A bilevel optimization model for load balancing in mobile networks through price incentives.

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1 A bieve optimization mode for oad baancing in mobie networs through price incentives. Jean Bernard Eytard, Marianne Aian, Mustapha Bouhtou, and Stéphane Gaubert INRIA, CMAP, Ecoe Poytechnique, CNRS Route de Sacay, Paaiseau, France Emai: {jean-bernard.eytard, marianne.aian, Orange Labs 44 avenue de a Répubique, Chation, France Emai: mustapha.bouhtou@orange.com Abstract We propose a mode of incentives for data pricing in arge mobie networs, in which an operator wishes to baance the number of connexions (active users) of different casses of users in the different ces and at different time instants, in order to ensure them a sufficient quaity of service. We assume that each user has a given tota demand per day for different types of appications, which he may assign to different time sots and ocations, depending on his own mobiity, on his preferences and on price discounts proposed by the operator. We show that this can be cast as a bieve programming probem with a specia structure aowing us to deveop a poynomia time decomposition agorithm suitabe for arge networs. First, we determine the optima number of connexions (which imizes a measure of baance); next, we sove an inverse probem and determine the prices generating this traffic. Our resuts expoit a recenty deveoped appication of tropica geometry methods to mixed auction probems, as we as agorithms in discrete convexity (minimization of discrete convex functions in the sense of Murota). We finay present an appication on rea data provided by Orange and we show the efficiency of the mode to reduce the peas of congestion. I. INTRODUCTION With the deveopment of new mobie data technoogies (3G, 4G), the demand for using the Internet with mobie phones has increased rapidy. Mobie service providers (MSP) have to confront congestion probems in order to guarantee a sufficient quaity of service (QoS). Severa approaches have been deveoped to improve the quaity of service, coming from different fieds of the teecommunication engineering and economics. For instance, one can refer to Bonad and Feuiet [1] for some modes of performance anaysis to optimize the networ in order to improve the QoS. One of the promising aternatives to sove such probems consists in using efficient pricing schemes in order to encourage customers to shift their mobie data consumption. In [2], Maié and Tuffin describe a mechanism of auctions based on game-theoretic methods for pricing an Internet networ, see aso [3]. In [4], Atman et a. study how to price different services by using a noncooperative game. These different approaches are based on congestion games. In the present wor, we are interested in how a MSP can improve This wor is supported in part by Orange Labs through INRIA Contract CRE the QoS by baancing the traffic in the networ. We wish to determine in which ocations, and at which time instants, it is reevant to propose price incentives, and to evauate the infuence of these incentives on the quaity of service. This ind of probem beongs to smart data pricing. We refer the reader to the survey of Sen et a. [5] and aso to the coection of artices [6]. Finding efficient pricing schemes is a revenue management issue. The first approach consists in usage-based pricing; the prices are fixed monthy by anaysing the use of the former months. It is possibe to improve this scheme by identifying pea hours and non-pea hours and proposing incentives in non-pea hours in order to decrease the demand at pea hours and to better use the networ capacity at non-pea hours. This eads to time-dependent pricing. Such a scheme for mobie data is deveoped by Ha et a. in [7]. The prices are determined at different time sots and based on the usage of the previous day in order to imize the utiity of the customers and the revenue of the MSP. This pricing scheme was concretey impemented by AT&T, showing the reevance of such a mode. In another approach, Tadrous et a. propose a mode in which the MSP anticipates pea hours and determines incentives for proactive downoads [8]. The atter modes concern ony the time aspects. One must aso tae into account the spatia aspect in order to optimize the demand between the different ocations. In [9], Ma, Liu and Huang present a mode depending on time and ocation of the customers where the MSP proposes prices and optimizes his profit taing into account the utiity of the customers. Here, we assume (as in [9]) that the MSP proposes incentives at different time and paces. Then, customers optimize their data consumption by nowing these incentives and the MSP optimizes a measure of the QoS. In this way, we introduce a bieve mode in which the provider proposes incentives in order to baance the traffic in the networ and to avoid as much as possibe the congestion (high eve probem), and customers optimize their own consumption for the given incentives (ow eve probem). Bieve programs have been widey studied, see the surveys of Coson, Marcotte and Savard [10] and of Dempe [11]. They represent an important cass of pricing probems in sense that they mode a eader wanting to imize his profit and

