Aggregation of Variables in Load Models for ... - Semantic Scholar

Aggregation of Variables in Load Models for Interference-Coupled Cellular Data Networks Albrecht J. Fehske, and Gerhard P. Fettweis Vodafone Stiftungslehrstuhl, Technische Universit¨at Dresden Email: {albrecht.fehske, fettweis}@ifn.et.tu-dresden.de

Abstract—In order to meet increasing traffic demands, future generations of cellular networks are characterized by decreasing cell sizes at full frequency reuse. Due to inevitable inter-cell interference, load conditions in neighboring cells can no longer be considered independent. Unfortunately, the adequate flow level model for such a setup is analytically intractable. Utilizing aggregation techniques, which were originally proposed to analyze large state models in economics, we propose a framework to compute the average base station loads based on an approximation of the joint stationary distribution of the number of active flows in all cells. The technique proposed requires solving a system of linear equations whose dimension increases exponentially with the number of cells. Since such a system is essentially intractable for large networks, we propose a fixed point algorithm to compute approximate base station loads based on the notion of average interference. Numerical results validate the accuracy of both modeling techniques. The modeling approach presented in this paper is essential for accurate characterization of cell throughput as well as base station energy consumption under varying load conditions.

I. I NTRODUCTION

Flow level models that capture the dynamic nature of arrivals and service periods of data flows have been successfully applied to characterize the performance of cellular networks. In such models, a cell is commonly represented by a queue and the main quantity of interest is the random process of the number of active data flows within the cell at any point in time. Scenarios with multiple interfering cells are naturally represented by a network of queues where the number of active flows is a vector process. In previous studies, the underlying assumption is that data flows always observe some given constant interference during their service. Such an assumption then allows for a network model with independent queues. The constant interference level is typically taken as either the maximal possible (e.g., [1]) or some kind of time average (e.g., [2]). While all base stations (BSs) in the former case are assumed to transmit perpetually, the latter case assumes that a sufficiently large number of interference scenarios are observed by each flow. Both assumptions, however, are accurate only in a fully loaded system. Realistically, each flow observes only a single interference condition and, as a consequence, the service rates in each cell depend on the network state, i.e., the number of active flows in all other cells. Unfortunately, the adequate queueing model, a so called processor sharing model, renders analytically intractable. Similar observations are already made by Bonald et al. in [3], where the authors provide certain bounds on the load of individual cells in a network based on minimal and maximal

interference assumptions in all other cells. The work presented here addresses the same problem, however, instead of finding bounds through minimal and maximal interferences we aim to approximate the joint stationary distribution of the network. Having obtained stationary probabilities for all possible network states, it is then straightforward to compute individual load conditions, i.e., the average resource utilization of all BSs. Since an analytical or straightforward numerical calculation of the joint stationary distribution is not possible, we employ the concept of aggregation of variables to derive an intuitive approximate model. Aggregation of variables is a versatile tool to reduce the complexity of analyzing systems with a large state space. Simon and Ando lay foundations for these principles in [4] with the original intention of modeling complex economic interactions. Later on, the concept was adapted by Courtois to analyze networks of queues arising in computer models (e.g., [5]). The general idea behind aggregation is to decompose the overall state space into groups or aggregates where strong interactions occur, and then characterize transitions within and amongst aggregates separately. The contributions of this paper are twofold. Firstly, we provide a tractable approach to approximate the stationary distribution of the number of active flows in interference coupled wireless networks from which individual BS loads and common network performance measures can be derived. These results are outlined in Section III. For N cells, the approach entails solving a system of 2N linear equations, which effectively becomes intractable for larger N . To overcome this problem we propose an alternative approximation to the BS loads that utilizes a fixed point iteration of dimension N in Section IV. We prove convergence of the proposed algorithm and uniqueness of the solution. Numerical results that compare the accuracy of the approximation are presented in Section V. The research presented in this paper is driven by recent interest in energy efficiency of wireless communications and considerable drive towards understanding and reducing the energy needs of BSs. While the overall power drawn by early BS generations was largely independent of the BSs’ load condition, modern equipment scales with load. Consequently, appropriate network management procedures that reduce network capacity in low load conditions, e.g., by adapting transmission bandwidth, offer considerable energy saving potential. Such techniques call for precise characterization and prediction of the network’s load condition in order to avoid transitioning into overload situation while still harnessing energy saving

