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 Multi-Stage Model of Cell Proliferation and Death: Tracking Cell Divisions with Erlang Distributions

Lymphocyte populations, stimulated in vitro or in vivo , grow as cells divide. Stochastic models are appropriate because some cells undergo multiple rounds of division, some die, and others of the same type in the same conditions do not divide at all. If individual cells behave independently, each can be imagined as sampling from a probability density of times to division. The most convenient choice of density in mathematical and computational work, the exponential density, overestimates the probability of short division times. We consider a multi-stage model that produces an Erlang distribution of times to division, and an exponential distribution of times to die. The resulting dynamics of competing fates is a type of cyton model. Using Approximate Bayesian Computation, we compare our model to published cell counts, obtained after CFSE-labelled OT-I and F5 T cells were transferred to lymphopenic mice. The death rate is assumed to scale linearly to the generation (number of divisions) and the number of stages of undivided cells (generation 0) is allowed to differ from that of cells that have divided at least once (generation greater than zero). Multiple stages are preferred in posterior distributions, and the mean time to first division is longer than the mean time to subsequent divisions.

DEVELOPMENT OF A MODEL FOR THE DYNAMICS OF PROBABILITIES OF STATES OF SEMI-MARKOV SYSTEMS

The subject is the study of the dynamics of probability distribution of the states of the semi-Markov system during the transition process before establishing a stationary distribution. The goal is to develop a technology for finding analytical relationships that describe the dynamics of the probabilities of states of a semi-Markov system. The task is to develop a mathematical model that adequately describes the dynamics of the probabilities of the states of the system. The initial data for solving the problem is a matrix of conditional distribution laws of the random duration of the system's stay in each of its possible states before the transition to some other state. Method. The traditional method for analyzing semi-Markov systems is limited to obtaining a stationary distribution of the probabilities of its states, which does not solve the problem. A well-known approach to solving this problem is based on the formation and solution of a system of integral equations. However, in the general case, for arbitrary laws of distribution of the durations of the stay of the system in its possible states, this approach is not realizable. The desired result can only be obtained numerically, which does not satisfy the needs of practice. To obtain the required analytical relationships, the Erlang approximation of the original distribution laws is used. This technique significantly increases the adequacy of the resulting mathematical models of the functioning of the system, since it allows one to move away from overly obligatory exponential descriptions of the original distribution laws. The formal basis of the proposed method for constructing a model of the dynamics of state probabilities is the Kolmogorov system of differential equations for the desired probabilities. The solution of the system of equations is achieved using the Laplace transform, which is easily performed for Erlang distributions of arbitrary order. Results. Analytical relations are obtained that specify the desired distribution of the probabilities of the states of the system at any moment of time. The method is based on the approximation of the distribution laws for the durations of the stay of the system in each of its possible states by Erlang distributions of the proper order. A fundamental motivating factor for choosing distributions of this type for approximation is the ease of their use to obtain adequate models of the functioning of probabilistic systems. Conclusions. A solution is given to the problem of analyzing a semi-Markov system for a specific particular case, when the initial distribution laws for the duration of its sojourn in possible states are approximated by second-order Erlang distributions. Analytical relations are obtained for calculating the probability distribution at any time.