2 proposing prices to some foowers who imize themseves their own utiity. Most casses of bieve programs are nown to be NP-hard. Severa methods have been introduced to sove such probems. For instance, if the ow eve program is convex, it can be repaced by its Karush-Kuhn-Tucer optimaity conditions and the bieve probem becomes a cassica one-stage optimization probem, which is however generay non convex. If some variabes are binary or discrete, and the objective function is inear, the goba bieve probem can be rewritten as a mixed integer program, as in Brotcorne et a. [12]. In the present wor, we optimize the consumption of each customer in a arge area (arge urban aggomerations) during typicay one day divided in time sots of one hour, taing into account the different types of customers and of appications that they use. Therefore, we have to confront both with the difficuties inherent to bieve programming and with the arge number of variabes (around 10 7 ). Hence, we need to find poynomia time agorithms, or fast approximate methods, for casses of probems of a very arge scae, which, if treated directy, woud ead to mixed integer inear or noninear programming formuations beyond the capacities of current off-the-sheve sovers. This motivated us to introduce a different approach, based on tropica geometry. Tropica geometry methods have been recenty appied by Badwin and Kemperer in [13] to an auction probem. This has been further deveoped by Yu and Tran [14]. In these approaches, the response of an agent to a price is represented by a certain poyhedra compex (arrangement of tropica hypersurfaces). This approach is intuitive since it aows one to vizuaize geometricay the behavior of the agents: each ce of the compex corresponds to the set of incentives eading to a given response. Then, we vizuaize the coective response of a group of customers by superposing (refining) the poyhedra compexes attached to every customer in this group. We appy here this idea to represent the response of the ow-eve optimizers in a bieve probem. This eads to the foowing decomposition method: first we compute, among a the admissibe consumptions of the customers, the one which imizes a measure of baance of the networ; then, we determine the price incentive which achieves this consumption. In this way, a bieve probem is reduced to the minimization of a convex function over a certain Minowsi sum of sets. We identify situations in which the atter probem can be soved in poynomia time, by expoiting the discrete convexity resuts deveoped by Murota [15]. In this approach, a critica step is to chec the membership of a vector to a certain Minowsi sum of sets of integer points of poytopes. In our present mode, these poytopes, which represent the possibe consumptions of one customer, have a remarabe combinatoria structure (they are hypersimpices). Expoiting this combinatoria structure, we show that this critica step can be performed quicy, by reduction to a shortest path probem in a graph. This eads to an exact soution method when there is ony one type of contract and one type of appication sensitive to price incentive, and to a fast approximate method in the genera case. We finay present the appication of this mode on rea data from Orange and show how price incentives can improve the QoS by baancing the number of active customers in an urban aggomeration during one day. These resuts indicate that a price incentive mechanism can effectivey improve the satisfaction of the users by dispacing their consumption from the most oaded regions of the space-time domain to ess oaded regions. The paper is organized as foows. In Section II, we present the bieve mode. In Section III, we expain how a certain poyhedra compex can be used to represent the user s responses. In Section IV, we describe the decomposition method. In Section V, we dea with the high eve probem and identify specia cases which are sovabe in poynomia time. In Section VI, we propose a genera reaxation method. The appication to the instance provided by Orange is presented in Section VII. II. A BILEVEL MODEL We consider a time horizon of one day, divided in T time sots numbered t [T ] = {1,... T }, and a networ divided in L different ces numbered [L]. We assume K customers, numbered [K], are in the networ. The customers have different types of contracts b [B] and they mae requests for different types of appications a [A] (web/mai, streaming, downoad,... ). We denote by K b the set of customers with the contract b. A given customer K b is characterized by the foowing data. We denote by L t [L] the position of the customer at each time t [T ], so that the sequence (L 1,..., L T ) represents the trajectory of this customer. We assume that this trajectory is deterministic, so we consider customers with a reguar daiy mobiity (for exampe, the trip between home and wor). We denote by ρ a (t) the incination of a customer to mae a request for an appication of type a at time t [T ]. We suppose that customer wishes to mae a fixed number of requests R a T using the appication a during the day. We consider a set of time sots I a [T ] in which the customer decides not to consume the appication a. We denote by u a (t) the consumption of the customer for the appication a at time t, setting u a (t) = 1 if is active at time t and maes a request of type a and u a (t) = 0 otherwise. Therefore, the number N a,b (t, ) of active customers with contract b for the appication a at time t and ocation is given by N a,b (t, ) = K b ua (t)1(l t = ), where 1 denotes the indicator function, and the tota number of active customers N(t, ) at time t and ocation is given b N a,b (t, ). by N(t, ) = a We consider the foowing two-stage mode of price incentives. The first stage consists for the operator in announcing a discount y a,b (t, ) at time t and ocation for the customers of contract b maing requests of type a. We consider ony nonnegative discounts, so y a,b (t, ) 0. The second stage modes the behavior of customers who modify their consumption by taing the discounts into account. We wi

