Stopping Time Theory Research Articles

The Latency Performance Analysis and Effective Relay Selection for Visible Light Networks.

In visible light communication (VLC), the precise latency evaluation of wireless access networks and the efficient forwarding strategy of core networks are the crux for end-to-end reliability provisioning. Leveraging martingale theory, an elegant latency-bounded reliability analysis framework is studied for the VLC network. Considering the characteristic that VLC links are easy to block, the martingale of the service process is constructed. Based on the time shift features, the martingale process related to latency is proposed for the VLC system, which models the influence of entanglement between aggregate arrivals and random service on latency. A stopping time event about latency is defined. Renting the stopping time theory, a tight upper bound of the unreliability with regard to latency is derived. In the core network, a dynamic forward backhaul framework is proposed, which uses the relay selection algorithm based on back-pressure theory to improve data transmission quality. The theoretical latency-bounded reliability matches the simulation results well, which verifies the effectiveness of the proposed analysis framework, and the proposed relay selection algorithm can also improve network performance under data-intensive transmission.

The first hitting time analysis of evolutionary algorithms based on renewal process

<abstract> <p>Running time analysis of evolutionary algorithms for continuous optimization is one research challenge in the field of evolutionary algorithms (EAs). However, the theoretical analysis results have rarely been applied to evolutionary algorithms for continuous optimization in practice, let alone their variants for evolution strategy. In this paper, we regarded the first hitting time of evolution strategy as the stopping time of the renewal process on the basis of the renewal process and in combination with Wald's inequality and stopping time theory. Afterwards, to demonstrate the application of the proposed model in the first hitting time analysis of (1 + 1) ES, we analyzed it with different mutation operators on the sphere function. First, we significantly improved the lower bound on the first hitting time of (1 + 1) ES with a uniform mutation operator, i.e., from $\Omega(n)$ to $\Omega\left(e^{c n}\right)$. Next, $O\left(n^{2} \sqrt{n}\right)$ was the upper bound on the first hitting time of (1 + 1) ES with a Gaussian mutation operator from the initial distance <italic>R</italic> to half of the initial distance <italic>R</italic>/2. The numerical experimental results showed that the theoretical calculation was consistent with the actual running time, which provides a novel method for analyzing the first hitting time of EAs.</p> </abstract>

Bandwidth abstraction and service rate instantiation for latency-bounded reliability provisioning in 5th generation wireless networks

The provisioning of the statistical reliability regard to latency is crux for the bursty traffic in the 5th generation (5G) wireless network slicing. Based on martingale theory, a novel analysis framework of the latency-bounded reliability is proposed. For the queuing system with bursty traffic, a Queuing Length Martingale is constructed to reveal the impact of the entanglement between the arrival process and the service process on the latency performance, where these two stochastic processes are sharply heterogeneous. The complex martingale parameters are determined. Relying on the time shift feature of the Queuing Length Martingale, a Latency Martingale is defined, which enables to analyze the latency-bounded unreliability in the martingale domain. Leveraging the stopping time theory, a tight upper bound of the unreliability regard to latency is obtained. A bandwidth abstraction and service rate instantiation scheme is designed for wireless network slicing. Bandwidth abstraction facilitates to decouple the statistical reliability requirements as the desired bandwidth of the wireless node. In the 5G access network, a transmission power allocation problem is formulated with the constraint of abstracted bandwidth. According to the access states and channel characteristics, the demanded transmission power is captured dynamically to instantiate the service rates. Simulation results highlight the effectiveness of the bandwidth abstraction and service rate instantiation scheme.

Fasys: Visible-Light-Based Communication and Positioning Services towards Smart Cities.

Visible-light-based transmission application plays an important role in various types of sensor services for the Internet of Things (IoTs). However, in big data scenarios, current visible-light-based systems cannot achieve concurrent high-speed communication, low-speed communication, and positioning. Therefore, in this article, we propose a smart visible-light-based fusion applications system, named Fasys, to solve the above problem for the big data traffic with heterogeneity. Specifically, for low-speed data services, we propose a novel linear block coding and bit interleaving mechanism, which enhances the LED positioning accuracy and recovers the lost data bits in the interframe gap (IFG). For high-speed data services with traffic possessing burstiness, an elegant statistical reliability analysis framework in regard to latency is proposed based on martingale theory. The backlog martingale process is constructed. Leveraging stopping time theory, a tight upper bound of unreliability is obtained. An arrival abstraction and traffic allocation scheme is designed, which contributes to decouple the reliability requirement as the maximum supportable arrival load. Finally, we implement our Fasys system, and extensive experimental results show that our system can achieve consistent high-precision positioning and low-BER data communication for low-speed data services. And the proposed martingale-based traffic allocation scheme can achieve the provisioning of reliability in regard to the latency for high-speed data services.