potential. Due to space constraints, we restrict this paper to presenting the modeling aspects. Application of the framework to, e.g., energy efficiency analysis and optimization is left for future publications. II. F LOW L EVEL M ODELING OF C ELLULAR N ETWORKS We consider the downlink of a cellular network consisting of N base stations covering a compact region L ⊆ R2 . Users are assumed R to be distributed according to some distribution δ(u) with L δ(u) du = 1. In the following, we shall use the terms BS and cell synonymously. A. Radio Link Model and Resource Sharing In wireless systems, the radio link quality generally depends on the users’ distance to the serving and the surrounding base stations, on the number of active BSs generating interference, as well as their transmit powers. Here, we make the following common yet important assumption: Assumption 1. If there is at least one active flow within a cell, the corresponding base station utilizes all available transmission resources and thus decidedly transmits at full power. We call a BS active if it serves at least one active flow. We further assume, that transmission resources (such as the number of subcarriers in OFDMA or channelization codes in CDMA systems) are shared evenly among all active flows in the cell. Let the terms Li and u denote the serving area of a cell i and the user position, respectively. We further denote by y ∈ Y := {0, 1}N a vector with yi = 1 if BS i is active and yi = 0 otherwise. The indices of inactive and active BS for each y are collected in the sets N0 (y) := {i = 1, . . . , N | yi = 0} and N1 (y) := {i = 1, . . . , N | yi = 1} .

B. Flow Level Model For a Single Base Station This section presents a common flow level model for a single BS, where we initially assume that the vector y of active BSs is fixed. We shall drop this assumption in the next section. In any case, we assume that the arrival of data flows to base station i takes place according to a Poisson process with intensity λi . Flow sizes are assumed to be exponentially distributed with common mean Ω. For a given vector y of active BSs and considering all points in its serving area, the mean serving rate for flows at BS i is given as Z −1 δi (u) Ri (y) , Ri (y) := du , (2) µi (y) = Ω Li ri (u, y) denotes the distribution of users where δi (u) := R δ(u) δ(u) du Li conditioned on being in cell i. For each i and y 6= 0 we further define the ratio ρi (y) :=

λi Ω λi = . µi (y) Ri (y)

(3)

Let Xi (t) ∈ N0 denote the number of active flows in some cell i at time t. With the above definitions, we can understand Xi (t) as continuous time random process, which omits a stationary distribution if and only if ρi (y) < 1 holds. C. Flow Level Model For Interfering Base Stations Let X(t) := (X1 (t), . . . , XN (t)) ∈ S := NN 0 for t ≥ 0 denote the vector process of active flows in N cells. In the following, we consider the process X(t) with transition rates   for x0 = x + ei , λ i (4) q(x, x0 ) = µi (y) for x0 = x − ei ,   0 else,

Assuming BS i to be active, the signal-to-interference-andnoise-ratio (SINR) and the achievable data rate at location u ∈ Li are given by pi (u) and (1a) γi (u, y) = P j∈N1 (y) pj (u) + θ j6=i
ri (u, y) = log2 1 + γi (u, y) , (1b)

where ei denotes an N -dimensional vector with the i’th component equal to one and all other components equal to zero. In addition, we define (more formally) the vector of active cells as y := sgn(x). Note that the transition rates µi (·) are state dependent in principal, but are equal in states where the same BSs are active and are strictly smaller in states where additional BSs are active, i.e.,

respectively. Here, pi (u) and θ denote the power received at location u from BS i and the power of the thermal noise added at the receiver, respectively. Without loss of generality we assume the noise power to be equal at all terminals. We use Shannon capacity to model the achievable rate ri , however, the results presented in this paper apply to any concave function of the SINR. It should be noted that the transmit powers pi are considered as the power per transmission resource. Note that path gains in wireless systems are subject to slow and fast fading. With good accuracy, we can assume that the former happens on a much larger and the latter on a much smaller time scale than a typical flow duration. Consequently, slow and fast fading are assumed to be incorporated as instantaneous and average values within the received powers pi (u), respectively.

sgn(x) = sgn(x0 ) ⇒ µi (x) = µi (x0 ) and 0

0

sgn(x) > sgn(x ) ⇒ µi (x) < µi (x ).

