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In this article we provide new results for the asymptotic behavior of a time-fractional birth and death process $N_{\alpha}(t)$ , whose transition probabilities $\mathbb{P}[N_{\alpha}(t)=\,j\mid N_{\alpha}(0)=i]$ are governed by a time-fractional system of differential equations, under the condition that it is not killed. More specifically, we prove that the concepts of quasi-limiting distribution and quasi-stationary distribution do not coincide, which is a consequence of the long-memory nature of the process. In addition, exact formulas for the quasi-limiting distribution and its rate convergence are presented. In the first sections, we revisit the two equivalent characterizations for this process: the first one is a time-changed classic birth and death process, whereas the second one is a Markov renewal process. Finally, we apply our main theorems to the linear model originally introduced by Orsingher and Polito [23]. [ABSTRACT FROM AUTHOR]

In this paper we introduce new birth-and-death processes with partial catastrophe and study some of their properties. In particular, we obtain some estimates for the mean catastrophe time, and the first and second moments of the distribution of the process at a fixed time t. This is completed by some asymptotic results. [ABSTRACT FROM AUTHOR]

This article deals with the emergence of a specific mating preference pattern called homogamy in a population. Individuals are characterized by their genotype at two haploid loci, and the population dynamics is modelled by a nonlinear birth-and-death process. The first locus codes for a phenotype, while the second locus codes for homogamy defined with respect to the first locus: two individuals are more (resp. less) likely to reproduce with each other if they carry the same (resp. a different) trait at the first locus. Initial resident individuals do not feature homogamy, and we are interested in the probability and time of invasion of a mutant presenting this characteristic under a large population assumption. To this aim, we study the trajectory of the birth-and-death process during three phases: growth of the mutant, coexistence of the two types, and extinction of the resident. We couple the birth-and-death process with simpler processes, like multidimensional branching processes or dynamical systems, and study the latter ones in order to control the trajectory and duration of each phase. [ABSTRACT FROM AUTHOR]

We prove under mild conditions that the Fleming-Viot process selects the minimal quasi-stationary distribution for Markov processes with soft killing on non-compact state spaces. Our results are applied to multi-dimensional birth and death processes, continuous time Galton-Watson processes and diffusion processes with soft killing. [ABSTRACT FROM AUTHOR]

Abstract Take a complete (sub-)critical Galton–Watson branching tree with finitely many leaves almost surely. Picking two distinct leaves at random, we ask for the height (as measured from the root) of their latest common ancestor. Upon conditioning on the first branching event we compute the distribution of this height, with calculations explicit in the case of a binary tree. [ABSTRACT FROM AUTHOR]

Abstract The conjecture of Valent about the type of Jacobi matrices with polynomially growing weights is proved. [ABSTRACT FROM AUTHOR]

Summary: In this paper, our interest is in the perturbation analysis of level‐dependent quasi‐birth‐and‐death (LD‐QBD) processes, which constitute a wide class of structured Markov chains. An LD‐QBD process has the special feature that its space of states can be structured by levels (groups of states), so that a tridiagonal‐by‐blocks structure is obtained for its infinitesimal generator. For these processes, a number of algorithmic procedures exist in the literature in order to compute several performance measures while exploiting the underlying matrix structure; among others, these measures are related to first‐passage times to a certain level L(0) and hitting probabilities at this level, the maximum level visited by the process before reaching states of level L(0), and the stationary distribution. For the case of a finite number of states, our aim here is to develop analogous algorithms to the ones analyzing these measures, for their perturbation analysis. This approach uses matrix calculus and exploits the specific structure of the infinitesimal generator, which allows us to obtain additional information during the perturbation analysis of the LD‐QBD process by dealing with specific matrices carrying probabilistic insights of the dynamics of the process. We illustrate the approach by means of applying multitype versions of the susceptible‐infective (SI) and susceptible‐infective‐susceptible (SIS) epidemic models to the spread of antibiotic‐sensitive and antibiotic‐resistant bacterial strains in a hospital ward. [ABSTRACT FROM AUTHOR]

