We consider an M/M/1 queue with a removable server that dynamically chooses its service rate from a set of finitely many rates. If the server is off, the system must warm up for a random, exponentially distributed amount of time, before it can begin processing jobs. We show under the average cost criterion, that work conserving policies are optimal. We then demonstrate the optimal policy can be characterized by a threshold for turning on the server and the optimal service rate increases monotonically with the number in system. Finally, we present some numerical experiments to provide insights into the practicality of having both a removable server and service rate control.

As more people move back into densely populated cities, bike sharing is emerging as an important mode of urban mobility. In a typical bike-sharing system (BSS), riders arrive at a station and take a bike if it is available. After retrieving a bike, they ride it for a while, then return it to a station near their final destinations. Since space is limited in cities, each station has a finite capacity of docks, which cannot hold more bikes than its capacity. In this paper, we study BSSs with stations having a finite capacity. By an appropriate scaling of our stochastic model, we prove a mean-field limit and a central limit theorem for an empirical process of the number of stations with k bikes. The mean-field limit and the central limit theorem provide insight on the mean, variance, and sample path dynamics of large-scale BSSs. We also leverage our results to estimate confidence intervals for various performance measures such as the proportion of empty stations, the proportion of full stations, and the number of bikes in circulation. These performance measures have the potential to inform the operations and design of future BSSs.

Authors of this paper were educated and have spent many years in Lomonosov Moscow State University (MSU), in the Interfaculty Laboratory of Statistical Methods (ILSM). Prof. Yu. K. Belyaev was student of Kolmogorov, and for several years his deputy in the ILSM, created and led by the great A. N. Kolmogorov. Now Prof. Yuri Belyaev is Emeritus Professor at Umea University (Sweden). Asaf Hajiyev was a PhD student in Kolmogorov's ILSM. In this paper some unusual incidents about the legendary Kolmogorov, and people around him are presented. Some stories were taken from the books and the internet (see references) dedicated to the memory of Kolmogorov; some were told by his students, and some happened with authors of this paper.

Men are cruel but Man is kind.

– A. N. Kolmogorov

We introduce the special issue on “Learning, Optimization, and the Theory of G-Networks” of the journal Probability in the Engineering and Informational Sciences that appears in 2019. We first outline some of the applications and developments of G-Networks which motivate the ongoing interest for this area, including some areas which could not be covered in this special issue. We then briefly discuss the contributions presented in the ten papers that are published in this special issue in the context of related work.

Motivated by the classical De Bruijn's identity for the additive Gaussian noise channel, in this paper we consider a generalized setting where the channel is modelled via stochastic differential equations driven by fractional Brownian motion with Hurst parameter H ∈ (0, 1). We derive generalized De Bruijn's identity for Shannon entropy and Kullback–Leibler divergence by means of Itô's formula, and present two applications. In the first application we demonstrate its equivalence with Stein's identity for Gaussian distributions, while in the second application, we show that for H ∈ (0, 1/2], the entropy power is concave in time while for H ∈ (1/2, 1) it is convex in time when the initial distribution is Gaussian. Compared with the classical case of H = 1/2, the time parameter plays an interesting and significant role in the analysis of these quantities.

The random neural network (RNN) is a mathematical model for an “integrate and fire” spiking network that closely resembles the stochastic behavior of neurons in mammalian brains. Since its proposal in 1989, there have been numerous investigations into the RNN's applications and learning algorithms. Deep learning (DL) has achieved great success in machine learning. Recently, the properties of the RNN for DL have been investigated, in order to combine their power. Recent results demonstrate that the gap between RNNs and DL can be bridged and the DL tools based on the RNN are faster and can potentially be used with less energy expenditure than existing methods.

We establish that a random sum of independent and identically distributed (i.i.d.) random quantities has a log-concave cumulative distribution function (cdf) if (i) the random number of terms in the sum has a log-concave probability mass function (pmf) and (ii) the distribution of the i.i.d. terms has a non-increasing density function (when continuous) or a non-increasing pmf (when discrete). We illustrate the usefulness of this result using a standard actuarial risk model and a replacement model.

