A Heuristic Decomposition Algorithm for Scheduling Problems on Mixed Graphs

We consider a scheduling problem where a set of n jobs has to be processed on a set of m machines and arbitrary precedence constraints between operations are given. Moreover, for any two operations i and j values a _ij >0 and a _ji >0 may be given where a _ij is the minimal difference between the starting times of operations i and j when operation i is processed first. Often, the objective is to minimize the makespan but we consider also arbitrary regular criteria. Even the special cases of the classical job shop problem J//C_max belong to the set of NP-hard problems. Therefore, approximation or heuristic algorithms are necessary to handle large-dimension problems. Based on the mixed graph model we give a heuristic decomposition algorithm for such a problem, i.e. the initial problem is partitioned into subproblems that can be solved exactly or approximately with a small error bound. These subproblems are obtained by a relaxation of a subset of the set of undirected edges of the mixed graph. The subproblems are successively solved and a proportion of the results obtained for one subproblem is kept for further subproblem definitions. Numerical results of the algorithm presented here are given.     

Availability Analysis of Repairable <Emphasis Type="Italic">N</Emphasis>-Server <Emphasis Type="Italic">n</Emphasis><Subscript><Emphasis Type="Italic">i</Emphasis></Subscript>-Replicated Systems

We develop an exact closed-form expression for the steady-state availability of a repairable, N-server system in which the ith server contains n _i identical, reconfigurable, breakdown-prone units. Our approach, which follows from the Markov chain balance equations and the recursive properties of Hessenberg matrix determinants, is simpler than previously proposed matrix geometric approaches, and can readily be adapted to the availability analysis of more complicated structures. We illustrate this by computing the steady-state availability of a mixed parallel-serial gracefully degrading replicated system.     

Frobenius-Perron operators and approximation of invariant measures for set-valued dynamical systems

A set-valued dynamical systemF on a Borel spaceX induces a set-valued operatorF onM(X) — the set of probability measures onX. We define arepresentation ofF, each of which induces an explicitly defined selection ofF; and use this to extend the notions of invariant measure and Frobenius-Perron operators to set-valued maps. We also extend a method ofS. Ulam to Markov finite approximations of invariant measures to the set-valued case and show how this leads to the approximation ofT-invariant measures for transformations , whereT corresponds to the closure of the graph of .     

Exponential weak Bernoulli mixing for collet-eckmann maps

We prove exponential weak Bernoulli mixing for invariant measures of certain piecewise monotone interval maps studied in [BK] and [KN]. In particular we prove this for unimodal maps with negative Schwarzian derivative satisfying lim , wherec is the unique critical point ofT.     

On the growth of meromorphic functions of infinite order

Letf be a meromorphic function of infinite order,T(r, f) its Nevanlinna (or Ahlfors-Shimizu) characteristic, andM(r, f) its maximum modulus. It is proved that $$\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{rT'(r,f)}} \leqslant \pi and\mathop {\lim \inf }\limits_{r \to \infty } \frac{{\log M(r,f)}}{{T(r,f)\psi (log T(r,f))}} = 0$$ . if ? (x)/x is non-decreasing, ?′(x)<-√?(x) and ∝^∞ dx/?(x) < ∞.     

Clumping Data in 2-Way Tables: A User-Orientated Perspective

Following a critique of existing algorithms, an algorithm is presented which will re-organize a 2-way data table to bring like rows together, and like columns together. Extensions of the method are described, and justified, to accommodate distances measured in modular arithmetic, and with bipolar columns/rows, as in repertory grid analysis. One value of the algorithms is that the user can see relationships in the tables without the data in the cells themselves ever having been transformed. Thus, users will continue to feel they own their data.     

Refining the Diffusion Approximation for the <Emphasis Type="Italic">GI</Emphasis><Superscript><Emphasis Type="Italic">x</Emphasis></Superscript><Emphasis Type="Italic">/G/c</Emphasis> Queue

This paper deals with the GI ^x /G/c queueing system in a steady state. We refine a diffusion approximation method incorporating the constraint of traffic conservation for general queueing systems. An approximate expression for the distribution of the number of customers is obtained. Numerical results are presented to show that the refined model provides improved performance.     

Local Borel summability of Euclidean 311-1311-1311-1a simple proof via differential flow equations

It is shown how the differential flow equation (or, equivalently, the continous renormalization group) method can be employed to give an astonishingly easy proof of the local Borel summability of the renormalized perturbative Euclidean massive ₄ ⁴ .Supported by NSF grant # DMS-9100383