We consider the distribution of the age of an individual picked uniformly at random at some fixed time in a linear birth-and-death process. By exploiting a bijection between the birth-and-death tree and a contour process, we derive the cumulative distribution function for this distribution. In the critical and supercritical cases, we also give rates for the convergence in terms of the total variation and other metrics towards the appropriate exponential distribution. [ABSTRACT FROM AUTHOR]

Abstract: Understanding macroevolutionary processes using phylogenetic trees is a challenging and complex process that draws on mathematics, computer science and biology. Given the development of complex mathematical models and the growing computational processing power, simulation tools are becoming increasingly popular. In order to simulate phylogenetic trees, most evolutionary biologists are forced to build their own algorithms or use existing tools built on different platforms and/or as standalone programmes. The absence of a simulation tool accommodating for user‐chosen model specifications limits, amongst others, model testing and pipelining with approximate Bayesian computation methods or other subsequent statistical analysis. We introduce “TreeSimGM,” an r‐package simulation tool for phylogenetic trees under a general Bellman and Harris model. This package allows the user to specify any desired probability distribution for the waiting times until speciation and extinction (e.g. age‐dependent speciation/extinction). Upon speciation, the user can specify whether one descendant species corresponds to the ancestor species inheriting its age or whether both descendant species are new species of age 0. Moreover, it is possible to scale the waiting time to speciation/extinction for newly formed species. Thus, “TreeSimGM” not only allows the user to simulate stochastic phylogenetic trees assuming several popular existing models, such as the Yule model, the constant‐rate birth–death model, and proportional to distinguishable arrangement models, but it also allows the user to formulate new models for exploration. A short explanation of the supported models and a few examples of how to use our package are presented here. As an r‐package, “TreeSimGM” allows flexible and powerful stochastic phylogenetic tree simulations. Moreover, it facilitates the pipelining of outputs or inputs with other functions in r. “TreeSimGM” contributes to the tools available to the r community in the fields of ecology and evolution, is freely available under the GPL‐2 licence and can be downloaded at https://cran.r-project.org/web/packages/TreeSimGM. [ABSTRACT FROM AUTHOR]

Single-type and multitype branching processes have been used to study the dynamics of a variety of stochastic birth-death type phenomena in biology and physics. Their use in epidemiology goes back to Whittle's study of a susceptible-infected-recovered (SIR) model in the 1950s. In the case of an SIR model, the presence of only one infectious class allows for the use of single-type branching processes. Multitype branching processes allow for multiple infectious classes and have latterly been used to study metapopulation models of disease. In this article, we develop a continuous time Markov chain (CTMC) model of infectious salmon anemia virus in two patches, two CTMC models in one patch and companion multitype branching process (MTBP) models. The CTMC models are related to deterministic models which inform the choice of parameters. The probability of extinction is computed for the CTMC via numerical methods and approximated by the MTBP in the supercritical regime. The stochastic models are treated as toy models, and the parameter choices are made to highlight regions of the parameter space where CTMC and MTBP agree or disagree, without regard to biological significance. Partial extinction events are defined and their relevance discussed. A case is made for calculating the probability of such events, noting that MTBPs are not suitable for making these calculations. [ABSTRACT FROM AUTHOR]

Continuous-time branching processes describe the evolution of a population whose individuals generate a random number of children according to a birth process. Such branching processes can be used to understand preferential attachment models in which the birth rates are linear functions. We are motivated by citation networks, where power-law citation counts are observed as well as aging in the citation patterns. To model this, we introduce fitness and age-dependence in these birth processes. The multiplicative fitness moderates the rate at which children are born, while the aging is integrable, so that individuals receives a finite number of children in their lifetime. We show the existence of a limiting degree distribution for such processes. In the preferential attachment case, where fitness and aging are absent, this limiting degree distribution is known to have power-law tails. We show that the limiting degree distribution has exponential tails for bounded fitnesses in the presence of integrable aging, while the power-law tail is restored when integrable aging is combined with fitness with unbounded support with at most exponential tails. In the absence of integrable aging, such processes are explosive. [ABSTRACT FROM AUTHOR]