(5) (6)

where, the inequality is taken component wise. Note that the left hand sides of (5) and (6) are vector relations and the property need not hold if reduced to a condition based merely on the number of active cells. D. Balance Equations When analyzing the performance of the network it is desirable to characterize the joint stationary distribution π(x) of the process of active flows X(t) from which different network performance measures, in particular the individual cell loads, can be computed. The distribution exists as the solution of a

system of balance equations, which can be derived from Eq. (4) for x ∈ S and y = sgn(x) as   X X  λn + λn + µn (y) π(x) = n∈N0 (y)

the network in a state x ∈ S(y) for all y as the common product form, which is defined up to a constant factor [6]. Since each S(y) constitutes only P a subset of the total state space we require the condition x∈S(y) π ˜ (x) = σ(y) to hold. This requirement yields

n∈N1 (y)

=

X

µn (y + en )π(x + en )+

n∈N0 (y)

+

X

λn π(x − en ) + µn (y)π(x + en ). (7)

n∈N1 (y)

Above, the summations over n ∈ N0 (y) are taken over only λn and µn (y + en ) on the left and right hand side, respectively, since an empty cell cannot transit to or have arrivals from lower states. Even though Eq. (7) is linear in π(x), obtaining an exact solution is difficult since the state space (and thereby the system of equations) is countably infinite. Unfortunately, since the service rates vary among states, the queueing network is not partially reversible and techniques to obtain a product form stationary distribution as proposed, e.g., in [6] are not applicable. In fact in [7], Fayolle et al. observed that in case of two queues a product form solution exists if and only if µ1 (0, 1) + µ2 (1, 0) = µ1 (1, 1) + µ2 (1, 1) which requires that the service rate of one BS must be larger in case of interference, which is contradictory to Eq. (6). Since an analytical expression for the stationary distribution π(x) is not attainable, we subsequently derive a framework to approximate π(x) using state aggregation techniques. III. AGGREGATION OF S TATES We start with partitioning the set of all possible states N S = NN disjoint subsets by grouping states corre0 into 2 sponding to the same set of active base stations, i.e.,  S(y) := x ∈ NN (8) 0 | sgn(x) = y . Considering the sets S(y) we can think of y as a collection or an aggregate of states x ∈ S(y). Let us denote the stationary probability of being in an aggregated state y as X X σ(y) := π(x) with σ(y) = 1. x∈S(y)

y∈{0,1}N

The terms σ(y), thus denote the probability that the network is found in a state where only BSs with indices in N1 (y) are active and BSs with indices in N0 (y) are inactive. In the following, we simplify the analysis of the stationary behavior of the process X(t) by considering transitions within and among aggregates separately. A. Approximating State Transitions Within Aggregates A key observation for subsequent derivations is that, conditioned on being in S(y), the network essentially behaves as a network of independent queues since the service rates are equal within each S(y). By neglecting the interactions between states from different aggregates, we can thereby derive the approximate stationary probabilities π ˜ (x) of finding

(Q


1 − ρn (y) ρxnn −1 (y) σ(y) for y 6= 0 π ˜ (x) = σ(0) for y = 0, (9) where the quantity ρn (y) is given by Eq. (3). n∈N1 (y)

B. Approximating Transitions Among Aggregates Assuming interactions among states within an aggregated set S(y) are captured by Eq. (9), we shall now characterize the transitions between aggregates. As a first step, we sum up Eq. (7) over all x ∈ S(y) for some y 6= 0, which yields 

 X

X

λn +

 n∈N0 (y)

=

λn + µn (y) σ(y) =

n∈N1 (y)

X

X

µn (y + en )π(x + en )+

n∈N0 (y) x∈S(y)

+

X

X

λn π(x − en ) + µn (y)π(x + en ). (10)

n∈N1 (y) x∈S(y)