3 assume the preference of a customer for consuming at time t becomes ρ a (t) + αa ya,b (t, L t ), where α a denotes the sensitivity of customer to price incentives for the appication a. It corresponds to cassica inear utiity functions, see e.g. [13]. We aso assume that the customers cannot mae more than one request at each time, that is t [T ], a ua (t) 1. Therefore, each customer determines his consumptions u a = (ua (t)) t [T ] {0, 1} T for the appications, as an optima soution of the inear program: Probem II.1 (Low-eve, customers). T [ ρ a (t) + αy a a,b (t, L t ) ] u a (t) (1) u a {0,1}T s.t. a [A], a [A] t=1 T u a (t) = R, a t=1 t I a, a [A], u a (t) = 0 t [T ], a [A] u a (t) 1 Consequenty, each price y a,b = (y a,b (t, )) t [T ], [L] determines the possibe individua consumptions u a for the users with contract b, and so the possibe cumuated traffic vectors N a,b = (N a,b (t, )) t [T ], [L] and N = a b N a,b. The aim of the operator is, through price incentives, to baance the oad in the networ into the different ocations and time sots to improve the quaity of service perceived by each customer. We introduce a coefficient γ b reative to the ind of contracts of the different customers in order to favor some casses of premium customers. In [16], Lee et a. suppose that the satisfaction of a customer depends on his perceived throughput, which can be considered as inversey proportiona to the number of customers in the ce. Here, we assume that the satisfaction of each customer in the ce [L] is a decreasing function s a,b of the tota number of active customers in the ce N(t, ), depending on the characteristics of the ce, of the type of appication the user wants to do (some appications ie streaming need a higher rate than other) and on the type of contract. We aso assume the satisfaction of a the customers with contract b using a given appication a in a given ce is ima unti the number of active customers reaches a certain threshod N a,b, then s a,b (N(t, )) = 1 for N(t, ) N a,b. After this threshod, the satisfaction decreases unti a critica vaue N C. We add the constraint t [T ], [L], N(t, ) N C to prevent the congestion. For non-rea time services ie web, mai, downoad, the satisfaction function can be viewed as a concave function of the throughput, ie 1 e δ/δc where δ denotes the throughput, see Moety et a. [17]. Hence, we wi consider that for contents ie web, mai and downoad, N( a,b = N 1 ), s a,b (n) = 1 for n N 1 and s a,b (n) = 1 λ b exp 2N C n N 1 for N 1 n N C where λ b is a positive parameter depending on the ind of contract of the customer. The more expensive the contract of the customer is, the arger is λ b. We can prove that this function is concave for 0 n N C. For rea time services ie video streaming, the customers need a more important throughput to ensure a good QoS [16]. We wi here consider the same type of functions s a,b but( with N) 1 repaced by N a,b = 0, that is s a,b (n) = 1 λ b exp 2N C n for 0 < n N C. s a,b 1 0 N 1 N C N(t, ) Fig. 1. Different ind of satisfaction functions of the number of active customers in a ce. The bue ones are those for streaming contents whereas the red ones are those for web, mai and downoad contents. The dashed ones corresponds to the satisfaction of standard customers, the continuous ones to the satisfaction of premium customers. So, the first stage consists in imizing the goba satisfaction function s which depends on the vectors N a,b N T L and is defined by: s(n a,b ) = = = T T t=1 a [A] b [B] t=1 a [A] b [B] K b =1 T L t=1 =1 a [A] b [B] γ b s a,b L K b t L (N(t, L t ))u a (t) γ b s a,b (N(t, ))1(L t = )u a (t) γ b N a,b (t, )s a,b (N(t, )) with b [B], γ b > 0. Our fina mode consists in soving the foowing bieve program: Probem II.2 (High-eve, provider). T L γ b N a,b (t, )s a,b y a,b R T L (N(t, )) (2) + t=1 =1 a [A] b [B] where t [T ], [L], N(t, ) = A B a=1 b=1 N a,b (t, ), and N(t, ) N C, t [T ], [L], a [A], b [B], N a,b (t, ) = K b ua (t)1(l t = ), and [K], the vectors u a are soutions of the probem II.1. III. A TROPICAL APPROACH FOR THE BILEVEL PROBLEM We wi present a decomposition method for soving the previous bieve probem. In this section, and in the two next ones, we suppose that there is ony one ind of appication and one ind of contract. This specia case is aready reevant in appications: it covers the case when, for instance, ony the downoad requests are infuenced by price incentives, whereas other requests ie streaming or web are fixed. Whereas the anaytica resuts of the present section carry over to the genera mode, the resuts of the next two sections (poynomia time sovabiity) are ony vaid under these restrictive assumptions.

