Optimizing Erlangization-based approximations for finite discrete distributions and discrete phase-type distributions

In this paper, we investigate a wear and mixed shock model in which the system can fail due to internal aging or external shocks. The lifetime of the system, due to internal wear, follows continuous phase-type (PH) distributions. The external random shocks arrive at the system according to a PH renewal process. The system will fail when the internal failure occurs or k1 consecutive external shocks, the size of at least d1 or k2 consecutive external shocks the size of at least d2 occur, where d1<d2, k1>k2. The failed system can be repaired immediately, and the repair times of the system are governed by continuous PH distributions. The system can be replaced by a new and identical one based on a bivariate replacement policy (L,N). The long-run average profit rate for the system is obtained by employing the closure property of the PH distribution. Finally, a numerical example is also given to determine the optimal replacement policy.

Considering maintenance resources assignments of the large-scale complex equipment, the policy that single repairman has multiple vacations is utilized to analyze a warm standby repairable system. The former models that the known conditions are usually assumed to follow exponential distribution or other typical distributions have many constraint conditions. It is assumed that the online component lifetime, standby components lifetime, repair time and repairman vacation time are subject to different continuous Phase-type (PH) distribution, respectively. The reliability model with high versatile is acquired. The reliability features, such as system reliability function, system stationary availability, stationary failure frequency and mean time between failures (MTBF), are provided by the model. Finally, the applicability of the model is implemented by numerical application, and the influences of standby components counts and vacations or non-vacation of a repairman on the system reliability laws are demonstrated.

Due to having useful properties in approximating to the other distributions and mathematically tractable, phase type distributions are commonly used in actuarial risk theory. Claim occurrence time and individual claim size distributions are modelled by phase type distributions in literature. This paper aims to calculate the survival probabilities of an insurance company under the assumption that compound binomial risk model where the individual claim sizes are distributed as discrete Phase Type distribution

This paper considers parameter estimation of a class of discrete multi-variate phase-type distributions (DMPH). Such discrete phase-type distributions are based on discrete Markov chains with marked transitions introduced by He and Neuts (Stoch Process Appl 74(1):37–52, 1998) and is a generalization of the discrete univariate phase–type distributions. Properties of the DMPHs are presented. An EM-algorithm is developed for estimating the parameters for DMPHs. A number of numerical examples are provided to address some interesting parameter selection issues and to show possible applications of DMPHs.

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. If in the quantization some of the elements in the finite support are preselected, then the quantization is called a conditional quantization. In this paper, we have determined the conditional quantization, first for two different finite discrete distributions with a same conditional set, and for a finite discrete distribution with two different conditional sets. Next, we have determined the conditional and unconditional quantization for an infinite discrete distribution with support {12n:n∈N}. We have also investigated the conditional quantization for an infinite discrete distribution with support {1n:n∈N}. At the end of the paper, we have given a conjecture and discussed about some open problems based on the conjecture.

Discrete power distributions are proposed and studied, by considering the positive jumps on the discontinuities of an original discrete distribution function. Inequalities in moments and distribution functions are studied, allowing the definition of discrete intermediate distributions that lie between an original distribution and a power distribution. Original uniform, binomial, Poisson, negative binomial, and hypergeometric distributions are considered, to propose new power and intermediate distributions. Stochastic orders and unimodality are discussed. Estimation problems using likelihood are investigated. Simulation experiments are performed, to evaluate the bias and the mean square error of the maximum likelihood estimates, that are numerically calculated, with classic tools for numerical optimization.

In this article, we consider a discrete-time inventory model in which demands arrive according to a discrete Markovian arrival process. The inventory is replenished according to an $$(s,S)$$ policy, and the lead time is assumed to follow a discrete phase-type distribution. The demands that occur during stock-out periods either enter a pool which has an infinite capacity or leave the system with a predefined probability. The demands in the pool are selected one by one, if the on-hand inventory level is above $$s+1,$$ and the interval time between any two successive selections is assumed to have a discrete phase-type distribution. The joint probability distribution of the number of customers in the pool and the inventory level is obtained in the steady-state case. We derive the system performance measures under steady state and using these measures, the total expected cost rate of the system is calculated. The impacts of arrival rate on the performance measures are graphically illustrated. Finally, we study the impact of cost on the optimal values of the total expected cost rate, inventory level and the reorder point.

