Mathematical Modeling of Product Rating: Sufficiency, Misbehavior and Aggregation Rules

Many web services like eBay, Tripadvisor, Epinions, etc, provide historical product ratings so that users can evaluate the quality of products. Product ratings are important since they affect how well a product will be adopted by the market. The challenge is that we only have {\em "partial information"} on these ratings: Each user provides ratings to only a "{\em small subset of products}". Under this partial information setting, we explore a number of fundamental questions: What is the "{\em minimum number of ratings}" a product needs so one can make a reliable evaluation of its quality? How users' {\em misbehavior} (such as {\em cheating}) in product rating may affect the evaluation result? To answer these questions, we present a formal mathematical model of product evaluation based on partial information. We derive theoretical bounds on the minimum number of ratings needed to produce a reliable indicator of a product's quality. We also extend our model to accommodate users' misbehavior in product rating. We carry out experiments using both synthetic and real-world data (from TripAdvisor, Amazon and eBay) to validate our model, and also show that using the "majority rating rule" to aggregate product ratings, it produces more reliable and robust product evaluation results than the "average rating rule".Comment: 33 page

Block-Structured Supermarket Models

Supermarket models are a class of parallel queueing networks with an adaptive control scheme that play a key role in the study of resource management of, such as, computer networks, manufacturing systems and transportation networks. When the arrival processes are non-Poisson and the service times are non-exponential, analysis of such a supermarket model is always limited, interesting, and challenging. This paper describes a supermarket model with non-Poisson inputs: Markovian Arrival Processes (MAPs) and with non-exponential service times: Phase-type (PH) distributions, and provides a generalized matrix-analytic method which is first combined with the operator semigroup and the mean-field limit. When discussing such a more general supermarket model, this paper makes some new results and advances as follows: (1) Providing a detailed probability analysis for setting up an infinite-dimensional system of differential vector equations satisfied by the expected fraction vector, where "the invariance of environment factors" is given as an important result. (2) Introducing the phase-type structure to the operator semigroup and to the mean-field limit, and a Lipschitz condition can be obtained by means of a unified matrix-differential algorithm. (3) The matrix-analytic method is used to compute the fixed point which leads to performance computation of this system. Finally, we use some numerical examples to illustrate how the performance measures of this supermarket model depend on the non-Poisson inputs and on the non-exponential service times. Thus the results of this paper give new highlight on understanding influence of non-Poisson inputs and of non-exponential service times on performance measures of more general supermarket models.Comment: 65 pages; 7 figure

Linking ventilation heterogeneity quantified via hyperpolarized He-3 MRI to dynamic lung mechanics and airway hyperresponsiveness

Advancements in hyperpolarized helium-3 MRI (HP 3He-MRI) have introduced the ability to render and quantify ventilation patterns throughout the anatomic regions of the lung. The goal of this study was to establish how ventilation heterogeneity relates to the dynamic changes in mechanical lung function and airway hyperresponsiveness in asthmatic subjects. In four healthy and nine mild-to-moderate asthmatic subjects, we measured dynamic lung resistance and lung elastance from 0.1 to 8 Hz via a broadband ventilation waveform technique. We quantified ventilation heterogeneity using a recently developed coefficient of variation method from HP 3He-MRI imaging. Dynamic lung mechanics and imaging were performed at baseline, post-challenge, and after a series of five deep inspirations. AHR was measured via the concentration of agonist that elicits a 20% decrease in the subject’s forced expiratory volume in one second compared to baseline (PC20) dose. The ventilation coefficient of variation was correlated to low-frequency lung resistance (R = 0.647, P < 0.0001), the difference between high and low frequency lung resistance (R = 0.668, P < 0.0001), and low-frequency lung elastance (R = 0.547, P = 0.0003). In asthmatic subjects with PC20 values <25 mg/mL, the coefficient of variation at baseline exhibited a strong negative trend (R = -0.798, P = 0.02) to PC20 dose. Our findings were consistent with the notion of peripheral rather than central involvement of ventilation heterogeneity. Also, the degree of AHR appears to be dependent on the degree to which baseline airway constriction creates baseline ventilation heterogeneity. HP 3He-MRI imaging may be a powerful predictor of the degree of AHR and in tracking the efficacy of therapy.This work was funded by the National Heart, Lung, and Blood Institute Grants R01 HL62269-04 and R01 HL-096797

A Matrix-Analytic Solution for Randomized Load Balancing Models with Phase-Type Service Times

In this paper, we provide a matrix-analytic solution for randomized load balancing models (also known as \emph{supermarket models}) with phase-type (PH) service times. Generalizing the service times to the phase-type distribution makes the analysis of the supermarket models more difficult and challenging than that of the exponential service time case which has been extensively discussed in the literature. We first describe the supermarket model as a system of differential vector equations, and provide a doubly exponential solution to the fixed point of the system of differential vector equations. Then we analyze the exponential convergence of the current location of the supermarket model to its fixed point. Finally, we present numerical examples to illustrate our approach and show its effectiveness in analyzing the randomized load balancing schemes with non-exponential service requirements.Comment: 24 page

Homotopy Method for the Large, Sparse, Real Nonsymmetric Eigenvalue Problem

A homotopy method to compute the eigenpairs, i.e., the eigenvectors and eigenvalues, of a given real matrix A1 is presented. From the eigenpairs of some real matrix A0, the eigenpairs of A(t) ≡ (1 − t)A0 + tA1 are followed at successive "times" from t = 0 to t = 1 using continuation. At t = 1, the eigenpairs of the desired matrix A1 are found. The following phenomena are present when following the eigenpairs of a general nonsymmetric matrix: • bifurcation, • ill conditioning due to nonorthogonal eigenvectors, • jumping of eigenpaths. These can present considerable computational difficulties. Since each eigenpair can be followed independently, this algorithm is ideal for concurrent computers. The homotopy method has the potential to compete with other algorithms for computing a few eigenvalues of large, sparse matrices. It may be a useful tool for determining the stability of a solution of a PDE. Some numerical results will be presented