Naturally, Eq. (10) and thus the terms σ(y) depend on different probabilities π(x) of states x belonging to aggregates y as well as neighbors y + en . As stated above, no closed form expressions are attainable for the probabilities π(x). The idea, now, is to replace probabilities π(·) with π ˜ (·) to arrive at a system of equations in σ(y) that approximates Eq. (10). Before we proceed along these lines, we introduce some results on summations over certain sets of stationary probabilities π ˜ (x) in order to simplify subsequent derivations. 1) Sums of Approximate Stationary State Probabilities: Let us define two collections of states as Sn+ (y) := {x ∈ S | x − en ∈ S(y)} and

(11a)

Sn− (y)

(11b)

:= {x ∈ S | x + en ∈ S(y)} .

In case yn = 0, the set Sn+ (y) is a subset of S(y +en ) containing all states x ∈ S(y + en ) with xn = 1, i.e. all states at the boundary of S(y). In case yn = 1, the set Sn+ (y) is a subset of S(y) consisting of all states x ∈ S(y) with xn > 1, i.e. all states except the boundary of S(y − en ). Furthermore, for y = 0, the set Sn− (y) is empty and we are no longer concerned with it. For y = 1 we have Sn− (y) = S(y) ∪ S(y − en ), i.e., the set Sn− (y) comprises all states corresponding to aggregates y and y − en . We can now state results for summation of approximate stationary state probabilities π ˜ (x) over the sets Sn+ (y) and

Sn− (y). In particular, for any y with yn = 1 it holds that X X π ˜ (x) = σ(y) − π ˜ (x) = ρn (y)σ(y), (12a) + x∈Sn (y)

X

x∈S(y) xn =1

X

π ˜ (x) =

− x∈Sn (y)

π ˜ (x) +

x∈S(y)

X

π ˜ (x) = σ(y) + σ(y − en ).

σP = 0,

x∈S(y−en )

(12b) Above, Eq. (12a) follows from the relation X
π ˜ (x) = 1 − ρn (y) σ(y) x∈S(y) xn =1

and Eq. (12b) follows from the fact that the set Sn− (y) can be represented as a union. Considering Eq. (12a), for the case yn = 0, we obtain X X π ˜ (x) = σ(yn + en ) − π ˜ (x) = + x∈Sn (y)

+ x∈Sn (y+en )


= 1 − ρ(yn + en ) σ(y + en ). (12c) 2) Approximate Aggregate Probabilities: Using the notation introduced in Eqs. (11) and replacing probabilities π(·) with π ˜ (·) we rewrite the right hand side of Eq. (10) as X X ... = µ(y + en )˜ π (x) +

X

X

λn π ˜ (x) +

 − (y) x∈Sn

X

µn (y)˜ π (x) .

+ (y) x∈Sn

After applying the summation results from Eqs. (12) to the terms on the right hand side we obtain X
... = µ(y + en ) − λn σ(y + en ) + n∈N0 (y)

+

X

λn σ(y) + σ(y − en )) + λn σ(y),

n∈N1 (y)

=

X

E[Xi ] ≈

X

X   E Xi |X ∈ S(y) σ(y) =

y∈Ai

y∈Ai

ρi (y) σ(y), 1 − ρi (y) D

where ρi (y) is given in Eq. (3) and Xi (t) −→ Xi . Further, t→∞ Ai := {y ∈ Y | yi = 1} denotes the set of all aggregates with active BS i. Using Little’s Law, the mean service time and mean flow throughput can be derived from the mean number of active flows.

n∈N1 (y)

X
µ(y+en )−λn σ(y+en )+

n∈N0 (y)

λn σ(y−en ).

n∈N1 (y)

(13) C. Computing Approximate Stationary Distributions Defining the transition rates   λn 0 p(y, y ) = µn (y) − λn   0

Obtaining values for activity levels of N BSs thus requires prior computation of 2N aggregate probabilities σ(y). Since the latter are the solutions of a linear system of size 2N , their computation might become prohibitively complex for larger networks. For instance, scenarios involving small indoor or outdoor BSs might easily feature more than 20 cells, which renders the solution of Eq. (15) essentially intractable. A. A Fixed Point Approximation of BS Activity Levels