4 We sha return to the genera case in Section VI, deveoping a fast approximate agorithm for the genera mode based on the present principes. In this specia case, the bieve mode can be rewritten: y R T L + T t=1 =1 L N(t, )s (N(t, )) where t, N(t, ) = [K] u (t)1(l t = ), and, the vectors u are soutions of the probem: s.t. u {0,1} T T [ ρ (t) + α y(t, L t ) ] u (t) t=1 T u (t) = R, t I, u (t) = 0, t=1 In order to dea more abstracty with the bieve mode, we introduce the notation u (t, ) = u (t)1(l t = ). Hence, we have u (t, ) = 0 if L t. By defining the set J = {(t, ) t I or L t } and ρ (t, ) = ρ (t)1(l t = )/α, we can rewrite each ow-eve probem: Probem III.1 (Abstract ow-eve probem). [ρ (t, ) + y(t, )] u (t, ) (3) u F t, where F = {x {0, 1} T L t, x(t, ) = R and (t, ) J, x(t, ) = 0}. Because the functions s are concave and decreasing, we can prove that the functions f : x xs (x) defined for x 0 are aso concave. The goba bieve probem is: Probem III.2 (Bieve probem). y R T L + K f (N(t, )) s.t. (t, ), N(t, ) = u (t, ) t, with u soutions of the probem III.1. t, =1 The ower-eve component of our bieve probem can be studied thans to tropica techniques. Tropica mathematics refers to the study of the -pus semified R, that is the set R { } endowed with two aws and defined by a b = (a, b) and a b = a + b, see [18], [19], [20], [21] for bacground. We first consider the reaxation in which the price vector y can tae any rea vaue, i.e. y R T L. Each customer defines his consumption u by soving the probem: [ρ (t, ) + y(t, )] u (t, ) = ρ + y, u, u F u F (5) The map P : y ρ + y, u is convex, piecewise affine, and the gradients of its inear parts are integer vaued. (4) It can be thought of as a tropica poynomia function in the variabe y. Indeed, with the tropica notation, we have P (y) = u F (ρ (1, 1) y(1, 1)) u(1,1) (ρ (T, L) y(t, L)) u (T,L), where z p := z z = p z denotes the pth tropica power. In this way, we see that a the monomias of P have degree t, u (t, ) = R, so that P is homogeneous of degree R, in the tropica sense. If we denote by e = (1... 1) R T L, it means that y R T L, β R, P (y + βe) = P (y). An important coroary is: Lemma III.3. The vaue of the bieve probem coincides with the vaue of the reaxed probem with y R T L. By definition, the tropica hypersurface associated to a tropica poynomia function is the nondifferentiabiity ocus of this function. Since the monomia P is homogeneous, its associated tropica hypersurface is invariant by the transation by a constant vector. Therefore, it can be represented as a subset of the tropica projective space TP T L 1. The atter is defined as the quotient of R T L by the equivaence reation which identifies two vectors which differ by a constant vector, and it can be identified to R T L 1 by the map TP T L 1 R T L 1, y (y(t, ) y(t, L)) (t,) [T ] [L]\{(T,L)}. Exampe III.4. Consider a simpe exampe with T = 3 time steps (for instance morning, afternoon and evening), L = 1, K = 5 and J = for each. For brevity, we wi write y t instead of y(t, ). The parameters of the customers are ρ 1 = [0, 0, 0], R 1 = 1, ρ 2 = [0, 1, 0], R 2 = 2, ρ 3 = [ 1, 1, 0], R 3 = 1 ρ 4 = [1/2, 1/2, 0], R 4 = 2, ρ 5 = [1/2, 2, 0], R 5 = 1. The tropica poynomia of the first customer is P 1 (y) = (y 1, y 2, y 3 ), meaning that this customer has no preference and consumes when the incentive is the best. Its associated tropica hypersurface is a tropica ine (since P 1 has degree 1), so it spits TP 2 in three different regions corresponding to a choice of the vector u 1 among (1, 0, 0), (0, 1, 0) and (0, 0, 1), see Figure 2. E.g., the ce abeed by (1, 0, 0) represents a consumption concentrated the morning, induced by a price y 1 > y 2 and y 1 > y 3. y 2 y 3 (0, 1, 0) (0, 0, 1) (1, 0, 0) y 1 y 3 Fig. 2. A customer response: a tropica ine spits the projective space into three ces. Each ce corresponds to a possibe customer response

5 To study jointy the responses of the five customers, we represent the arrangement of the tropica hypersurfaces associated to the P, [5], with P 2 (y) = (y 1 + y 2 1, y 1 + y 3, y 2 + y 3 1), P 3 (y) = (y 1 1, y 2 + 1, y 3 ), P 4 (y) = (y 1 + y 2 + 1, y 1 + y 3 + 1/2, y 2 + y 3 + 1/2), P 5 (y) = (y 1 + 1/2, y 2 + 2, y 3 ). y 2 y 3 (a) y 1 y 3 Fig. 3. Arrangement of tropica hypersurfaces: each tropica hypersurface corresponds to a customer response. For exampe, the ce (a) corresponds to discounts y with responses (1,0,0) for customer 1, (1,0,1) for customer 2, (0,1,0) for customer 3, (1,1,0) for customer 4 and (0,1,0) for customer 5. Hence, the tota number of customers in the networ with these discounts is (3,3,1). Lemma III.5 (Coroary of [14, 4, Lemma 3]). Each ce of the arrangement of tropica hypersurfaces corresponds to a coection of customers responses (u 1,..., u K ) and to an unique traffic vector N, defined by N = u. IV. DECOMPOSITION THEOREM We next show that the present bieve probem can be soved by decomposition. We note that the function to