This paper considers a discrete time priority queue formed by the customers induced interruption during the service time. The arrivals of customers are modeled by the Markovian Arrival Process (MAP) and the service time follows discrete phase type distribution. During the service, the customer may interrupt with a pre-assigned probability and the interrupted customers are moved to lower priority queue. The interrupted customers are not allowed any further interruptions except in the case of preemptive priority. Preemptive and non- preemptive disciplines to lower priority customers are considered. Matrix analytic method is used to analyse the model. The marginal distributions of both queue lengths in each discipline are analysed.

The probability distribution of the data entering a recourse problem is replaced by finite discrete distributions. It is proved that the convergence of the objective functions of the approximating problems to that one of the original problem can be achieved by choosing the discrete distributions in quite a natural way. For bounded feasible sets this implies the convergence of the optimal values. Finally some error bounds are derived.

A risk model with paying dividends and random environment

In this paper, a new RF energy harvesting (RF-EH) policy is proposed for Internet of Things (IoT) networks using cooperative communication approach. The proposed policy is evaluated in three devices network which are source, relay, and destination. Such a network model can be adopted in most IoT networks. Three scenarios are studied, where the relay has RF-EH capability with limited battery in all scenarios. However, the source has unlimited energy in the first scenario, but the limited energy in the second and third ones. In addition, in the third scenario source has RF-EH capability. New analyses based on discrete phase-type distribution (D-PHD) model as well as a simple analytical method (SAM) are presented to derive performance metrics which are network effective lifetime, average delay of source/relay packets, and source packets overflow. The effect of energy harvesting on such metrics are investigated to find the optimum relay policy. Results show that the proposed policy can maximize network effective lifetime in comparison with time switching (TS) and power splitting (PS)-based protocols.

PhFit, a new Phase-type fitting tool is presented in this paper. The presented tool is novel from several aspects. It allows for approximating distributions or set of samples not only by continuous but by discrete Phase-type distributions as well. The implemented algorithms separate the fitting of the body and the tail part of the distribution which results in satisfactory fitting also for heavy-tail distributions. Moreover, PhFit allows the user to choose the distance measure according to which the fitting is performed.

Strong and Weak Multivariate First-Order Stochastic Dominance

In this research, we contribute to the development of original discrete distributions for count data analysis. We elaborate four new and different discrete trigonometric distributions with two parameters based on a discretization of the so-called sin-Weibull, cos-Weibull, tan-Weibull, and type II tan-Weibull distributions. We investigate the advantages of the proposed distributions and their trigonometric nature for capturing count data. In particular, some fundamental distributional properties of the discrete sin-Weibull distribution are derived. Subsequently, we turn it into a statistical model. Parameter estimation by the maximum likelihood and proportion of zeros and ones methods are then discussed. A simulation study is carried out to evaluate some frequentist properties of the developed methodology. As the study focuses mainly on the practical aspect, the applicability of the proposed models is evaluated using real-count datasets from various fields.

For the problem of constructing a portfolio of finitely many assets whose return rates are described by a discrete joint distribution, we discuss an approach based on stochastic dominance. The portfolio return rate in this model is required to stochastically dominate a random benchmark, such as an index, or reference portfolio return. We formulate optimality conditions and duality relations for these models and construct equivalent optimization models with utility functions and law invariant coherent measures of risk. Two different formulations of the stochastic dominance constraint–primal and inverse–lead to two dual problems which involve von Neuman–Morgenstern utility functions for the primal formulation and rank dependent (or dual) utility functions for the inverse formulation. We also analyze relations of this approach to value at risk and conditional value at risk. Numerical illustration is provided.