J. George Shanthikumar

Convex separable optimization is not much harder than linear optimization

The polynomiality of nonlinear separable convex (concave) optimization problems, on linear constraints with a matrix with “small” subdeterminants, and the polynomiality of such integer problems, provided the inteter linear version of such problems ins polynomial, is proven. This paper presents a general-purpose algorithm for converting procedures that solves linear programming problems. The conversion is polynomial for constraint matrices with polynomially bounded subdeterminants. Among the important corollaries of the algorithm is the extension of the polynomial solvability of integer linear programming problems with totally unimodular constraint matrix, to integer-separable convex programming. An algorithm for finding a ε-accurate optimal continuous solution to the nonlinear problem that is polynomial in log(1/ε) and the input size and the largest subdeterminant of the constraint matrix is also presented. These developments are based on proximity results between the continuous and integral optimal solutions for problems with any nonlinear separable convex objective function. The practical feature of our algorithm is that is does not demand an explicit representation of the nonlinear function, only a polynomial number of function evaluations on a prespecified grid.

STOCHASTIC CONVEXITY AND ITS APPLICATIONS

Abstract : Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady state mean waiting time in a GI/D/c queue, and as a function of the arrival and service rates in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case as a function of the arrival rate is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff on the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In nonparametric statistics, the stochastic concavity (convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

Recursive Algorithm to Evaluate the Reliability of a Consecutive-k-out-of-n:F System

A recursive algorithm computes the reliability of a consecutive-k-out-of-n:F system with unequal component failure probabilities.

Multiclass queueing systems: polymatroidal structure and optimal scheduling control

In many multiclass queueing systems, certain performance measures of interest satisfy strong conservation laws. That is, the total performance over all job types is invariant under any nonidling service control rule, and the total performance over any subset (say A) of job types is minimized or maximized by offering absolute priority to the types in A over all other types. We develop a formal definition of strong conservation laws, and show that as a necessary consequence of these strong conservation laws, the state space of the performance vector is a (base of a) polymatroid. From known results in polymatroidal theory, the vertices of this polyhedron are easily identified, and these vertices correspond to absolute priority rules. A wide variety of multiclass queueing systems are shown to have this polymatroidal structure, which greatly facilitates the study of the optimal scheduling control of such systems. When the defining set function of the performance space belongs to the class of generalized symmet...

Two Variability Orders

In this paper we study a new variability order that is denoted by ≤ st:icx . This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤ st:icx Y implies that Var[ h ( X )] ≤ Var[ h ( Y )] for all increasing convex functions h . The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤ st:icx Y are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤ st:icx are derived, and some typical applications in queueing theory are described.

On stochastic decomposition in M / G /1 type queues with generalized server vacations

Recently, S. W. Fuhrmann and R. B. Cooper showed that the stationary distribution of the number of customers in an M/G/1 queueing system with generalized server vacation is a convolution of the distribution functions of two independent positive random variables. One of these is the stationary distribution of the number of customers in an ordinary M/G/1 queueing system without server vacations. They use an elegant, intuitive approach to establish this result. In this paper, a mechanistic (analytic) proof of this result is given for systems more general than that discussed in Fuhrmann and Cooper. Such systems allow customer arrivals in bulk and some variations with reneging, balking, and in arrival rate that is dependent on system state.

On server allocation in multiple center manufacturing systems

We study the problem of allocating a given number of identical servers among the work centers of a manufacturing system. The problem is formulated as a nonlinear integer program of allocating servers in a closed queueing network to maximize throughput. We show that the throughput of the closed queueing network has a monotonicity property, such that any optimal allocation must give more servers to stations with a higher workload. The number of allocations that satisfy this property is much smaller than the total number of feasible allocations. This property and a bounding technique for the throughput of the closed queueing network are combined to develop a search algorithm to obtain an optimal allocation of servers. A greedy heuristic is also developed, and its optimality proven in the special case of a two-center system in the general case, its optimality remains a conjecture.

Multivariate Imperfect Repair

In this paper, we consider models of systems whose components have dependent lifelengths and are imperfectly repaired upon failure until they are scrapped. First, assuming that no more than one component can fail at a time, we study two models that describe imperfect repairs, and derive the resulting density and other probabilistic quantities of interest. We then generalize the models to cover applications in which more than one component can fail at the same time, and obtain various properties of the resulting distributions. Finally, we illustrate the theory through some examples.

Design of manufacturing systems using queueing models

Design issues in various types of manufacturing systems such as flow lines, automatic transfer lines, job shops, flexible machining systems, flexible assembly systems and multiple cell systems are addressed in this paper. Approaches to resolving these design issues of these systems using queueing models are reviewed. In particular, we show how the structural properties that are recently derived for single and multiple stage queueing systems can be used effectively in the solution of certain design optimization problems.

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(θ), θ∈Θ} is said to be parametrically stochastically increasing and convex (concave) in θ∈Θ if X(θ) is stochastically increasing in θ, and if for any increasing convex (concave) function ϕ, Eϕ(X(θ)) is increasing and convex (concave) in θ∈Θ whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {Xn(θ), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {Xn(θ), θ∈Θ}, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.