Time Random Variables Research Articles

On Some Stochastic Ordering Comparisons for Renewal Processes

After a brief introduction to ordinary renewal process, we have defined compound renewal process generated by an ordinary renewal process followed by the distributional properties of the corresponding random variables. A few results are obtained by comparing independent renewal processes with respect to several stochastic orderings between the generating inter-arrival time random variables, like, stochastic order, hazard rate order, likelihood ratio order and variability order, as well as some ageing classes of the generating random variables. Some numerical illustrations are given. The results obtained here appear to be new.

Resilient airline scheduling to minimize delay risks

Airlines tend to design their flights schedules with the primary concern of the minimization of operational costs. However, the recently emerging idea of resilient scheduling defined as staying operational in case of unexpected disruptions and adaptability should be of great importance for airlines as well due to the high opportunity costs caused by the flight cancellations and passenger inconvenience caused by delays in the schedule. In this study, we integrate resilient airline schedule design, aircraft routing and fleet assignment problems with uncertain non-cruise times and controllable cruise times. We follow a data-driven method to estimate flight delay probabilities to calculate the airport congestion coefficients required for the probability distributions of non-cruise time random variables. We formulate the problem as a bi-criteria nonlinear mixed integer mathematical model with chance constraints. The nonlinearity caused by the fuel consumption and CO2 emission function associated with the controllable cruise times in our first objective is handled by second order conic inequalities. We minimize the total absolute deviation of the aircraft path variability’s from the average in our second objective to generate balanced schedules in terms of resilience. We compare the recovery performances of our proposed schedules to the minimum cost schedules by a scenario-based posterior analysis.

The Interaction of Time with Split-Plot Using Survival Function

It seems that most work has been done for continuous time random variables. Our grouped time models that are used for inference are chosen to relate to these we known continuous time models. We have generalized by the cox (1972) model to include main unit variability to time model.

Computing Multi-Modal Journey Plans under Uncertainty

Multi-modal journey planning, which allows multiple types of transport within a single trip, is becoming increasingly popular, due to a strong practical interest and an increasing availability of data. In real life, transport networks feature uncertainty. Yet, most approaches assume a deterministic environment, making plans more prone to failures such as missed connections and major delays in the arrival.
 This paper presents an approach to computing optimal contingent plans in multi-modal journey planning. The problem is modeled as a search in an and/or state space. We describe search enhancements used on top of the AO* algorithm. Enhancements include admissible heuristics, multiple types of pruning that preserve the completeness and the optimality, and a hybrid search approach with a deterministic and a nondeterministic search. We demonstrate an NP-hardness result, with the hardness stemming from the dynamically changing distributions of the travel time random variables. We perform a detailed empirical analysis on realistic transport networks from cities such as Montpellier, Rome and Dublin. The results demonstrate the effectiveness of our algorithmic contributions, and the benefits of contingent plans as compared to standard sequential plans, when the arrival and departure times of buses are characterized by uncertainty.

Global sensitivity analysis of reliability of structural bridge system

The article deals with the analysis of the probability of failure of a load-bearing steel bridge member under bending. The focus is on fatigue failure caused by stress cycles from multiple repeated traffic loading on the bridge. Failure is defined by the occurrence of a fatigue crack of critical size. Crack propagation and the fatigue limit state are described using linear fracture mechanics. The failure probability is a function of the equivalent stress range, initial crack length, Paris exponent, number of load cycles (stress changes) increasing over time and other input random variables. The failure probability is evaluated in time steps and then studied using a new type of global sensitivity analysis subordinated to contrasts. The results of the sensitivity analysis show that the first (second) dominant variable is the equivalent stress range (initial crack length) at any given point in time of the bridge operation. Strong main effect of equivalent stress range is associated with higher values of failure probability at the end of the lifetime of the bridge. Small values of failure probability are strongly influenced by interactions among input variables, which cannot be expressed as the sum of main effects of the individual input variables. The main and higher-order indices of each input variable are supplemented by displaying its total index. The direct goal of probability and sensitivity analysis is structural reliability. Sensitivity analysis confirms and deepens the knowledge gained from the time-dependent probability analysis. The numerical example illustrates the rationality of probability-oriented sensitivity indices and the feasibility of their estimation using Latin Hypercube Sampling (LHS). In addition, structural reliability is studied using Bayesian probability, which identifies the times for planning bridge inspections.