Qualitative analysis of a two-group SVIR epidemic model with random effect

In this paper, we investigate the dynamical behavior of a two-group SVIR epidemic model with random effect. Firstly, the two-group SVIR epidemic model with random perturbation of natural death rate is established. The existence and uniqueness of positive solution are proved by using stopping time theory and the Lyapunov analysis method. Secondly, a property of the system solution is obtained by using the law of strong numbers and the continuous local martingale. Finally, a new combination of Lyapunov functions is applied. The solution of the model we obtained is oscillating around a steady state if the basic reproduction number is less than one, which is the disease-free equilibrium of the corresponding deterministic model. A numerical simulation is presented to verify our theoretical results.

Two-Stage Bayesian Sequential Change Diagnosis

In this paper, we formulate and solve a two-stage Bayesian sequential change diagnosis (SCD) problem. Different from the one-stage sequential change diagnosis problem considered in the existing work, after a change has been detected, we can continue to collect low-cost samples so that the post-change distribution can be identified more accurately. The goal of a two-stage SCD rule is to minimize the total cost including delay, false alarm probability, and misdiagnosis probability. To solve the two-stage SCD problem, we first convert the problem into a two-ordered optimal stopping time problem. Using tools from optimal multiple stopping time theory, we obtain the optimal SCD rule. Moreover, to address the high computational complexity issue of the optimal SCD rule, we further propose a computationally efficient threshold-based two-stage SCD rule. By analyzing the asymptotic behaviors of the delay, false alarm, and misdiagnosis costs, we show that the proposed threshold SCD rule is asymptotically optimal as the per-unit delay costs go to zero.

Modelling and computation of optimal multiple investment timing in multi-stage capacity expansion infrastructure projects

<p style='text-indent:20px;'>So far, the optimal investment timing to maximize the total profit of multi-stage capacity expansion infrastructure projects is not clear. In the case of uncertain demands, the optimal multiple stopping time theory is adopted to model the optimal decision-making of investment timing for multi-stage expansion infrastructure projects in a finite time horizon. In this context, the first-stage of the project involves a dedicated asset investment for later expansion, and the capacity of the project at each stage is constrained, which makes the cash flow of the project exhibit the characteristic of bull call spread. The upwind finite difference method and multi-least squares Monte Carlo simulation are combined to solve the project value and determine the optimal exercise boundaries at all stages described by a sequence of demand thresholds. A multi-stage power plant project is taken as an example to validate the model. Through the example, the optimal investment strategies and the value of the multi-stage project are provided; the effects of the dedicated asset and capacity constraint are illustrated. This study novelly reveals the effect of the capacity constraints on the project value using the bull call spread theory.</p>

Bandwidth abstraction and instantiation under closed-loop latency constraint for tactile slice based on martingale theory

The closed-loop latency quality of service (QoS) provisioning for tactile services is a pending problem. Based on martingale theory, this paper studies the bandwidth abstraction and service rate instantiation under closed-loop latency constraint for the tactile slice. The demanded bandwidth for arrivals is abstracted by considering the complex features of arrival traffic. The uplink delay is mapped into the analysis of downlink queuing system, which enables the closed-loop latency analysis. A martingale analytical framework is proposed so that the precise delay performance for downlink with bursty arrivals can be evaluated. We construct a Backlog Martingale process for the buffer backlog. A stopping time event that considers uplink and downlink jointly is defined under the closed-loop latency QoS constraint. Leveraging stopping time theory, a tight bound of delay violation probability is derived to guide the bandwidth abstraction. Service rates are instantiated by activating RRHs to form a dynamic MIMO system. Simulation results verify the effectiveness of the proposed strategy.

