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Article

Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users

1
Department of System Analysis and Information Technologies, Kazan Federal University, Kazan 420008, Russia
2
Institute of Computational Mathematics and Information Technologies, Kazan Federal University, Kazan 420008, Russia
3
Faculty of Science, University of Oulu, FI-90014 Oulu, Finland
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2018, 23(2), 31; https://doi.org/10.3390/mca23020031
Submission received: 6 May 2018 / Revised: 11 June 2018 / Accepted: 14 June 2018 / Published: 17 June 2018
(This article belongs to the Special Issue Applied Modern Mathematics in Complex Networks)

Abstract

:
We consider a general problem of optimal allocation of limited resources in a wireless telecommunication network. The network users are divided into several different groups (or classes), which correspond to different levels of service. The network manager must satisfy these different users’ requirements. This approach leads to a convex optimization problem with balance and capacity constraints. We present several decomposition type methods to find a solution to this problem, which exploit its special features. We suggest applying first the dual Lagrangian method with respect to the total capacity constraint, which gives the one-dimensional dual problem. However, calculation of the value of the dual cost function requires solving several optimization problems. Our methods differ in approaches for solving these auxiliary problems. We consider three basic methods: Dual Multi Layer (DML), Conditional Gradient Dual Multilayer (CGDM) and Bisection (BS). Besides these methods we consider their modifications adjusted to different kind of cost functions. Our comparison of the performance of the suggested methods on several series of test problems show satisfactory convergence. Nevertheless, proper decomposition techniques enhance the convergence essentially.

1. Introduction

Efficient allocation of limited resources in communication networks require flexible mechanisms, which are based on proper mathematical models, since the conventional fixed allocation rules may lead to congestion effects and additional expenses from the inefficient utilization of network resources; see e.g., [1,2,3]. In particular, spectrum sharing is now one of the most critical issues and various adaptive mechanisms for allocation of resources in wireless telecommunication networks have been suggested. Most papers in this field are devoted to game-theoretic models and implementation of decentralized iterative methods for finding the Nash equilibrium points or their generalizations; see e.g., [4,5]. At the same time, various optimization-based mechanisms are also suggested; see e.g., [3,5,6,7]. Further, the allocation of energy and computing resources are considered in [8,9]. Management of these highly complicated systems are often based on proper decomposition approaches, which can involve zonal, time, frequency and other decomposition techniques.
In [10,11,12,13,14,15], several optimal resource allocation problems in telecommunication networks and proper zonal decomposition-based methods were suggested. They assumed that the network manager can satisfy all the varying user requirements or can buy additional volumes of the resource. This approach leads to constrained convex optimization problem for some selected time period. However, these models do not take into account possible differentiation of users with respect to service levels, which yields different service costs and somewhat different optimization problems.
In this paper, we consider problems of optimal allocation of a homogeneous resource in a telecommunication network with the differentiation of users. In such a way, we give a new formulation of this problem as an optimization problem and present several dual decomposition type methods for the affine and convex cases. We also compare the performance of the suggested methods on several series of test problems. The comparison shows the advantage of proposed methods over the conventional ones.

2. Problem Formulation

Let us consider a single telecommunication network with nodes (users). A network manager offers users m levels of network service (classes), which is reflected by expenses and prices. Within some selected time period, the network manager can offer a limited total amount C of a homogeneous resource of the network. An amount of resource allocated to the i-th class service is supposed to be equal to φ i ( x i ) if x i is an unknown consumed traffic volume at this level ( 0 x i β i ). The cost of implementation (network expense) of the amount x i of the i-th service level is supposed to be equal to μ i ( x i ) . Each user can choose only one level of service. Let N = { 1 , , n } denote a set of users, and N i a set of users of the i-th class (level) for i = 1 , , m . Let y j denote the unknown traffic volume offered to the j-th user with 0 y j α j and η j ( y j ) is the fee (incentive) value paid by the j-th user for this traffic. If all the users are attributed to the classes, we can calculate the total traffic volume for each i-th level as follows:
x i = j N i y j .
The general problem of the network manager is to find an optimal allocation of the limited homogeneous resource among the users in order to maximize the total payment received from the users and minimize the total network implementation expenses. This problem is now formulated as follows:
max ( x , y ) W , i = 1 m φ i ( x i ) C f ( x , y ) ,
where
f ( x , y ) = i = 1 m j N i η j ( y j ) μ i ( x i )
and
W = ( x , y ) | x i = j N i y j , 0 y j α j , j N i , 0 x i β i , i = 1 , , m .
In what follows we shall suppose that all the functions μ i ( x i ) , φ i ( x i ) and η j ( y j ) are convex, then (1)–(3) is a convex optimization problem.

