A bilevel optimization model for load balancing in mobile networks through price incentives
We propose a model of incentives for data pricing in large mobile networks, in which an operator wishes to balance the number of connections (active users) of different classes of users in the different cells and at different time instants, in order to ensure them a sufficient quality of service. We assume that each user has a given total demand per day for different types of applications, which he may assign to different time slots and locations, depending on his own mobility, on his preferences and on price discounts proposed by the operator. We show that this can be cast as a bilevel programming problem with a special structure allowing us to develop a polynomial time decomposition algorithm suitable for large networks. First, we determine the optimal number of connections (which maximizes a measure of balance); next, we solve an inverse problem and determine the prices generating this traffic. Our results exploit a recently developed application of tropical geometry methods to mixed auction problems, as well as algorithms in discrete convexity (minimization of discrete convex functions in the sense of Murota). We finally present an application on real data provided by Orange and we show the efficiency of the model to reduce the peaks of congestion.
Keywords:Bilevel programming Mobile data networks Tropical geometry Discrete convexity Graph algorithms
With the development of new mobile data technologies (3G, 4G), the demand for using the Internet with mobile phones has increased rapidly. Mobile service providers (MSP) have to confront congestion problems in order to guarantee a sufficient quality of service (QoS).
Several approaches have been developed to improve the quality of service, coming from different fields of the telecommunication engineering and economics. For instance, one can refer to Bonald and Feuillet bonald2013network () for some models of performance analysis to optimize the network in order to improve the QoS. One of the promising alternatives to solve such problems consists in using efficient pricing schemes in order to encourage customers to shift their mobile data consumption. In maille2006pricing (), Maillé and Tuffin describe a mechanism of auctions based on game-theoretic methods for pricing an Internet network, see also maille2014telecommunication (). In altman2006pricing (), Altman et al. study how to price different services by using a noncooperative game. These different approaches are based on congestion games. In the present work, we are interested in how a MSP can improve the QoS by balancing the traffic in the network. We wish to determine in which locations, and at which time instants, it is relevant to propose price incentives, and to evaluate the influence of these incentives on the quality of service.
This kind of problem belongs to smart data pricing. We refer the reader to the survey of Sen et al. sen2013survey () and also to the collection of articles sen2014smart (). Finding efficient pricing schemes is a revenue management issue. The first approach consists in usage-based pricing; the prices are fixed monthly by analysing the use of the former months. It is possible to improve this scheme by identifying peak hours and non-peak hours and proposing incentives in non-peak hours in order to decrease the demand at peak hours and to better use the network capacity at non-peak hours. This leads to time-dependent pricing. Such a scheme for mobile data is developed by Ha et al. in ha2012tube (). The prices are determined at different time slots and based on the usage of the previous day in order to maximize the utility of the customers and the revenue of the MSP. This pricing scheme was concretely implemented by AT&T, showing the relevance of such a model. In another approach, Tadrous et al. propose a model in which the MSP anticipates peak hours and determines incentives for proactive downloads tadrous2013pricing ().
The latter models concern only the time aspects. One must also take into account the spatial aspect in order to optimize the demand between the different locations. In ma2014time (), Ma, Liu and Huang present a model depending on time and location of the customers where the MSP proposes prices and optimizes his profit taking into account the utility of the customers.
Here, we assume (as in ma2014time ()) that the MSP proposes incentives at different time and places. Then, customers optimize their data consumption by knowing these incentives and the MSP optimizes a measure of the QoS. In this way, we introduce a bilevel model in which the provider proposes incentives in order to balance the traffic in the network and to avoid as much as possible the congestion (high level problem), and customers optimize their own consumption for the given incentives (low level problem).
Bilevel programs have been widely studied, see the surveys of Colson, Marcotte and Savard colson2007overview () and of Dempe dempe2003bilevel (). They represent an important class of pricing problems in the sense that they model a leader wanting to maximize his profit and proposing prices to some followers who maximize themselves their own utility. Most classes of bilevel programs are known to be NP-hard. Several methods have been introduced to solve such problems. For instance, if the low level program is convex, it can be replaced by its Karush-Kuhn-Tucker optimality conditions and the bilevel problem becomes a classical one-stage optimization problem, which is however generally non convex. If some variables are binary or discrete, and the objective function is linear, the global bilevel problem can be rewritten as a mixed integer program, as in Brotcorne et al. brotcorne2000bilevel ().