0

for y = y + en , for y 0 = y − en , else,

In many applications we are not directly interested in actual values of all stationary probabilities σ(y), but rather in the probability that a certain base station in the network is transmitting, which we refer to as its activity level. Note that, due to Assumption 1 the BS activity levels correspond to the average BS resource utilization, i.e., their loads. Let ηi denote the activity level of BS i. We obtain the value of ηi by summing over probabilities of all aggregates y where BS i is active, i.e., X ηi := σ(y). (16) y∈Ai

which after sorting terms yields the final relation   X X  λn + µn (y) − λn  σ(y) = n∈N0 (y)

where σ denotes the row vector collecting the 2N probabilities P σ(y) in corresponding order. Along with the condition y∈Y σ(y) = 1, Eq. (15) can be solved for σ by standard numerical tools. After the aggregate probabilities σ(y) have been obtained for all y, approximate stationary probabilities π ˜ (x) can be obtained for all x from Eq. (9). Moreover, in the stationary regime, the mean number of active flows in cell i is well approximated by the weighted sum



 n∈N1 (y)

(15)

IV. ACTIVITY L EVELS OF C ELLULAR BASE S TATIONS

+ n∈N0 (y) x∈Sn (y)

+

over time. Furthermore, assuming the aggregates to be arranged in any order allows us to identify the transition rates
with elements of a matrix P = [pij ] by pij := p y(i), y(j) . P2N We define pii := − j=1 pij and write Eq. (13) equivalently as the system

(14)

we can interpret Eq. (13) as balance equations of a random process describing the transitions between the aggregates y

In order to obtain an approximate solution for the BS activity levels in cases where the number of BSs is large, we propose a dedicated algorithm. In Eqs. (1)(2)(3), we define a link model and corresponding BS loads for each aggregated state y, more specifically for each possible interference condition. Instead in the following, we consider a link model and respective loads that occur when data flows experience the

where σ once again denotes the vector collecting the σ(y) in some order. By re-ordering the terms in the denominator of Eq. (17) we can express the mean SINRs as functions of η = (η1 , . . . , ηN )T in the form pi (u) γi (u, η) := P . j6=i pj (u)ηj + θ

(18)

Observe that, based on mean interference, the SINRs can be expressed as functions of N BS activity levels ηi instead of 2N aggregate probabilities σ(y). Moreover, if exposed to constant interference, the network of BSs behaves as a network of independent queues and all BS loads ηi are given by the ratios !−1 Z δi (u) λi Ω
du ρi (η) = , Ri (η) = . Ri (η) Li log2 1 + γi (u, η) B. Algorithmic Solution The above formulation suggests computation of the BS activity levels via the fixed point iteration 
 η k+1 := min ρ η k , 1 (19)
T where ρ(·) = ρ1 (·), . . . , ρN (·) . The min(·) operation is to be taken component wise. Applying the min(·) operation ensures that any η ≥ RN + is a feasible starting point for the iteration. We state general convergence results of the iteration in Eq. (19) in the following theorem.

Simulation ηsim n

0.9 Load ηn of a randomly selected BS

j6=i

1 Aggregation ηagg n

0.8

Fixed point ηfp n

0.7

min

Min interf. ηn

0.6

max

Max interf. ηn

0.5 0.4 0.3 0.2 0.1 0 0

0.2 0.4 0.6 0.8 Normalized arrival intensity λn / λmax

1

(a) Simulated and estimated BS loads 0

10

Relative error of load estimate

mean interference over all aggregates y. We write the SINR in this case as p (u) Pi , (17) γi (u, σ) := P σ(y) j∈N1 (y) pj (u) + θ y∈Y

−1

10

−2

10

Aggregation ηagg n Fixed point ηfp n

−3

10

min

Min interf. ηn

Max interf. ηmax n

−4

10

0

0.2 0.4 0.6 0.8 Normalized arrival intensity λn / λmax

1

(b) Relative error between estimated and simulated loads

Theorem 1. For anyinitial activity vector η 0 ∈ RN + , the se
k+1 k quence η := min ρ η , 1 for k = 0, 1, 2, . . . converges to a unique fixed point η ∗ .

