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For an e-hailing taxi operation, we analyse a driver's profit-maximising reactive strategy (to either accept or refuse a ride request) in response to the ride request broadcast by the platform. We analyse four operating modes, each of which is a combination of either of two reactive strategies: no refusal and refusal based on proximity, and either of two broadcasting methods. In an operating mode, our objective is to evaluate the expected total profit in a shift. We adopt a two-stage methodology to answer the research questions. In the first stage, we develop an agent-based simulation model to capture the effect of multiple taxis on driver's reactive strategy. Using real trip data, we find that a driver could follow a strategy of refusal based on proximity and earn approximately 25% more than the baseline no refusal strategy. In the second stage, we develop an approximate analytical model for a single taxi operation and compare the performance against the agent-based simulation model. We develop closed-form expressions of the expected total profit for each operating mode and topology of the service region. We find that our approximate analytical model provides an upper bound, and the profit deviation lies within 20% of the agent-based simulation model. [ABSTRACT FROM AUTHOR]

This paper studies the N-player gambler's ruin problem where two players are involved in an even-money bet during each round. The objective is to solve for each player's final placing probabilities given their initial wealths, as well as the expected time until ruin. Multigraphs were constructed to model the transitions between chip states. Linear systems were constructed based on these graphs. Solutions for the placing probabilities of each player were obtained using these linear systems. A numerical algorithm is developed to solve the general N-player gambler's ruin for any positive integer chip total. Expected time until ruin for any initial state can be solved using absorbing Markov chains. The solution leads to exact values and equities in this model depend on the exact number, not just proportion, of chips each player holds. Compared with the usual lattice model, the multigraph model uses smaller matrices in computations, yielding considerable savings in the number of operations and program run time. [ABSTRACT FROM AUTHOR]

Cyber risk has emerged as a significant threat to businesses that have increasingly relied on new and existing information technologies (IT). Across various businesses in different industries and sectors, a distinct pattern of IT network architectures, such as the client‐server network architecture, may, in principle, expose those businesses, which share it, to similar cyber risks. That is why in this article, we propose a probabilistic structural framework for loss assessments of cyber risks on the class of client‐server network architectures with K$$ K $$ different client types. To our knowledge, there exist no theoretical models of an aggregate loss distribution for cyber risk in this setting. With this structural framework via the exact mean and variance of losses, we demonstrate how the changing cybersecurity environment of a business's IT network impacts the loss distribution. Furthermore, our framework provides insights into better investment strategies for cybersecurity protection on the client‐server network. Motivated by cyberattacks across industries, we apply our framework to four case studies that utilize the client‐server network architecture. Our first application is implantable medical devices in healthcare. Our second application is the smart buildings domain. Third, we present an application for ride‐sharing services such as Uber and Lyft. The fourth is the application of vehicle‐to‐vehicle cooperation in traffic management. The results are corresponding exact means and variances of cyber risk loss distributions parameterized by various cybersecurity parameters allowing for liability assessments and decisions in cybersecurity protection investments. [ABSTRACT FROM AUTHOR]

We propose to introduce a new class of bivariate probability distributions, which we believe is of great interest to statisticians and data scientists. However different from the conventional Weibull it might be, the density function posited herein allows to generalize its properties in two dimensions (2D). This new function, essentially, has structure characteristics and properties different from those of the various bivariate Weibull-type functions found in the literature. The main features, such as the marginal distributions, moments, characteristic functions of this bivariate density are defined. Two related maximum likelihood estimation algorithms are also explicated, tested, and compared. Numerical simulations show the practicality of these algorithms as well as the interest of the new density in several areas of data analysis and extreme values modeling. [ABSTRACT FROM AUTHOR]

We propose a statistical framework for quantifying the importance of single events that do not provide intermediate rewards but offer implicit incentives through scheduling and the reward structure at the end of a multi-event contest. Applying the framework to primary elections in the US, where earlier elections have greater importance and influence, we show that schedule variations can mitigate the problem of front-loading elections. When applied to European football, we demonstrate the utility and meaningfulness of quantified event importance in relation to the in-match performance of contestants to improve outcome prediction and to provide an early indication of public interest. [ABSTRACT FROM AUTHOR]

In this paper, we present research results that extend and supplement our article recently published by MDPI. We derive the closed-form relations among the queue-length probabilities observed in the pre-arrival, random, and post-departure epochs for a complex, bulk-service, multi-server queueing system GI/M a , b /c. [ABSTRACT FROM AUTHOR]