optimize for the higher eve probem, i.e. the optimization probem of the provider, depends ony on N. The variabes y(t, ) aow one to generate the different possibe vectors N. So we wi characterize the feasibe vectors N in order to optimize directy the satisfaction function on the set of feasibe N. This idea is motivated by the tropica approach thans to Lemma III.5. Most of the foowing resuts are appications of cassica notions of convex anaysis which can be found in [22]. It is convenient to define for every the poytope as the convex hu of F, together with the convex function ϕ defined by ϕ (u) = ρ, u if u, and ϕ (u) = + otherwise. The vaue of each ow eve probem (5) can therefore be viewed as the vaue of the Legendre-Fenche transform of ϕ at point y, i.e., ϕ (y) = sup [ y, u ϕ (u )]. u Lemma IV.1. = {x [0, 1] T L t, x(t, ) = R and (t, ) J, x(t, ) = 0}. Proof. The first part of this emma is a cassica resut of inear programming: the soution of a inear program on a poytope is reached at one vertex of this poytope. Because is convex, it is cear that the function ϕ is convex and so that the optima vaue is ϕ (y). Ceary, the poytope = {x [0; 1] T L t, x(t, ) = R and (t, ) J, x(t, ) = 0} contains F and F is incuded in the set of vertices of. Consider another point x of which is not in F. There exists one index (t 1, 1 ) such that 0 < x(t 1, 1 ) < 1. In particuar x(t 1, 1 ) / N. However, t, x(t, ) N. So, there exists another index (t 2, 2 ) such that 0 < x(t 2, 2 ) < 1. Hence, there exists ε > 0 such that the points x ans x + defined by: x (t 1, 1 ) = x(t 1, 1 ) ε and x + (t 1, 1 ) = x(t 1, 1 ) + ε x (t 2, 2 ) = x(t 2, 2 ) + ε and x (t 2, 2 ) = x(t 2, 2 ) ε x (t, ) = x + (t, ) = x(t, ) otherwise are in. Because x = x +x+ 2 with x x and x x +, x is not a vertex of. The set of vertices of is exacty F. So, a vector u is a soution of the ow-eve probem iff u ϕ (y) where ϕ denotes the subdifferentia of the convex function ϕ. A feasibe N is a sum of such vectors u. We have, by [22, Th. 23.8], that N is feasibe iff y R T L, N ϕ (y) = ( ϕ ) (y), i.e., y, N ψ (y), or equivaenty y, y ψ(n), where ψ = ϕ is the infconvoution of the functions ϕ. The function ψ is poyhedra (as the inf-convoution of poyhedra convex functions) and it is finite at every point N. So, N, ψ(n) is a non-empty poyhedra convex set [22, Th ] and N is feasibe. Moreover, we have the foowing emma: Lemma IV.2. Let N = u with u. The foowing assertions are equivaent: 1) There exists y R T + L such that each u is a soution of the ow-eve probem of customer with discount vector y; 2) The vectors u 1,..., u K reaize the minimum in the infconvoution ψ, i.e. ψ(n) = ρ, u. Proof. : (1) (2) : For every : ρ + y, u = sup v [ ρ + y, v ] By summing those equaities, we have: y, N + ρ, u = sup [ ρ + y, v ] v = sup v 1 1,...,v K K [ ρ + y, v ] By considering ony the vectors v 1 1,..., v K K such that v = N, we can write ρ, u = sup ρ, v which is exacty the v 1 1,..., v K K v = N assertion (2).

6 (2) (1): The set ψ(n) is non-empty. Consider y ψ(n), that is N ψ (y). We can write: N, y, N ψ(n) y, N ψ(n ) 2) Find optima requests vectors u by soving the infconvoution probem: ρ, u. u 1 F 1,...,u K F K u =N So: 3) Find a vector y such that, u ρ + y, u = y, N + is a soution of the ow eve probem. ρ, u = y, N ψ(n) Proof. The function to optimize in the high-eve probem [ ] = sup y, N ψ(n depends ony on N. The possibe vaues of N are characterized ) N by N F as sum of integer optima soutions of the ow eve probem and by the high-eve constraint N(t, ) N 2. = sup N y, N So, a necessary condition for a vector y to be an optima + sup ρ, v soution of the bieve probem is that for every, there exists v 1 1,..., v K K u ϕ (y ) such that N = u is an optima soution v of the probem: = N = sup [ ρ + y, v ] N N(t, )s (N(t, ) F = v 1 1,...,v K K sup [ ρ + y, v ] v Consequenty, if one u is not an optima soution of the ow-eve probem, the previous equaity cannot be true. In our probem, we are not interested in the vectors N which are sums of optima soutions of each ow eve probem, but in the ones which are sums of integer optima soutions of each ow eve probem. These vectors beong to F. Let N F. We have N, so it can be written as the sum of optima soutions u of each ow eve probem. According to the previous emma, we have (u 1,..., u K ) optima soution of: ρ, v, t,, 0 v (t, ) 1,, s.t. t, v (t, ) = R,, (t, ) J, v (t, ) = 0, t,, v (t, ) = N(t, ). The poytope defined by the constraint can be written Av b where A is a totay unimoduar matrix and b is an integer vector. So, the optima soutions of this probem are integer vectors and each vector u beongs to F. Hence, the feasibe set of the high-eve probem is exacty F. We arrive at the foowing method. Theorem IV.3. (Decomposition) The bieve program can be soved as foows: 1) Find an optima soution N to the high eve probem with unnown N: N f (N(t, )) s.t. N(t, ) N F C t,. (6) t, t, s.t. N(t, ) N 2 After finding N, it is possibe to find u ϕ (y ) by