Rate control for communication networks: shadow prices,proportional fairness and stability

This paper analyses the stability and fairness of two classes of rate control algorithm for communication networks. The algorithms provide natural generalisations to large-scale networks of simple additive increase/multiplicative decrease schemes, and are shown to be stable about a system optimum characterised by a proportional fairness criterion. Stability is established by showing that, with an appropriate formulation of the overall optimisation problem, the network's implicit objective function provides a Lyapunov function for the dynamical system defined by the rate control algorithm. The network's optimisation problem may be cast in primal or dual form: this leads naturally to two classes of algorithm, which may be interpreted in terms of either congestion indication feedback signals or explicit rates based on shadow prices. Both classes of algorithm may be generalised to include routing control, and provide natural implementations of proportionally fair pricing.     

The medium access control problem for wireless communication networks modelled as hybrid dynamical systems

The paper considers the medium access control problem for a class of wireless communication networks. We apply mathematical models in terms of hybrid dynamical systems to analyze such networks. We introduce the concept of stabilizability for such systems which guarantees that any data packet in the wireless network will reach its destination within finite time. We obtain conditions for a wireless communication network to be stabilizable.     

The Delay-Capacity product for store-and-forward communication networks: Tree networks

A store-and-forward communication network under a maximal message delay criterion is considered. It is shown that the overall channel capacityC and the associated minimal maximal delayγ, as well as the maximal delayγ and the associated minimal overall capacityC, are characterized by a unique Delay-Capacity (γC) product number. The latter is related to a Delay-Capacity product (γC)⁺ number, uniquely determined solely by the topological structure of the communication network. Basic characteristics of the optimal delay and capacity assignment, a useful algoritm for the calculation of (γC)⁺ and simple upper and lower bounds on (γC)⁺, are derived for store-and-forward tree networks. Synthesis considerations and applications to hierarchical communication networks are noted.     

Perturbation analysis of communication networks with feedback control using stochastic hybrid models

Communication networks may be abstracted through Stochastic Fluid Models (SFM) with the node dynamics described by switched flow equations as various events take place, thus giving rise to hybrid automaton models with stochastic transitions. The inclusion of feedback mechanisms complicates these dynamics. In a tandem setting, a typical feedback mechanism is the control of a node processing rate as a threshold-based function of the downstream node’s buffer level. We consider the problem of controlling the threshold parameters so as to optimize performance metrics involving average workload and packet loss and show how Infinitesimal Perturbation Analysis (IPA) can be used to analyze congestion propagation through a network and develop gradient estimators of such metrics.     

Large homogeneous communication networks with Markovian access control II. Global behavior and residence time

A solution to the problem of dynamic analysis of large homogeneous communication networks with Markovian access control disciplines is given. For the case of multiple steady states and buffered users it is shown that: (i) the state space portrait of the system contains a one-dimensional stable manifold; (ii) the dynamics of the system can be reduced to a birth and death process defined on this manifold; and (iii) the residence time of each metastable steady state of the above birth and death process can be calculated on the basis of a new asymptotic procedure. Simulations show that this procedure results in acceptable precision.     

Path delays in communication networks

A communication network is modelled by a weighted graph. The vertices of the graph represent stations with storage capabilities, while the edges of the graph represent communication channels (or other information processing media). Channel capacity weights are assigned to the edges of the network. The network is assumed to operate in a store-and-forward manner, so that when a channel is busy the messages directed into it are stored at the station, joining there a queue which is governed by a first-come first-served service discipline. Assuming messages, with fixed length, to arrive at random at the network, following the statistics of a Poisson point process, we calculate the statistical characteristics of the message time-delays along a path in a communication network. We solve for the steadystate distributions of the message waiting-times along the path, for the distribution of the overall message delay-time, for the average memory size requirements at the stations, as well as for other statistical characteristics of the message flow and the queueing processes along a communication path.     

Planning for optimal expansion of leased line communication networks

We report a new heuristic algorithm useful for developing least-cost expansion plans for leased telecommunications networks in which there is a hierarchy of possible transmission facilities. The problem combines network topology design and capacity expansion problems. Implemented on a PC, the algorithm has successfully developed topology growth plans for twenty periods on networks with up to 31 nodes.     