Imagine, you enter a grocery store to buy food. How many people do you overlap with in this store? How much time do you overlap with each person in the store? In this paper, we answer these questions by studying the overlap times between customers in the infinite server queue. We compute in closed form the steady-state distribution of the overlap time between a pair of customers and the distribution of the number of customers that an arriving customer will overlap with. Finally, we define a residual process that counts the number of overlapping customers that overlap in the queue for at least δ time units and compute its distribution. [ABSTRACT FROM AUTHOR]

This paper treats a single server Markovian queueing system with changeable service rates and customer choice. When a customer arrives at an empty system, he is served by the server at a low service rate. This low service rate continues until the queue length reaches a threshold. Then the server switches to a high service rate to serve customers who join the system. The high service rate continues until the system becomes empty and a low service rate starts again. Under a service reward and waiting cost structure, customers decide to join or to balk based on their net utility of joining the system. The customers' equilibrium and socially optimal joining strategies are studied under two information scenarios which are observable and unobservable queues, respectively. Such a model fits a stochastic service system where the server can make his service rate decision and customers can make their joining or baking decisions and thus has wide applications in service industries. Our focus is on examining the equilibrium performance of the queue under the interaction between the server's service rate and customers' joining decisions. We develop the steady-state distribution and performance measures of the system by formulating two-dimensional continuous-time Markov chains. [ABSTRACT FROM AUTHOR]

We consider a collection of statistically identical two-state continuous time Markov chains (channels). A controller continuously selects a channel with the view of maximizing infinite horizon average reward. A switching cost is paid upon channel changes. We consider two cases: full observation (all channels observed simultaneously) and partial observation (only the current channel observed). We analyze the difference in performance between these cases for various policies. For the partial observation case with two channels or an infinite number of channels, we explicitly characterize an optimal threshold for two sensible policies which we name "call-gapping" and "cool-off." Our results present a qualitative view on the interaction of the number of channels, the available information, and the switching costs. [ABSTRACT FROM AUTHOR]

The Devylder–Goovaerts conjecture is probably the oldest conjecture in actuarial mathematics and has received a lot of attention in recent years. It claims that ruin with equalized claim amounts is always less likely than in the classical model. Investigating the validity of this conjecture is important both from a theoretical aspect and a practical point of view, as it suggests that one always underestimates the risk of insolvency by replacing claim amounts with the average claim amount a posteriori. We first state a simplified version of the conjecture in the discrete-time risk model when one equalizes aggregate claim amounts and prove that it holds. We then use properties of the Pareto distribution in risk theory and other ideas to target candidate counterexamples and provide several counterexamples to the original Devylder–Goovaerts conjecture. [ABSTRACT FROM AUTHOR]

We consider a competition involving $r$ teams, where each individual game involves two teams, and where each game between teams $i$ and $j$ is won by $i$ with probability $P_{i,j} = 1 - P_{j,i}$. We suppose that $i$ and $j$ are scheduled to play $n(i,j)$ games and say that the team that wins the most games is the winner of the competition. We show that the conditional probability that $i$ is the winner, given that $i$ wins $k$ games, is increasing in $k$. We bound the tail probability of the number of wins of the winning team. We consider the special case where $P_{i,j} = {v_i}/{(v_i + v_j)}$ , and obtain structural results on the probability that team $i$ is the winner. We give efficient simulation approaches for computing the probability that team $i$ is the winner, and the conditional probability given the number of wins of $i$. [ABSTRACT FROM AUTHOR]

Standard server assignment policies for multi-server queueing stations include the noncollaborative policy, where the servers work in parallel on different jobs; and the fully collaborative policy, where the servers work together on the same job. However, if each job can be decomposed into subtasks with no precedence relationships, then we consider a form of server coordination named task assignment where the servers work in parallel on different subtasks of the same job. We identify the task assignment policy that maximizes the long-run average throughput of a queueing station with finite internal buffers when blocked servers can be idled or reassigned to either replace or collaborate with other servers on unblocked subtasks. We then compare the server coordination policies and show that the task assignment is best when the servers are highly specialized; otherwise, the fully collaborative or noncollaborative policies are preferable depending on whether the synergy level among the servers is high or not. We also provide numerical results that quantify our previous comparison. Finally, we address buffer allocation in longer lines where there are precedence relationships between some of the tasks, and present numerical results that suggest our comparisons for one queueing station generalize to longer lines. [ABSTRACT FROM AUTHOR]

