Modelling inventories in a network

If each inventory in a supply chain, comprising a network of production and transportation facilities, is optimized separately by some EOQ formula, the overall result may be far from optimal. If the objective of inventories is to reduce the variability of production and delivery rates, then this can be modelled as a network system with feedback links. The propagation of demands through the network is described by certain “propagation vectors”, and the optimization is with respect to certain feedback factors.     

The Deregulated Electricity Market Viewed as a Bilevel Programming Problem

In this paper, we present a bilevel programming formulation of a deregulated electricity market. By examining the electricity market in this format, we achieve two things. First, the relation of the deregulated electricity market to general economic models that can be formulated as bilevel programming problems (e.g. Stackelberg leader-follower games and principal-agency models) becomes clear. Secondly, it provides an explanation of the reason why the so-called “folk theorems” can be proven to be false for electricity networks. The interpretation of the deregulated electricity market as a bilevel program also indicates the magnitude of the error that can be made if the electricity market model studied does not take into account the physical constraints of the electric grid, or oversimplifies the electricity network to a radial network.     

Application of a dynamic market inverse elasticity law with linear and log-linear demand models

In this paper we will evaluate the significance of the inclusion of “dynamics” in profit maximization for widely used demand functions. Specifically we will consider both linear and log-linear demand models. Using these demand functions we will obtain closed form solutions for optimum prices (dynamic market inverse elasticity laws). The optimum price in a market governed by dynamic demand response is different from the one within a static response framework; we will relate the differences to specific characteristics of the demand function. One focus of this work will be to develop intuitive explanations for our conclusion regarding the relative size of the optimum price in static and dynamic markets. This work was completed when the author was with Bell Laboratories, USA.     

A multiclass network with non-linear, non-convex, non-monotonic stability conditions

We consider a stochastic queueing network with fixed routes and class priorities. The vector of class sizes forms a homogeneous Markov process of countable state space Z⁶ ₊. The network is said “stable” (resp.“unstable”) if this Markov process is ergodic (resp. transient). The parameters are the traffic intensities of the different classes. An unusual condition of stability is obtained thanks to a new argument based on the characterization of the “essential states”. The exact stability conditions are then detected thanks to an associated fluid network: we identify a zone of the parameter space in which diverging, fluid paths appear. In order to show that this is a zone of instability (and that the network is stable outside this zone), we resort to the criteria of ergodicity and transience proved by Malyshev and Menshikov for reflected random walks in Z⁶ ₊ (Malyshev and Menshikov, 1981). Their approach allows us to neglect some “pathological” fluid paths that perturb the dynamics of the fluid model. The stability conditions thus determined have especially unusual characteristics: they have a quadratic part, the stability domain is not convex, and increasing all the service rates may provoke instability (Theorem 1.1 and section 7). This revised version was published online in June 2006 with corrections to the Cover Date.     

On the Borel-Cantelli lemma

A new variant of the “divergent” part of the Borel-Cantelli lemma for events derived from a Markov chain is given. Further two applications are considered. One of the applications refers to the denumerable Markov chain and the second is a new proof of the “strong” theorem corresponding to the “arc sine law”.     

“Miniaturized” Linearizations for Quadratic 0/1 Problems

In order to solve a quadratic 0/1 problem, some techniques, consisting in deriving a linear integer formulation, are used. Those techniques, called “linearization”, usually involve a huge number of additional variables. As a consequence, the exact resolution of the linear model is, in general, very difficult. Our aim, in this paper, is to propose “economical” linear models. Starting from an existing linearization (typically the so-called “classical linearization”), we find a new linearization with fewer variables. The resulting model is called “Miniaturized” linearization. Based on this approach, we propose a new linearization scheme for which numerical tests have been performed.     

“Rescale and modify” implementation of IRKS methods

In this paper, we implement the “rescale and modify” approach in variable step size mode with both fixed and variable orders for stiff and nonstiff ODEs. A comparison of this approach and the “rescale” approach from the point of stability behavior and also step size and order selection provides very interesting results. To illustrate the efficiency of the method we have considered some standard test problems and report very useful tables and figures for step size and order changes, number of rejected or accepted steps, and also global error. As an optimal implementation, the numerical experiments suggest the application of this approach with both fixed and variable orders for nonstiff, and in variable order for stiff ODEs.     