Fig. 1. Simulated and estimated loads of a randomly selected BS for increasing arrival intensity in a network of six interfering BSs.

Proof: The proof essentially relies on the framework of interference functions introduced by Yates in [8]. We argue that the function ρ(·) is a standard interference function (SIF), which implies Theorem 1 holds. A function f : RN → RN is called an SIF if f has the following properties

V. N UMERICAL E VALUATION

Positivity:

f (x) ≥ 0 for x ≥ 0,

Monotonicity:

x ≥ x0 ⇒ f (x) ≥ f (x0 ),

Scalability:

αf (x) > f (αx) for α ∈ R, α > 1,

where all inequalities are taken component wise. First we observe from Theorem 7 in [8] that if the function ρ(·) is
an SIF, then so is the function min ρ(·), 1 , and thus we focus only on the former. Showing positivity and monotonicity for the function ρ(·) can be done by inspection. In order to show scalability, we argue that each component ρi (·) is strictly concave in its argument. Concavity and positivity then imply scalability. Since these parts of the proof are of rather technical nature, we detail them in the appendix. If ρ(·) is a standard interference function, Theorem 1 follows from Theorem 2 in


[8] if there is at least one η with η ≥ min ρ(η), 1 . The latter condition, however, is fulfilled for any η ≥ 1.

In this section we demonstrate the ability of the aggregation technique and the fixed point iteration to approximate BS loads by comparing them to results from a discrete event simulation. In the following, let ηnsim , ηnagg , and ηnfp denote the BS activity levels obtained from simulation, via aggregation technique in Eq. (16), and from the fixed point iteration in Eq. (19), respectively. We study a network of N = 6 interfering cells evolving over time according to the transition probabilities in Eq. (4). For given service rates µn (y) and mean flow size Ω, we gradually increase the arrival intensities λn and determine the BS activity levels. For comparison we also show the load estimates assuming all flows experience constant maximal and minimal interference, i.e., ηnmin :=

λn , miny∈Y µn (y)

ηnmax :=

λn maxy∈Y µn (y)

.

Fig. 1(a) depicts the simulated load as well as different load estimates of a randomly chosen cell index n for increasing arrival intensity. Naturally, the BS loads η max and η min computed under maximal and minimal interference assumptions increase linearly with arrival intensity. For very low and very high load regimes η sim is well approximated by η min and η max , respectively. In the broad middle region, only η agg obtained via the aggregation approach provides a highly accurate estimate of the simulated BS load. Fig. 1(b) displays the relative errors between η sim and different approximations, confirming the above observations. The fixed point approximation η fp provides a better approximation than maximal and minimal interference assumptions over the entire load range. VI. C ONCLUSIONS In this paper we present a framework to approximate the joint stationary distribution of the number of active flows in a network of interfering BSs through aggregation of variables. Here, we make the common assumption that BSs utilize all available transmission resources as long as there is at least one active flow in the cell. The stationary probabilities of aggregated states are used to compute the average load conditions of all BSs. The average load conditions are of key importance to assess the average energy consumption of cellular BSs and design effective network management algorithms. In case the number of interacting cells becomes large, we propose a fixed point algorithm to obtain an approximation on the BS loads. Numerical examples demonstrate the accuracy of the proposed techniques compared to estimates based maximum and minimum interference. VII. A PPENDIX In order to derive strict concavity of ρi (·) for any i, note that theR former can be expressed as the composition ρi (·) = λi Ω Li (g ◦ hi )(·, u) δ(u) du with −1   1 , and g(x) := log2 1 + x   X hi (η, u) := pi (u)−1  ηj pj (u) + θ . j6=i