3. Solution Methods

It is well known that many efficient solution methods for convex optimization problems exist; see e.g., [16,17]. However, due to large dimensionality and inexact data of the optimal resource allocation problems in telecommunication networks one can meet serious difficulties when solving these problems with conventional general iterative solution methods. To create an efficient method just for problem (1)–(3), we have to take into account its separability and apply certain decomposition approach. Moreover, the standard duality scheme using the Lagrangian function with respect to all the functional constraints leads to the multi-dimensional dual optimization problem. We will apply another approach, which was suggested in [12,18] and leads to solution of one-dimensional problems.
Let us first define the Lagrange function of problem (1)–(3) as follows:
L ( x , y , λ ) = f ( x , y ) λ i = 1 m φ i ( x i ) C .
This means we will utilize the Lagrangian multiplier λ only for the total resource bound. We can now replace problem (1)–(3) with its dual:
min λ 0 ψ ( λ ) ,
where
ψ ( λ ) = max ( x , y ) W L ( x , y , λ ) = λ C + max ( x , y ) W i = 1 m j N i η j ( y j ) ( μ i ( x i ) + λ φ i ( x i ) ) .
By duality (see e.g., [16,17]), problems (1)–(3) and (4) have the same optimal value. However, solution of (4) can be found by one of the well-known single-dimensional optimization algorithms; see e.g., [17]. The main problem is to implement these algorithms properly.
To calculate the value of ψ ( λ ) we have to solve the inner problem:
max ( x , y ) W i = 1 m j N i η j ( y j ) ( μ i ( x i ) + λ φ i ( x i ) ) .
This problem clearly decomposes into m independent class problems
max j N i η j ( y j ) ( μ i ( x i ) + λ φ i ( x i ) ) ,
subject to
x i = j N i y j , 0 y j α j , j N i , 0 x i β i , f o r i = 1 , , m .
Our methods for problem (1)–(3) will differ in approaches to problem (5)–(6).
We first describe the decomposition approach, which follows in general that from [10,11,12]. Denote by ν i ( x i ) the optimal value of the i-th service optimization problem:
max j N i η j ( y j )
subject to
j N i y j = x i , 0 y j α j , j N i .
Then (5)–(6) reduces to the one-dimensional problem:
min 0 x i β i ν i ( x i ) μ i ( x i ) λ φ i ( x i ) .
It is easy to see that ν i ( x i ) is a convex, but non differentiable function in general.
Thus, the initial problem (1)–(3) is replaced by its one-dimensional dual (4) with the cost function ψ ( λ ) , such that calculation of its value reduces to solution of m independent problems of form (5)–(6), whose calculation again reduces to solution of one-dimensional problems of form (9).
However, each function ν i is given algorithmically, i.e., via solution of problem (7)–(8). In the general case we can apply again a dual type method to find the value of ν i ( x i ) . Let us to introduce the the Lagrange function
L ˜ j ( y , θ i ) = j N i η j ( y j ) θ i j N i y j x i ,
and then to solve the one-dimensional dual:
min θ i 0 ζ i ( θ i ) ,
where
ζ i ( θ i ) = θ i x i + j N i max 0 y j α j [ η j ( y j ) θ i y j ] .
Therefore, we can use here only algorithms for a set of hierarchical one-dimensional problems. Let us denote this method as (DML).
Please note that this approach involves several levels of hierarchical problems requiring certain concordance in the accuracies of the solution of all these problems, besides, each solution of one upper level problem requires solution of all the lower level problems many times, which entails large computational costs. However, they can be reduced for some special types of functions.
For instance, consider the case where the functions η j ( y j ) , j N i are affine, whereas the functions φ i ( x i ) and μ i ( x i ) are convex and differentiable. Then we can find an exact solution of problems (7) and (8) by a simple ordering algorithm in a finite number of iterations; see [19] for more detail.