soving the inf-convoution probem as a consequence of Lemma IV.2. Because u ϕ (y ) is equivaent to y ϕ (u ), each point of ϕ (u ) is an optima soution of the bieve probem. The second step of this theorem consists in soving a inear program. We next show that the third step reduces to a inear feasibiity probem. Lemma IV.4. u is an optima soution of the ow eve probem iff the set of indices (t, ) such that u (t, ) = 1 coincides with the set of indices of the R highest coordinates not incuded in J of the vector ρ +y, i.e. (t, ), (t, ) / J such that u (t, ) = 1, u (t, ) = 0, we have ρ (t, ) + y(t, ) ρ (t, ) + y(t, ). For every, the atter inequaities define a poytope, and we have to find y in the intersection of a these poytopes. V. ALGORITHM FOR SOLVING THE BILEVEL PROBLEM A. Soving the high-eve probem We next expain how to sove Probem (III.2). We wi use some eements of discrete convexity deveoped by Murota [15]. An integer set B Z n is M-convex [15, Ch. 4, p.101] if x, y B, i [n] such that x i > y i, j [n] such that x j < y j, x e i + e j B and y + e i e j B, where e i is the i-th vector of the canonica basis in R n. Lemma V.1. The feasibe domain of the high-eve program B = {N F t, N(t, ) N C } is a M-convex set of Z T L. Proof. We can chec easiy that, the sets E and F are M-convex. Taing two different vectors u and v in F, there exist (t, ) and (t, ) such that u (t, ) = 1, v (t, ) = 0 and

7 u (t, ) = 0, v (t, ) = 1. These indices (t, ) and (t, ) do not beong to J. The vectors u e (t,) + e (t, ) and v + e (t,) e (t, ) have coordinates in {0, 1} with a sum equa to R and a coordinates in J equa to 0. It is nown that a Minowsi sum of M-convex sets is M-convex [15, Th. 4.23, p.115], and so the set F is M- convex. Finay, consider two vectors N and N of B. They beong to F, so for each (t, ) with N(t, ) > N (t, ), we can find (t, ) with N(t, ) < N (t, ) such that N e (t,) + e (t, ) and N + e (t,) e (t, ) are in F. The (t, ) coordinate of N e (t,) + e (t, ) is N(t, ) 1 < N(t, ) N C and the (t, ) coordinate of N e (t,) + e (t, ) is N(t, ) + 1 N (t, ) N C. So N e (t,) + e (t, ) B and simiary N + e (t,) e (t, ) B, which proves the M- convexity of B We have to imize a separabe concave function on a M- convex set. This is easy, because oca optimaity is equivaent to goba optimaity, as shown by the foowing resut: Theorem V.2 ([15, Th. 6.26, p.148]). Let f be a separabe concave function on Z n, B a M-convex set, and N B. Then, N is a imum point of f over B iff i, j [n] such that N e i + e j B, f(n e i + e j ) f(n ). Moreover, Murota ([15], ch.10, p.281) gives an agorithm which runs in pseudo-poynomia time to imize separabe concave functions on M-convex sets. Agorithm 1 Murota s agorithm to minimize a M-convex function f on a M-convex set B. 1) Find N B; 2) Find i, j arg f(n e + e );, [n] s.t. N e +e B 3) If f(n e i + e j ) f(n) then N = N is a goba minimizer of f over B; 4) Ese N := N e i + e j and go bac to Step 2; B. A poynomia time agorithm for the bieve probem Agorithm V-A can be appied to the high-eve probem (6) of Theorem IV.3, with f(n) = t, f (N(t, )) and B = F. The most critica part of this agorithm is to chec for a given N F whether N e i + e j for i, j [T ] [L] beongs to F. However, it can be easiy done. Lemma V.3. Let u F for each [K] such that ψ(n) = ρ, u. Consider the quantity wαβ for [K] and α, β [T ] [L] defined by wαβ = ρ (α) ρ (β) if u (α) = 1 and u (β) = 0 and wαβ = + otherwise. The optima v F such that ψ(n e i + e j ) = ρ, v can be obtained by soving the shortest path probem between i and j in a graph of T L nodes with edges weighted by w αβ = min wαβ. Proof. By emma IV.2, we now that if N e i +e j F, then N e i +e j = v with each v F and ψ(n e i + e j ) = ρ, v. We consider ψ(n) = ρ, u with each u F. Hence, ψ(n e i + e j ) ψ(n) is equa to: min ρ, u v. v F and v =N e i+e j We have u v = e i e j. When v describes F, the possibe u v are the vectors of { 1; 0; 1} T L with sum equa to 0 and verifying (u v )(α) {0; 1} if u (α) = 1 and (u v )(β) { 1; 0} if u (β) = 0. Consider the quantity wαβ for [K] and α, β [T ] [L] defined by wαβ = ρ (α) ρ (β) if u (α) = 1 and u (β) = 0 and wαβ = + otherwise. Hence, ρ, u v is either equa to 0 other can be written as a sum of wαβ for certain α, β. Because of the condition u v = e i e j, we have ρ, u v = w 1 iα +w2 αβ + +wp γj. Consider a graph of T L nodes with weights of the edge between α and β of vaue w αβ = min wαβ. The vaue ψ(n e i + e j ) ψ(n) corresponds to the shortest path between i and j in the former graph. Consider an edge of this path of vaue wαβ for a certain. The optima consumption of is v defined by v (α) = 0, v (β) = 1 and the other coordinates of v equa to those of u. We can prove that there exists no cyce with negative weight in this graph. Consider such a cyce. It can be written wα 1 1α 2 + wα 2 2α 3 + +wα p pα 1 < 0. For a i {1... p}, we consider v i defined ie u i with v i (α) = 0 and v i (β) = 1, and v = u for the other