Dual methods for optimal allocation of total network resources

We consider a general problem of optimal allocation of a homogeneous resource (bandwidth) in a wireless communication network, which is decomposed into several zones (clusters). The network manager must satisfy different users requirements. However, they may vary essentially from time to time. This makes the fixed allocation rules inefficient and requires certain adjustment procedure for each selected time period. Besides, sometimes users requirements may exceed the local network capacity in some zones, hence the network manager can buy additional volumes of this resource. This approach leads to a constrained convex optimization problem. We discuss several ways to find a solution of this problem, which exploit its special features. We suggest the dual Lagrangian method to be applied to selected constraints. This in particular enables us to replace the initial problem with one-dimensional dual one. We consider the case of the affine cost (utility) functions, when each calculation of the value of the dual function requires solution of a special linear programming problem. We can also utilize the zonal resource decomposition approach, which leads to a sequence of onedimensional optimization problems. The results of the numerical experiments confirm the preferences of the first method. Keywords—Resource allocation, wireless networks, bandwidth, zonal network partition, dual Lagrange method, linear search, zonal resource decomposition, linear programming. I. I NTRODUCTION THE current development of telecommunication systems creates a number of new challenges of efficient management mechanisms involving various aspects. One of them is the efficient allocation of limited communication networks resources. In fact, despite the existence of powerful processing and transmission devices, increasing demand of different communication services and its variability in time, place, and quality, leads to serious congestion effects and inefficient utilization of significant network resources (e.g., bandwidth and batteries capacity), especially in wireless telecommunication networks. This situation forces one to replace the fixed allocation rules with more flexible mechanisms; see e.g. [1]–[4]. Naturally, treatment of these very complicated systems is often based on a proper decomposition/clustering approach, which can involve zonal, time, frequency and other decomposition procedures for nodes/units; see e.g. [5], [6], [7], [8], [9]. I.V. Konnov is with the Department of System Analysis and Information Technologies, Kazan Federal University, ul. Kremlevskaya, 18, Kazan 420008, Russia. E-mail: konn-igor@ya.ru A.Yu. Kashuba is with LLC ”AST Povolzhye”, ul.Sibirskiy trakt, 34A, Kazan, 420029, Russia. E-mail: leksser@rambler.ru E. Laitinen is with the Department of Mathematical Sciences, University of Oulu, Oulu, Finland. E-mail: erkki.laitinen@oulu.fi In [10], [11], several optimal resource allocation problems in telecommunication networks and proper decomposition based methods were suggested. They assumed that the network manager can satisfy all the varying users requirements. However, zonal resource amounts may be not sufficient in some time periods due to instable behavior of many users, hence the network manager can buy additional volumes of the resource. We note that such a strategy is rather typical for contemporary wireless communication networks, where WiFi or femtocell communication services are utilized in addition to the usual network resources; see e.g. [12]. This approach leads to a constrained convex optimization problem for some selected time period. We discuss several ways to find a solution of this problem, which exploit its special features. We suggest the dual Lagrangian method to be applied to selected constraints. The utilization of the dual decomposition in this problem was also considered in [13]. It was based on an explicit volume resource allocation procedure with a a sequence of one-dimensional optimization problems and gave a multilevel iterative procedure. In this paper, we discuss several possible approaches to the zonal resource allocation problem and give some other way to enhance the performance of the solution method. It consists in utilization of the Lagrangian multipliers only for the total resource bound, which yields an one-dimensional dual optimization problem. We consider the case of the affine cost (utility) functions, when each calculation of the value of the dual function requires solution of a special linear programming problem. The results of the numerical experiments confirms the preferences of the new method over the previous ones. II. PROBLEM DESCRIPTION Let us consider a network with nodes (attributed to users), which is divided inton zones (clusters) within some fixed time period. For the k-th zone (k= 1, . . . , n), Ik denotes the index set of nodes (currently) located in this zone, bk is the maximal fixed resource value. We suppose that users can move but that all the assignments of users to zones are fixed within this time period. The network manager satisfies users resource requirements in the k-th zone by allocation of the own (inner) resource valuexk ∈ [0, bk] and also by taking the external resource value zk ∈ [0, ck]. Clearly, these values require proper maintenance expenses fk(xk) and side paymentshk(zk) for eachk = 1, . . . , n. We suppose also that there exists the upper boundB for the total amount of the inner resource INTERNATIONAL JOURNAL OF MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES Volume 10, 2016