Next, consider the particular case where all the functions η j ( y j ) , μ i ( x i ) , and φ i ( x i ) are affine, i.e.,
η j ( y j ) = η j , 1 y j + η j , 0 , η j , 1 > 0 , j N i , i = 1 , , m , μ i ( x i ) = μ i , 1 x i + μ i , 0 , μ i , 1 > 0 , i = 1 , , m , φ i ( x i ) = φ i , 1 x i + φ i , 0 , φ i , 1 > 0 , i = 1 , , m .
Then the cost function in (5) can be rewritten equivalently as
η j , 1 y j ( μ i , 1 + λ φ i , 1 ) x i .
This means that problems (5) and (6) reduces to a two-side auction market with fixed prices (see [19]) and also is solved in a finite number of iterations by a simple ordering algorithm; see also [13,18]. Let us denote this method as (SDM).
We can extend this approach to the case where the functions η j ( y j ) are affine as in (10), whereas the functions φ i ( x i ) and μ i ( x i ) are only convex and differentiable. This means that the prices (marginal utilities) η j , 1 of the users are fixed, but the marginal expenses and prices depend on volumes, so that they are non-decreasing.
Set y ( i ) = ( y j ) j N i and
W i = ( x i , y ( i ) ) | x i = j N i y j , 0 y j α j , j N i , 0 x i β i .
The necessary and sufficient optimality condition for problem (5) and (6) is now written in the form of the variational inequality: find ( x ¯ i , y ¯ ( i ) ) W i such that
( μ i ( x ¯ i ) + λ φ i ( x ¯ i ) ) ( x i x ¯ i ) j N i η j , 1 ( y j y ¯ j ) 0 , ( x i , y ( i ) ) W i .
This is a two-sided market equilibrium problem with one seller and several buyers; see e.g., [19]. It is equivalent to the problem of finding a vector ( x ¯ i , y ¯ ( i ) ) W i and a cutting price p ¯ i such that
μ i ( x ¯ i ) + λ φ i ( x ¯ i ) p ¯ i , if x ¯ i = 0 , = p ¯ i , if x ¯ i [ 0 , β i ] , p ¯ i , if x ¯ i = β i ,
and
η j , 1 p ¯ i , if y ¯ j = 0 , = p ¯ i , if y ¯ j [ 0 , α j ] , p ¯ i , if y ¯ j = α j .
Since buyers prices are fixed, we can re-arrange them to be non-increasing and then find easily an intersection point of the staircase-wise inverse common demand and offer price μ i ( x i ) + λ φ i ( x i ) lines; see also [13]. Therefore, the exact solution of problem (11) or (5) and (6) can be also found directly by simple ordering type algorithms applying to (12) and (13) although (5) and (6) contains a non-linear function. In other words, calculation of values of ψ ( λ ) can be now accomplished by several independent simple ordering type algorithms. Notice that the re-arrangement of bid prices η j , 1 in each class should be made only one time that reduces the computational expenses essentially in comparison with the general duality approach. Let us denote this method also as (SDM).
We now again consider the general case where all the functions μ i ( x i ) , φ i ( x i ) , and η j ( y j ) are convex and differentiable. For these problems there exist many rather efficient solution methods; see e.g., [20] and references therein. In view of the above properties we can replace each problem (5) and (6) with a sequence of linearized problems of the form:
min ( x i , y ( i ) ) W i ( μ i ( x i k ) + λ φ i ( x i k ) ) x i j N i η j ( y j k ) y j
if we apply the conventional conditional gradient method (CGM) as suggested in [21]. For the sake of clarity, we describe (CGM) applied to the general optimization problem
min v V ϕ ( v ) ,
where V is a convex closed set, ϕ is a convex and differentiable function.
(CGM) Take an arbitrary initial point v 0 V and a number δ > 0 . At the s-th iteration, s = 0 , 1 , , we have a point v s V and calculate u s V as a solution of the linear programming problem
min u V ϕ ( v s ) , u .
Then we set p s = u s v s . If p s δ , stop, we have an approximate solution. Otherwise we find the next iterate v s + 1 as follows:
v s + 1 = σ s u s + ( 1 σ s ) v s ,