Myocardial infarction evaluation from stopping time decision toward interoperable algorithmic states in reinforcement learning

BackgroundThe Elliot wave principle commonly characterizes the impulsive and corrective wave trends for both financial market trends and electrocardiograms. The impulsive wave trends of electrocardiograms can annotate several wave components of heart-beats including pathological heartbeat waveforms. The stopping time inquires which ordinal element satisfies the assumed mathematical condition within a numerical set. The proposed work constitutes several algorithmic states in reinforcement learning from the stopping time decision, which determines the impulsive wave trends. Each proposed algorithmic state is applicable to any relevant algorithmic state in reinforcement learning with fully numerical explanations. Because commercial electrocardiographs still misinterpret myocardial infarctions from extraordinary electrocardiograms, a novel algorithm needs to be developed to evaluate myocardial infarctions. Moreover, differential diagnosis for right ventricle infarction is required to contraindicate a medication such as nitroglycerin.MethodsThe proposed work implements the stopping time theory to impulsive wave trend distribution. The searching process of the stopping time theory is equivalent to the actions toward algorithmic states in reinforcement learning. The state value from each algorithmic state represents the numerically deterministic annotated results from the impulsive wave trend distribution. The shape of the impulsive waveform is evaluated from the interoperable algorithmic states via least-first-power approximation and approximate entropy. The annotated electrocardiograms from the impulsive wave trend distribution utilize a structure of neural networks to approximate the isoelectric baseline amplitude value of the electrocardiograms, and detect the conditions of myocardial infarction. The annotated results from the impulsive wave trend distribution consist of another reinforcement learning environment for the evaluation of impulsive waveform direction.ResultsThe accuracy to discern myocardial infarction was found to be 99.2754% for the data from the comma-separated value format files, and 99.3579% for those containing representative beats. The clinical dataset included 276 electrocardiograms from the comma-separated value files and 623 representative beats.ConclusionsOur study aims to support clinical interpretation on 12-channel electrocardiograms. The proposed work is suitable for a differential diagnosis under infarction in the right ventricle to avoid contraindicated medication during emergency. An impulsive waveform that is affected by myocardial infarction or the electrical direction of electrocardiography is represented as an inverse waveform.

Delay analysis for aggregate traffic based on martingales theory

Based on martingale theory, the authors explore a novel method to analyse the bound of delay violation probability for aggregate traffic in this study. The proposed arrival model, four-state Markov modulated multinomial process, describes the behaviour of mixed service flows, which depicts the packet arrival process of two interweaved services and highlights the arrival characteristics of the batch, bursty, and heterogeneous. The traffic of data service is modelled by the interrupted multinomial process to depict the randomness and discontinuity. Relying on martingale theory, arrival-supermartingales provide a model for integrating multiple heterogeneous services. They construct a supermartingale process of the backlog in the buffer, which enables the analysis of delay for the queuing system with multiple arrivals and a single server. Finally, the delay of aggregate traffic is featured by the parameters of its corresponding supermartingales. Based on the stopping time theory of martingales, the delay violation probability bound is derived, which matches the simulation results well and is tight.

Router Buffer Behavior Analysis for Aggregate Traffic Based on Martingale Method

The buffer behavior of routers has an important influence on the quality of service of network services. Exploring theory and methods to analyze the characteristics of the buffer is an interesting job, especially when aggregate traffic is transmitted. This letter investigates the buffer behavior of a single server queuing system with multiple arrival flows, which models the operations of a router. A martingale analytical framework is proposed, which could evaluate buffer performance for aggregate traffic arrivals with different distributions. We use the processes of arrival and service to construct martingales, and analyze the queuing system with heterogeneous arrivals in martingale domain. Applying the stopping time theory of martingale, the transient performance of queue buffer is presented. The probability of buffer overflow and the expectation of buffer stopping time are derived together in this model. The simulations are given to validate the analytical results. This investigation provides a valuable reference for router buffer design.

Bandwidth estimation for aggregate traffic under delay QoS constraint based on supermartingale theory

Bandwidth estimation for network traffic plays an important role in various communication network applications. Due to the diversity presented by the network services, exploring theory and methods to measure bandwidth of superposition of traffic flows has great practical significance. In this paper, we propose a supermartingale approach to calculate bandwidth for aggregate traffic under delay Quality of Service (QoS) constraint. The concept of supermartingale bandwidth is put forward to define the required bandwidth of traffic, which measures bandwidth more accurately. The communication network carrying mixed services is modeled as a queuing system with aggregate arrival traffic. Supermartingale method enables the complex queuing system analysis. Based on the stopping time theory of supermartingale, remarkably tight bound of delay-violation probability for aggregate traffic is derived. Subsequently, we estimate the bandwidth according to delay QoS requirement and traffic characteristics. Searching program for bandwidth estimation is designed. Simulation results show that the estimated bandwidth through supermartingale model is more accurate and smaller than effective bandwidth, especially when arrival traffic has bursty feature. Although this approach is presented for the analysis of two typical arrivals, the proposed model is applicable to general aggregate traffic.

Critical level policy for a production-inventory model with lost sales

We consider a storage process under the generalised order-up-to-level policy, based on a continuous-time Markov chain (CTMC). Specifically, the process starts at level S; whenever it drops to s, an order is sent, which is carried out after an exponential lead time. If during the lead time level S is reached, the order is cancelled, incurring some fee. This paper is written as an extension of Barron [2016. “An Fluid Inventory Model with Exponential Lead Times and Order Cancellations.” Stochastic Models 32 (2): 301–332]. While the latter paper considered a fluid inventory model with backlogging and focused on discounted analysis only, the case of lost sales was not solved. The present paper generalises the analysis to incorporate unsatisfied demand for the expected discounted costs and for the average costs per time unit. We consider four costs. There is a fixed nonzero ordering cost or a fee for each order cancellation, a purchase cost for each ordered item, a storage cost for the stock, and a penalty cost due to the unmet demand. Applying renewal theory, multi-dimensional martingales, and stopping time theory, we obtain explicit expressions of the cost components. Numerical study provides several guidelines on the optimal controls.

Introduction to Stopping Time in Stochastic Finance Theory. Part II

Summary We start proceeding with the stopping time theory in discrete time with the help of the Mizar system [1], [4]. We prove, that the expression for two stopping times k1 and k2 not always implies a stopping time (k1 + k2) (see Theorem 6 in this paper). If you want to get a stopping time, you have to cut the function e.g. (k1 + k2) ⋂ T (see [2, p. 283 Remark 6.14]). Next we introduce the stopping time in continuous time. We are focused on the intervals [0, r] where r ∈ ℝ. We prove, that for I = [0, r] or I = [0,+∞[ the set {A ⋂ I : A ∈ Borel-Sets} is a σ-algebra of I (see Definition 6 in this paper, and more general given in [3, p.12 1.8e]). The interval I can be considered as a timeline from now to some point in the future. This set is necessary to define our next lemma. We prove the existence of the σ-algebra of the τ -past, where τ is a stopping time (see Definition 11 in this paper and [6, p.187, Definition 9.19]). If τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we can prove, that the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past (see Theorem 9 in this paper and [6, p.187 Lemma 9.21]). Suppose, that you want to use Lemma 9.21 with some events, that never occur, see as a comparison the paper [5] and the example for ST(1)={+∞} in the Summary. We don’t have the element +1 in our above-mentioned time intervals [0, r[ and [0,+1[. This is only possible if we construct a new σ-algebra on ℝ {−∞,+∞}. This construction is similar to the Borel-Sets and we call this σ-algebra extended Borel sets (see Definition 13 in this paper and [3, p. 21]). It can be proved, that {+∞} is an Element of extended Borel sets (see Theorem 21 in this paper). Now we use the interval [0,+∞] as a basis. We construct a σ-algebra on [0,+∞] similar to the book ([3, p. 12 18e]), see Definition 18 in this paper, and call it extended Borel subsets. We prove for stopping times with this given σ-algebra, that for τ1 and τ2 are stopping times with τ1 is smaller or equal than τ2 we have the σ-algebra of the τ1-past is a subset of the σ-algebra of the τ2-past, see Theorem 25 in this paper. It is obvious, that {+∞} 2 extended Borel subsets. In general, Lemma 9.21 is important for the proof of the Optional Sampling Theorem, see 10.11 Proof of (i) in [6, p. 203].

Quickest Sequential Multiband Spectrum Sensing with Mixed Observations.

Spectrum sensing is a key technology enabling the cognitive radio system. In this paper, the problem of how to quickly and accurately find an unoccupied channel from a large amount of potential channels is considered. The cognitive radio system under consideration is equipped with a narrow band sensor, hence it can only sense those potential channels in a sequential manner. In this scenario, we propose a novel two-stage mixed-observation sensing strategy. In the first stage, which is named as scanning stage, the sensor observes a linear combination of the signals from a pair of channels. The purpose of the scanning stage is to quickly identify a pair of channels such that at least one of them is highly likely to be unoccupied. In the second stage, which is called refinement stage, the sensor only observers the signal from one of those two channels identified from the first stage, and selects one of them as the unoccupied channel. The problem under this setup is an ordered two concatenated Markov stopping time problem. The optimal solution is solved using the tools from the multiple stopping time theory. It turns out that the optimal solution has a rather complex structure, hence a low complexity algorithm is proposed to facilitate the implementation. In the proposed low complexity algorithm, the cumulative sum test is adopted in the scanning stage and the sequential probability ratio test is adopted in the refinement stage. The performance of this low complexity algorithm is analyzed when the presence of unoccupied channels is rare. Numerical simulation results show that the proposed sensing strategy can significantly reduce the sensing time when the majority of potential channels are occupied.

A Note on the Kou’s Continuity Correction Formula

This article introduces a hyper-exponential jump diffusion process based on the continuity correction for discrete barrier options under the standard B-S model, using measure transformation and stopping time theory to prove the correction, thus broadening the conditions of the continuity correction of Kou.

Pricing Corporate Debt with Finite Maturity and Chapter 11 Proceedings

We are concerned with the valuation of the finite horizon corporate debt under Chapter 11 of the U.S. bankruptcy code. Broadie and Kaya (2007) set up a binomial model by assuming that determining the bankruptcy boundary is restricted to be at initial time. We distinguish the so-called bankruptcy declaring boundary from the bankruptcy boundary, which enables us to develop a continuous-time pricing model by optimal stopping time theory. The resulting free boundary corresponds to the optimal bankruptcy declaring strategy. Since the hybrid feature of finite horizon and optimal stopping time makes analytical pricing formulas unavailable, we propose a numerical procedure based on the associated partial differential equation formulations. Our numerical results reveal that the optimal bankruptcy declaring boundary, usually increasing with time, becomes subtle as time goes to maturity. Also, our model yields higher equity values than Broadie and Kaya's model because the strategy implied by our model is indeed optimal.

Research on Stock Analysis Based on Stochastic Process

In stock market, the stock prices directly reflects market condition, therefore, the research on stock price process is one of the research contents of mathematical finance. In this paper by using the election model of statistical physics model to study the stock price fluctuation . This paper first applying stochastic process theory to establish election model, then the election model and stopping time theory are applied to establish stock profit process, we get the stock price process.

Joint Channel Probing and Proportional Fair Scheduling in Wireless Networks

The design of a scheduling scheme is crucial for the efficiency and user-fairness of wireless networks. Assuming that the channel quality information (CQI) of all users is available to a central controller, a simple scheme which maximizes the sum-log utility function has been shown to guarantee proportional fairness. This work studies a more general problem which takes both the CQI acquisition and the user scheduling into account. First, in case the statistics of the channel quality is available to the controller, a joint channel probing and proportional fair scheduling scheme is developed based on the optimal stopping time theory. The convergence and optimality of the scheme is proved. Next, the problem is further studied in the case where the channel statistics are not available to the controller, and a joint learning, probing and scheduling scheme is designed by solving a generalized bandit problem. Furthermore, it is shown that the multiuser diversity gain does not always increase as the number of users increases. Numerical results demonstrate that the proposed scheduling schemes can provide significant gain over existing schemes.

Analytical Valuation of Contingent Claims by Stochastic Interacting Systems for Stock Market

In the present paper, by applying the theory of stochastic processes and interacting particle systems and models, including stopping time theory and stochastic voter model, we model a financial stock price model that contains two types of investors, and we use this financial model to describe the behavior and fluctuations of a stock price process in a stock market. In the financial model, besides the professional investors, we also consider the general investors or nonprofessional investors, where the stopping time and the voter model are applied to model and study the statistical properties of investment of the nonprofessional investors. By using the stochastic methods of statistical analysis, we show that the probability distribution of the normalized random price process for this financial model converges to the corresponding distribution of the Black-Scholes model. Further, we discuss the valuation and hedging of European contingent claims for this price process model.