In this paper, theoretical solutions for degree distribution of decreasing random birth-and-death networks are provided. First, we prove that the degree distribution has the form of Poisson summation, for which degree distribution equations under steady state and probability generating function approach are employed. Then, based on the form of Poisson summation, we further confirm the tail characteristic of degree distribution is Poisson tail. Finally, simulations are carried out to verify these results by comparing the theoretical solutions with computer simulations. [ABSTRACT FROM AUTHOR]

For any distribution π on [ABSTRACT FROM AUTHOR]

This paper investigates the long-term behavior of an interacting particle system of interest in the hot topic of evolutionary game theory. Each site of the d-dimensional integer lattice is occupied by a player who is characterized by one of two possible strategies. Following the traditional modeling approach of spatial games, the configuration is turned into a payoff landscape that assigns a payoff to each player based on her strategy and the strategy of her 2d neighbors. The payoff is then interpreted as a fitness, by having each player independently update their strategy at rate one by mimicking their neighbor with the largest payoff. With these rules, the mean-field approximation of the spatial game exhibits the same asymptotic behavior as the popular replicator equation. Except for a coexistence result that shows an agreement between the process and the mean-field model, our analysis reveals that the two models strongly disagree in many aspects, showing in particular that the presence of a spatial structure in the form of local interactions plays a key role. More precisely, in the parameter region where both strategies are evolutionary stable in the replicator equation, in the spatial model either one strategy wins or the system fixates to a configuration where both strategies are present. In addition, while defection is evolutionary stable for the prisoner's dilemma game in the replicator equation, space favors cooperation in our model. [ABSTRACT FROM AUTHOR]

In this paper, we study the Fisher Information for the birth rate of a partially observable simple birth process involvingnobservations. We suppose that at each observation time, each individual in the population can be observed independently with known fixed probabilityp. Finding an analytical form of the Fisher Information in general appears intractable. Nonetheless, we find a very good approximation for the Fisher Information by exploiting the probabilistic properties of the underlying stochastic process. Both numerical and theoretical results strongly support the latter approximation and confirm its high level of accuracy. [ABSTRACT FROM PUBLISHER]

We introduce two models for random trees with multiple states motivated by studies of trait dependence in the evolution of species. Our discrete time model, the multiple state ERM tree, is a generalization of Markov propagation models on a random tree generated by a binary search or 'equal rates Markov' mechanism. Our continuous time model, the multiple state Yule tree, is a generalization of the tree generated by a pure birth or Yule process to the tree generated by multi-type branching processes. We study state dependent topological properties of these two random tree models. We derive asymptotic results that allow one to infer model parameters from data on states at the leaves and at branch-points that are one step away from the leaves. [ABSTRACT FROM AUTHOR]

In this paper, we discuss about some applications to the birth and death process in biology and in the theory of waiting. First, we introduce some theoretical notions about the stochastic processes and we define the Markov chain and the Poisson process. Afterwards, we consider two processes of birth: "The Poissonian tide of particles" and " The Yule-Furry process", that have multiple applications, the theory of waiting being one of them. Finally, by using the theoretical concept presented above, we solve two examples: the first is a problem from veterinary medicine, and the second one is an example of the model M/M/1 from agriculture, which means if the entrance, the time of serving and the number of stations are given, we'll calculate the number of the biological units in system, the medium number of the applicants in line and the medium time of waiting. [ABSTRACT FROM AUTHOR]