where σ s ( 0 , 1 ) is a step-size parameter.
In particular, we can utilize the inexact line search procedure: Find m as the minimal non-negative integer such that
ϕ ( v s + θ m p s ) ϕ ( v s ) + α θ m ϕ ( v s ) , p s ,
for some α ( 0 , 1 ) and θ ( 0 , 1 ) , and set σ s = θ m ; see [22].
It is easy to see that (15) gives then (14). Hence, (14) can be solved by simple ordering type algorithms as in (SDM). We denote this method as (CGDM). However, this approach requires application of (CGM) many times at each iteration of a single-dimensional optimization algorithm applied to the upper problem (4). At the same time, we can apply the same dual decomposition method to problem (5) and (6). For the sake of simplicity, we rewrite (5) and (6) as follows:
max ( x , y ) D j J η j ( y j ) u ( x ) ,
where
D = ( x , y ) | x = j J y j , 0 y j α j , j J , 0 x β , x = x i , y = ( y j ) j J , J = N i , β = β i , u ( x ) = μ i ( x ) + λ φ i ( x ) .
Let g ( x ) = u ( x ) and w j ( y j ) = η j ( y j ) . The necessary and sufficient optimality condition for problem (16) is given in (11) and re-written now as the variational inequality: find ( x ¯ , y ¯ ) D such that
g ( x ¯ ) ( x x ¯ ) j J w j ( y ¯ j ) ( y j y ¯ j ) 0 , ( x , y ) D .
The optimality conditions in (12) and (13) have the form: find ( x ¯ , y ¯ ) D and p such that
g ( x ¯ ) p , if x ¯ i = 0 , = p , if x ¯ i [ 0 , β ] , p , if x ¯ i = β ,
and
w j ( y ¯ j ) p , if y ¯ j = 0 , = p , if y ¯ j [ 0 , α j ] , p , if y ¯ j = α j ; for j J .
Following the dual approach, we write the Lagrange function of problem (16) with the negative sign:
M ( x , y , p ) = u ( x ) j J η j ( y j ) p x j J y j = ( u ( x ) p x ) j J ( η j ( y j ) p y j ) .
To find a value of the dual cost function
θ ( p ) = min x [ 0 , β ] , y [ 0 , α ] M ( x , y , p ) ,
where α = ( α j ) j J , we have to solve the one-dimensional problems:
min 0 x β ( u ( x ) p x ) ,
and
min 0 y j α j ( η j ( y j ) + p y j ) , for j J .
For the sake of simplicity, we also suppose that the functions u and η j are strictly convex. Then solutions of the above problems denoted by x ( p ) and y j ( p ) , j J , respectively, are defined uniquely. It follows that the function θ ( p ) is concave and differentiable with the derivative
θ ( p ) = j J y j ( p ) x ( p ) .
Besides, the one-dimensional dual problem
max p θ ( p )
coincides with the simple equation
θ ( p ) = 0 ,
where θ ( p ) is non-increasing. Therefore, if p is the solution of (19), then we can find the solution of problem (16) from (17) to (18) by setting p = p , which gives a solution of the initial problem (5) and (6).
To find a solution of (17) we can apply bisection type algorithms. Let γ = g ( 0 ) and γ = g ( β ) . Then γ < γ . Let δ j = w ( 0 ) and δ j = w ( α j ) . If we set p = max j J δ j and p = γ , then the case p p gives immediately the zero solutions in accordance with (17) and (18). So we can consider only the non-trivial case where p < p . Then by (17) and (18) we must have θ ( p ) 0 and θ ( p ) 0 . These properties enable us to utilize the simplest bisection algorithm; see e.g., [14,15].
Algorithm (BS). Given an accuracy ε > 0 and the initial segment [ p , p ] , we take p ˜ = 0.5 ( p + p ) , calculate θ ( p ˜ ) . Then we set p = p ˜ if θ ( p ˜ ) > 0 and p = p ˜ otherwise, until ( p p ) < ε .
Also, if all the functions are quadratic, we can utilize a heuristic method similar to that in [14].
Algorithm (SQ). Let ω j = η j , 1 + λ φ i , 1 . Define J a = { j J | ω j > p } , set y j = 0 for j J a and re-arrange the indices in J a to have the descending order for the values of ω j . Then find two sequential indices j l and j l + 1 in J a such that Δ l < 0 and Δ l + 1 > 0 , where
Δ l = s = 1 l y j s ( ω j l ) x ω j l ) .
Then find p such that θ ( p ) = 0 in the segment [ ω j l , ω j l + 1 ] .