Since, the integration over u is a linear functional and hi (·, u) are affine functions for all u it remains to show strict concavity of g(·) for positive arguments. We proceed by showing that its second derivative is negative, i.e., g 00 (x) < 0 for x > 0. Calculation of the second derivative reveals that this condition is equivalent to ln(y) > 2

y−1 for y > 1, y+1

(20)

where we substituted y := 1 + x1 in the second equation. Observe that for y = 1, both sides of Eq. (20) are equal to 0. Comparing the derivatives of the left and right hand 4 2 sides of Eq. (20) yields y1 > (y+1) 2 ⇔ (y − 1) > 0, i.e., the derivative on the left is greater than the one on the right for y 6= 1 and both derivatives are equal for y = 1. Thus we

conclude the relations in Eq. (20) hold, which implies that the functions g(·) and thus ρi (·) are strictly concave for positive arguments. In addition we show that, if the function ρi (·) is strictly concave for positive arguments, the scalability condition holds, i.e., αρi (η) > ρi (αη) for α > 1, η ≥ 0. Since the function f is strictly concave in η, its hypograph, that is the set hypo(ρi ) := {(η, t) | t ≤ ρi (η)}, is a convex set and for each +1 η there exists a vector a ∈ RN defining a supporting + hyperplane to the hypograph such that the relation

aT η, ρi (η) > aT η 0 , ρi (η 0 ) for all η 0 ∈ RN (21) + holds ([9], p. 51). Considering specifically η 0 := αη for some η and some α > 1 we obtain the inequalities


αaT η, ρi (η) ≥ aT η, ρi (η) > aT αη, ρi (αη) . (22) Since all addends in the summations on the left and right hand sides are equal, except for the terms αρi (η) and ρi (αη) we conclude αρi (η) > ρi (αη). ACKNOWLEDGEMENT This work was supported in part by European Community’s Seventh Framework Programme (FP7/2007- 2013) under grant agreement n◦ 247733. R EFERENCES [1] T. Bonald and N. Hegde, “Capacity Gains of Some Frequency Reuse Schemes in OFDMA Networks,” in GLOBECOM 2009 - 2009 IEEE Global Telecommunications Conference. IEEE, Nov. 2009, pp. 1–6. [Online]. Available: http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=5425225 [2] H. Kim, G. Veciana, X. Yang, and M. Vengkatchalam, “Distributed α -Optimal User Association and Cell Load Balancing in Wireless Networks,” IEEE/ACM Transactions on Networking, 2011. [Online]. Available: http://wncg.org/publications/dl.php?file=de+Veciana1268149085.pdf [3] T. Bonald, S. Borst, N. Hegde, and A. Prouti´ere, “Wireless data performance in multi-cell scenarios,” in SIGMETRICS’04, Proceedings of the joint international conference on Measurement and modeling of computer systems, vol. 32, no. 1, Jun. 2004, p. 378. [Online]. Available: http://portal.acm.org/citation.cfm?id=1012888.1005730 [4] H. A. Simon and A. Ando, “Aggregation of Variables in Dynamic Systems,” Econometrica, vol. 29, no. 2, pp. 111–138, Nov. 1961. [Online]. Available: http://www.jstor.org/stable/info/1909285 [5] P. J. Courtois, “Decomposability, instabilities, and saturation in multiprogramming systems,” Communications of the ACM, vol. 18, no. 7, pp. 371–377, Jul. 1975. [Online]. Available: http://doi.acm.org/10.1145/360881.360887 [6] T. Bonald, “Insensitive queueing models for communication networks,” in Proceedings of the 1st international conference on Performance evaluation methodolgies and tools - valuetools ’06. New York, New York, USA: ACM Press, Oct. 2006, p. 57. [Online]. Available: http://portal.acm.org/citation.cfm?id=1190095.1190168 [7] G. Fayolle and R. Iasnogorodski, “Two coupled processors: The reduction to a Riemann-Hilbert problem,” Zeitschrift fuer Wahrscheinlichkeitstheorie und Verwandte Gebiete, vol. 47, no. 3, pp. 325–351, 1979. [Online]. Available: http://www.springerlink.com/content/h564g3hr0173u987/ [8] R. D. Yates, “A framework for uplink power control in cellular radio systems,” IEEE Journal on Selected Areas in Communications, vol. 13, no. 7, pp. 1341–1347, 1995. [9] S. Boyd and L. Vandenberghe, Introduction to convex optimization. Stanfor, CA: Cambridge University Press, 2003.