Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

This text presents a modern theory of analysis, control, and optimization for dynamic networks. Mathematical techniques of Lyapunov drift and Lyapunov optimization are developed and shown to enable constrained optimization of time averages in general stochastic systems. The focus is on communication and queueing systems, including wireless networks with time-varying channels, mobility, and randomly arriving traffic. A simple drift-plus-penalty framework is used to optimize time averages such as throughput, throughput-utility, power, and distortion. Explicit performance-delay tradeoffs are provided to illustrate the cost of approaching optimality. This theory is also applicable to problems in operations research and economics, where energy-efficient and profit-maximizing decisions must be made without knowing the future. Topics in the text include the following: - Queue stability theory - Backpressure, max-weight, and virtual queue methods - Primal-dual methods for non-convex stochastic utility maximization - Universal scheduling theory for arbitrary sample paths - Approximate and randomized scheduling theory - Optimization of renewal systems and Markov decision systems Detailed examples and numerous problem set questions are provided to reinforce the main concepts. Table of Contents: Introduction / Introduction to Queues / Dynamic Scheduling Example / Optimizing Time Averages / Optimizing Functions of Time Averages / Approximate Scheduling / Optimization of Renewal Systems / Conclusions

Stochastic Network Optimization with Application to Communication and Queueing Systems

Direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives. Direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis. Nonetheless, users remained loyal to these methods, most of which were easy to program, some of which were reliable. In the past fifteen years, these methods have seen a revival due, in part, to the appearance of mathematical analysis, as well as to interest in parallel and distributed com- puting. This review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited. Our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results. The underlying principles allow general- ization to handle bound constraints and linear constraints. We also discuss extensions to problems with nonlinear constraints.

https://www.cs.wm.edu/~va/research/sirev.pdf

Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods ∗

Information flow in a telecommunication network is accomplished through the interaction of mechanisms at various design layers with the end goal of supporting the information exchange needs of the applications. In wireless networks in particular, the different layers interact in a nontrivial manner in order to support information transfer. In this text we will present abstract models that capture the cross-layer interaction from the physical to transport layer in wireless network architectures including cellular, ad-hoc and sensor networks as well as hybrid wireless-wireline. The model allows for arbitrary network topologies as well as traffic forwarding modes, including datagrams and virtual circuits. Furthermore the time varying nature of a wireless network, due either to fading channels or to changing connectivity due to mobility, is adequately captured in our model to allow for state dependent network control policies. Quantitative performance measures that capture the quality of service requirements in these systems depending on the supported applications are discussed, including throughput maximization, energy consumption minimization, rate utility function maximization as well as general performance functionals. Cross-layer control algorithms with optimal or suboptimal performance with respect to the above measures are presented and analyzed. A detailed exposition of the related analysis and design techniques is provided.

/pdf/resource-allocation-and-cross-layer-control-in-wireless-1bikwdtjww.pdf

Resource Allocation and Cross Layer Control in Wireless Networks

https://homepages.fhv.at/hgb/New-Papers/CMAME07_BS07b.pdf

Robust Optimization - A Comprehensive Survey

The Nelder--Mead simplex algorithm, first published in 1965, is an enormously popular direct search method for multidimensional unconstrained minimization. Despite its widespread use, essentially no theoretical results have been proved explicitly for the Nelder--Mead algorithm. This paper presents convergence properties of the Nelder--Mead algorithm applied to strictly convex functions in dimensions 1 and 2. We prove convergence to a minimizer for dimension 1, and various limited convergence results for dimension 2. A counterexample of McKinnon gives a family of strictly convex functions in two dimensions and a set of initial conditions for which the Nelder--Mead algorithm converges to a nonminimizer. It is not yet known whether the Nelder--Mead method can be proved to converge to a minimizer for a more specialized class of convex functions in two dimensions.

/pdf/convergence-properties-of-the-nelder-mead-simplex-method-in-1e36h79lmd.pdf

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

A communications system and method of ordered borrowing which facilitates dynamic access to a global channel set that has been partitioned into subsets, with each cell of the system being assigned a particular subset of the channel set. The assignment of channel subsets is performed in such a way as to respect various constraints imposed by the physical layout. Calls originating in a cell are first assigned to the channels allocated to the base station of that cell, in an order determined by the cell. Once the allotted channels are exhausted, i.e. in the busy state, the cell attempts to borrow channels from those allotted to the base stations of neighboring cells in a specified order. The borrowing cell borrows a specified number of channels from each neighbor before returning to a particular cell to borrow additional channels. The channels borrowed from a neighbor are accessed in an order which is substantially the reverse of the order in which they are accessed by the owner cell. This prescribes, for each cell, a prespecified order in which the entire set of channels may be accessed by calls originating in that cell.

Apparatus and method for dynamic resource allocation in wireless communication networks utilizing ordered borrowing

In a master/slave multiprocessor system, process execution tasks are allocated on the basis of anticipated execution time. Work which can be performed by either the master or the slave is entered on a double ended queue in a non-descending order of anticipated execution times. Processes having high anticipated execution times are selected for execution by the slave processor and processes having low anticipated execution times are selected for execution by the master processor. Anticipated holding times are computed on the basis of the time spent by the slave computer in the execution of a segment of a process without requiring interaction with the master processor. The slave processor records the time spent in execution each time a segment of the process is completed and the process has to be returned to the master. The master uses the recorded time as a measure of the anticipated execution time for a next segment of the same process.

Adaptive job scheduling for multiprocessing systems with master and slave processors executing tasks with opposite anticipated execution times respectively

We consider schemes for reuse-efficient packet access in wireless data networks. We show that computing the maximum ergodic packet arrival rate is NP-hard. We give an upper bound on the maximum ergodic throughput in terms of the eigenvalues of matrices related to the path-gain matrix. We present simple, practical heuristic algorithms which exhibit good throughput and packet delay and report on results of preliminary simulations. More sophisticated algorithms that yield optimal throughput are also presented. A recent result of McKeown, Anantharam and Walrand (1996) on scheduling of input-queued switches is obtained as a by-product.

Dynamic global packet routing in wireless networks

We consider the double queue arising from a system consisting of two processors serving three job streams generated by independent Poisson sources. The central job stream of rate v consists of jobs which place resource demands on both processors, which are handled separately by each processor once the request is made. In addition, the first processor receives background work at a rate of A while the second receives similar tasks at a rate r1. Each processor has exponentially distributed service times with rates ao and P respectively. A functional equation is found for P(z, w), the generating function of the joint queue-length distribution, which leads to a relation between P(z, 0) and P(0, w) in the region Izl, Iwl < 1 of a complex algebraic curve associated with the problem. The curve is parametrized by elliptic functions z(?) and w(?) and the relation between P(z(?), 0) and P(0, w(?)) persists on their analytic continuation as elliptic functions in the a-plane. This leads to their eventual determination by an appeal to the theory of elliptic functions. From this determination we obtain asymptotic limit laws for the expectations of the mean number of jobs in each queue conditioned on the other, as the number of jobs in both processors tends to oo. Transitions are observed in the asymptotic behavior of these quantities as one crosses various boundaries in the parameter space. An interpretation of these results via the theory of large deviations is presented.

Paul E. Wright

Papers

Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions

Apparatus and method for dynamic resource allocation in wireless communication networks utilizing ordered borrowing

Adaptive job scheduling for multiprocessing systems with master and slave processors executing tasks with opposite anticipated execution times respectively

Dynamic global packet routing in wireless networks

Two parallel processors with coupled inputs