customers. We have u v = 0 and ρ, v = ρ, u +wα 1 1α 2 +wα 2 2α 3 + +wα p pα 1 < ρ, u which refutes the optimaity of the vectors u in the definition of ψ(n). This eads to the foowing agorithm. Note that the pseudo- Agorithm 2 Soving the bi-eve probem, for one appication and one type of contract 1) Find N F with this optima decomposition N = u ; 2) For each i, j [T ] [L], cacuate the shortest path between i and j in the graph of weights w αβ defined in the former emma and deduce if N e i + e j F and the optima decomposition N e i + e j = v ; 3) Find i, j arg (,) s.t. N e +e f(n e + e ); F 4) If f(n e i + e j ) f(n) then N = N and go to Step 6; 5) Ese N := N e i + e j and go bac to Step 2; 6) Find y R T L verifying the property of Lemma IV.4 and return y. poynomia time bound for Murota agorithm eads in this specia case to a poynomia time bound. Theorem V.4. Agorithm 2 returns a goba optimizer in poynomia time. Exampe V.5. Consider again Exampe III.4 together with the concave function f : N t, N(t, )2. We suppose that, J =. Hence, we can prove that F = {N N 3 3 i=1 N i = 7 and (N i ) 5}. First, we

8 want to sove N F (N N2 2 + N3 2 ). We start from N (0) = (5, 2, 0), a feasibe point. Foowing Agorithm V-A, we compute N (1) = (4, 2, 1) and N (2) = (3, 2, 2) which is a minimizer. We tae N = (3, 2, 2). Now, we sove u1 F 5 1,...,u 5 F 5, =1 u =N ρ, u. We obtain u 1 = [1, 0, 0], u 2 = [1, 0, 1], u 3 = [0, 1, 0], u 4 = [1, 0, 1], u 5 = [0, 1, 0]. Appying Lemma IV.4, we obtain the inear inequaities y1 y2 3/2, 0 y1 y3 and 1 y2 y3 1/2. In particuar, y = (3/4, 0, 3/4) is an optima soution. VI. THE GENERAL ALGORITHM In this section, we come bac to the genera bieve probem II.2 proposed in Section II, and extend the Agorithm of Section V to it. In the ow eve probem of each customer, the consumptions for different contents verify the constraints a [A], T t=1 ua (t) = Ra, t Ia, a [A], ua (t) = 0 and t [T ], a [A] ua (t) 1. We mae the assumption that for each customer, the sets of possibe instants at which this customer maes a request for the different appications are disjoint, meaning that for any two appications a a, the compements of I a and Ia in [T ] have an empty intersection. Then the constraint t [T ], a [A] ua (t) 1 is automaticay verified and the ow-eve probem of each customer can be separated into different optimization probems corresponding to the consumption vector u a of each customer for each appication a. Each of these probems taes the foowing form: Probem VI.1. s.t. u a {0,1}T T [ ρ a (t) + αy a a,b (t, L t ) ] u a (t) (7) t=1 T u a (t) = R, a t I a, a [A], u a (t) = 0. t=1 We denote by F a the feasibe set of this probem. The above assumption (that the compements of I a and Ia have an empty intersection) is reevant in particuar if ony one ind of appication is sensitive to price incentives. For instance, requests for downoading data can be anticipated (see [8]) and it maes sense to assume that customers are ony sensitive to incentives for this ind of contents. In this case, the assumption means that customers wanting to downoad data can shift their consumption ony at instants when they do not request another ind of content. Under this assumption, the decomposition theorem is sti vaid. The high-eve probem consists in imizing the separabe function ( ) t, a [A] b [B] γ bn a,b (t, )s a,b (N(t, )) where each vector N a,b beongs to a M-convex set K F a b according to Theorem IV.3 and Lemma V.1. Because each function s a,b is concave decreasing and each N a,b (t, ) is positive, we notice that a [A], b [B], the function wich sends N a,b (t, ) to a [A] b [B] γ bn a,b (t, )s a,b (N(t, )) is sti concave. Consequenty, the function to optimize in the high eve probem is M-concave in each vector N a,b considered separatey. This eads to a boc descent method (Agorithm 3), in which we imize the objective function, successivey, over every vector N a,b. We denote by f(n 1,1,..., N A,B ) the objective function of the higheve probem. Step 2 of this agorithm can be impemented Z T L Agorithm 3 Soving the bieve probem for an arbitrary number of types of contracts. 1) Find a, b, N a,b K b F a 2) Find, for each a [A], b [B], (i a,b, j a,b ) beonging to arg (,) s.t. N a,b e +e f(n 1,1 e i 1,1 + e j 1,1,..., K b F a N a,b e + e,..., N A,B ); 3) If f(n 1,1 e i 1,1 + e j 1,1,..., N A,B e i A,B + e j A,B) f(n 1,1,..., N A,B ) then a, b, return the optima soution N,a,b = N a,b. 4) Ese for each a, b, N a,b := N a,b e i a,b + e j a,b and go bac to Step 2; by soving the shortest path probem of certain graphs as in Lemma V.3. Unie Agorithm 2, Agorithm 3 is not guaranteed to give a gobay optima soution. VII. EXPERIMENTAL RESULTS We consider an appication based on rea data provided by Orange. It invoves the data consumptions in an area of L = 43 ces, during one day divided in time sots of one hour, that is T = 24 time sots. We wi focus here our study on price incentives ony for downoad contents. During this day, a number