Queues with Simultaneous Arrival of Customers and the Dependence Structure of the Waiting Times

We consider a queueing system where some customers decide to simultaneously wait in two queues, rather than in a single queue, to receive their service. In practice, when there exists a number of queues rendering the same service, the customers may tend to simultaneously wait in more than one queue in order to receive the service sooner and thus scale down their waiting time. In this framework, the customers may abandon one of the queues when they are called to receive the service from the other. We treat this situation as customer reneging or abandonment. We study the customer’s waiting time under this model for the cases of independence and dependence of the waiting time random variables of the queues. In conducting this study, a Copula approach is applied to take into account the dependence structure of the waiting time random variables. We compare the numerical results of the dependence with that of the independence.

Stochastic modeling of non-identical redundant systems with priority, preventive maintenance, and Weibull failure and repair distributions

The ever-increasing demand of automation in appliances and economic constraints in purchasing capacity of consumers motivate to analyse the non-identical redundant system. For this analysis, a stochastic model has been designed for a non-identical redundant system having two units: one operative and other cold standby. Each unit has two possible states: either operative or failure. Original unit gets priority for repair over repair and preventive maintenance of duplicate unit. The repairs, preventive maintenance, and switches used to shift standby unit into operation are perfect. All failure and repair time random variables are arbitrary distributed. The system is observed using SMP and RPT to obtain various characteristics of interest to system designers. At last, some numerical results have been obtained by considering exponential, Rayleigh, and Weibull distributions, and compared with the results obtained by Kumar et al. (Rev Invest Oper 37(3):247–257, 2016a; Cogent Math 3(1):1135721, 2016b).

Waiting time for consecutive repetitions of a pattern and related distributions

Let k be a positive integer. Some exact distributions of the waiting time random variables for k consecutive repetitions of a pattern are derived in a sequence of independent identically distributed trials. It is proved that the number of equations of conditional probability generating functions for deriving the distribution can be reduced to less than or equal to the length of the basic pattern to be repeated consecutively. By using the result, various properties of the distributions of usual runs are extended to those of consecutive repetitions of a pattern. These results include some properties of the geometric distribution of order k and those of the waiting time distributions of the $$(k_1,k_2)$$ -events. Further, the probability generating function of the number of non-overlapping occurrences of k consecutive repetitions of a pattern can be written in an explicit form with k as a parameter. Some recurrence relations, which are useful for evaluating the probability mass functions, are also given.

Simultaneous arrival of customers to two different queues and modeling dependence via copula approach

ABSTRACTIn classical queueing systems, a customer is allowed to wait only in one queue to receive the service. In practice, when there exist a number of queues rendering the same service, some customers may tend to simultaneously take turn in more than one queue with the aim to receive the service sooner and thus reduce their waiting time. In this article, we introduce such a model and put forward a methodology to deal with the situation. In this regard, we consider two queues and assume that if a customer, who has turn in both queues, receives the service from one of the queues, the other turn is automatically withdrawn. This circumstance for the model brings about some abandonment in each queue as some customers receive the service from the other one. We study the customer’s waiting time in the mentioned model, which is defined as the minimum of waiting times in both queues and obtain probability density function of this random variable. Our approach to obtain probability density function of each of the waiting time random variables is to rely on the existing results for the abandonment case. We examine the situation for the cases of independence and dependence of the waiting time random variables. The latter is treated via a copula approach.

Estimating the Parameters of Mixed Shifted Negative Binomial Distributions via an EM Algorithm

Discrete phase-type (DPH) distributions have one property that is not shared by continuous phase-type (CPH) distributions, i.e., representing a deterministic value as a DPH random variable. This property distinguishes the application of DPH in stochastic modeling of real-life problems such as stochastic scheduling where service time random variables should be compared with a deadline that is usually a constant value. In this paper, we consider a restricted class of DPH distributions, called Mixed Shifted Negative Binomial (MSNB) and show its flexibility in producing a wide range of variances as well as its adequacy in fitting fat-tailed distributions. These properties render MSNB applicable to represent data on certain types of service time. Therefore, we adapt an expectation-maximization (EM) algorithm to estimate the parameters of MSNB distributions that accurately fit trace data. To present the applicability of the proposed algorithm, we use it to fit real operating room times as well as a set of benchmark traces generated from continuous distributions as case studies. Finally, we illustrate the efficiency of the proposed algorithm by comparing its results to that of two existing algorithms in the literature. We conclude that our proposed algorithm outperforms other DPH algorithms in fitting trace data and distributions.

Limited Failure Censored Life Test Sampling Plan in Burr Type X Distribution

The Burr type X distribution is considered as a life time random variable of a product whose lots are to be decided for acceptance or otherwise on the basis of sample lifetimes drawn from the lot. The sample is divided into various groups in order to develop a group sampling plan in such a way that the life testing experiment is terminated as soon as the first failure in each group is observed. The acceptance criterion based on the theory of order statistics is proposed and is shown to be more economical than a criterion proposed in the earlier similar works.