Equilibria of a two-stage market with chance outcome in the spot market

We address the problem of how to improve the efficiency of markets of similar goods (electric power, gas, and other resources). One way to undermine the market dominance of some companies is the possibility of forward contracts. Here a model of the spot and forward markets functioning as Curnout auctions is studied using the example of symmetrical oligopoly. Suppliers try to maximize their profit by this two-stage game’s strategies of traded subgame equilibrium (TSE). The conditions for equilibrium achieved by correlated mixed strategies are elucidated: either a “bull” or “bear” market is established according to a chance factor. The optimum strategies of rational bidders are found to depend on the reserve price and a risk-avoiding parameter. TSE is compared to the Nash equilibria for one-stage models.     

Packet reordering in networks with heavy-tailed delays

An important characteristic of any TCP connection is the sequencing of packets within that connection. Out-of sequence packets indicate that the connection suffers from loss, duplication or reordering. More generally, in many distributed applications information integrity requires that data exchanges between different nodes of a system be performed in a specific order. However, due to random delays over different paths in a system, the packets may arrive at the receiver in a different order than their chronological order. In such a case, a resequencing buffer at the receiver has to store disordered packets temporarily. We analyze both the waiting time of a packet in the resequencing buffer and the size of this resequencing queue. We derive the exact asymptotics for the large deviation of these quantities under heavy-tailed assumptions. In contrast with results obtained for light-tailed distributions, we show that there exists several “typical paths” that lead to the large deviation. We derive explicitly these different “typical paths” and give heuristic rules for an optimal balancing.     

Stochastic programming: Potential hazards when random variables reflect market interaction

There are two types of random phenomena modeled in stochastic programs. One type is what we may term “external” or “natural” random variables, such as temperature or the roll of a dice. But in many other cases, random variables are used to reflect the behavior of other market participants. This is the case for such as price and demand of a product. Using simple game theoretic models, we demonstrate that stochastic programming may not be appropriate in these cases, as there may be no feasible way to replace the decisions of others by a random variable, and arrive at the correct decision. Hence, this simple note is a warning against certain types of stochastic programming models. Stochastic programming is unproblematic in pure forms of monopoly and perfect competition, and also with respect to external random phenomena. But if market power is involved, such as in oligopolies, the modeling may not be appropriate.     

An analytical study of the sensitivity of a discrete stochastic model for sales dynamics

We consider a discrete model for sales dynamics in the case of a stochastic model of the market. The model includes “fast” and “slow” components of the market situation described by a stochastic process of “white noise” type and the correlated stochastic process. By using an integral representation of the main characteristics of the Kalman filter, we obtain expressions for stochastic parameters of additional errors of the estimate that arise in the case where the characteristics of noises are inexact. We make an asymptotical analysis of these expressions and give recommendations for the price-forming strategy in the case of uncertainty of the market situation. Bibliography: 2 titles. Translated fromObchyslyuval'na ta Prykladna Matematyka, No. 81, 1997, pp. 110–116.     

Stochastic measures of arbitrage

Empirical research has provided evidence supporting the existence of arbitrage opportunities in real financial markets although market imperfections are often the main reason to explain these empirical deviations. Consequently, recent literature has turned the attention to imperfect markets in order to extend the most significant results on asset pricing. This paper develops several stochastic measures providing relative arbitrage earnings available in a financial market. The measures allow us to take into account different type of frictions. They are introduced by means of several dual pairs of vector optimization problems. Primal problems permit us to characterize the arbitrage absence even in an imperfect market and they also provide optimal arbitrage portfolios if the arbitrage absence fails. Dual ones allow us to extend the risk-neutral valuation methodology for imperfect and noarbitrage free markets and provide new interpretations for the measures in terms of “frictions effect” or “committed errors” in the valuation process. Partially funded by Comunidad Autónoma de Madrid (ref: CAM 07T/0027/2000) and Spanish Ministry of Science and Technology (ref: BEC2000-1388-C04)     

Lebesgue numbers and Atsuji spaces in subsystems of second-order arithmetic

We study Lebesgue and Atsuji spaces within subsystems of second order arithmetic. The former spaces are those such that every open covering has a Lebesgue number, while the latter are those such that every continuous function defined on them is uniformly continuous. The main results we obtain are the following: the statement “every compact space is Lebesgue” is equivalent to ; the statements “every perfect Lebesgue space is compact” and “every perfect Atsuji space is compact” are equivalent to ; the statement “every Lebesgue space is Atsuji” is provable in ; the statement “every Atsuji space is Lebesgue” is provable in . We also prove that the statement “the distance from a closed set is a continuous function” is equivalent to . Received: February 2, 1996     