4. Numerical Experiments

The methods were implemented in C++ with a PC with the following facilities: Intel(R) Core(TM) i7-4500, CPU 1.80 GHz, RAM 6 Gb.
The initial interval for the dual variable λ were chosen as [ 0 , 1000 ] . The parameters β i ( i = 1 , , m ) and α j ( j N i , i = 1 , , m ) were chosen as values of trigonometric functions in [ 1 , 51 ] and [ 1 , 2 ] , respectively. We set the constant C to be equal 1000. The number of classes was varied from 3 to 45, the number of users was varied from 210 to 1010. Users were distributed among classes either uniformly or according to the normal distribution.
We considered the following kinds of functions in test problems of form (1) to (3):
  • [Case L] All the functions μ i ( x i ) , φ i ( x i ) , and η j ( y j ) are affine;
  • [Case QL] All the functions η j ( y j ) are affine, all the functions μ i ( x i ) and φ i ( x i ) are quadratic;
  • [Case EQ] All the functions η j ( y j ) are convex quadratic, all the functions μ i ( x i ) and φ i ( x i ) are convex exponential;
  • [Case Q] All the functions η j ( y j ) , μ i ( x i ) , and φ i ( x i ) are convex quadratic;
  • [Case E] All the functions η j ( y j ) , μ i ( x i ) , and φ i ( x i ) are convex exponential;
  • [Case LG] All the functions η j ( y j ) , μ i ( x i ) , and φ i ( x i ) are convex logarithmic.
Let J denote the total number of users. The test functions were determined as follows:
1.
Linear functions
η j ( y j ) = η j , 1 y j + η j , 0 , η j , 1 > 0 , j = 1 , , J , μ i ( x i ) = μ i , 1 x i + μ i , 0 , μ i , 1 > 0 , i = 1 , , m , φ i ( x i ) = φ i , 1 x i + φ i , 0 , φ i , 1 > 0 , i = 1 , , m ,
where
η j , 1 = 2 | sin ( j + 1 ) | + 1 , η j , 0 = 2 | sin ( 2 j ) | + 1 , j = 1 , , J , μ i , 1 = | cos ( i ) | + 1 , μ i , 0 = 2 | cos ( 2 i ) | + 1 , i = 1 , , m , φ i , 1 = μ i , 1 , φ i , 0 = μ i , 0 , i = 1 , , m .
2.
Quadratic functions
η j ( y j ) = 0.5 η j , 2 y j 2 + η j , 1 y j , η j , 2 < 0 , j = 1 , , J , μ i ( x i ) = 0 . 5 μ i , 2 x i 2 + μ i , 1 x i , μ i , 2 > 0 , i = 1 , , m , φ i ( x i ) = 0.5 φ i , 2 x i 2 + φ i , 1 x i , φ i , 2 > 0 , i = 1 , , m ,
where
η j , 2 = 4 | cos ( 2 j 1 ) | 4 , η j , 1 = | sin ( j + 1 ) | + 1 , j = 1 , , J , μ i , 2 = | sin ( 2 i ) | + 1 , μ i , 1 = | cos ( i ) | + 3 , i = 1 , , m , φ i , 2 = μ i , 2 , φ i , 1 = μ i , 1 , i = 1 , , m .
3.
Exponential functions
η j ( y j ) = η j , 0 + η j , 1 y j η j , 2 e η j , 3 y j , η j , 1 , η j , 2 > 0 , j = 1 , , J , μ i ( x i ) = μ i , 0 e μ i , 1 x i , μ i , 1 , μ i , 0 > 0 , i = 1 , , m , φ i ( x i ) = φ i , 0 e φ i , 1 x i , φ i , 1 , φ i , 0 > 0 , i = 1 , , m ,
where
η j , 3 = | sin ( j + 1 ) | + 1 , η j , 2 = 2 | sin ( 2 j ) | + 1 , η j , 1 = 2 | sin ( j + 1 ) | + 8 , η j , 0 = 2 | sin ( 2 j ) | + 9 , j = 1 , , J , μ i , 1 = | cos ( i ) | + 1 , μ i , 0 = 2 | cos ( 2 i ) | + 1 , i = 1 , , m , φ i , 1 = μ i , 1 , φ i , 0 = μ i , 0 , i = 1 , , m .
4.
Logarithmic functions
η j ( y j ) = η j , 2 ln ( 1 + η j , 0 + η j , 1 y j ) , η j , 0 , η j , 1 , η j , 2 > 0 , j = 1 , , J , μ i ( x i ) = μ i , 0 + μ i , 1 x i ln ( 1 + μ i , 2 + μ i , 3 x i ) , μ i , 1 , μ i , 2 , μ i , 3 > 0 , i = 1 , , m , φ i ( x i ) = φ i , 0 + φ i , 1 x i ln ( 1 + φ i , 2 + φ i , 3 x i ) , φ i , 1 , φ i , 2 , φ i , 3 > 0 , i = 1 , , m ,