A self-tuning heuristic for the design of communication networks

This paper addresses the design of communication networks that has a large application area. The problem is to design a minimum cost network subject to a given reliability level. Complexity of the problem is twofold: (1) finding a minimum-cost network topology that every pair of nodes can communicate with each other and (2) computing overall reliability to provide the reliability constraint. Over the last two decades, metaheuristic algorithms have been widely applied to solve this problem due to its NP-hardness. In this study, a self-tuning heuristic (STH), which is a new approach free from parameter tuning, is applied to the design of communication networks. Extensive computational results confirm that STH generates superior solutions to the problem in comparison to some well-known local search metaheuristics, and also more sophisticated metaheuristics proposed in the literature. The practical advantage of STH lies in both its effectiveness and simplicity in application to the design problem.     

Upper bounds for the forwarding indices of communication networks

The decision version of the forwarding index problem is, given a connected graph G and an integer ξ, to find a way of connecting each ordered pair of vertices by a path so that every vertex is an internal point of at most such paths. The optimization version of the problem is to find the smallest ξ for which a routing of this kind exists. Such a problem arises in the design of communication networks and distributed architectures. A model of parallel computation is represented by a network of processors, or machines processing and forwarding (synchronous) messages to each other, subject to physical constraints bearing on either the number of messages that can be processed by a single machine or the number of messages that can be sent through a connection. It was in this context that the problem was first introduced by Chung et al. (IEEE Trans. Inform. Theory 33 (1987) 224). The aim of this paper is to establish upper bounds for the optimal ξ as a function of the connectivity of the graph.     

A simple technique in Markovian control with applications to resource allocation in communication networks

The purpose of this note is to demonstrate that for semi-Markov decision problems with exponentially distributed transition times considerable computational improvements on the value-iteration algorithm can be obtained. This can be achieved by the simple trick of introducing fictitious decision epochs so that sparse transition matrices are created. Applications to optimal sharing of resources in communication networks are given.     

Jamming communication networks under complete uncertainty

This paper describes a problem of interdicting/jamming wireless communication networks in uncertain environments. Jamming communication networks is an important problem with many applications, but has received relatively little attention in the literature. Most of the work on network interdiction is focused on preventing jamming and analyzing network vulnerabilities. Here, we consider the case where there is no information about the network to be jammed. Thus, the problem is reduced to jamming all points in the area of interest. The optimal solution will determine the locations of the minimum number of jamming devices required to suppress the network. We consider a subproblem which places jamming devices on the nodes of a uniform grid over the area of interest. The objective here is to determine the maximum grid step size. We derive upper and lower bounds for this problem and provide a convergence result. Further, we prove that due to the cumulative effect of the jamming devices, the proposed method produces better solutions than the classical technique of covering the region with uniform circles.     

Efficient structure of noisy communication networks

In the canonical network model, the connections model, only three specific network structures are generically efficient: complete, empty, and star networks. This renders many plausible network structures inefficient. We show that requiring robustness with respect to stochastic information transmission failures rehabilitates incomplete, redundant network structures. Specifically, we show that star and complete networks are not generally robust to transmission failures, that circular and quasi-circular networks are efficient at intermediate costs in four-player networks, and that if either of them is efficient, then at least one of them is pairwise stable even without reallocation. Thus, incomplete, redundant networks are efficient and stable at intermediate costs.     

Optimal and suboptimal capacity allocation in communication networks

Gomory and Hu (Ref. 1) formulated the optimal allocation of capacities to the links of a communication networks as a problem in linear programming. The application of this formulation to the solution of problems of realistic size does, however, require an excessive amount of computation. In the present paper, a slightly different formulation is given. The resulting optimality conditions readily lend themselves to the construction of problems with known optimal solutions, thereby providing suitable examples for the assessment of the efficiencies of approximate methods. An approximate method that has been found highly efficient in many cases is illustrated by an example.     

