAL in (9) to obtain (using power ) with the initial load set as , it can be easily checked that the load vector decreases in every iteration. Since , the iterations must converge to . By Theorem 3, any is implementable. As , there is an implementable neighbourhood of . That is, there exists , for which any load vector satisfying is implementable. Since the result holds for any in , it follows that is open. \qed
VD Algorithm for Optimal Power Vector
By Theorem 2, we can use the IAP to compute the optimal power for (11) in Theorem 1 for any given implementable load . We recall that to minimize the energy we set (Theorem 1). To obtain the output of the function in each step of the IAP, bisection search is able to determine the power such that (see Remark 4). Putting together the theoretical insights results in the following formal algorithmic description (Algorithm 1) for computing .
Algorithm 1 solves the NPCE for given , by iteratively updating the power vector and reevaluating the resulting load . The bulk of the algorithm starts at Line 2. The outer loop terminates if the load vector has converged to . For each outer iteration, the inner loop is run starting at Line 3, for which the power vector for each cell is updated. In each update, the power range is first initialized to , where , such that and . Since the function is a strictly decreasing function, the bisection search from Lines 712 ensures convergence to the unique solution for , or equivalently, the value of . Load reevaluation is then carried out in Line 15.
Vi Numerical Evaluation
Via Simulation Setup
In this section, we provide numerical results to illustrate the theoretical findings. The simulations have been performed for a reallife based cellular network scenario, with publicly available data provided by the European MOMENTUM project [24]. The channelgain data are derived from a pathloss model and calibrated with real measurements of signal strength in the network of a subarea of Alexanderplatz in the city of Berlin. The pathloss model takes into account the terrain and environment, preoptimized antenna configuration (height, azimuth, mechanical tilt, electrical tilt); fast fading is not part of the data made available. Further details are available in [24].
The scenario is illustrated in Fig. 2. The scenario has 50 base station sites, sectorized into 148 cells. In Fig. 2, the red dots indicate base station sites and the green dots represent the location of users. Most of the sites have three sectors (cells) equipped with directional antennas. The blue short lines represent the antenna directions of the cells. The entire service area of the Berlin network scenario is divided into 22500 pixels as shown in Fig. 2. That is, each pixel represents a small square area, with resolution 50 50 m, for which signal propagation is considered uniform. Users located in the same pixel are assumed to have the same channel gains. In our simulations, each cell serves up to ten randomly distributed users in its serving area as defined in the MOMENTUM data set. The total bandwidth of each cell is 4.5 MHz. Following the LTE standards, we use one resource block to represent a resource unit with 180 kHz bandwidth each in the simulation. Network and simulation parameters are summarized in Table I.
Parameter  Value 

Service area size  7500 7500 m 
Pixel resolution  50 50 m 
Number of sites  50 
Number of cells  148 
Number of pixels  22500 
Number of users  1480 
Thermal noise spectral density  145.1 dBm/Hz 
Total bandwidth per cell  4.5 MHz 
Bandwidth per resource unit  180 kHz 
Tolerance in IAP  
Initial power vector in IAP  W 
ViB Results
Our objective is to numerically illustrate the relationship among the load, power, and sum transmission energy. First, we consider the use of uniform load with for various , with being the case of full load. Given the load vector , the optimal power solution is then obtained by using the IAP described by Algorithm 1. Next, for benchmarking, we consider the conventional scheme that employs uniform power allocation . We choose that results in the minimum sum energy subject to the constraint that the corresponding load satisfies , as follows. From the proof of Lemma 2, the energy (given by the product of load and power) for each cell strictly decreases as the power strictly decreases. Thus, to minimize the sum energy, we choose the smallest such that ; this can be obtained by a bisection search starting with sufficiently small and large values of . For any under consideration, the IAL is used to obtain the load corresponding to the power .
In the first numerical experiment, we consider the sum energy for rate demand with being successively increased, while keeping satisfiable. Fig. 3 compares the sum energy for various uniform load levels, including full load, and that obtained by uniform power allocation. From Fig. 3, the sum energy for all cases appears to grow exponentially fast as the rate demand increases, approaching infinity as the rate demand increases. The vertical dotted line in Fig. 3 corresponds to the boundary when the rate demand is not satisfiable, i.e., , and hence represents the upper bound for which the system can support. This behaviour is consistent with Lemma 1. Deploying full load achieves the smallest sum energy, in accordance with Lemma 2. The reduction in sum energy is particularly evident in comparison to the scheme of uniform power – the relative saving is or higher for the rate demand shown in Fig. 3. Conversely, for a fixed amount of sum energy, deploying full load and optimizing the corresponding power allows for maximizing the rate demand that can be served.