where
η j , 0 = 2 | sin ( 2 j ) | , η j , 1 = | sin ( j + 1 ) | + 1 , η j , 2 = 3 | sin ( 2 j ) | + 1 , j = 1 , , J , μ i , 0 = 2 | cos ( 2 i ) | + 1 , μ i , 1 = | cos ( i ) | + 1 , μ i , 2 = 2 | cos ( 2 i ) | , μ i , 3 = | cos ( i ) | + 1 , i = 1 , , m , φ i , 0 = μ i , 0 , φ i , 1 = μ i , 1 , φ i , 2 = μ i , 2 , φ i , 3 = μ i , 3 , i = 1 , , m .
For all the methods of solving problem (1)–(3) the accuracy of solution of upper dual problem (4) was varied from 10 1 to 10 4 . The accuracy of solution of lower level problems was fixed to be equal 10 2 . For each set of the parameters 50 tests were made. The aim of the numerical experiments is to calculate the time complexity (total processor time) of the methods with different kind of cost functions. In the tables, T ε denotes the total processor time in seconds. The averaged results of computations for Case L are given in Table 1, Table 2 and Table 3, for Case QL are given in Table 4, Table 5 and Table 6, for Case Q are given in Table 7, Table 8 and Table 9, for Case EQ are given in Table 10, Table 11 and Table 12, for Case E are given in Table 13, Table 14 and Table 15, for Case LG are given in Table 16, Table 17 and Table 18.
Together with the three basic methods for problem (1)–(3) named (DML), (CGDM), and (BS), we tested also their modifications adjusted mainly to some particular classes of problems. We applied the method (DML) with adaptive strategy of choosing the inner accuracies and named it (DMLA). In the case where the functions η j ( y j ) are affine, we applied also the simplified versions of these methods named (DMLS) and (DMLAS), respectively. They solve auxiliary problems (7) and (8) by a simple ordering algorithm in a finite number of iterations and require only one arrangement of buyers’ prices. Methods (DML), (DMLA), (DMLS), (DMLAS) and (SDM) were applied for cases L and QL, where (DMLS) and (DMLAS) showed better performance than (DML) and (DMLA), but (SDM) showed the best results here.
Next, in the nonlinear case, we applied (CGDM), where (CGDM0) denotes the version with zero initial point for any (CGM), (CGDMB) denotes the version with taking the initial point for any (CGM) in the boundary of the feasible set. We utilized these methods with the inexact line search procedure. Therefore, methods (DML), (DMLA), (CGDM0), (CGDMB), (BS), and (SQ) were applied for Case Q. Here (BS) and (SQ) showed the best performance, and the results of (CGDM0) and (CGDMB) were better than those of (DML) and (DMLA).
Methods (DML), (DMLA), (CGDM0), (CGDMB), and (BS) were applied for cases EQ, E, and LG. Here (BS) showed the essentially better results than the other methods. Also, (DMLA) showed better performance than (DML), (CGDM0), and (CGDMB) in most test experiments.
In general, all the suggested methods were rather efficient for these classes of problems. However, we should notice that special decomposable versions of the same methods which exploits peculiarities of each class appeared more efficient that general iterative methods. In particular, proper decomposition of the problem to a set of one-dimensional problems can enhance the convergence essentially. In fact, (BS) showed the best results for the nonlinear problems.

