Paul E. Wright

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

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.

Dynamic global packet routing in wireless networks

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.

TWO PARALLEL PROCESSORS WITH COUPLED INPUTS

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.

Routing in the presence of breakdowns

Abstract Jobs generated by a single Poisson source can be routed through N alternative gateways, modelled as parallel M / M /1 queues. The servers at those queues are subject to random breakdowns and repairs. When a breakdown occurs, all jobs present in the corresponding queue are lost; moreover, no incoming jobs are directed to that queue during the subsequent repair period. The marginal queue size distributions are determined by finding the roots of a polynomial inside the unit disc, and solving a set of simultaneous linear equations. The optimal splitting of the input stream between the servers, so as to minimize the job loss rate, is examined. In the case N = 2, it is also possible to find the joint equilibrium distribution of the numbers of jobs in the two queues, by a reduction to a Dirichlet boundary value problem on a circle.

On sublattices of the hexagonal lattice

How many sublattices of index N are there in the planar hexagonal lattice? Which of them are the best from the point of view of packing density, signal-to-noise ratio, or energy? We answer the first question completely and give partial answers to the other questions.

Processor-shared buffers with reneging

Abstract Motivated by buffer design questions in data networks, we consider a processor shared queue with finite buffer capacity and reneging. Reneging models events such as the timing-out of a communication path due to excessive delays, or decisions to preempt a path in order to re-route it according to some congestion control mechanism. The focus of the paper is on the asymptotic behavior of this system in a variety of regimes that correspond to large service rates (bandwidths). Congestion is measured by the blocking probability P , the mean number in system Q , and a cost function L giving the sum of the blocking and reneging rates normalized by the arrival rate. Convenient parameters of the analysis are the respective ratios ϱ and η of the arrival rate to the service rate and the service rate to the reneging rate. Fixing the relative rates at which η → ∞ and the buffer capacity N → ∞ in an exponential model, asymptotic estimates of P , Q , and L are derived as functions of η with ϱ a parameter. Similar results are obtained under a suitable scaling leading to heavy-traffic diffusion limits. These latter results are extended to more general distributional assumptions, and used in the solution of a problem in which bandwidth and reneging parameters must be chosen so as to minimize L .

Survival reliability of some double-loop networks and chordal rings

Three of the most well known double-loop networks are the distributed double-loop computer network (DDLCN), the daisy chain, and the braided ring. We compute the exact reliabilities of all three networks. We also extend the results on double-loop networks to directed chordal rings.

Performance analysis of concurrent-read exclusive-write

We analyze the concurrent-read exclusive-write protocol for access to a shared resource, such as occurs in database and distributed operating systems. Readers arrive according to a Poisson process and acquire shareable i.e., non-exclusive, locks which, once granted, are released after a generally distributed random period. Writers arrive according to an arbitrary renewal process and acquire exclusive locks which, once granted, are held for a random time which is also generally distributed. Locks are granted in the order in which requests are received.We derive necessary and sufficient conditions under which the queue is stable i.e., the Iatencies for reader/writer lock acquisition have a limiting distribution. In the unstable case, the delays of successive readers/writers become unbounded. The stability condition is sensitive to the interarrival-time distribution of the writers and the lock holding-time distribution of the readers but depends only on the mean lock holding-time of the writers.Distributional and moment bounds are given for the latencies of read/write requests.

GATED, EXHAUSTIVE, PARALLEL SERVICE

We analyze gated, exhaustive service of an infinite-server system with vacations. Customers enter a queue in a Poisson stream. The servers, working in parallel, serve customers in stages. A stage begins with all customers transferred from the queue to the servers (the gate opens). The servers then begin serving these customers, all simultaneously. The stage ends when their services are completed. Service is exhaustive because the servers must again examine the queue to see if any new customers arrived during the last stage. If there are any, a new stage begins. If there are none, the servers move on to other work. The time spent away from the queue is called vacation time . The queue may represent a node or station in a data transmission network and the servers may be communication channels. We analyze the equilibrium behavior of the number of requests served during a stage for general service and vacation time distributions. This analysis leads to the solution of a Fredholm integral equation of the second kind. We find conditions under which the system is stable and compute bounds on performance metrics of interest. Approximate techniques are introduced and tested. Finally, an extension to polling systems is studied.

Design and analysis of master/slave multiprocessors

A simple model of master/slave processors is presented, along with two simple, practical scheduling algorithms. An approximate analysis of the model yields simple formulas for performance measures in terms of the hardware and workload parameters, and gives insight into the power and the limitations of master/slave systems. In particular, formulae are obtained for the maximal processing power (throughput) of the system, a quantity that remains bounded as the number of slave processors increases. This analysis is applicable to symmetric multiprocessors, where performance considerations such as cache performance may dictate asymmetric assignment of system tasks to the processors. >