This paper studies a probabilistic characteristic called buffered probability of exceedance (bPOE). It is a function of a random variable and a real-valued threshold. By definition, bPOE is the probability of a tail such that the average of this tail equals the threshold. This characteristic is an extension of the so-called buffered failure probability and it is equal to one minus inverse of the conditional value-at-risk (CVaR). bPOE is a quasi-convex function of the random variable w.r.t. the regular addition operation and a concave function w.r.t. the mixture operation; it is a monotonic function of the random variable; it is a strictly decreasing function of the threshold on the interval between the expectation and the essential supremum. The multiplicative inverse of the bPOE is a convex function of the threshold, and a piecewise-linear function in the case of discretely distributed random variable. The paper provides efficient calculation formulas for bPOE. Minimization of bPOE is reduced to a convex program for a convex feasible region and to linear programming for a polyhedral feasible region and discretely distributed random variables. A family of bPOE minimization problems and corresponding CVaR minimization problems share the same set of optimal solutions.

The concept of Conditional Value-at-Risk (CVaR) is used in various applications in uncertain environment. This paper introduces CVaR (superquantile) norm for a random variable, which is by definition CVaR of absolute value of this random variable. It is proved that CVaR norm is indeed a norm in the space of random variables. CVaR norm is defined in two variations: scaled and non-scaled. L-1 and L-infinity norms are limiting cases of the CVaR norm. In continuous case, scaled CVaR norm is a conditional expectation of the random variable. A similar representation of CVaR norm is valid for discrete random variables. Several properties for scaled and non-scaled CVaR norm, as a function of confidence level, were proved. Dual norm for CVaR norm is proved to be the maximum of L-1 and scaled L-infinity norms. CVaR norm, as a Measure of Error, is related to a Regular Risk Quadrangle. Trimmed L1-norm, which is a non-convex extension for CVaR norm, is introduced analogously to function L-p for p ¡ 1. Linear regression problems were solved by minimizing CVaR norm of regression residuals.

We propose a new characteristic for counting the number of large outcomes in a data set that are considered to be large w.r.t. some fixed threshold x. A popular characteristic used for this purpose is the Cardinality of Upper Tail (CUT), which counts the number of outcomes with magnitude larger than the threshold. We propose a similar characteristic called the Cardinality of Upper Average (CUA), defined as the number of largest data points which have average value equal to the threshold. CUA not only assesses the number of outcomes that are large, but also their overall magnitude. CUA also has superior mathematical properties: it is a continuous function of the threshold, its reciprocal is piece-wise linear w.r.t. threshold, and it is directly optimizable via convex and linear programming. This is in contrast to CUT, which does not asses the severity of large outcomes, is discontinuous as a function of threshold, and is such that direct optimization yields numerically difficult non-convex problems. We show that CUA can be used to formulate meaningful optimization problems containing counters of the largest components of a vector without introduction of binary variables, leading to large improvement in computation speeds. In particular, we apply the CUA concept to create new formulations of network optimization problems involving overloaded nodes or edges, where we aim to minimize the number of most burdened nodes or edges.

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PurposeThis study aims to explore virtual site visit adoption patterns of US convention facilities based on the diffusion of innovation (DOI) theory. Additionally, it offers predictive models of virtual site visit tool adoption by applying probability distributions.Design/methodology/approachThe study used content analysis of 369 US convention facility websites. Data collected from the websites recorded the presence or absence of the following tools facilitating virtual site visits: photos, floor plans, videos, 360-photos, 360-tours and virtual reality (VR)-optimized tours. The website content analysis was followed by application of the DOI theory and predictive modeling.FindingsAccording to the DOI theory, the use of VR-optimized tours (4.34%) is still in the early adoption stage, followed by 360-degree tours (12.74%) and standard videos (17.89%) that have transitioned into the early majority stage of adoption and photos (72.09%) and floor plans (84.82%) that represent a late majority stage. Three predictive models with shifted Gompertz, Gumbel and Bass distributions forecasted that convention centers would achieve a 50% adoption rate of 360-degree tools (photos and tours) in 4.67, 4.2 and three years, respectively. The same models predicted a 50% adoption rate of 360-degree tours in 6.62, 5.81 and 4.42 years.Practical implicationsThe research indicates that most US convention facilities have not taken full advantage of their websites as a sales and marketing tool.Originality/valueThis study is the first comprehensive attempt to evaluate the adoption rate of VR and other technologies enabling virtual site visits by using content analysis of US convention facility websites. Additionally, it is the first attempt to apply probability distributions to predict technology adoption in the convention industry context.