5. Conclusions

We considered a general problem of optimal allocation of a homogeneous resource in a wireless telecommunication network with several levels of service. By using the dual Lagrangian method with respect to the capacity constraint, we suggest to reduce the initial problem to a single-dimensional optimization problem. Calculation of the resulting cost function value leads to independent solutions of optimal allocation problems for each kind of service, which can be solved by simple solution methods. The results of computational experiments confirm the efficiency and applicability of the new methods presented.

Author Contributions

I.K. and E.L. are responsible for conceptualization, methodology and draft preparation. A.K. is responsible for software and computational results. Moreover, I.K. and E.L. are responsible for project administration and funding acquisition. All authors contributed to the writing of the manuscript.

Funding

I.K. and A.K. were supported by the RFBR grant, project No. 16-01-00109a. Also, I.K. and E.L. were supported by grants No. 315471 and No. 315366 from Academy of Finland. The results of the first author in this work were obtained within the state assignment of the Ministry of Science and Education of Russia, project No. 1.460.2016/1.4. The work of the second author was performed within the Russian Government Program of Competitive Growth of Kazan Federal University.

Conflicts of Interest

The authors declare no conflict of interest.

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Table 1. Results for Case L with J = 510 , m = 25 .
Table 1. Results for Case L with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
10 1 0.16800.08970.00250.00240.0003
10 2 0.19190.11590.00310.00250.0004
10 3 0.23710.15970.00470.00280.0009
10 4 0.27280.21280.00620.00460.0012
Table 2. Results for Case L with m = 25 , ε = 10 2 .
Table 2. Results for Case L with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
2100.08490.04890.00250.00120.0004
3100.11920.07120.00210.00280.0004
4100.15490.09280.00120.00090.0005
5100.19190.11590.00310.00250.0005
6100.23400.13680.00500.00280.0005
7100.27160.15900.00470.00280.0003
8100.31000.18340.00440.00350.0004
9100.34870.20560.00500.00620.0009
10100.38720.22870.00720.00420.0010
Table 3. Results for Case L with J = 510 , ε = 10 2 .
Table 3. Results for Case L with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
30.17810.09880.00240.00210.0004
90.19660.11300.00190.00120.0004
150.19760.11560.00180.00090.0001
210.20040.11570.00180.00180.0003
270.19530.11400.00270.00120.0007
330.19580.11530.00280.00210.0007
390.19940.11740.00240.00220.0004
450.20100.11750.00400.00220.0005
Table 4. Results for Case QL with J = 510 , m = 25 .
Table 4. Results for Case QL with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
10 1 0.16650.09060.00250.00210.0003
10 2 0.19590.11330.00280.00240.0004
10 3 0.23460.16030.00490.00280.0006
10 4 0.27620.21440.00530.00410.0007
Table 5. Results for Case QL with m = 25 , ε = 10 2 .
Table 5. Results for Case QL with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
2100.08140.04780.00220.00060.0001
3100.11550.07030.00250.00120.0003
4100.15930.09180.00150.00270.0002
5100.19590.11330.00280.00240.0003
6100.23310.13790.00310.00210.0004
7100.27360.15980.00400.00150.0005
8100.31150.18550.00550.00310.0004
9100.35220.20750.00440.00190.0004
10100.39040.23050.00500.00280.0006
Table 6. Results for Case QL with J = 510 , ε = 10 2 .
Table 6. Results for Case QL with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (DMLS) T ε : (DMLAS) T ε : (SDM)
30.17530.09840.00560.00090.0001
90.19530.11530.00310.00160.0003
150.19810.11870.00310.00190.0002
210.19650.11960.00210.00090.0002
270.19240.11410.00250.00190.0001
330.19580.11630.00340.00250.0003
390.19770.11620.00410.00150.0001
450.19640.11580.00310.00150.0006
Table 7. Results for Case Q with J = 510 , m = 25 .
Table 7. Results for Case Q with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (SQ) T ε : (BS)
10 1 0.28150.14880.02830.07620.00120.0018
10 2 0.32710.19290.04740.11160.00150.0019
10 3 0.39090.26690.10490.19190.00260.0040
10 4 0.45750.35700.14860.25340.00280.0058
Table 8. Results for Case Q with m = 25 , ε = 10 2 .
Table 8. Results for Case Q with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (SQ) T ε : (BS)
2100.13770.08330.01130.03670.00060.0009
3100.20040.11790.02080.05800.00120.0019
4100.26570.15640.03210.08280.00090.0018
5100.32710.19290.04740.11160.00150.0019
6100.39000.23020.06410.13960.00270.0030
7100.45400.26820.07280.15680.00190.0034
8100.51810.30520.08620.18040.00220.0028
9100.58090.34370.09340.19600.00280.0043
10100.64340.37930.10210.21100.00250.0046