Based on an assumption of an existing scoring probability difference between two teams engaged in a football match, the conditional probability distribution given the number of goals in the match for the low quality team beating the high quality team, is derived. Furthermore, similar distributions for the high quality team beating the low quality team as well as a draw are also derived. Based on a Poisson distribution of goals, we discuss a potential for optimizing expected uncertainty of outcome (UO) by adjusting intra-match rules or league rules for team sports. The main variables in this regard are (i) the typical number of goals scored, and (ii) the evenness of competing teams. We identify a curve defining the optimal expected number of goals as a function of team quality difference. [ABSTRACT FROM AUTHOR]

We consider a stochastic cash balance problem in which cash receipts and cash disbursements change linearly and form a fluid process. The firm has to decide how much cash to hold in order to meet its obligations. The cash balance is monitored continuously and a four thresholds (0, S, M, m) policy is used to control the cash ( 0 < S < M < m ). That is, (a) If, at any time, the cash level drops to an order-level 0, a cash transfer of size S is requested such that the cash level is immediately raised to level S. (b) Whenever the cash level lies within the interval (0, m), no action is taken. (c) Every time the cash balance reaches the salvage-level m, it is restored to a level M and the excess amount (m - M) is invested in other earning assets that provide a suitable return. We further study the extended (s, S, M) policy, in which s be an order level and each request for a cash transfer takes some random time (called the lead-time) until it is approved. Using fluid-flow results and multi-dimensional martingales, we construct a closed-form expression for the discounted total cost of running the cash balance. Numerical study provides several guidelines for the optimal control. For example, we show that even when the holding cost is high and receipts occur more frequently, it would be wise for the firm to hold on some extra cash. [ABSTRACT FROM AUTHOR]

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ($N$) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone. [ABSTRACT FROM AUTHOR]

There is a set of n bandits and at every stage, two of the bandits are chosen to play a game, with the result of a game being learned. In the "weak regret problem," we suppose there is a "best" bandit that wins each game it plays with probability at least p > 1/2, with the value of p being unknown. The objective is to choose bandits to maximize the number of times that one of the competitors is the best bandit. In the "strong regret problem", we suppose that bandit i has unknown value vi, i = 1, ..., n, and that i beats j with probability vi/(vi + vj). One version of strong regret is interested in maximizing the number of times that the contest is between the players with the two largest values. Another version supposes that at any stage, rather than choosing two arms to play a game, the decision maker can declare that a particular arm is the best, with the objective of maximizing the number of stages in which the arm with the largest value is declared to be the best. In the weak regret problem, we propose a policy and obtain an analytic bound on the expected number of stages over an infinite time frame that the best arm is not one of the competitors when this policy is employed. In the strong regret problem, we propose a Thompson sampling type algorithm and empirically compare its performance with others in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider a two-stage call center staffing model. In the first stage, the interval staffing levels are set under arrival rate uncertainty. In the second stage, these initial staffing levels are corrected to the right value based on more precise arrival rate information. We show that this problem is of newsvendor type, where the costs are the initial staffing costs plus the second stage adaptation costs. We show that we should initially staff according to a quantile of the distributional forecast, rather than the mean. It is also shown that the errors in staffing are approximately linear in the forecasting errors. This leads to the conclusion that the weighted sum of errors should be the error measurement in call center forecasting, since minimizing, it minimizes the total staffing costs. In special cases where the costs are symmetric for over- and understaffing, this is equivalent to minimizing the weighted absolute percentage error. [ABSTRACT FROM AUTHOR]

The paper deals with prognosis estimation for industrial systems in a series configuration, modeled by superimposed renewal processes (SRP), when the cause of failures is not available. In the presence of missing information, an SRP is commonly approximated by a Poisson process or a virtual age model. The performance of the approximations was assessed in the ideal configuration where all parameters of the models are known. The current article adopts a practitioner's perspective by assuming that the parameters of the models are unknown and must be estimated. In addition to inference procedures, the assessment of the prognosis indicators, such as the remaining useful life, is discussed. Finally, we investigate a fleet of infrastructure components of the Norwegian railway network operated by Bane NOR. [ABSTRACT FROM AUTHOR]