Next, we examine the energy consumption by progressively increasing the uniform load for four rate demand levels with taking the values of 350 kbps, 450 kbps, 550 kbps and 600 kbps. The results are shown in Fig. 4. We observe that the sum energy decreases monotonically by increasing the load. The reduction of sum energy appears to be exponentially fast in the lowload regime, but is much slower in the highload regime. In addition, the numerical results reinforce the fact that some load vectors are not implementable. In particular, it is not always possible to obtain a power vector for a load vector with very small . From Fig. 4, the sum energy surges to infinity when the load approaches some fixed (small) value, which suggests that, for any , the load cannot become arbitrarily small, irrespective of power.
Furthermore, we numerically investigate the convergence behavior of the IAP. The theoretical analysis on the convergence speed of the IAP depends on whether further satisfies some property such as contractivity [25], for which linear rate of convergence can be shown to hold. In Fig. 5, we set the target load vector and the initial power vector Watt, with rate demand where kbps. The Euclidean distance between the iterate and target is given by the 2norm . The evolution of for the four different rate demand cases is illustrated in Fig. 5. We consider the algorithm converged if the largest error between the load iterate and the target load is less than , i.e., if . For the four rate demand cases, convergence is reached after 11, 19, 36 and 59 iterations, respectively. Given the size of the network (148 cells), the values are moderate. Also, we notice that when the rate demand increases, more iterations are required for convergence with a longer tailoff. This is mainly because a high rate demand means that, in general, the NPCE is operating in the high SINR regime. The amount of progress in load in an IAP iteration is mainly dependent on the denominator in (3). For high SINR regime, the relative change in load is lesser due to the logarithm operator, thus slowing down the progress. Moreover, the number of iterations depends on the initial power point. In general, fewer iterations are required if the starting power point is closer to the optimum. Note that no matter what the rate of convergence is, the convergence of the IAP is guaranteed by Theorem 2. The convergence speed depends also on other factors, e.g., the scale of the network (in terms of the number of BS and users), the rate demand and the choice of . An explicit characterization of this dependence is beyond the scope of this paper.
In case of the presence of some time constraints in a practical application, the IAP may be terminated before full convergence is reached. Thus, the capability of delivering a loadfeasible and closetoconvergent solution within few iterations is of significance. It can be seen in Fig. 5 that a majority of the iterations is due to the tailingoff effect – the load vector is in fact close to the target value within about half of the iterations. For all the rate demand levels, convergence is in effect achieved in less than 20 iterations; this is promising for the practical relevance of the proposed IAP scheme. Finally, to ensure that the load is strictly less than full load for practical implementation, we may set with .
Fig. 6 illustrates the the number of iterations required for the load to converge, i.e., until , with . The total number of users is increased from the current (corresponding to users per cell) to (corresponding to users per cell). We have set the rate demand as with kbps, because the original case of is no longer satisfiable for users. From Fig. 6 , we see that the number of iterations increases as the number of users increases, and the rate of increase is higher if the number of users is large or the demand is high. Hence, for systems that support high data rate or large number of users, more computational resources are needed to implement the algorithm.
Vii Conclusion
We have obtained some fundamental properties for the cellular network modeled by a nonlinear load coupling equation (NLCE), from the perspective of minimizing the energy consumption of all the base stations. To obtain analytical results on the optimality of full load, and the computation and existence of the power allocation, we have investigated a dual to the NLCE, given by a nonlinear power coupling equation (NPCE). Interestingly, although the NPCE cannot be stated in closedform, we have obtained useful properties that are instrumental in proving the analytical results. Our analytical results suggest that in loadcoupled OFDMA networks or more specifically LTE networks, the maximal use of bandwidth and time resources over power leads to the highest energy efficiency. In the literature, the maximal use of resources is typically suggested to maximize the network throughput; our work gives a similar conclusion but from a different and complementary approach of minimizing energy. To implement the solution, the load and power solutions have to be computed and sent to all base stations for implementation. Hence, some level of coordination has to be set up in practice. In this paper, we have assumed the use of ideal power amplifier and that the users’ associations to the base stations are given. The effects of nonlinear power amplifier and the problem of user association may be considered as future work.
Acknowledgements
We would like to thank the anonymous reviewers for their valuable comments and suggestions. The work of the second author has been supported by the LinköpingLund Excellence Center in Information Technology (ELLIIT), Sweden. The work of the third author has been supported by the Chinese Scholarship Council (CSC) and the overseas PhD research internship scheme from Institute for Infocomm Research (IR), A*STAR, Singapore.
Lemma 6 (Theorem 2, [22])
Consider the NLCE (3) with power fixed. Given the rate vectors and with and , the corresponding load vectors and satisfy .
Lemma 7
For Problem , the optimal rate vector satisfies .
Proof:
Suppose that at optimality, there exists at least one rate element that is strictly greater than its corresponding (minimum) rate demand . Taking the power to be fixed as , if we decrease to , then the load will strictly decrease while satisfying the constraint (6a) by Lemma 6. Thus, the objective function value decreases. This contradicts the optimality of . Thus . \qed
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