Table 9. Results for Case Q with J = 510 , ε = 10 2 .
Table 9. Results for Case Q with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (SQ) T ε : (BS)
30.30280.16630.04650.09370.00030.0031
90.32030.18810.05460.09860.00150.0021
150.32500.19560.06680.12050.00120.0019
210.32780.19520.05410.11310.00150.0022
270.32710.19570.04620.10750.00120.0028
330.33000.19460.04620.11110.00060.0034
390.33530.19720.03650.10370.00030.0028
450.33370.19740.03330.09680.00060.0021
Table 10. Results for Case EQ with J = 510 , m = 25 .
Table 10. Results for Case EQ with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
10 1 0.27810.14780.08330.32250.0040
10 2 0.32810.19380.14970.41530.0043
10 3 0.39130.26660.25470.60680.0088
10 4 0.45340.35720.35960.75530.0103
Table 11. Results for Case EQ with m = 25 , ε = 10 2 .
Table 11. Results for Case EQ with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
2100.13820.08450.02930.08240.0030
3100.20280.11870.06740.17530.0018
4100.26440.15550.11270.28570.0021
5100.32810.19380.14970.41530.0043
6100.39060.22940.16930.54990.0053
7100.45100.26690.20310.67420.0050
8100.51610.30750.23120.76560.0059
9100.57660.34490.27180.86460.0097
10100.64210.38190.31840.95790.0085
Table 12. Results for Case EQ with J = 510 , ε = 10 2 .
Table 12. Results for Case EQ with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
30.30350.16710.07620.37030.0040
90.31880.18710.07460.37530.0030
150.32310.19320.12280.43450.0046
210.32780.19490.14590.42690.0053
270.32780.19240.11930.34790.0047
330.33090.19550.12150.34110.0041
390.33030.19810.11240.29960.0053
450.33250.19490.11660.29280.0052
Table 13. Results for Case E with J = 510 , m = 25 .
Table 13. Results for Case E with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
10 1 0.27870.14681.98622.31460.0071
10 2 0.31900.18843.27663.63430.0083
10 3 0.38870.26425.19065.65610.0146
10 4 0.44460.35226.92357.46600.0175
Table 14. Results for Case E with m = 25 , ε = 10 2 .
Table 14. Results for Case E with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
2100.13340.07970.99621.06430.0028
3100.19660.11431.64071.78650.0078
4100.25590.15062.38432.62250.0081
5100.31900.18843.27663.63430.0083
6100.37930.22314.14814.69260.0090
7100.43990.26124.77695.43120.0136
8100.50570.29815.69896.51400.0140
9100.56590.33536.20937.10380.0156
10100.63090.37157.09198.03950.0168
Table 15. Results for Case E with J = 510 , ε = 10 2 .
Table 15. Results for Case E with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
30.29810.16622.41912.84960.0072
90.31280.18282.82313.28030.0068
150.32030.18963.69254.13870.0106
210.32160.19183.65654.02790.0103
270.32250.18813.20273.50530.0089
330.32470.19071.70651.96720.0109
390.32780.19162.95063.21690.0099
450.32640.19062.94523.19390.0113
Table 16. Results for Case LG with J = 510 , m = 25 .
Table 16. Results for Case LG with J = 510 , m = 25 .
ε λ T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
10 1 0.36170.19210.14380.22130.0028
10 2 0.42400.25280.23180.31500.0046
10 3 0.50900.34520.39340.51330.0051
10 4 0.59000.46270.52590.67840.0053
Table 17. Results for Case LG with m = 25 , ε = 10 2 .
Table 17. Results for Case LG with m = 25 , ε = 10 2 .
J T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
2100.17800.10540.04270.07030.0016
3100.25850.15330.08110.12400.0019
4100.33910.20090.15650.22470.0034
5100.42400.25280.23180.31500.0046
6100.51090.30130.31360.42810.0041
7100.59620.35270.41900.55070.0043
8100.68190.40190.50870.64940.0046
9100.76750.45460.60410.74870.0046
10100.85400.50380.72930.88800.0064
Table 18. Results for Case LG with J = 510 , ε = 10 2 .
Table 18. Results for Case LG with J = 510 , ε = 10 2 .
m T ε : (DML) T ε : (DMLA) T ε : (CGDM0) T ε : (CGDMB) T ε : (BS)
30.39750.22030.63330.53570.0012
90.42620.24930.43460.49970.0015
150.43020.25440.36050.44220.0027
210.43030.25690.26070.34520.0031
270.42340.24940.18630.27400.0019
330.42460.25220.13910.19680.0040
390.42530.25280.15150.24440.0021
450.42780.25270.13750.23460.0040

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Konnov, I.; Kashuba, A.; Laitinen, E. Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users. Math. Comput. Appl. 2018, 23, 31. https://doi.org/10.3390/mca23020031

AMA Style

Konnov I, Kashuba A, Laitinen E. Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users. Mathematical and Computational Applications. 2018; 23(2):31. https://doi.org/10.3390/mca23020031

Chicago/Turabian Style

Konnov, Igor, Aleksey Kashuba, and Erkki Laitinen. 2018. "Dual Methods for Optimal Allocation of Telecommunication Network Resources with Several Classes of Users" Mathematical and Computational Applications 23, no. 2: 31. https://doi.org/10.3390/mca23020031

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