COMPARISON OF TWO FORMS OF ERLANGIAN DISTRIBUTION LAW IN QUEUING THEORY

Context. For modeling various data transmission systems, queuing systems G/G/1 are in demand, this is especially important because there is no final solution for them in the general case. The problem of the derivation in closed form of the solution for the average waiting time in the queue for ordinary system with erlangian input distributions of the second order and for the same system with shifted to the right distributions is considered. Objective. Obtaining a solution for the main system characteristic – the average waiting time for queue requirements for three types of queuing systems of type G/G/1 with usual and shifted erlangian input distributions. Method. To solve this problem, we used the classical method of spectral decomposition of the solution of Lindley integral equation, which allows one to obtain a solution for average the waiting time for systems under consideration in a closed form. For the practical application of the results obtained, the well-known method of moments of the theory of probability was used. Results. For the first time, spectral expansions of the solution of the Lindley integral equation for systems with ordinary and shifted Erlang distributions are obtained, with the help of which the calculation formulas for the average waiting time in the queue for the above systems in closed form are derived. Conclusions. The difference between the usual and normalized distribution is that the normalized distribution has a mathematical expectation independent of the order of the distribution k, therefore, the normalized and normal Erlang distributions differ in numerical characteristics. The introduction of the time shift parameter in the laws of input flow distribution and service time for the systems under consideration turns them into systems with a delay with a shorter waiting time. This is because the time shift operation reduces the coefficient of variation in the intervals between the receipts of the requirements and their service time, and as is known from queuing theory, the average wait time of requirements is related to these coefficients of variation by a quadratic dependence. The system with usual erlangian input distributions of the second order is applicable only at a certain point value of the coefficients of variation of the intervals between the receipts of the requirements and their service time. The same system with shifted distributions allows us to operate with interval values of coefficients of variations, which expands the scope of these systems. This approach allows us to calculate the average delay for these systems in mathematical packages for a wide range of traffic parameters.

Spectral expansion method for analysis of a system with shifted Erlang and hyper-Erlang distributions

In this paper, we obtained a spectral expansion of the solution to the Lindley integral equation for a queuing system with a shifted Erlang input flow of customers and a hyper-Erlang distribution of the service time. On its basis, a calculation formula is derived for the average waiting time in the queue for this system in a closed form. As you know, all other characteristics of the queuing system are derivatives of the average waiting time. The resulting calculation formula complements and expands the well-known unfinished formula for the average waiting time in queue in queuing theory for G/G/1 systems. In the theory of queuing, studies of private systems of the G/G/1 type are relevant due to the fact that they are actively used in the modern theory of teletraffic, as well as in the design and modeling of various data transmission systems.

Refining One Theorem For The Romanovsky Distribution

The paper considered a refinement of the theorem for a negative-hypergeometric distribution( the Romanovsky distribution), i.e., convergence over variation of the Romanovsky distribution by Erlang distributions. The theorem is proved by the direct asymptotic method.

Erlang mixture distribution with application on COVID-19 cases in egypt

In this paper, a finite mixture of m-Erlang distributions is proposed. Different moments, shape characteristics and parameter estimates of the proposed model are also provided. The proposed mixture has the property that it has a bounded hazard function. A special case of the mixed Erlang distribution is introduced and discussed. In addition, a predictive technique is introduced to estimate the needed number of mixture components to fit a certain data. A real data concerning the confirmed COVID-19 cases in Egypt is introduced to utilize the predictive estimation technique. Two more real datasets are used to examine the flexibility of the proposed model.

Time Is Of The Essence: Incorporating Phase-Type Distributed Delays And Dwell Times Into ODE Models

Ordinary differential equations models have a wide variety of applications in the fields of mathematics, statistics, and the sciences. Though they are widely used, these models are sometimes viewed as inflexible with respect to the incorporation of time delays. The Generalized Linear Chain Trick (GLCT) serves as a way for modelers to incorporate much more flexible delay or dwell time distribution assumptions than the usual exponential and Erlang distributions. In this paper we demonstrate how the GLCT can be used to generate new ODE models by generalizing or approximating existing models to yield much more general ODEs with phase-type distributed delays or dwell times.

Upper Bounds and Explicit Formulas for the Ruin Probability in the Risk Model with Stochastic Premiums and a Multi-Layer Dividend Strategy

This paper is devoted to the investigation of the ruin probability in the risk model with stochastic premiums where dividends are paid according to a multi-layer dividend strategy. We obtain an exponential bound for the ruin probability and investigate conditions, under which it holds for a number of distributions of the premium and claim sizes. Next, we use the exponential bound to construct non-exponential bounds for the ruin probability. We show that the non-exponential bounds turn out to be tighter than the exponential one in some cases. Moreover, we derive explicit formulas for the ruin probability when the premium and claim sizes have either the hyperexponential or the Erlang distributions and apply them to investigate how tight the bounds are. To illustrate and analyze the results obtained, we give numerical examples.