Slowdowns in lineage accumulation in phylogenies suggest that speciation rates decline as diversity increases. Likelihood methods have been developed to detect such diversity dependence. However, a thorough test of whether such approaches correctly infer diversity dependence is lacking., Here, we simulate phylogenetic branching under linear negative diversity-dependent and diversity-independent models and estimate from the simulated phylogenies the maximum-likelihood parameters for three different conditionings - on survival of the birth-death process given the crown age, on tree size ( N) and on tree size given the crown age. We report the accuracy of recovering the simulation parameters and the reliability of the model selection based on the χ2 likelihood ratio test., Parameter estimate accuracy: Conditioning on survival given the crown age yields a severe bias of the carrying capacity K towards N and an upward bias of the speciation rate, particularly in clades where diversity-dependent feedbacks are still weak ( N « K). Conditioning on N yields an overestimate of K and an underestimate of speciation rate, particularly when saturation has been reached. Dual conditioning yields relatively unbiased parameter estimates on average, but the deviation from the true value for any single estimate may be large., Model selection reliability: The frequency of incorrectly rejecting a diversity-independent model when the simulation was diversity-independent (type I error) differs substantially from the significance level α used in the likelihood ratio test, rendering the likelihood ratio test inappropriate. The frequency of correctly rejecting the diversity-independent model when the simulation was diversity-dependent (power) is larger when the clade is closer to equilibrium and for conditioning on crown age., We conclude that conditioning on crown age has the best statistical properties overall, but caution that parameter estimates may be biased. To assess parameter uncertainty in future studies of diversity dependence on real data, we recommend parametric bootstrapping, examination of the likelihood surface and comparison of estimates across the types of conditioning. To assess model selection reliability, we discourage the use of the χ2 likelihood ratio test or AIC (which are equivalent in this case), but recommend a likelihood ratio test based on parametric bootstrap. We illustrate this method for the diversification of Dendroica warblers. [ABSTRACT FROM AUTHOR]

We present a unified framework to identify spectra of Jacobi matrices. We give applications of the long-standing problem of Chihara (Mt J Math 21(1):121-137, , J Comput Appl Math 153(1-2):535-536, ) concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent (SIAM J Appl Math 42(5):1020-1046, ) concerning continuous spectra of generators of birth and death processes, and to spectral properties of operators studied by Janas and Moszyńki (Integral Equ Oper Theory 43(4):397-416, ) and Pedersen (Proc Am Math Soc 130(8):2369-2376, ). [ABSTRACT FROM AUTHOR]

The signs of deterioration in worldwide infrastructure and the associated socio-economic and environmental losses call for sustainable resource management and policy-making. To this end, this work presents an enhanced variant of partially observable Markov decision processes (POMDPs) for the life cycle assessment and maintenance planning of infrastructure. POMDPs comprise a method, commonly employed in the field of robotics, for decision-making on the basis of uncertain observations. In the work presented herein, a continuous-state POMDP formulation is presented which is adapted to the problem of decision-making for optimal management of civil structures. The aforementioned problem may comprise non-linear and non-deterministic action and observation models. The continuous-state POMDP is herein coupled with a normalised unscented transform (NUT) in order to deliver a framework able to tackle non-linearities that likely characterise action models. The capabilities of this enhanced framework and its applicability to the maintenance planning problem are presented via two applications. In a first illustrative example, the use of the NUT is demonstrated within the framework of the value iteration algorithm. Next, the proposed continuous-state framework is compared against a discrete-state formulation for implementation on a life cycle assessment problem. [ABSTRACT FROM PUBLISHER]

In this paper, we obtain a general achievable rate region for discrete memoryless multiple-access relay channel (DM-MARC) with common message via partial decode and forward strategy and regular block Markov encoding/backward decoding. This general achievable rate region, as an extension of the achievability part of the relay channel via partial decode and forward strategy to the MARC, subsumes the previous known achievable rate regions for multiple-access channel with common message and MARC. Also, by applying Fano's inequality, we derive an outer bound for DM-MARC with common message. By extending the discrete alphabet results to the continuous alphabet, an inner bound for Gaussian MARC (GMARC) is achieved. As expected, our results for the Gaussian MARC includes the previous results for the Gaussian multiple-access channel with and without common message and the Gaussian relay channel as special cases. Finally, by considering a specific topology for MARC, performance of GMARC with common message versus the position of the relay is investigated numerically. Also, it is shown that our achievable rate region is larger than the previous rate region of GMARC with common message with irregular encoding/successive decoding. Copyright © 2014 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