Generalized waiting time distributions associated with runs

Let \(\left\{ X_{t},t\ge 1\right\} \) be a sequence of random variables with two possible values as either “1” (success) or “0” (failure). Define an independent sequence of random variables \(\left\{ D_{i},i\ge 1\right\} \). The random variable \(D_{i}\) is associated with the success when it occupies the ith place in a run of successes. We define the weight of a success run as the sum of the D values corresponding to the successes in the run. Define the following two random variables: \(N_{k}\) is the number of trials until the weight of a single success run exceeds or equals k, and \(N_{r,k}\) is the number of trials until the weight of each of r success runs equals or exceeds k in \(\left\{ X_{t},t\ge 1\right\} \). Distributional properties of the waiting time random variables \(N_{k}\) and \(N_{r,k}\) are studied and illustrative examples are presented.

Channel Model for Satellite Communication Links Above 10GHz Based on Weibull Distribution

Modern satellite communication networks will employ frequencies above 10GHz. At these frequency bands, rain attenuation is the dominant fading mechanism. In this paper, a novel channel model, a synthesizer for generating rain attenuation time series for satellite links operating at 10GHz and above is presented. The proposed channel model modifies Maseng-Bakken (M-B) model since it generates rain attenuation time series that follow the Weibull distribution. The new stochastic dynamic model is based on the first-order Stochastic Differential Equations (SDEs) and considers rain attenuation induced on a slant path as a Weibull-based stochastic process. Moreover, the theoretical expressions for the computation of the exceedance probability of hitting time random variable are presented. The hitting time statistics may be employed for the optimum design of Fade Mitigation Techniques (FMTs). The synthesizer is verified in terms of the exceedance probability and the theoretical CCDF of hitting time comparing to these derived from the simulations in the numerical results section.

Computing probability generating functions of coin flipping and its generalizations

A classical probability question asks for the expected waiting time for flipping a coin (fair or not) until a series of consecutive k heads occur. Now instead of k heads, we can ask for the expected waiting time for a prescribed string such as HTHHTT (H for ‘head’ and T for ‘tail’), and furthermore, the following more general setting: replacing coin flipping by taking a letter, one at a time, what is the expected waiting time until a prescribed string (a series of letters) is reached? Here we allow different probabilities for the occurrence of different letters. We give an exposition to this problem by offering an elementary algorithm and implementing it to compute the corresponding probability generating function: we show that there exists a universal program taking as inputs the choice of letters with given probabilities and the prescribed string, and as output, returning the probability generating function for the waiting time. The same method is applied to solve the problem of several competing strings, which asks for the probability (or more generally the probability generating function) of one of the given strings occurring before the remaining strings. In particular, this solves the problem of finding the expectation and variance for the waiting time random variable of the first problem.

대기행렬 네트워크 시뮬레이션에서 분지확률 통제변수의 응용

본 연구는 대기행렬 네트워크 시뮬레이션에서 통제변수를 활용하여 목표 반응변수를 보다 더 정확히 추정하는 기법을 탐색한다. 반응변수 추정에서 통제변수기법의 효율성은 반응변수와 높은 상관관계를 가지는 통제변수의 선택과 선택된 통제변수를 이용하여 통제추정량을 어떻게 정의하는가에 따라 달라진다. 대기행렬 네트워크 시뮬레이션 모형에서 확률적 모형의 발전과정은 확률적 서비스시간과 분지 확률에 의하여 재현된다. 대부분의 통제변수기법은 통제추정치 구성에서 서비스 시간 확률변수를 사용한다. 본 연구는 서비스 시간 확률변수와 분지 확률변수를 동시에 사용하는 통제 추정량을 제안하고 이를 컴퓨터 네트워크 시스템의 관심 반응변수 추정에 응용하여 그 효율성을 탐색하고자 한다. 시뮬레이션 결과는 반응변수 추정에 있어서 분지확률 통제변수의 활용 가능성을 제시하고 있으며, 서비스-시간과 분지확률을 동시에 이용하는 결합 통제변수의 활용은 향후 연구가 필요한 분야로 판단된다. This research discusses the application of the control variables to achieve a more precise estimation for the target response in queueing network simulation. The efficiency of control variable method in estimating the response depends upon how we choose a set of control variables strongly correlated with the response and how we construct a function of selected control variables. For a class of queuing network simulations, the random variables that drive the simulation are basically the service-time and routing probability random variables. Most of applications of control variable method focus on utilization of the service time random variables for constructing a controlled estimator. This research attempts to suggest a controlled estimator which uses these two kinds of random variables and explore the efficiency of these estimators in estimating the reponses for computer network system. Simulation experiments on this model show the promising results for application of routing probability control variables. We consider the applications of the routing probability control variables to various simulation models and combined control variables using information of service time and routing probability together in constructing a control variable as future researches.