Estimation of the Defect Status on Visual Field Longitudinal Data

In this paper, we show the use of Multivariate Time Series models, Markov Random Fields and Bayesian methodologies to solve an applied ophthalmological problem related to the study of glaucoma. Glaucoma is a very serious and widely extended eye disease characterized by a gradual decrease in the intensity of the patient’s sight. It is not, however, homogeneous over all the visual field, and starts at one or several sites and gradually spreads to nearby sites. Measurement of the patient’s “seeing threshold” at different points in the visual field is an important diagnostic tool for glaucoma and other diseases. It results in a map with 52 numerical values, each of which represents the level of intensity perceived by the patient at that site, and ranges from 0 (complete blindness) to 35 (exceptional vision). Additionally a “defect status” variable can be attached at each site in the visual field. This variable would indicate whether the site is normal or defective. Using Bayesian methodologies, the “defect status” process can be regarded as a parameter of the probability distribution of the thresholds and can be estimated as the maximum of its posterior distribution. The stochastic model assumed for the observed “threshold”, given the “defect status”, is a first order autoregressive integrated model (VARI(1,1)) in time, with first order homogeneous spatial correlation. The defect status is modeled by using a Spatiotemporal Autologistic Model with non-homogeneous spatial dependence. This dependence assumes that the propagation of the lesions follows the directions taken by the nerve fibers. MCMC methods are used to jointly estimate the defect status, and parameters and hyperparameters of the model.     

A Probabilistic Approach to Carne’s Bound

Carne’s bound is a sharp inequality controlling the transition probabilities for a discrete reversible Markov chain (Section 1). Its ordinary proof uses spectral techniques which look as efficient as miraculous. Here we present a new proof, comparing a “drift” for ways “out” and “back”, to get the gaussian part of the bound (Section 2), and using a conditioning technique to get the flight factor (Section 4). Moreover we show how our proof is more “supple” than Carne’s one and may generalize (Section 3.2).      

Reduction of hugoniot-maslov chains for trajectories of solitary vortices of the “shallow water” equations to the hill equation

According to Maslov’s idea, many two-dimensional, quasilinear hyperbolic systems of partial differential equations admit only three types of singularities that are in general position and have the property of “structure self-similarity and stability.” Those are: shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by an infinite chain of ordinary differential equations (ODE) that generalize the well-known Hugoniot conditions for shock waves. After some reasonable closure of the chain for the case of solitary vortices in the “shallow water” equations, we obtain a nonlinear system of sixteen ODE, which is exactly equivalent to the (linear) Hill equation with a periodic potential. This means that, in some approximations, the trajectory of a solitary vortex can be described by the Hill equation. This result can be used to predict the trajectory of the vortex center if we know its observable part. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 112, No. 1, pp. 47–66.     

Ols parameter estimation for a linear paired confluent model

We investigate OLS parameter estimation for a linear paired model in the case of a passive experiment with errors in both variables. The explicit form of the OLS estimates is obtained, their equivalence to maximum likelihood estimates is demonstrated in the presence of normal errors, and estimate consistency is proved. The OLS estimates are compared analytically and numerically with known parameter estimates of “direct,” “orthogonal,” and “diagonal” regression models.     

Conditions of reverse bullwhip effect in pricing for price-sensitive demand functions

Supply chain mechanisms that exacerbate price variation needs special attention, since price variation is one of the root causes of the bullwhip effect. In this study, we investigate conditions that create an amplification of price variation moving from the upstream suppliers to the downstream customers in a supply chain, which is referred as the “reverse bullwhip effect in pricing” (RBP). Considering initially a single-stage supply chain in which a retailer faces a random and price-sensitive demand, we derive conditions on a general demand function for which the retail price variation is higher than that of the wholesale price. The investigation is extended to a multi-stage supply chain in which the price at each stage is determined by a game theoretical framework. We illustrate the use of the conditions in identifying commonly used demand functions that induce RBP analytically and by means of several numerical examples.     

The semantics of knowledge attributions

The basic idea of conversational contextualism is that knowledge attributions are context sensitive in that a given knowledge attribution may be true if made in one context but false if made in another, owing to differences in the attributors’ conversational contexts. Moreover, the context sensitivity involved is traced back to the context sensitivity of the word “know,” which, in turn, is commonly modelled on the case either of genuine indexicals such as “I” or “here” or of comparative adjectives such as “tall” or “rich.” But contextualism faces various problems. I argue that in order to solve these problems we need to look for another account of the context sensitivity involved in knowledge attributions and I sketch an alternative proposal.