K of more than 2500 customers mae some requests for downoading data in this area and we are interested in baancing the number of active customers in the networ. Even though they are insensitive to price incentives, other ind of requests (web, mai, etc.) have to be satisfied and they are taen into account in the high eve optimization probem. We consider two casses of users: standard and premium customers. The premium ones demand a better quaity of service. Hence, they are ess satisfied than the standard customers if they share their ce with a given number of active customers. We therefore define the satisfaction function as in Section II. The provider wants to favor the premium customers. Hence, we tae γ b = 2 for the atter ones and γ b = 1 for the standard customers, in the high-eve optimization probem. We aso assume that the premium customers are ess sensitive to the incentives, and thus tae α a = 1/2 for a standard customers and α a = 1 for a premium customers in the oweve probem II.1. We estimate very simpy the parameters ρ. We tae ρ (t) = 1 when the customer consumes downoad at time t without incentives, ρ (t) = 0 when he does not mae any request without incentives but maes a request for downoad at times t 1 or t + 1 (we assume he coud shift his consumption of one hour) and ρ (t) = otherwise. We sove the bieve probem using Agorithm 3, impemented in Sciab. The computation too 9526 seconds on a singe core of an Inte i GHz.

9 Fig. 4. Satisfaction of premium customers for streaming without (eft) and with (right) incentives. The grey eve indicates the satisfaction: critica unsatisfaction, s < 0.3 (bac), 0.3 < s < 0.7 (dar grey), 0.7 < s < 0.9 (grey), 0.9 < s < 0.99 (ight grey) and compete satisfaction 0.99 < s (white). Fig. 6. Satisfaction of premium customers for web, mai or downoad without (eft) and with (right) incentives Fig. 5. Satisfaction of standard customers for streaming without (eft) and with (right) incentives On Figures 4 7, we show the evoution of the satisfaction of different ind of customers for different ind of contents without and with incentives. These resuts show that price incentives have an effective infuence on the oad, especiay in the most oaded ces (the number of bac regions in the space-time coordinates, in which the unsatisfaction of the users is critica, is consideraby reduced). Moreover, Figure 8 reveas that the consumption of users is not ony moved in time, but aso in space: not ony some consumption is moved from the pea hour to the night (off pea), but the surface of the dar grey region, representing the tota downoad consumption in the ce over the whoe day, is decreased, indicating that some part of the consumption has been shifted to other ces. VIII. CONCLUSION We presented here a bieve mode for price incentives in data mobie networs. We soved this probem by a decomposition method based on discrete convexity and tropica geometry. We finay appied our resuts to rea data. In further wor, we sha consider more genera modes: unfixed number of requests, noninear preferences of the customers, satisfaction functions of the provider taing into account the profit. Stochastic modes sha aso be considered in particuar to tae into account the partia information of the provider about the customers preferences and trajectories. Fig. 7. Satisfaction of standard customers for web, mai or downoad without (eft) and with (right) incentives IX. ACKNOWLEDGEMENTS We than the reviewers for their remars and comments, heping us to improve this wor. REFERENCES [1] T. Bonad and M. Feuiet, Networ performance anaysis. John Wiey & Sons, [2] P. Maié and B. Tuffin, Pricing the internet with mutibid auctions, IEEE/ACM transactions on networing, vo. 14, no. 5, pp , [3], Teecommunication networ economics: from theory to appications. Cambridge University Press, [4] E. Atman, D. Barman, R. E Azouzi, D. Ros, and B. Tuffin, Pricing differentiated services: A game-theoretic approach, Computer Networs, vo. 50, no. 7, pp , [5] S. Sen, C. Joe-Wong, S. Ha, and M. Chiang, A survey of smart data pricing: Past proposas, current pans, and future trends, ACM Computing Surveys (CSUR), vo. 46, no. 2, p. 15, [6], Smart Data Pricing. John Wiey & Sons, [7] S. Ha, S. Sen, C. Joe-Wong, Y. Im, and M. Chiang, Tube: timedependent pricing for mobie data, ACM SIGCOMM Computer Communication Review, vo. 42, no. 4, pp , [8] J. Tadrous, A. Eryimaz, and H. E Gama, Pricing for demand shaping and proactive downoad in smart data networs, in INFOCOM, 2013 Proceedings IEEE. IEEE, 2013, pp [9] Q. Ma, Y.-F. Liu, and J. Huang, Time and ocation aware mobie data pricing, in Communications (ICC), 2014 IEEE Internationa Conference on. IEEE, 2014, pp [10] B. Coson, P. Marcotte, and G. Savard, An overview of bieve optimization, Annas of operations research, vo. 153, no. 1, pp , [11] S. Dempe, Bieve programming: A survey. Dean der Fa. für Mathemati und Informati, 2003.

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