This paper seeks to establish the stability of the birth-death process in relation to the Keller-Segel Model. As well, it attempts to describe the stability of non-linear diffusion for chemotaxis. Attention will be on mass criticality results applying to the chemotaxis model. Afterwards, the analysis of the relative stability that stationary states exhibit is undertaken using the Keller-Segel system for the chemotaxis having linear diffusion. Standard linearization and separation of variables are the techniques employed in the analysis. The stability or instability of the analysed cases is demonstrated by the graphics. By using the critical results obtained for the models, the graphics are then compared with the rest. [ABSTRACT FROM AUTHOR]

This paper deals with a single-server queue with modulated arrivals, service requirements and service capacity. In our first result, we derive the mean of the total workload assuming generally distributed service requirements and any service discipline which does not depend on the modulating environment. We then show that the workload is exponentially distributed under heavy-traffic scaling. In our second result, we focus on the discriminatory processor sharing (DPS) discipline. Assuming exponential, class-dependent service requirements, we show that the joint distribution of the queue lengths of different customer classes under DPS undergoes a state-space collapse when subject to heavy-traffic scaling. That is, the limiting distribution of the queue-length vector is shown to be exponential, times a deterministic vector. The distribution of the scaled workload, as derived for general service disciplines, is a key quantity in the proof of the state-space collapse. [ABSTRACT FROM AUTHOR]

In an M/ M/1/ C queue, customers are lost when they arrive to find C customers already present. Assuming that each arriving customer brings a certain amount of revenue, we are interested in calculating the value of an extra waiting place in terms of the expected amount of extra revenue that the queue will earn over a finite time horizon [0, t]. There are different ways of approaching this problem. One involves the derivation of Markov renewal equations, conditioning on the first instance at which the state of the queue changes; a second involves an elegant coupling argument; and a third involves expressing the value of capacity in terms of the entries of a transient analogue of the deviation matrix. In this paper, we shall compare and contrast these approaches and, in particular, use the coupling analysis to explain why the selling price of an extra unit of capacity remains the same when the arrival and service rates are interchanged when the queue starts at full capacity. [ABSTRACT FROM AUTHOR]

In this paper we consider time-changed models of population evolution Xf (t) = X(Hf (t)), where Xf is a counting process and Hf is a subordinator with Laplace exponent f. In the case where X is a pure birth process, we study the form of the distribution, the intertimes between successive jumps, and the condition of explosion (also in the case of killed subordinators). We also investigate the case whereXrepresents a death process (linear or sublinear) and study the extinction probabilities as a function of the initial population size n0. Finally, the subordinated linear birth-death process is considered. Special attention is devoted to the case where birth and death rates coincide; the sojourn times are also analysed. [ABSTRACT FROM AUTHOR]

This paper defines themultivariate Krawtchouk polynomials, orthogonal on themultinomial distribution, and summarizes their properties as a review. The multivariate Krawtchouk polynomials are symmetric functions of orthogonal sets of functions defined on each of N multinomial trials. The dual multivariate Krawtchouk polynomials, which also have a polynomial structure, are seen to occur naturally as spectral orthogonal polynomials in a Karlin and McGregor spectral representation of transition functions in a composition birth and death process. In this Markov composition process in continuous time, there are N independent and identically distributed birth and death processes each with support 0, 1, . . .. The state space in the composition process is the number of processes in the different states 0, 1, . . .. Dealing with the spectral representation requires new extensions of the multivariate Krawtchouk polynomials to orthogonal polynomials on a multinomial distribution with a countable infinity of states. [ABSTRACT FROM AUTHOR]