We apply this fundamental result to establish that a compound renewal process observed during a random time interval has a log-concave cdf if the observation time interval and the inter-renewal time distribution have log-concave densities, while the compounding distribution has a decreasing density or pmf. We use this second result to establish the optimality of a so-called (s,S) policy for various inventory models with a stock-out cost coefficient of dimension [$/unit], significantly generalizing the conditions for the demand and leadtime processes, in conjunction with the cost structure in these models. We also identify the implications of our results for various algorithmic approaches to compute optimal policy parameters.

In this paper, several properties of a class of trees presenting preferential attachment phenomenon—plane-oriented recursive trees (PORTs) are uncovered. Specifically, we investigate the degree profile of a PORT by determining the exact probability mass function of the degree of a node with a fixed label. We compute the expectation and the variance of degree variable via a Pólya urn approach. In addition, we study a topological index, Zagreb index, of this class of trees. We calculate the exact first two moments of the Zagreb index (of PORTs) by using recurrence methods. Lastly, we determine the limiting degree distribution in PORTs that grow in continuous time, where the embedding is done in a Poissonization framework. We show that it is exponential after proper scaling.

Suppose there are n players in an ongoing competition, with player i having value vi, and suppose that a game between i and j is won by i with probability vi/(vi + vj). Consider the winner plays competition where in each stage two players play a game, and the winner keeps playing in the next game. We consider two models for choosing its opponent, analyze both models as Markov chains, and determine their stationary probabilities as well as other quantities of interest.

Many cryptocurrencies including Bitcoin are susceptible to a so-called double-spend attack, where someone dishonestly attempts to reverse a recently confirmed transaction. The duration and likelihood of success of such an attack depends on the recency of the transaction and the computational power of the attacker, and these can be related to the distribution of time for counts from one Poisson process to exceed counts from another by some desired amount. We derive an exact expression for this distribution and show how it can be used to obtain efficient simulation estimators. We also give closed-form analytic approximations and illustrate their accuracy.

For a system that is subject to shocks, it is assumed that the distribution of the magnitudes of shocks changes after the first shock of size at least d1, and the system fails upon the occurrence of the first shock above a critical level d2 (> d1). In this paper, the distribution of the lifetime of such a system is studied when the times between successive shocks follow matrix-exponential distribution. In particular, it is shown that the system's lifetime has matrix-exponential distribution when the intershock times follow Erlang distribution. The model is extended to the case when the system fails upon the occurrence of l consecutive critical shocks.

Falls are a major health concern in older adults. Falls lead to mortality, immobility and high costs to social and health care services. Early detection and classification of falls is imperative for timely and appropriate medical aid response. Traditional machine learning models have been explored for fall classification. While newly developed deep learning techniques have the ability to potentially extract high-level features from raw sensor data providing high accuracy and robustness to variations in sensor position, orientation and diversity of work environments that may skew traditional classification models. However, frequently used deep learning models like Convolutional Neural Networks (CNN) are computationally intensive. To the best of our knowledge, we present the first instance of a Hybrid Multichannel Random Neural Network (HMCRNN) architecture for fall detection and classification. The proposed architecture provides the highest accuracy of 92.23% with dropout regularization, compared to other deep learning implementations. The performance of the proposed technique is approximately comparable to a CNN yet requires only half the computation cost of the CNN-based implementation. Furthermore, the proposed HMCRNN architecture provides 34.12% improvement in accuracy on average than a Multilayer Perceptron.

Artificial Neural Networks (ANNs)-based techniques have dominated state-of-the-art results in most problems related to computer vision, audio recognition, and natural language processing in the past few years, resulting in strong industrial adoption from all leading technology companies worldwide. One of the major obstacles that have historically delayed large-scale adoption of ANNs is the huge computational and power costs associated with training and testing (deploying) them. In the mean-time, Neuromorphic Computing platforms have recently achieved remarkable performance running the bio-realistic Spiking Neural Networks at high throughput and very low power consumption making them a natural alternative to ANNs. Here, we propose using the Random Neural Network, a spiking neural network with both theoretical and practical appealing properties, as a general purpose classifier that can match the classification power of ANNs on a number of tasks while enjoying all the features of being a spiking neural network. This is demonstrated on a number of real-world classification datasets.

We study a single server queue under a processor-sharing type of scheduling policy, where the weights for determining the sharing are given by functions of each job's remaining service (processing) amount, and obtain a fluid limit for the scaled measure-valued system descriptors.

