Sojourn Time Distribution Research Articles

Embedding and elimination for performance analysis in stochastic process algebra dtsdPBC

ABSTRACT dtsdPBC extends the well-known algebra of parallel processes, Petri box calculus (PBC), by incorporating discrete time stochastic and deterministic delays. To analyze performance in this extended calculus, the underlying semi-Markov chains, and the related (complete) and reduced discrete time Markov chains of the process expressions are built. The semi-Markov chains are extracted using the embedding method, which constructs the embedded discrete time Markov chains and calculates the sojourn time distributions in the states. The reductions of the discrete time Markov chains are obtained through the elimination method, which removes the vanishing states (those with zero sojourn times) and recalculates the transition probabilities among the tangible states (those with positive sojourn times). We prove that the reduced semi-Markov chain coincides with the reduced discrete time Markov chain, by demonstrating that an additional embedding into the reduced semi-Markov chain is needed for the reduced embedded discrete time Markov chain to match the embedded reduced discrete time Markov chain, and by comparing the respective sojourn times.

Evaluation of readiness of the technical system using the semi-Markov model with selected sojourn time distributions

Mechanization of forestry work is crucial in forest management, and the specific nature of the tasks performed requires reliable machines with a high level of technical readiness. Therefore, models describing the exploitation process of multi-operational machines used to obtain wood raw materials, and especially assessing the level of their technical readiness, are extremely important. For multi-tasking objects performing random activities in given time intervals, calculating readiness measures is a complex issue. Forecasting subsequent operational states and their durations allows to predict the behavior of technical objects and schedule work. In addition to identifying the sequence of states, it is also important to identify the sojourn time in these states. This article presents a method for identifying the semi-Markov process and then assessing the technical readiness of a Harvester machine. This allowed for conclusions regarding the timely completion of assigned tasks, and also made it possible to adjust the activities carried out to the requirements of forest management.

A study on N-policy BMAP/G/1 queueing system

This study presents a simple approach for analyzing the BMAP/G/1 queueing system under N policy. In this policy, the server goes idle at the end of a busy period and the queue length is checked at every arrival moment. The idle server starts serving customers when the queue length reaches or above a predetermined number, namely N, and keeps doing so until the system is completely empty. We first use a simple sequential substitution approach to derive the system length distribution at departure epoch. A comparative study is also conducted to highlight the benefits and strength of our straightforward approach to that of the RG-factorization technique and the roots finding method. We extract the distribution of system length at a random time point by utilizing the remaining service time of a customer who is currently being served as the supplementary variable. The probability density function of the sojourn time distribution for an arbitrary customer of an incoming batch is also computed. We propose an expected linear cost function to estimate the optimal value of N at minimum cost. The validity of our analytic technique has been shown through a variety of numerical examples involving different service time distributions.

Degradation modeling and remaining life prediction of multi-state long-life systems under random environmental influences

Abstract For large-scale systems such as bridges, which have long operating lifetimes, the operating states are usually categorized into multiple levels, and they are also subjected to various random environmental influences during operation. However, due to the significant granularity in the categorization of system states, it is difficult to assess the system state transitions influenced by random environmental factors, which compromises the accuracy of remaining life predictions. In this study, we focus on long-life systems with multiple states and investigate the degradation modeling and remaining life prediction considering the impact of random environmental factors. The system degradation process, based on the semi-Markov process and multi-state modeling, was decomposed into states using the sub-exponential approximation method. A state transition probability model considering exponential environmental influences was constructed. Furthermore, based on the developed model for calculating the distribution of sojourn times in multiple states, a reliability and remaining life prediction model for the system was derived. By taking the bridge deck as a case study, the verification and analysis of remaining life prediction for the bridge deck were conducted under the influences of average daily traffic volume and bridge age. The results indicate that both the average daily traffic volume and bridge age have a significant impact on the degradation of the bridge deck. The relative error of the predicted results considering the above effects falls within the range of 1.77%–12.18%.

Almost sure synchronization of stochastic multi-links semi-Markov jump systems via aperiodically intermittent control

This paper concentrates on the almost sure synchronization for a class of stochastic multi-links coupled semi-Markov jump systems through aperiodically intermittent control. For these stochastic switching systems, almost sure synchronization is investigated by employing mode-dependent multiple Lyapunov-like function method and stochastic analysis. Notably, mode-dependent multiple Lyapunov-like function method is designed to provide more flexibility. In addition, when considering the sojourn time distribution of the semi-Markov jump, dependence on both the current state and the next state can effectively reduce unnecessary restrictions compared with previous assumption of the sojourn time distribution function. Ultimately, the Chua’s circuit along with numerical simulations are provided to validate the effectiveness of the theoretical results.

Estimation of age of onset and progression of breast cancer by absolute risk dependent on polygenic risk score and other risk factors.