In the present work, we optimize the consumption of each customer in a large area (large urban agglomeration) during typically one day divided in time slots of one hour, taking into account the different types of customers and of applications that they use. Therefore, we have to confront both with the difficulties inherent to bilevel programming and with the large number of variables (around ). Hence, we need to find polynomial time algorithms, or fast approximate methods, for classes of problems of a very large scale, which, if treated directly, would lead to mixed integer linear or nonlinear programming formulations beyond the capacities of current off-the-shelve solvers.
This motivated us to introduce a different approach, based on tropical geometry. Tropical geometry methods have been recently applied by Baldwin and Klemperer in baldwin2012tropical () to an auction problem. This has been further developed by Yu and Tran tran2015product (). In these approaches, the response of an agent to a price is represented by a certain polyhedral complex (arrangement of tropical hypersurfaces). This approach is intuitive since it allows one to vizualize geometrically the behavior of the agents: each cell of the complex corresponds to the set of incentives leading to a given response. Then, we vizualize the collective response of a group of customers by “superposing” (refining) the polyhedral complexes attached to every customer in this group. We apply here this idea to represent the response of the low-level optimizers in a bilevel problem. This leads to the following decomposition method: first we compute, among all the admissible consumptions of the customers, the one which maximizes a measure of balance of the network; then, we determine the price incentive which achieves this consumption. In this way, a bilevel problem is reduced to the minimization of a convex function over a certain Minkowski sum of sets. We identify situations in which the latter problem can be solved in polynomial time, by exploiting the discrete convexity results developed by Murota murota2003discrete (). In this approach, a critical step is to check the membership of a vector to a certain Minkowski sum of sets of integer points of polytopes. In our present model, these polytopes, which represent the possible consumptions of one customer, have a remarkable combinatorial structure (they are hypersimplices). Exploiting this combinatorial structure, we show that this critical step can be performed quickly, by reduction to a shortest path problem in a graph. This leads to an exact solution method when there is only one type of contract and one type of application sensitive to price incentive, and to a fast approximate method in the general case.
We finally present the application of this model on real data from Orange and show how price incentives can improve the QoS by balancing the number of active customers in an urban agglomeration during one day. These results indicate that a price incentive mechanism can effectively improve the satisfaction of the users by displacing their consumption from the most loaded regions of the space-time domain to less loaded regions.
The paper is organized as follows. In Section 2, we present the bilevel model. In Section 3, we explain how a certain polyhedral complex can be used to represent the user’s responses, and we describe the decomposition method. In Section 4, we deal with the high level problem and identify special cases which are solvable in polynomial time. In Section 5, we develop accelerated algorithms which enable to solve bilevel problems with a large number of customers. In Section 6, we propose a general relaxation method. The application to the instance provided by Orange is presented in Section 7.
The first results of this article (without proofs) were published in the proceedings of the conference WiOpt 2017 eytard2017bilevel ().
2 A bilevel model
We consider a time horizon of one day, divided in time slots numbered , and a network divided in different cells numbered . We assume that customers, numbered , are in the network. The customers have different types of contracts and they make requests for different types of applications (web/mail, streaming, download, …). We denote by the set of customers with the contract . A given customer is characterized by the following data. We denote by the position of the customer at each time , so that the sequence represents the trajectory of this customer. We assume that this trajectory is deterministic, so we consider customers with a regular daily mobility (for example, the trip between home and work). We denote by the inclination of a customer to make a request for an application of type at time . We suppose that customer wishes to make a fixed number of requests using the application during the day. We consider a set of time slots in which the customer decides not to consume the application .
We denote by the consumption of the customer for the application at time , setting if is active at time and makes a request of type and otherwise. Therefore, the number of active customers with contract for the application at time and location is given by , where denotes the indicator function, and the total number of active customers at time and location is given by .
We consider the following two-stage model of price incentives. The first stage consists for the operator in announcing a discount at time and location for the customers of contract making requests of type . We consider only nonnegative discounts, so . The second stage models the behavior of customers who modify their consumption by taking the discounts into account. We will assume the preference of a customer of contract for consuming at time becomes , where denotes the sensitivity of customer to price incentives for the application . It corresponds to classical linear utility functions, see e.g. baldwin2012tropical (). We also assume that the customers cannot make more than one request at each time, that is , . Therefore, each customer determines his consumptions for the applications, as an optimal solution of the linear program:
Problem 1 (Low-level, customers).