Analysis of inverse stochastic resonance and the long-term firing of Hodgkin–Huxley neurons with Gaussian white noise

In order to explain the occurrence of a minimum in firing rate which occurs for certain mean input levels μ as noise level σ increases (inverse stochastic resonance, ISR) in Hodgkin–Huxley (HH) systems, we analyze the underlying transitions from a stable equilibrium point to limit cycle and vice-versa. For a value of μ at which ISR is pronounced, properties of the corresponding stable equilibrium point are found. A linearized approximation around this point has oscillatory solutions from whose maxima spikes tend to occur. A one dimensional diffusion is also constructed for small noise. Properties of the basin of attraction of the limit cycle (spike) are investigated heuristically. Long term trials of duration 500000 ms are carried out for values of σ from 0 to 2.0. The graph of mean spike count versus σ is divided into 4 regions R1,…,R4, where R3 contains the minimum associated with ISR. In R1 transitions to the basin of attraction of the rest point are not observed until a small critical value of σ=σc1 is reached, at the beginning of R2. The sudden decline in firing rate when σ is just greater than σc1 implies that there is only a small range of noise levels 0<σ<σc1 where repetitive spiking is safe from annihilation by noise. The firing rate remains small throughout R3. At a larger critical value σ=σc2 which signals the beginning of R4, the probability of transitions from the basin of attraction of the equilibrium point to that of the limit cycle apparently becomes greater than zero and the spike rate thereafter increases with increasing σ. The quantitative scheme underlying the ISR curve is outlined in terms of the properties of exit time random variables. In the final subsection, several statistical properties of the main random variables associated with long term spiking activity are given, including distributions of exit times from the two relevant basins of attraction and the interspike interval.

Exact Distributions of Waiting Time Problems of Mixed Frequencies and Runs in Markov Dependent Trials

We study waiting time problems for first-order Markov dependent trials via conditional probability generating functions. Our models involve α frequency cells and β run cells with prescribed quotas and an additional γ slack cells without quotas. For any given and , in our Model I we determine the waiting time until at least frequency cells and at least run cells reach their quotas. For any given τ ≤ α + β, in our Model II we determine the waiting time until τ cells reach their quotas. Computer algorithms are developed to calculate the distributions, expectations and standard deviations of the waiting time random variables of the two models. Numerical results demonstrate the efficiency of the algorithms.

Accelerated failure time (AFT) modeling for the development and survival of Russian wheat aphid, Diuraphis noxia (Mordvilko)

Accelerated failure time model (AFT) and Cox’s proportional hazards model (PHM) are considered the two most significant models in survival analysis, which has become a de facto standard for biomedical data analysis and modeling. AFT not only plays an extremely significant role in survival analysis but also finds extensive applications in engineering reliability. Survival analysis studies a special type of random variables: time-to-event (also known as failure time, lifetime or survival time) random variables. Examples of time-to-event random variables include survival times of patients in a clinical trial and failure times of machine components. Since molting and death times of insect individuals are also perfect examples of time-to-event random variables, we argue that survival analysis including AFT modeling is ideal for analyzing insect development and survival data, and further for building dynamic models of insect development and survival. Here we demonstrate such an application with data collected by observing stage-to-stage development and survival of 1,800 Russian wheat aphids (RWA), Diuraphis noxia, reared in laboratory growth chambers arranged in 25 treatments (each with 72 individuals). The main advantages of survival analysis, including the unified modeling of survival and development as well as handling of information censoring, are also discussed.

Waiting times of exceedances in random threshold models

Waiting time distributions associated with a sequence of binary trials have been widely studied and applied in various areas of probability and statistics. In this paper we study simple and compound waiting time random variables associated with random threshold models which make the elements of sequence dependent. The exact distributions are derived and the results are illustrated for random thresholds based on order statistics.

SOME TIME RANDOM VARIABLES RELATED TO A GOMPERTZ-TYPE DIFFUSION PROCESS

In the context of a Gompertz-type diffusion process (associated with a particular expression of the Gompertz curve for which its limit value depends on the initial value), we consider the study of two time-random variables: the inflection time of the model and the time at which it achieves a certain percentage of the total growth. Once we have shown the difficulties in the approach of the former, we deal with it as a particular case of the second. Furthermore, because of the peculiar characteristics of the considered process, we prove that these two problems can be formulated as a first-passage-time problem through a constant boundary. Finally, we conclude with an application to the growth of rabbits, in which we obtain the density functions of the inflection time and the time at which a rabbit achieves the half of its total growth.