This work demonstrates that the card shuffling procedure commonly performed in casino poker rooms is insufficient for randomizing a deck of cards. We explore this in the context of Texas Hold'em, which has established itself as the most popular form of poker worldwide over the past few decades. We show the degree to which any given card may be more (or less) likely to show up as an important card in the subsequent hand. Additionally, we find that the shuffling procedure does not sufficiently separate cards from their starting point; that is, cards are more likely to stay close together after shuffling than they should by chance. Thus, in this work, we demonstrate that Texas Hold'em players can gain an advantage over their opponents by recognizing these deficiencies in the shuffling procedure. [ABSTRACT FROM AUTHOR]

Nondominated sorting is a discrete process that sorts points in Euclidean space according to the coordinatewise partial order and is used to rank feasible solutions to multiobjective optimization problems. It was previously shown that nondominated sorting of random points has a Hamilton--Jacobi equation continuum limit. We prove quantitative error estimates for the convergence of nondominated sorting to its continuum limit Hamilton--Jacobi equation. Our proof uses the maximum principle and viscosity solution machinery, along with new semiconvexity estimates for domains with corner singularities. [ABSTRACT FROM AUTHOR]

In this article, we focus on reliability properties of a k-out-of-n system equipped with two cold standby components. We describe the lifetime of the system and compute the reliability function of the system, which will allow us to investigate the risk of system failure. Further, three different mean residual life functions of the system are derived. We also provide some stochastic ordering results. The theoretical results established in this article addresses a problem proposed by Eryilmaz and Wang. We have presented a few numerical results to illustrate the importance of our analysis. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

We propose a novel time discretization for the log-normal SABR model which is a popular stochastic volatility model that is widely used in financial practice. Our time discretization is a variant of the Euler–Maruyama scheme. We study its asymptotic properties in the limit of a large number of time steps under a certain asymptotic regime which includes the case of finite maturity, small vol-of-vol and large initial volatility with fixed product of vol-of-vol and initial volatility. We derive an almost sure limit and a large deviations result for the log-asset price in the limit of a large number of time steps. We derive an exact representation of the implied volatility surface for arbitrary maturity and strike in this regime. Using this representation, we obtain analytical expansions of the implied volatility for small maturity and extreme strikes, which reproduce at leading order known asymptotic results for the continuous time model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we review different definitions that multi-state k-out-of-n systems have received along the literature and study them in a unified way using the algebra of monomial ideals. We thus obtain formulas and algorithms to compute their reliability and bounds for it. We provide formulas and computer experiments for simple and generalized multi-state k-out-of-n systems and for binary k-out-of-n systems with multi-state components. [ABSTRACT FROM AUTHOR]

Reliability theory and survival analysis, the residual entropy is known as a measure suitable to describe the dynamic information content in stochastic systems conditional on survival. Aiming to analyze the variability of such information content, in this paper we introduce the variance of the residual lifetimes, "residual varentropy" in short. After a theoretical investigation of some properties of the residual varentropy, we illustrate certain applications related to the proportional hazards model and the first-passage times of an Ornstein–Uhlenbeck jump-diffusion process. [ABSTRACT FROM AUTHOR]

This article considers the interval availability and instant availability of the k-system. A certain relationship between the two types of availability is established. Some lower and upper bounds to interval availability are derived. It also provides a couple of conditions under which the availability of two systems can be compared. Several examples are given to show the complexity of comparisons of availability. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

In this paper, we display methods for the computation of convergence and perturbation bounds for Mt/Mt/1 system with balking, catastrophes, server failures and repairs. Based on the logarithmic norm of linear operators, the bounds on the rate of convergence, perturbation bounds, and the main limiting characteristics of the queue-length process are obtained. Finally, we consider the application of all obtained estimates to a specific model. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

For rare blood groups the recruitment of donor relatives, for example siblings, is expected to be effective, since the probability of a similar rare blood group is likely. However, the likelihood differs between blood groups and is not commonly available. This paper provides a unified mathematical formulation to calculate such likelihoods. From a mathematical and probabilistic point of view, it is shown that these likelihoods can be obtained from the computation of a stationary genotype distribution. This, in turn, can be brought down to a system of quadratic stochastic operators. A generic mathematical approach is presented which directly leads to a stationary genotype distribution for arbitrary blood groups. The approach enables an exact computation for the effectiveness of recruiting next of kin for blood donorship. Next to an illustration of computations for 'standard' ABO and Rhesus-D blood groups, it is particularly illustrated for the extended Rhesus blood group system. Also other applications requiring next of kin blood group associations can be solved directly by using the unified mathematical formulation. [ABSTRACT FROM AUTHOR]

