A Cox model for gradually disappearing events

Innovations in medicine provide us longer and healthier life, leading lower mortality. Sooner rather than later, much greater longevity would be possible for us due to artificial intelligence advances in health care. Similarly, Advanced Driver Assistance Systems (ADAS) in highly automated vehicles may reduce or even eventually eliminate accidents by perceiving dangerous situations, which would minimize the number of accidents and lead to fewer loss claims for insurance companies. To model the survivor function capturing greater longevity as well as the number of claims reflecting less accidents in the long run, in this paper, we study a Cox process whose intensity process is piecewise-constant and decreasing. We derive its ultimate distributional properties, such as the Laplace transform of intensity integral process, the probability generating function of point process, their associated moments and cumulants, and the probability of no more claims for a given time point. In general, this simple model may be applicable in many other areas for modeling the evolution of gradually disappearing events, such as corporate defaults, dividend payments, trade arrivals, employment of a certain job type (e.g., typists) in the labor market, and release of particles. In particular, we discuss some potential applications to insurance.

Poisson size-biased Lindley distribution and its applications

In this paper, a new model for count data is introduced by compounding the Poisson distribution with size-biased three-parameter Lindley distribution. Statistical properties, such as reliability, hazard rate, reverse hazard rate, Mills ratio, moments, shewness, kurtosis, moment genrating function, probability generating function and order statistics, have been discussed. Moreover, the collective risk model is discussed by considering the proposed distrubution as the primary distribution and the expoential and Erlang distributions as the secondary ones. Parameter estimation is done using maximum likelihood estimation (MLE). Finally a real dataset is discussed to demonstrate the suitability and applicability of the proposed distribution in modeling count dataset.

A New Neutrosophic Negative Binomial Distribution: Properties and Applications

Many problems in real life exist that are full of confusion, vagueness, and ambiguity. The quantification of such issues in a scientific way is the need of time. The negative binomial distribution is an important discrete probability distribution from the account of classical probability distribution theory. The distribution was used to study the chance of kth success in n trials before n − 1 failures for crisp data. The literature lacks in dealing with the situations for interval-valued data under negative binomial distribution. In this research, the neutrosophic negative binomial distribution is proposed to generalize the classical negative binomial distribution. The generalized proposed distribution considers the indeterminacy and crisp form from interval-valued. Several properties of the proposed distribution, such as moment generating function, characteristic function, and probability generating function, are also derived. Furthermore, the derivation of reliability analysis properties such as survival, hazard rate, reversed hazard rate, cumulative hazard rate, mills ratio, and odds ratio are also presented. In addition, order statistics for the proposed distribution, including w th , joint, median, minimum, and maximum order statistics are part of the paper. The proposed distribution is discussed from the real data applications perspective by considering the different case studies. This research opens the way to deal with the problems that follow conventional conveyances and include nonprecisely determined details simultaneously.

Analysis of a bulk arrival N-policy queue with two-service genre, breakdown, delayed repair under bernoulli vacation and repeated service policy.

This article deals with an unreliable bulk arrival single server queue rendering two-heterogeneous optional repeated service (THORS) with delayed repair, under Bernoulli Vacation Schedule (BVS) and N-policy. For this model, the joint distribution of the server's state and queue length are derived under both elapsed and remaining times. Further, probability generating function (PGF) of the queue size distribution along with the mean system size of the model are determined for any arbitrary time point and service completion epoch, besides various pivotal system characteristics. A suitable linear cost structure of the underlying model is developed, and with the help of a difference operator, a locally optimal N-policy at a lower cost is obtained. Finally, numerical experiments have been carried out in support of the theory.

Discrete Pseudo Lindley Distribution: Properties, Estimation and Application on INAR(1) Process

In this paper, we introduce a discrete version of the Pseudo Lindley (PsL) distribution, namely, the discrete Pseudo Lindley (DPsL) distribution, and systematically study its mathematical properties. Explicit forms gathered for the properties such as the probability generating function, moments, skewness, kurtosis and stress–strength reliability made the distribution favourable. Two different methods are considered for the estimation of unknown parameters and, hence, compared with a broad simulation study. The practicality of the proposed distribution is illustrated in the first-order integer-valued autoregressive process. Its empirical importance is proved through three real datasets.

Queue Size Distribution on a New ND Policy Geo/G/1 Queue and Its Computation Designs

This paper examines a new ND policy in the discrete-time Geo/G/1 queue. Under this ND policy, the idle server restarts its service when the N and D policies are simultaneously satisfied. By two classifications of the customers, the probability-generating function and the probabilistic analysis, the steady-state queue size distributions at a departure time and an arbitrary time t + are studied. Finally, the theoretical results are applied to the power-saving problem of a wireless sensor network. To improve model universality and numerical slowness, some computation designs are carried out. Under the N, D, and two ND policies, the numerical experiments are presented to obtain the optimal policy thresholds and the corresponding minimum power consumptions are compared.

Performance Analysis of Retrial Queueing Model with Working Vacation, Interruption, Waiting Server, Breakdown and Repair

In this present paper, an M/M/1 retrial queueing model with a waiting server subject to breakdown and repair under working vacation, vacation interruption is considered. Customers are served at a slow rate during the working vacation period, and the server may undergo breakdowns from a normal busy state. The customer has to wait in orbit for the service until the server gets repaired. Steady-state solutions are obtained using the probability generating function technique. Probabilities of different server states and some other performance measures of the system are developed. The variation in mean orbit size, availability of the server, and server state probabilities are plotted for different values of breakdown parameter and repair rate with the help of MATLAB software. Finally, cost optimization of the system is also discussed, and the optimal value of the slow service rate for the model is obtained.

Computing minimal signature of coherent systems through matrix-geometric distributions

Signatures are useful in analyzing and evaluating coherent systems. However, their computation is a challenging problem, especially for complex coherent structures. In most cases the reliability of a binary coherent system can be linked to a tail probability associated with a properly defined waiting time random variable in a sequence of binary trials. In this paper we present a method for computing the minimal signature of a binary coherent system. Our method is based on matrix-geometric distributions. First, a proper matrix-geometric random variable corresponding to the system structure is found. Second, its probability generating function is obtained. Finally, the companion representation for the distribution of matrix-geometric distribution is used to obtain a matrix-based expression for the minimal signature of the coherent system. The results are also extended to a system with two types of components.

The switch of DNA states filtering the extrinsic noise in the system of frequency modulation

There is a special node, which the large noise of the upstream element may not always lead to a broad distribution of downstream elements. This node is DNA, with upstream element TF and downstream elements mRNA and proteins. By applying the stochastic simulation algorithm (SSA) on gene circuits inspired by the fim operon in Escherichia coli, we found that cells exchanged the distribution of the upstream transcription factor (TF) for the transitional frequency of DNA. Then cells do an inverse transform, which exchanges the transitional frequency of DNA for the distribution of downstream products. Due to this special feature, DNA in the system of frequency modulation is able to reset the noise. By probability generating function, we know the ranges of parameter values that grant such an interesting phenomenon.

Period-Life of a Branching Process with Migration and Continuous Time

The homogeneous branching process with migration and continuous time is considered. We investigated the distribution of the period-life τ, i.e., the length of the time interval between the moment when the process is initiated by a positive number of particles and the moment when there are no individuals in the population for the first time. The probability generating function of the random process, which describes the behavior of the process within the period-life, was obtained. The boundary theorem for the period-life of the subcritical or critical branching process with migration was found.