Consequently, each price determines the possible individual consumptions for the users with contract , and so the possible cumulated traffic vectors and . The aim of the operator is, through price incentives, to balance the load in the network into the different locations and time slots to improve the quality of service perceived by each customer. We introduce a coefficient relative to the kind of contracts of the different customers in order to favor some classes of premium customers. In lee2005non (), Lee et al. suppose that the satisfaction of a customer depends on his perceived throughput, which can be considered as inversely proportional to the number of customers in the cell. Here, we assume that the satisfaction of each customer in the cell is a nonincreasing function of the total number of active customers in the cell , depending on the characteristics of the cell, of the type of application the user wants to do (some applications like streaming need a higher rate than others) and on the type of contract. We also assume that the satisfaction of all the customers with contract using a given application in a given cell is maximal until the number of active customers reaches a certain threshold , then for . After this threshold, the satisfaction decreases until a critical value . We add the constraint to prevent the congestion. For non-real time services like web, mail, download, the satisfaction function can be viewed as a concave function of the throughput, like where denotes the throughput, see Moety et al. moety2016satisfaction (). Hence, we will consider that for contents like web, mail and download, , for and for where is a positive parameter depending on the kind of contract of the customer. The more expensive the contract of the customer is, the larger is . We can prove that this function is concave for . For real time services like video streaming, the customers need a more important throughput to ensure a good QoS lee2005non (). We will here consider the same type of functions but with replaced by , that is for .
So, the first stage consists in maximizing the global satisfaction function which depends on the vectors and is defined by:
with . Our final model consists in solving the following bilevel program:
Problem 2 (High-level, provider).
where , and , , and , the vectors are solutions of Problem 1.
3 A decomposition approach for solving the first model
We will present a decomposition method for solving the previous bilevel problem. In this section, and in the next two ones, we suppose that there is only one kind of application and one kind of contract. This special case is already relevant in applications: it covers the case when, for instance, only the download requests are influenced by price incentives, whereas other requests like streaming or web are fixed. Whereas the analytical results of the present section carry over to the general model, the results of the next two sections (polynomial time solvability) are only valid under these restrictive assumptions. We shall return to the general case in Section 6, developing a fast approximate algorithm for the general model based on the present principles.
In the above special case, the bilevel model can be rewritten:
where and , and for each the vectors are solutions of the problem:
In order to deal more abstractly with the bilevel model, we introduce the notation . Hence, we have if . By defining the set or , we have that implies that . We can then define if and otherwise. Then, we can rewrite each low-level problem as:
where , and the global bilevel problem becomes:
with . Notice that the set corresponds to the set of couples such that . It is possible to enumerate all the couples . Let us define and associate each couple to an integer . The quantities , , and can be respectively denoted by , , and . The function and the integer can be respectively denoted by and . It means that for two indices and associated to two couples and with the same , we have and . The low-level problem can be rewritten:
Problem 3 (Abstract low-level problem).
where and .
The global bilevel problem is:
Problem 4 (Bilevel problem).
with for all , solution of Problem 3.
Suppose that the functions are nonincreasing and concave on . Then, the functions are also concave on .
The result comes easily if we suppose that the functions are twice differentiable, because we have:
We could deduce that the same is true without the differentiability assumption by a density argument, writing a concave function as a pointwise limit of smooth concave functions. However, we prefer to provide the following elementary argument. Consider and . Because is nonincreasing, we have . We have:
Because of the well-known inequality , we have:
Then, because is nonincreasing, we have:
and is concave. ∎
3.1 A tropical representation of customers’ response
The lower-level component of our bilevel problem can be studied thanks to tropical techniques. Tropical mathematics refers to the study of the max-plus semifield , that is the set endowed with two laws and defined by and , see bcoq (); itenberg2009tropical (); butkovicbook (); MacLaganSturmfels () for background. We first consider the relaxation in which the price vector can take any real value, i.e. . Each customer defines his consumption by solving the problem:
The map is convex, piecewise affine, and the gradients of its linear parts are integer valued. It can be thought of as a tropical polynomial function in the variable . Indeed, with the tropical notation, we have
where denotes the th tropical power. In this way, we see that all the monomials of have degree , so that is homogeneous of degree , in the tropical sense. This remark leads to the following lemma:
Consider a solution of the relaxed problem. Because is homogeneous of degree , we have for all , . In particular:
Hence, leads to the same repartition of the customers and corresponds also to an optimal solution of the relaxed bilevel problem. ∎
The bilevel problem 4 has the same value as its relaxation .