On routing in VLSI design and communication networks

In this paper, we study the global routing problem in VLSI design and the multicast routing problem in communication networks. First we propose new and realistic models for both problems. In the global routing problem in VLSI design, we are given a lattice graph and subsets of the vertex set. The goal is to generate trees spanning these vertices in the subsets to minimize a linear combination of overall wirelength (edge length) and the number of bends of trees with respect to edge capacity constraints. In the multicast routing problem in communication networks, a graph is given to represent the network, together with subsets of the vertex set. We are required to find trees to span the given subsets and the overall edge length is minimized with respect to capacity constraints. Both problems are APX-hard. We present the integer linear programming (LP) formulation of both problems and solve the LP relaxations by the fast approximation algorithms for min-max resource-sharing problems in [K. Jansen, H. Zhang, Approximation algorithms for general packing problems and their application to the multicast congestion problem, Math. Programming, to appear, doi:10.1007/s10107-007-0106-8] (which is a generalization of the approximation algorithm proposed by Grigoriadis and Khachiyan [Coordination complexity of parallel price-directive decomposition, Math. Oper. Res. 2 (1996) 321-340]). For the global routing problem, we investigate the particular property of lattice graphs and propose a combinatorial technique to overcome the hardness due to the bend-dependent vertex cost. Finally, we develop asymptotic approximation algorithms for both problems with ratios depending on the best known approximation ratio for the minimum Steiner tree problem. They are the first known theoretical approximation bound results for the problems of minimizing the total costs (including both the edge and the bend costs) while spanning all given subsets of vertices.     

Sub-optimal constant-volume control for open channel networks

A state variable mathematical model for use in the synthesis of automatic control systems for open-channel networks is presented. The system considered here consists of n-cascaded reaches joined by control gates.The linear time invariant model consists of a controllable and observable representation where the state variables are the stored water volume variations in each reach and the control signals are the variations of the control gates opening sections. The model derives, through appropriate simplifications, from a more complex one in terms of transfer functions which was derived by linearizing the Saint-Venant equations.The problem of a linear quadratic optimal regulator is formulated in classical terms for the canal system and the constant-volume control laws obtained for the simplified model have been imposed on the complex one: such a control is therefore to be considered sub-optimal.The results of a digital simulation of the controlled system behaviour indicate that the system operates with practically constant stored water volumes in each reach and that such behaviour is fairly close to that of a pressure-water pipe system.     

Gaussian neural networks for control function implementation

The systematic storage in neural networks of prior information to be used in the design of various control subsystems is investigated. Assuming that the prior information is available in a certain form (namely, input/output data points and specifications between the data points), a particular neural network using Gaussian nonlinearities and a corresponding parameter design method are introduced. The proposed neural network addresses the issue of effectively using prior information, and its use is illustrated in an implementation of a control law scheduler.     

Topological design of wide area communication networks

This paper develops a model for designing a backbone network. It assumes the location of the backbone nodes, the traffic between the backbone nodes and the link capacities are given. It determines the links to be included in the design and the routes used by the origin destination pairs. The objective is to obtain the least cost design where the system costs consist of connection costs and queueing costs. The connection costs depend on link capacity and queueing costs are incurred by users due to the limited capacity of links. The Lagrangian relaxation embedded in a subgradient optimization procedure is used to obtain lower bounds on the optimal solution of the problem. A heuristic based on the Lagrangian relaxation is developed to generate feasible solutions.     

Routing optimization in packet switching communication networks

For routing assignments a special model and an optimization algorithm are proposed. The efficiency of the routing assignments is evaluated by the average value of the total cost of delays for all packets in the network. It is the objective function. The main idea is that traffic, which is transmitted from the source node to the destination node, can be split between two or more logical paths. The minimum of the objective function can be found by varying the traffic on every path and simultaneously from all the source nodes to the destination nodes. If this approach is applied, then the objective function is nonseparable and nonlinear. Because its shape is unknown in advance, an adaptive nonlinear optimization algorithm is proposed. For evaluating its efficiency a special set of test functions has been used.     

A model for stochastic hybrid systems with application to communication networks

We propose a model for stochastic hybrid systems (SHSs) where transitions between discrete modes are triggered by stochastic events much like transitions between states of a continuous-time Markov chains. However, the rate at which transitions occur is allowed to depend both on the continuous and the discrete states of the SHS. Based on results available for piecewise-deterministic Markov process (PDPs), we provide a formula for the extended generator of the SHS, which can be used to compute expectations and the overall distribution of the state.As an application, we construct a stochastic model for on-off TCP flows that considers both the congestion-avoidance and slow-start modes and takes directly into account the distribution of the number of bytes transmitted. Using the tools derived for SHSs, we model the dynamics of the moments of the sending rate by an infinite system of ODEs, which can be truncated to obtain an approximate finite-dimensional model. This model shows that, for transfer-size distributions reported in the literature, the standard deviation of the sending rate is much larger than its average. Moreover, the later seems to vary little with the probability of packet drop. This has significant implications for the design of congestion control mechanisms.