Genetic, lifestyle, reproductive, and anthropometric factors are associated with the risk of developing breast cancer. However, it is not yet known whether polygenic risk score (PRS) and absolute risk based on a combination of risk factors are associated with the risk of progression of breast cancer. This study aims to estimate the distribution of sojourn time (pre-clinical screen-detectable period) and mammographic sensitivity by absolute breast cancer risk derived from polygenic profile and the other risk factors. The authors used data from a population-based case-control study. Six categories of 10-year absolute risk based on different combinations of risk factors were derived using the Breast and Ovarian Analysis of Disease Incidence and Carrier Estimation Algorithm. Women were classified into low, medium, and high-risk groups. The authors constructed a continuous-time multistate model. To calculate the sojourn time, they simulated the trajectories of subjects through the disease states. There was little difference in sojourn time with a large overlap in the 95% confidence interval (CI) between the risk groups across the six risk categories and PRS studied. However, the age of entry into the screen-detectable state varied by risk category, with the mean age of entry of 53.4years (95% CI, 52.2-54.1) and 57.0years (95% CI, 55.1-57.7) in the high-risk and low-risk women, respectively. In risk-stratified breast screening, the age at the start of screening, but not necessarily the frequency of screening, should be tailored to a woman's risk level. The optimal risk-stratified screening strategy that would improve the benefit-to-harm balance and the cost-effectiveness of the screening programs needs to be studied.

Hypothesis testing for Panels of Semi-Markov Processes with parametric sojourn time distributions

This work deals with the asymptotic properties of maximum likelihood estimators for semi-Markov processes with parametric sojourn time distributions. It is motivated by the comparison, via a two-sample test procedure, of the distribution of two panels of qualitative trajectories modeled by semi-Markov processes and observed over a random number of transitions. Considering first one panel of growing size, we derive, under classical conditions, the convergence in probability of the estimators of the transition probabilities and the parameters of the sojourn time distributions as well as their asymptotic normality. We then consider panels of semi-Markov processes drawn from two different populations and study two-sample tests based on likelihood ratio. We also introduce a two-sample Wald type test. The finite sample performances of the proposed two-sample tests are evaluated with a brief simulation study.

Effect of sojourn time distributions on the early dynamics of COVID-19 outbreak.

Compartmental models are commonly used in practice to investigate the dynamical response of infectious diseases such as the COVID-19 outbreak. Such models generally assume exponentially distributed latency and infectiousness periods. However, the exponential distribution assumption fails when the sojourn times are expected to distribute around their means. This study aims to derive a novel S (Susceptible)-E (Exposed)-P (Presymptomatic)-A (Asymptomatic)-D (Symptomatic)-C (Reported) model with arbitrarily distributed latency, presymptomatic infectiousness, asymptomatic infectiousness, and symptomatic infectiousness periods. The SEPADC model is represented by nonlinear Volterra integral equations that generalize ordinary differential equation-based models. Our primary aim is the derivation of a general relation between intrinsic growth rate r and basic reproduction number with the help of the well-known Lotka-Euler equation. The resulting equation includes separate roles of various stages of the infection and their sojourn time distributions. We show that estimates are considerably affected by the choice of the sojourn time distributions for relatively higher values of r. The well-known exponential distribution assumption has led to the underestimation of values for most of the countries. Exponential and delta-distributed sojourn times have been shown to yield lower and upper bounds of the values depending on the r values. In quantitative experiments, values of 152 countries around the world were estimated through our novel formulae utilizing the parameter values and sojourn time distributions of the COVID-19 pandemic. The global convergence, , has been estimated through our novel formulation. Additionally, we have shown that increasing the shape parameter of the Erlang distributed sojourn times increases the skewness of the epidemic curves in entire dynamics.

The Tiny-Tasks Granularity Trade-Off: Balancing Overhead Versus Performance in Parallel Systems

Models of parallel processing systems typically assume that one has <inline-formula><tex-math notation="LaTeX">$l$</tex-math></inline-formula> workers and jobs are split into an equal number of <inline-formula><tex-math notation="LaTeX">$k=l$</tex-math></inline-formula> tasks. Splitting jobs into <inline-formula><tex-math notation="LaTeX">$k &gt; l$</tex-math></inline-formula> smaller tasks, i.e. using “tiny tasks”, can yield performance and stability improvements because it reduces the variance in the amount of work assigned to each worker, but as <inline-formula><tex-math notation="LaTeX">$k$</tex-math></inline-formula> increases, the overhead involved in scheduling and managing the tasks begins to overtake the performance benefit. We perform extensive experiments on the effects of task granularity on an Apache Spark cluster, and based on these, develop a four-parameter model for task and job overhead that, in simulation, produces sojourn time distributions that match those of the real system. We also present analytical results which illustrate how using tiny tasks improves the stability region of split-merge systems, and analytical bounds on the sojourn and waiting time distributions of both split-merge and single-queue fork-join systems with tiny tasks. Finally we combine the overhead model with the analytical models to produce an analytical approximation to the sojourn and waiting time distributions of systems with tiny tasks which include overhead. We also perform analogous tiny-tasks experiments on a hybrid multi-processor shared memory system based on MPI and OpenMP which has no load-balancing between nodes. Though no longer strict analytical bounds, our analytical approximations with overhead match both the Spark and MPI/OpenMP experimental results very well.