By definition, the tropical hypersurface associated to a tropical polynomial function is the nondifferentiability locus of this function. Since the monomial is homogeneous, its associated tropical hypersurface is invariant by the translation by a constant vector. Therefore, it can be represented as a subset of the tropical projective space . The latter is defined as the quotient of by the equivalence relation which identifies two vectors which differ by a constant vector, and it can be identified to by the map
Consider a simple example with time steps (for instance morning, afternoon and evening), (that is ), and for each . The parameters of the customers are
The tropical polynomial of the first customer is , meaning that this customer has no preference and consumes when the incentive is the best. Its associated tropical hypersurface is a tropical line (since has degree ), so it splits in three different regions corresponding to a choice of the vector among , and , see Figure 2. E.g., the cell labeled by represents a consumption concentrated the morning, induced by a price and .
To study jointly the responses of the five customers, we represent the arrangement of the tropical hypersurfaces associated to the (see Figure 3), with
Lemma 3 (Corollary of (tran2015product, , §4, Lemma 3.1)).
Each cell of the arrangement of tropical hypersurfaces corresponds to a collection of customers responses and to a unique traffic vector , defined by .
3.2 Decomposition theorem
We next show that the present bilevel problem can be solved by decomposition. We note that the function to optimize for the higher level problem, i.e. the optimization problem of the provider, depends only on . The variables allow one to generate the different possible vectors .
A vector is said to be feasible if there exists vectors such that and there exists such that for each , .
So, we will characterize the feasible vectors in order to optimize directly the satisfaction function on the set of feasible . We define the relaxation of Problem 4 to the case .
Problem 5 (Bilevel problem with real discounts).
with and for all , solution of:
According to Lemma 1, Problem 4 has the same value than the relaxation problem 5. Moreover, according to Lemma 2, if is an optimal solution of Problem 5, then is also an optimal solution of Problem 5 for every . We recall that is a vector defined by . Then, if we find an optimal solution of Problem 5, then with is a solution of Problem 5 such that . Consequently, is a solution of Problem 4. Hence, a solution of Problem 5 (with real discounts) provides a solution of Problem 4 (with nonnegative discounts). In the sequel, we will study the bilevel problem 5.
Most of the following results are applications of classical notions of convex analysis which can be found in rockafellar1970convex (). It is convenient to introduce the convex characteristic function of a set , defined by if , and otherwise. If is a convex set, then is a convex function. We define also for every the polytope as the convex hull of , together with the convex function defined by .
and and is exactly the set of vertices of .
Let us define the polytope and . Clearly, . Then, .
Consider a point of which is not in . There exists an index such that . In particular . However, . So, there exists another index such that . Hence, there exists such that the points ans defined by:
are in . Because with and , is not a vertex of . Consequently, the set of vertices of is included in . Because is the convex hull of its vertices, we have .
The polytope is such that , with if and otherwise, and . Then, because is a totally unimodular matrix, the vertices of are exactly its integer points, that is . ∎
The value of each low level problem 3 is the value of the Legendre-Fenchel transform of at point , i.e. .
The vertices of are . Hence:
We want to characterize the feasible vectors. We have first the following result.
Let be a real vector. Then, there exists and such that and for every , if and only if .
Such vectors belong to , so .
Let and . A vector is such that if and only if , where denotes the subdifferential of the convex function . Then, a vector if and only if . By (rockafellar1970convex, , Th. 23.8), , where is the inf-convolution of the functions .
Let be a real vector. Then, there exists and such that and for every , if and only if , or equivalenty (because is convex), that is if and only if . The function is polyhedral (as the inf-convolution of polyhedral convex functions) and it is finite at every point in . So, is a non-empty polyhedral convex set (rockafellar1970convex, , Th. 23.10). The result comes straightforwardly. ∎
It is now possible to characterize the feasible vectors.
A vector is feasible if and only if .
According to Definition 1, a vector is feasible if and only if there exists and vectors such that and . As a consequence of Lemma 4, . Then, by Lemma 5, a vector is feasible if and only if . We have now to prove . Because , the inclusion is obvious. Conversely, consider . Then, the set is a non-empty polytope. A vector belongs to if it satisfies the following constraints:
, that is , with such that for every , and and defined by: