Stationary Probability Vector Research Articles

An investigation into the abnormal dynamic connection mechanism of generalized anxiety disorders based on non-homogeneous Markov models

BackgroundThe dynamic and hierarchical nature of the functional brain network. The neural dynamical systems tend to converge to multiple attractors (stable fixed points or dynamical states) in long run. Little is known about how the changes in this brain dynamic “long-term” behavior of the connectivity flow of brain network in generalized anxiety disorder (GAD). MethodsThis study recruited 92 patients with GAD and 77 healthy controls (HC). We applied a reachable probability approach combining a Non-homogeneous Markov model with transition probability to quantify all possible connectivity flows and the hierarchical structure of brain functional systems at the dynamic level and the stationary probability vector (10-step transition probabilities) to describe the steady state of the system in the long run. A random forest algorithm was conducted to predict the severity of anxiety. ResultsThe dynamic functional patterns in distributed brain networks had larger possibility to converge in bilateral thalamus, posterior cingulate cortex (PCC), right superior occipital gyrus (SOG) and smaller possibility to converge in bilateral superior temporal gyrus (STG) and right parahippocampal gyrus (PHG) in patients with GAD compared to HC. The abnormal transition probability pattern could predict anxiety severity in patients with GAD. LimitationsSmall samples and subjects taking medications may have influenced our results. Future studies are expected to rule out the potential confounding effects. ConclusionOur results have revealed abnormal dynamic neural communication and integration in emotion regulation in patients with GAD, which give new insights to understand the dynamics of brain function of patients with GAD.

Performance evaluation of non-repairable cyclic phased-mission systems using evidential reasoning rule and multi-valued decision diagrams

Phased-mission systems (PMS) often perform different missions that have multiple consecutive, non-overlapping phases. Many recent studies have focused on the reliability analysis for PMS. However, not much attention has been paid to the PMS performance evaluation and the phase cycle characteristic of PMS. In this paper, a performance evaluation method of non-repairable cyclic phased-mission systems (CPMS) is proposed. Considering the uncertainty caused by the change of phase duration and component performance degradation, the average duration of each phase and the fuzzy state probability vector of each component are obtained by the evidential reasoning rule (ER rule). Then, the multi-valued decision diagrams (MDD) manipulation rules suitable for cyclic phase conditions are proposed, and the component state composition path generation method based on cycle-end component states is developed. On the above basis, MDD models for CPMS (CPMS-MDD) in different states are established, and the system performance is evaluated by the continuous time Markov chains (CTMC). A phase duration adjustment strategy is further proposed based on the evaluation results. The effectiveness and correctness of the proposed method are verified by case studies on performance evaluation of satellite turntable system. Comparative study shows the concise and efficient of the proposed method.

An Optimization of Home Delivery Services in a Stochastic Modeling with Self and Compulsory Vacation Interruption

This study presents and discusses the home delivery services in stochastic queuing-inventory modeling (SQIM). This system consists of two servers: one server manages the inventory sales processes, and the other server provides home delivery services at the doorstep of customers. Based on the Bernoulli schedule, a customer served by the first server may opt for a home delivery service. If any customer chooses the home delivery option, he hands over the purchased item for home delivery and leaves the system immediately. Otherwise, he carries the purchased item and leaves the system. When the delivery server returns to the system after the last home delivery service and finds that there are no items available for delivery, he goes on vacation. Such a vacation of a delivery server is to be interrupted compulsorily or voluntarily, according to the prefixed threshold level. The replenishment process is executed due to the (s,Q) reordering policy. The unique solution of the stationary probability vector to the finite generator matrix is found using recursive substitution and the normalizing condition. The necessary and sufficient system performance measures and the expected total cost of the system are computed. The optimal expected total cost is obtained numerically for all the parameters and shown graphically. The influence of parameters on the expected number of items that need to be delivered, the probability that the delivery server is busy, and the expected rate at which the delivery server’s self and compulsory vacation interruptions are also discussed.

Markov modeling and performance analysis of infectious diseases with asymptomatic patients.

After over three years of COVID-19, it has become clear that infectious diseases are difficult to eradicate, and humans remain vulnerable under their influence in a long period. The presence of presymptomatic and asymptomatic patients is a significant obstacle to preventing and eliminating infectious diseases. However, the long-term transmission of infectious diseases involving asymptomatic patients still remains unclear. To address this issue, this paper develops a novel Markov process for infectious diseases with asymptomatic patients by means of a continuous-time level-dependent quasi-birth-and-death (QBD) process. The model accurately captures the transmission of infectious diseases by specifying several key parameters (or factors). To analyze the role of asymptomatic and symptomatic patients in the infectious disease transmission process, a simple sufficient condition for the stability of the Markov process of infectious diseases is derived using the mean drift technique. Then, the stationary probability vector of the QBD process is obtained by using RG-factorizations. A method of using the stationary probability vector is provided to obtain important performance measures of the model. Finally, some numerical experiments are presented to demonstrate the model's feasibility through analyzing COVID-19 as an example. The impact of key parameters on the system performance evaluation and the infectious disease transmission process are analyzed. The methodology and results of this paper can provide theoretical and technical support for the scientific control of the long-term transmission of infectious diseases, and we believe that they can serve as a foundation for developing more general models of infectious disease transmission.

Existence and Uniqueness of Stationary Probability Vector for Stochastic Matrices using Farkas’ Lemma

The purpose of this note is to provide a simple proof of existence of stationary probability vectors (fixed points) for stochastic matrices using Farkas’ lemma. This result as well as the uniqueness of stationary probability vectors also holds for a certain subclass of quasi-stochastic matrices.

Graph-based algorithms for phase-type distributions

Phase-type distributions model the time until absorption in continuous or discrete-time Markov chains on a finite state space. The multivariate phase-type distributions have diverse and important applications by modeling rewards accumulated at visited states. However, even moderately sized state spaces make the traditional matrix-based equations computationally infeasible. State spaces of phase-type distributions are often large but sparse, with only a few transitions from a state. This sparseness makes a graph-based representation of the phase-type distribution more natural and efficient than the traditional matrix-based representation. In this paper, we develop graph-based algorithms for analyzing phase-type distributions. In addition to algorithms for state space construction, reward transformation, and moments calculation, we give algorithms for the marginal distribution functions of multivariate phase-type distributions and for the state probability vector of the underlying Markov chains of both time-homogeneous and time-inhomogeneous phase-type distributions. The algorithms are available as a numerically stable and memory-efficient open source software package written in C named ptdalgorithms. This library exposes all methods in the programming languages C and R. We compare the running time of ptdalgorithms to the fastest tools using a traditional matrix-based formulation. This comparison includes the computation of the probability distribution, which is usually computed by exponentiation of the sub-intensity or sub-transition matrix. We also compare time spent calculating the moments of (multivariate) phase-type distributions usually defined by inversion of the same matrices. The numerical results of our graph-based and traditional matrix-based methods are identical, and our graph-based algorithms are often orders of magnitudes faster. Finally, we demonstrate with a classic problem from population genetics how ptdalgorithms serves as a much faster, simpler, and completely general modeling alternative.

Stochastic performance modeling for practical byzantine fault tolerance consensus in the blockchain

The practical Byzantine fault tolerant (PBFT) consensus mechanism is one of the most basic consensus algorithms (or protocols) in blockchain technologies. Thus its performance evaluation is an interesting and challenging topic due to the higher complexity of its consensus work in a peer-to-peer network. This study describes a simple stochastic performance model of the PBFT consensus mechanism. This model is refined not only as a queuing system with complicated service times but also as a level-independent quasi-birth-and-death (QBD) process. With regard to the level-independent QBD process, we apply the matrix-geometric solution to obtain the necessary and sufficient condition under which the PBFT consensus system is stable and then numerically compute the stationary probability vector of the QBD process. Thus, we provide four useful performance measures for the PBFT consensus mechanism, and we can numerically calculate these performance measures. Finally, we use numerical examples to verify the validity of our theoretical results and demonstrate how the four performance measures are influenced by certain key parameters of the PBFT consensus. Considering theory of multi-dimensional Markov processes, we are optimistic that the methodology and results presented in this study are applicable to a wide range of PBFT consensus mechanism and even other types of consensus mechanisms.

Analysis of Stochastic State-Dependent Arrivals in a Queueing-Inventory System with Multiple Server Vacation and Retrial Facility

This article analyses a four-dimensional stochastic queueing-inventory system with multiple server vacations and a state-dependent arrival process. The server can start multiple vacations at a random time only when there is no customer in the waiting hall and the inventory level is zero. The arrival flow of customers in the system is state-dependent. Whenever the arriving customer finds that the waiting hall is full, they enter into the infinite orbit and they retry to enter the waiting hall. If there is at least one space in the waiting hall, the orbital customer enters the waiting hall. When the server is on vacation, the primary (retrial) customer enters the system with a rate of λ1(θ1). If the server is not on vacation, the primary (retrial) arrival occurs with a rate of λ2(θ2). Each arrival rate follows an independent Poisson distribution. The service is provided to customers one by one in a positive time with the rate of μ, which follows exponential distribution. When the inventory level drops to a fixed s, reorder of Q items is triggered immediately under (s,Q) ordering policy. The stability of the system has been analysed, and using the Neuts matrix geometric approach, the stationary probability vectors have been obtained. Moreover, various system performance measures are derived. The expected total cost analysis explores and verifies the characteristics of the assumed parameters of this model. The average waiting time of a customer in the waiting hall and orbit are investigated using all the parameters. The monotonicity of the parameters is verified with its characteristics by the numerical simulation. The discussion about the fraction time server being on vacation suggests that as the server’s vacation duration reduces, its fraction time also reduces. The mean number of customers in the waiting hall and orbit is reduced whenever the average service time per customer and average replenishment time are reduced.

A method to minimize the maintenance costs of a broadband access network based on optimal steady-state probability mixture distribution of proactive and reactive maintenance

• Minimizing the maintenance cost of access lines in a broadband multimedia network. • A large set of line failure data collected over a period of 10 years is considered. • Line failure occurrences can be described by a probability mixture distribution. • Optimal maintenance; reactive and proactive approaches included in a balanced way. • Optimal share of proactive maintenance calculated by a discrete-time Markov chain. Paper presents an analysis of the long-term maintenance of access lines in a real broadband network. The dynamics of the maintenance process were recorded by using fault management systems over a ten-year period. The data collected belong to three groups: data that represent the stochastic nature of the occurrence of failures on lines, data related to the type and quality of detector used for remote detection of faults and degradation based on Quality of Service (QoS) measures, and data on user habits that connect QoS and Quality of Experience (QoE). We devised our model proposal that enables the calculation of the optimal amount of proactive maintenance, taking into consideration the characteristics of the network, characteristics of the detector system and QoE. The intention was to facilitate decision making about optimal ratio of reactive and proactive maintenance, which reduces costs. Ratios are calculated as a stationary probability vector of a Markov chain, which is finally used to calculate the optimal maintenance cost. Maintenance cost reductions can vary, depending on the different variable settings of the model, but in the case covered in the paper, with 8.33% of proactive repairs introduced, maintenance costs were reduced by 3.71%.

Analysis of Stochastic M/M/c/N Inventory System with Queue-Dependent Server Activation, Multi-Threshold Stages and Optional Retrial Facility

The purpose of this article is to examine the server activation policy (SAP) in a multi-server queuing-inventory system (MQIS). The queue has a total of c number of multi-threshold stages as well as c-homogeneous servers. The activation of each server begins one by one if there is an adequate queue length and inventory in the system; otherwise, they remain idle. The server deactivation process continues until the queue length exceeds the manageable level (predetermined stages) or there is insufficient stock. In addition, when we assume the length of the two successive threshold levels is one, the server activation policy model becomes a regular multi-server model. The Neuts matrix geometric approach is used to discuss the stability condition, stationary probability vector. The Laplace–Stieltjes transform (LST) is used to analyse the waiting time distributions of the queue and orbital customers. Additionally, significant system performance metrics and sensitivity analysis are used to investigate the effects of various parameters and cost values. In the comparative result between the server activation model (SAM) and without the server activation model (WSAM) on the expected total cost, we obtain the minimised cost in the SAM. Moreover, the results are obtained by assuming that the length of the intervals between the two successive threshold levels is to be taken into account as the non-uniform length. The expected inventory level, reorder rate, and waiting time of a customer in the waiting hall and orbit were explored numerically by the parameter analysis.

Optimal Window-Symbolic Time Series Analysis for Pattern Classification and Anomaly Detection

This article proposes the optimal window-symbolic time series analysis (OW-STSA) methodology to optimize parameters of feature extraction and pattern classification in industrial processes. The underlying theory is built upon minimization of an empirical risk function to discriminate between nominal and anomalous operations of the physical process under consideration. In particular, the proposed methodology produces: optimized windows of the time series used for pattern classification and anomaly detection, and optimized identification of feature extractors and classifier parameters. The algorithm is realized by segmenting a given time series into windows of equal size. Then, the stationary state probability vector is computed for each window in the sense of OW-STSA for anomaly prediction with locally optimal accuracy of detection performance. The proposed methodology has been experimentally validated in laboratory environment with different classifiers for two distinct industrial processes. The first experiment addresses detection of fatigue failure in polycrystalline alloy structures using time series of ultrasonic signals. The second experiment investigates detection of thermoacoustic instability in an emulated combustion system using time series of pressure-wave signals. In both experiments, the proposed OW-STSA methodology yielded excellent detection performance of anomalous behavior with multiple classification techniques.

Point-kinetics neutron noise modeling and analysis via probabilistic finite state automata

This paper addresses two issues for safety & performance analysis of nuclear reactors; these issues are: (i) Modeling of the neutron noise process, which is constructed as stochastic differential equations by augmenting deterministic dynamic models of point kinetics, thermal hydraulics, and mechanical vibrations in the reactor core, and (ii) Time-series analysis of nuclear reactor internals for early-stage anomaly detection by making use of the neutron noise model. The model is used to generate an ensemble of time series of neutron noise under both normal and anomalous operating conditions, where an anomaly in the postulated probability distribution of neutron noise is caused by abnormal vibration of the fuel assembly. A symbolic time series analysis (STSA) tool, called probabilisitc finite state automata (PFSA), has been selected to detect the anomalies, if any. The PFSA method symbolizes a time-series and then encodes the resulting symbol string into a state transition probability matrix, which is both stochastic and ergodic by construction. The discrimination function for anomaly detection is built upon the residual error of the principal (left) eigenvector of the state transition probability matrix, which is also the state probability vector of the PFSA. The accuracy of the proposed analysis and its capability for online anomaly detection is demonstrated by simulation on a low-order dynamic model of a 2,500 MWt PWR of the Three-Mile-Island type.

Absorption kinetics of vacancies by cavities in aluminum: Numerical characterization of sink strengths and first-passage statistics through Krylov subspace projection and eigenvalue deflation

Modeling the microstructural evolution of metal and alloys, specifically under irradiation, is essential to predict the aging properties of materials. Many models are based on a transition rate matrix describing the jump frequencies of defects and involve a master equation governing the time evolution of a state probability vector. Here, we present non-stochastic numerical techniques to characterize the motion of individual defects migrating over long distances prior to recombining or being absorbed by another defect, resorting to the theory of absorbing Markov chains. These important events are fully determined by their first-passage time distribution to distant locations, no-passage distribution, and walker fluxes to the sinks. We show that these functions can be efficiently computed using a method combining Krylov subspace projection and eigenvalue deflation. For a model system describing the absorption of a vacancy by a cavity in aluminum, the use of a small Krylov subspace deflated by the unique eigenmode corresponding to the quasi-stationary distribution is sufficient to capture the kinetics of the defect absorption faithfully. This method can be used in kinetic Monte Carlo simulations to perform stochastic non-local moves or in cluster dynamics simulations to compute sink strengths.

Markovian queueing model with single working vacation and catastrophic

A markovian queueing system with a single working vacation and different service rates have been considered. Service times during the vacation period, service times during the service period as well as vacation times are all exponentially distributed. If the queue length is increasing, the server converts the service parameter from µv to µb, and a normal working time begins. According to the Poisson process, a catastrophe event with parameter α occurs only if there is a customer in the system. This type of queue model has been analysed using Matrix Geometric Method (MGM) to find steady state probability vectors. Using it some performance measure is also determined.

Steady State Analysis of Impulse Customers and Cancellation Policy in Queueing-Inventory System

This article discusses the queueing-inventory model with a cancellation policy and two classes of customers. The two classes of customers are named ordinary and impulse customers. A customer who does not plan to buy the product when entering the system is called an impulse customer. Suppose the customer enters into the system to buy the product with a plan is called ordinary customer. The system consists of a pool of finite waiting areas of size N and maximum S items in the inventory. The ordinary customer can move to the pooled place if they find that the inventory is empty under the Bernoulli schedule. In such a situation, impulse customers are not allowed to enter into the pooled place. Additionally, the pooled customers buy the product whenever they find positive inventory. If the inventory level falls to s, the replenishment of Q items is to be replaced immediately under the (s, Q) ordering principle. Both arrival streams occur according to the independent Markovian arrival process (MAP), and lead time follows an exponential distribution. In addition, the system allows the cancellation of the purchased item only when there exist fewer than S items in the inventory. Here, the time between two successive cancellations of the purchased item is assumed to be exponentially distributed. The Gaver algorithm is used to obtain the stationary probability vector of the system in the steady-state. Further, the necessary numerical interpretations are investigated to enhance the proposed model.

A New Theoretical Framework of Pyramid Markov Processes for Blockchain Selfish Mining

In this paper, we provide a new theoretical framework of pyramid Markov processes to solve some open and fundamental problems of blockchain selfish mining under a rigorous mathematical setting. We first describe a more general model of blockchain selfish mining with both a two-block leading competitive criterion and a new economic incentive mechanism. Then we establish a pyramid Markov process and show that it is irreducible and positive recurrent, and its stationary probability vector is matrix-geometric with an explicitly representable rate matrix. Also, we use the stationary probability vector to study the influence of orphan blocks on the waste of computing resource. Next, we set up a pyramid Markov reward process to investigate the long-run average mining profits of the honest and dishonest mining pools, respectively. As a by-product, we build one-dimensional Markov reward processes and provide some new interesting interpretation on the Markov chain and the revenue analysis reported in the seminal work by Eyal and Sirer (2014). Note that the pyramid Markov (reward) processes can open up a new avenue in the study of blockchain selfish mining. Thus we hope that the methodology and results developed in this paper shed light on the blockchain selfish mining such that a series of promising research can be developed potentially.

Early detection of lean blowout in a combustor using symbolic analysis of colour images

Recently, lean combustion has been accepted as one of the preferable modes to run both small and large scale energy sectors as it reduces the level of NOx (oxides of nitrogen) emissions. However, ultra-lean combustion is more susceptible to lean blowout (LBO) which significantly reduces engine life and operating reliability. Therefore, early detection of LBO is important so that operator can get sufficient lead time to initiate precautionary actions. The current investigation proposes a novel technique based on the conversion of flame colour matrix into symbolic strings. The generation of symbolic strings is done based on the partitioning of the hue space obtained from transformation of RGB (Red-Green-Blue) to HSV (Hue-Saturation-Value) using uniform partitioning method. The partitions are determined at the reference state and kept fixed for other equivalence ratios (anomalous or non-reference states). Thus, the hue value at each pixel of the image is assigned a symbol depending on its belonging to a particular partition. Thereafter, an array of symbolic state probability vector at reference combustion state is determined based on the probability of pixel hue value being in a particular symbolic state or symbol (number of symbolic states is considered as 8). Also, for other anomalous or non-reference combustion states, symbolic state probability vectors are estimated using the partition formed at the reference state. The difference in symbolic state probability vectors between reference and anomalous combustion states gives the anomaly measure, which is used here to detect the flame blowout. The early increasing trend of anomaly towards the blowout for both premixed and partially premixed flames indicate that the measure can be helpful to detect the approach towards LBO. The estimation of anomaly is shown for using the flame images of different quantities. The random selection of five images from a flame video shows to be very effective in capturing the trend of anomaly which also ensures low computational effort. Thus, the tool based on symbolic colour image analysis (SCIA) may also be helpful in the mitigation of LBO.

Analysis of Interconnected Arrivals on Queueing-Inventory System with Two Multi-Server Service Channels and One Retrial Facility

Present-day queuing inventory systems (QIS) do not utilize two multi-server service channels. We proposed two multi-server service channels referred to as T1S (Type 1 n-identical multi-server) and T2S (Type 2 m-identical multi-server). It includes an optional interconnected service connection between T1S and T2S, which has a finite queue of size N. An arriving customer either uses the inventory (basic service or main service) for their demand, whom we call T1, or simply uses the service only, whom we call T2. Customer T1 will utilize the server T1S, while customer T2 will utilize the server T2S, and T1 can also get the second optional service after completing their main service. If there is a free server with a positive inventory, there is a chance that T1 customers may go to an infinite orbit whenever they find that either all the servers are busy or no sufficient stock. The orbital customer can request for T1S service under the classical retrial policy. Q(=S−s) items are replaced into the inventory whenever it falls into the reorder level s such that the inequality always holds n<s. We use the standard (s,Q) ordering policy to replace items into the inventory. By varying S and s, we investigate to find the optimal cost value using stationary probability vector ϕ. We used the Neuts Matrix geometric approach to derive the stability condition and steady-state analysis with R-matrix to find ϕ. Then, we perform the waiting time analysis for both T1 and T2 customers using Laplace transform technique. Further, we computed the necessary system characteristics and presented sufficient numerical results.

A MAP/PH/1 Perishable Inventory System with Dependent Retrial Loss

This paper considers a continuous review perishable inventory system with an infinite orbit. The replenishment of items in the inventory is governed by (s, Q) policy. The Markovian Arrival Process characterizes the arrival pattern of primary customers and the service time follows phase-type distribution. Inventoried items are subject to decay with some preassigned linear rate. The items are purchased from an outside source and lead time is exponentially distributed. An unsatisfied primary customer may enter an orbit or exit the system with a predetermined probability. The retrial customers, those who entered into the orbit, try to access the idle server with a linear rate. An unsatisfied retrial customer may enter or exit the orbit with prefixed probability based on the inventory level. Stationary probability vectors are obtained using the Matrix Analytic method. Numerical optimization of the profit function is carried out for different parameter values.

Stationary Probability Vectors of Higher-Order Two-Dimensional Symmetric Transition Probability Tensors

In this paper, we investigate stationary probability vectors of higher-order two-dimensional symmetric transition probability tensors. We show that there are two special symmetric transition probability tensors of order [Formula: see text] dimension 2, which have and only have two stationary probability vectors; and any other symmetric transition probability tensor of order [Formula: see text] dimension 2 has a unique stationary probability vector. As a byproduct, we obtain that any symmetric transition probability tensor of order [Formula: see text] dimension 2 has a unique positive stationary probability vector, and that any symmetric irreducible transition probability tensor of order [Formula: see text] dimension 2 has a unique stationary probability vector.