SMC for Discrete-Time Networked Semi-Markovian Switching Systems With Random DoS Attacks and Applications

This article concentrates on the discrete-time sliding mode control (DTSMC) problem for uncertain networked semi-Markovian switching systems (S-MSSs) under random denial-of-service (DoS) attacks. The semi-Markovian kernel (SMK) approach is utilized such that the switching among different modes is mutually governed by the transition probability and the sojourn-time distribution function. Considering randomly occurring DoS attacks, a sliding mode function related to the attack probability is constructed to analyze the impact of malicious attacks. Then, sufficient conditions under the equivalent DTSMC law are derived in view of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma $ </tex-math></inline-formula> -error mean square stability criterion. Furthermore, the synthesis problem of the proposed DTSMC law ensures that the resulting closed-loop system dynamics can be driven onto the prespecified sliding region within a limited time. Finally, an electronic throttle control system is shown to validate the proposed algorithm.

Evaluations of heterogeneous epidemic models with exponential and non-exponential distributions for latent period: the Case of COVID-19.

Most of heterogeneous epidemic models assume exponentially distributed sojourn times in infectious states, which may not be practical in reality and could affect the dynamics of the epidemic. This paper investigates the potential discrepancies between exponential and non-exponential distribution models in analyzing the transmission patterns of infectious diseases and evaluating control measures. Two SEIHR models with multiple subgroups based on different assumptions for latency are established: Model Ⅰ assumes an exponential distribution of latency, while Model Ⅱ assumes a gamma distribution. To overcome the challenges associated with the high dimensionality of GDM, we derive the basic reproduction number ($ R_{0} $) of the model theoretically, and apply numerical simulations to evaluate the effect of different interventions on EDM and GDM. Our results show that considering a more realistic gamma distribution of latency can change the peak numbers of infected and the timescales of an epidemic, and GDM may underestimate the infection eradication time and overestimate the peak value compared to EDM. Additionally, the two models can produce inconsistent predictions in estimating the time to reach the peak. Our study contributes to a more accurate understanding of disease transmission patterns, which is crucial for effective disease control and prevention.

Stochastic Geometry Analysis of Sojourn Time in RF/VLC Hybrid Networks

The spectrum scarcity in the radio frequency (RF) communication in indoor environments motivates the integration of an alternative technology like visible light communication (VLC) with the existing RF architecture that results in a hybrid RF/VLC network. While VLC helps offloading the congested RF spectrum by offering capacity-per-area improvements, the resulting heterogeneity and narrow coverage areas of optical base stations (BSs) impose several challenges for user mobility such as unnecessary handovers. To help addressing these challenges, in this paper, we derive the mean and the distribution of sojourn time in RF/VLC hybrid networks. The mathematical analysis conducted in this paper makes use of the tools from stochastic geometry and abstracting the BSs’ locations via two independent homogeneous Poisson point processes (PPPs). Since PPP modeling is yet to be well established for RF/VLC hybrid networks, we compare the PPP based analytical results to those obtained for an actual deployment, a Matérn hard-core point process (MHCPP) based deployment, and a deterministic square lattice deployment of VLC luminaries. Furthermore, we utilize the sojourn time distribution to calculate the unnecessary handover probability. Our numerical results show the interplay between the sojourn time and the receiver field of view as a function of BS density and they highlight the cost of BS densification in terms of unnecessary handovers.

Semi-Markov models of inspection-based maintenance with empirical data from case studies on hydrant pumps

Considerable benefits have been gained from condition-based maintenance (CBM) utilizing continuous monitoring integrated with information technology. However, periodic inspection for CBM is still used widely as a practically helpful method to know the condition of the equipment. This paper starts from a case study where a maintenance log recorded by periodic inspection from five hydrant pumps is used to estimate the required parameter for maintenance modeling. To process the data for CBM, two schemes are taken into consideration: Inference of condition indicator through repair activities and reflection of non-observable events with virtual nodes. A CBM model of inspection-based preventive maintenance with discrete data is developed using the Markov model. The semi-Markov process is adopted then with more flexibility allowing the Weibull distributed sojourn times and the Multiphase Markov process is suggested to reflect the periodic inspection. Thus, the model for pumps takes into account both SMP and multiphase Markov process. Monte-Carlo simulations are generated to calculate state probability and the number of maintenances. An analytical solution is proposed by the transition probability of embedded Markov chain (EMC) and sojourn time of SMP. The developed CBM models are verified and compared based on analysis results and empirical data.

