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Problem definition: Inspired by new developments in dynamic spectrum access, we study the dynamic pricing of wireless Internet access when demand and capacity (bandwidth) are stochastic. Academic/practical relevance: The demand for wireless Internet access has increased enormously. However, the spectrum available to wireless service providers is limited. The industry has, thus, altered conventional license-based spectrum access policies through unlicensed spectrum operations. The additional spectrum obtained through these operations has stochastic capacity. Thus, the pricing of this service by the service provider has novel challenges. The problem considered in this paper is, therefore, of high practical relevance and new to the academic literature. Methodology: We study this pricing problem using a Markov decision process model in which customers are posted dynamic prices based on their bandwidth requirement and the available capacity. Results: We characterize the structure of the optimal pricing policy as a function of the system state and of the input parameters. Because it is impossible to solve this problem for practically large state spaces, we propose a heuristic dynamic pricing policy that performs very well, particularly when the ratio of capacity to demand rate is low. Managerial implications: We demonstrate the value of using a dynamic heuristic pricing policy compared with the myopic and optimal static policies. The previous literature has studied similar systems with fixed capacity and has characterized conditions under which myopic policies perform well. In contrast, our setting has dynamic (stochastic) capacity, and we find that identifying good state-dependent heuristic pricing policies is of greater importance. Our heuristic policy is computationally more tractable and easier to implement than the optimal dynamic and static pricing policies. It also provides a significant performance improvement relative to the myopic and optimal static policies when capacity is scarce, a condition that holds for the practical setting that motivated this research. 

Brain age (BA), distinct from chronological age (CA), can be estimated from MRIs to evaluate neuroanatomic aging in cognitively normal (CN) individuals. BA, however, is a cross-sectional measure that summarizes cumulative neuroanatomic aging since birth. Thus, it conveys poorly recent or contemporaneous aging trends, which can be better quantified by the (temporal) pace P of brain aging. Many approaches to map P, however, rely on quantifying DNA methylation in whole-blood cells, which the blood–brain barrier separates from neural brain cells. We introduce a three-dimensional convolutional neural network (3D-CNN) to estimate P noninvasively from longitudinal MRI. Our longitudinal model (LM) is trained on MRIs from 2,055 CN adults, validated in 1,304 CN adults, and further applied to an independent cohort of 104 CN adults and 140 patients with Alzheimer’s disease (AD). In its test set, the LM computes P with a mean absolute error (MAE) of 0.16 y (7% mean error). This significantly outperforms the most accurate cross-sectional model, whose MAE of 1.85 y has 83% error. By synergizing the LM with an interpretable CNN saliency approach, we map anatomic variations in regional brain aging rates that differ according to sex, decade of life, and neurocognitive status. LM estimates of P are significantly associated with changes in cognitive functioning across domains. This underscores the LM’s ability to estimate P in a way that captures the relationship between neuroanatomic and neurocognitive aging. This research complements existing strategies for AD risk assessment that estimate individuals’ rates of adverse cognitive change with age.