We consider the problem of estimating the rate of defects (mean number of defects per item), given the counts of defects detected by two independent imperfect inspectors on one sample of items. In contrast with the setting for the well-known method of Capture–Recapture, we do not have information regarding the number of defects jointly detected by both inspectors. We solve this problem by constructing two types of estimators—a simple moment-type estimator, and a complicated maximum-likelihood (ML) estimator. The performance of these estimators is studied analytically and by means of simulations. It is shown that the ML estimator is superior to the moment-type estimator. A systematic comparison with the Capture–Recapture method is also made.

The present paper is concerned with reliability economics, considering a certain performance-per-cost criterion for coherent and mixed systems, as introduced in [Dugas, M.R. & Samaniego, F.J. (2007). On optimal system designs in reliability-economics frameworks. Naval Research Logistics 54, 568–582]. We first present a new comparison result for performance-per-cost of systems with independent and identically distributed component lifetimes under certain stochastic orderings. We then consider optimization of the performance-per-cost criterion, first reconsidering and refining results from the above cited paper, and then considering mixtures of given subsets of coherent systems.

The sample squared Sharpe ratio (SSR) is a critical statistic of the risk-return tradeoff. We show that sensitive upper-tail probabilities arise when the sample SSR is employed to test the mean-variance efficiency under different test statistics. Assuming the error's normality with a nonzero mean, we integrate the sample SSR and the arbitrage regression into a noncentral chi-square (χ2) test. We find that the distribution of the sample SSR based on the regression error is to the left of the F-distribution when assuming the returns' normality. Compared to two benchmarks that use the noncentral F-distribution and the central F-statistic, the χ2-statistic is more effective, competitive, significant, and locally robust when used to reject the upper-tailed mean-variance efficiency test using the usual parameters (sample size, portfolio size, and SSR).

Supply chains are fundamental to the economy of the world and many supply chains focus on perishable items, such as food, or even clothing that is subject to a limited shelf life due to fashion and seasonable effects. G-networks have not been previously applied to this important area. Thus in this paper, we apply G-networks to supply chain systems and investigate an optimal order allocation problem for a N-node supply chain with perishable products that share the same order source of fresh products. The objective is to find an optimal order allocation strategy to minimize the purchase price per object from the perspective of the customers. An analytical solution based on G-networks with batch removal, together with optimization methods are shown to produce the desired results. The results are illustrated by a numerical example with realistic parameters.

We consider a two-echelon production inventory system with a manufacturer having limited production capacity and a distribution center (DC). There is a positive transportation time between the manufacturer and the DC. Customers gain a value by receiving the product and incur a waiting cost when facing a delay. We assume that customers' waiting cost depends on their degree of impatience with respect to delay, which is captured by a convex waiting cost function. Customers are strategic with respect to joining the system and either place an order or balk from the system upon their arrival depending on their expected waiting time. We study the Stackelberg equilibrium assuming that the DC acts as a Stackelberg leader and customers are the followers. We first obtain the total expected revenue and then provide a heuristic to derive the optimal base-stock levels in the warehouse and the DC as well as the optimal price of the product.

The reliability study of k-out-of-n systems is of interest both from theoretical and practical points of view. Applications of such models can be seen in many real-world phenomena, including telecommunication, transmission, transportation, manufacturing, and services. A probabilistic study of a real-world k-out-of-n system often helps to develop an optimal strategy for maintaining high system-level reliability. There are many investigations devoted to the reliability-centric analysis of such systems. We consider a mathematical model of a repairable k-out-of-n system that works until k of its n components have failed. During the system's life cycle, its components are repaired with the help of a single repair facility. It is supposed that the components' lifetimes have an exponential distribution and their repair times have a general distribution. The proposed model is intended to be applied to the description of operation of unmanned rotorcraft high-altitude platforms and to be validated with the help of an experimental prototype. For the considered system, we propose an algorithm for calculation of the reliability function, and for special cases, k = 2 and k = 3, its closed-form representation is given. A numerical investigation is performed for special cases. The obtained results are a first step toward the sensitivity analysis of reliability characteristics of k-out-of-n systems to the shape of the repair time distributions of their components.