Statistical Inference for Multi State Systems under the Generalized Modified Weibull Class

Multi state systems can be seen as semi-Markov processes by considering an arbitrary distribution function for sojourn times. Especially, in this work, the Modified Weibull distribution is employed to be the distribution of sojourn times with a shape parameter λ such that is member of a distributions family that is closed under minima. Parameters estimators are provided and the proposed methodology is evaluated using a detailed simulation procedure.

Multistate models for the natural history of cancer progression.

Multistate models can be effectively used to characterise the natural history of cancer. Inference from such models has previously been useful for setting screening policies. We introduce the basic elements of multistate models and the challenges of applying these models to cancer data. Through simulation studies, we examine (1) the impact of assuming time-homogeneous Markov transition intensities when the intensities depend on the time since entry to the current state (i.e., the process is time-inhomogenous semi-Markov) and (2) the effect on precancer risk estimation when observation times depend on an unmodelled intermediate disease state. In the settings we examined, we found that misspecifying a time-inhomogenous semi-Markov process as a time-homogeneous Markov process resulted in biased estimates of the mean sojourn times. When screen-detection of the intermediate disease leads to more frequent future screening assessments, there was minimal bias induced compared to when screen-detection of the intermediate disease leads to less frequent screening. Multistate models are useful for estimating parameters governing the process dynamics in cancer such as transition rates, sojourn time distributions, and absolute and relative risks. As with most statistical models, to avoid incorrect inference, care should be given to use the appropriate specifications and assumptions.

Sojourn-time Distribution for $$M/G^a/1$$ Queue with Batch Service of Fixed Size - Revisited

Sojourn-time Distribution for $$M/G^a/1$$ Queue with Batch Service of Fixed Size - Revisited

Finite-frequency control for nonlinear semi-Markov jump systems with piecewise transition probabilities

Considering the frequency effect of external disturbances, this paper concerns the finite-frequency control problem for nonlinear semi-Markov jump systems (SMJSs) with piecewise transition probabilities (TPs) via the Takagi–Sugeno (T–S) fuzzy modeling approach. More precisely, the piecewise TPs are assumed to switch stochastically within limits, which implies that the corresponding distributions of sojourn time (ST) also vary randomly. Furthermore, another upper semi-Markov chain is utilized to characterize TPs variation in a finite set. With the aid of Finsler’s lemma and Parseval’s theorem, sufficient criteria for the controlled SMJS are constructed to meet the desired disturbance attenuation performance in the frequency domain. Then, the mode-dependent controllers are developed to use the TP information more efficiently. In particular, a novel matrix method is proposed to decouple the mode-dependent variables and controller gains by selecting slack matrices. Eventually, a numerical example is transmitted to demonstrate the validity and merits of established results.

Stationary workload and service times for some nonwork-conserving M/G/1 preemptive LIFO queues

We analyze two nonwork-conserving variations of the M/G/1 preemptive last-in first-out (LIFO) queue with emphasis on deriving explicit expressions for the limiting (stationary) distributions of service times found in service by an arrival, workload and a variety of related quantities of interest. Workload is also used as a tool to derive the proportion of time that the system is busy, and stability conditions. In the first model, known as preemptive-repeat different (PRD), preempted customers are returned to the front of the queue with a new independent and identically distributed service time. In the second, known as preemptive-repeat identical (PRI), they are returned to the front of the queue with their original service time. Our analysis is based on queueing theory methods such as the Rate Conservation Law, PASTA, regenerative process theory and Little’s Law (). For the second model we even derive the joint distribution of age and excess of the service time found in service by an arrival, and find they are quite different from what is found in standard work-conserving models. We also give heavy-traffic limits and tail asymptotics for stationary workload for both models, as well as deriving an implicit representation for the distribution of sojourn time by introducing an alternative effective service time distribution.

Two sample tests for Semi-Markov processes with parametric sojourn time distributions: an application in sensory analysis

Two sample tests for Semi-Markov processes with parametric sojourn time distributions: an application in sensory analysis

On the asymptotic properties of some kernel estimators for continuous-time semi-Markov processes

This paper provides kernel estimators of the main characteristics of a continuous-time semi-Markov process, like conditional and unconditional sojourn times in a state, semi-Markov kernel, etc. The main goal of this paper is to establish asymptotic properties of the semi-Markov kernel estimators and of the sojourn time distribution estimators (conditional and unconditional), as well as of the estimators of the associated Radon-Nikodym derivatives, when the sample size becomes large. The approach is illustrated by considering a three state example with detailed calculus and numerical evaluations.