Logistics planners often need to estimate length of a path through n random points. The problem has connections to geographical surveys, Steiner tree problems, traveling salesman problems, Amazon.com and Alibaba.cn drone aircraft multi-package outbound delivery paths, and aircraft sorties. In this effort, we utilize two applied probability tenets in concert to derive a closed form model for the expected distance between orderly pairs of randomly distributed delivery locations across some region. Indeed, by themselves each of the two tenets offer little help but when utilized simultaneously they elucidate the problem. Then, this expected distance is multiplied by (n − 1) to estimate the total length of the associated orderly path (albeit not the optimal path) through the n points. This total path length estimate, in turn, provides decision makers with a better information for distribution and logistics planning. [ABSTRACT FROM AUTHOR]

In quantitative finance, it is often necessary to analyze the distribution of the sum of specific functions of observed values at discrete points of an underlying process. Examples include the probability density function, the hedging error, the Asian option, and statistical hypothesis testing. We propose a method to calculate such a distribution, utilizing a recursive method, and examine it using various examples. The results of the numerical experiment show that our proposed method has high accuracy. [ABSTRACT FROM AUTHOR]

The method of aliasing vastly expands the palette of discrete random variate generation methodologies while providing excellent speed. However, its application is limited to finitely supported distributions. We demonstrate that an application of moment preserving finitization called the Negative Taylor Series Finitization (NTSF) method for the power series family of discrete distributions, when coupled with the method of aliasing, can greatly improve infinitely supported discrete random variate generation speed with certain limitations. We illustrate this with the logarithmic power series distribution, and we compare four published algorithms designed to generate random variates from a logarithmic distribution to the aliasing method of random variate generation from an NTSF version of the same distribution. We compare the accuracy and speed (user‐time) of these various methods for generating variates from a logarithmic distribution. [ABSTRACT FROM AUTHOR]

The Shannon entropy based on the probability density function is a key information measure with applications in different areas. Some alternative information measures have been proposed in the literature. Two relevant ones are the cumulative residual entropy (based on the survival function) and the cumulative past entropy (based on the distribution function). Recently, some extensions of these measures have been proposed. Here, we obtain some properties for the generalized cumulative past entropy. In particular, we prove that it determines the underlying distribution. We also study this measure in coherent systems and a closely related generalized past cumulative Kerridge inaccuracy measure. [ABSTRACT FROM AUTHOR]

Negative Taylor Series Finitization (NTSF) is a moment preserving method that transforms a power series distribution into another having smaller support of size n with moments coinciding with the first n moments of the parent distribution. We present algorithms for addressing two issues that arise when developing an NTSF. The first is based on the Kolmogorov–Smirnov statistic for choosing n. The second produces the maximum feasible parameter space (MFPS), or largest set of parameters yielding a proper distribution, which is necessary because the parameter space of an NTSF distribution is constrained. Both are essential for finitization applications such as fast variate generation. Under well‐defined conditions, (1) the MFPS is determined uniquely via the roots obtained by setting the (n−1)st finitized probability to 0, (2) the MFPS monotonically decreases as a function of increasing n except for the Poisson (where the MFPS is constant for all n), and (3) the Poisson is unique in this regard. [ABSTRACT FROM AUTHOR]

In many settings, people must give numerical scores to entities from a small discrete set--for instance, rating physical attractiveness from 1-5 on dating sites, or papers from 1-10 for conference reviewing. We study the problem of understanding when using a different number of options is optimal. We consider the case when scores are uniform random and Gaussian. We study computationally when using 2, 3, 4, 5, and 10 options out of a total of 100 is optimal in these models (though our theoretical analysis is for a more general setting with k choices from n total options as well as a continuous underlying space). One may expect that using more options would always improve performance in this model, but we show that this is not necessarily the case, and that using fewer choices--even just two--can surprisingly be optimal in certain situations. While in theory for this setting it would be optimal to use all 100 options, in practice, this is prohibitive, and it is preferable to utilize a smaller number of options due to humans' limited computational resources. Our results could have many potential applications, as settings requiring entities to be ranked by humans are ubiquitous. There could also be applications to other fields such as signal or image processing where input values from a large set must be mapped to output values in a smaller set. [ABSTRACT FROM AUTHOR]