We study a general class of birth-and-death processes with state space $${\mathbb {N}}$$ that describes the size of a population going to extinction with probability one. This class contains the logistic case. The scale of the population is measured in terms of a 'carrying capacity' $$K$$ . When $$K$$ is large, the process is expected to stay close to its deterministic equilibrium during a long time but ultimately goes extinct. Our aim is to quantify the behavior of the process and the mean time to extinction in the quasi-stationary distribution as a function of $$K$$ , for large $$K$$ . We also give a quantitative description of this quasi-stationary distribution. It turns out to be close to a Gaussian distribution centered about the deterministic long-time equilibrium, when $$K$$ is large. Our analysis relies on precise estimates of the maximal eigenvalue, of the corresponding eigenvector and of the spectral gap of a self-adjoint operator associated with the semigroup of the process. [ABSTRACT FROM AUTHOR]

We consider a state-dependent $$M_{n}$$ / $$G_{n}$$ /1 queueing system with both finite and infinite buffer sizes. We allow the arrival rate of customers to depend on the number of people in the system. Service times are also state dependent and service rates can be modified at both arrivals and departures of customers. We show that the steady-state solution of this system at arbitrary times can be derived using the supplementary variable method, and that the system's state at arrival epochs can be analyzed using an embedded Markov chain. For the system with infinite buffer size, we first obtain an expression for the steady-state distribution of the number of customers in the system at both arbitrary and arrival times. Then, we derive the average service time of a customer observed at both arbitrary times and arrival epochs. We show that our state-dependent queueing system is equivalent to a Markovian birth-and-death process. This equivalency demonstrates our main insight that the $$M_{n}$$ / $$G_{n}$$ /1 system can be decomposed at any given state as a Markovian queue. Thus, many of the existing results for systems modeled as an M / M / 1 queue can be carried through to the much more practical M / G / 1 model with state-dependent arrival and service rates. Then, we extend the results to the $$M_{n}$$ / $$G_{n}$$ /1 queueing systems with finite buffer size. [ABSTRACT FROM AUTHOR]

This paper investigates the reliable mixed passive and [ABSTRACT FROM AUTHOR]

Service life of many real-life systems cannot be considered infinite, and thus the systems will be eventually stopped or will break down. Some of them may be re-launched after possible maintenance under likely new initial conditions. In such systems, which are often modelled by birth and death processes, the assumption of stationarity may be too strong and performance characteristics obtained under this assumption may not make much sense. In such circumstances, timedependent analysis is more meaningful. In this paper, transient analysis of one class of Markov processes defined on non-negative integers, specifically, inhomogeneous birth and death processes allowing special transitions from and to the origin, is carried out. Whenever the process is at the origin, transition can occur to any state, not necessarily a neighbouring one. Being in any other state, besides ordinary transitions to neighbouring states, a transition to the origin can occur. All possible transition intensities are assumed to be non-random functions of time and may depend (except for transition to the origin) on the process state. To the best of our knowledge, first ergodicity and perturbation bounds for this class of processes are obtained. Extensive numerical results are also provided. [ABSTRACT FROM AUTHOR]

In collaborative wireless networks with low infrastructure, the presence of misbehaving nodes can have a negative impact on network performance. In particular, we are interested in dealing with this nasty presence in road safety applications, based on vehicular ad hoc networks (VANETs). In this paper, we consider as harmful the presence of malicious nodes, which spread false and forged data, and selfish nodes, which cooperate only for their own benefit. To deal with this, we propose a distributed trust model (DTM^2), which is adapted from the job market signaling model. DTM^2 is based on allocating credits to nodes and securely managing these credits. To motivate selfish nodes to cooperate more, our solution establishes the cost of reception to access data, forcing them to earn credits. Moreover, to detect and exclude malicious nodes, DTM^2 requires the cost of sending, using signaling values inspired from economics and based on the node's behavior so that the more malicious a node is, the higher its sending cost, thus limiting their participation in the network. Similarly, rewards are given to nodes whose sent messages are considered truthful and that paid a sending cost considered correct. The latter is a guarantee for the receivers about the truthfulness of the message since, in the case of message refusal, the source node is not rewarded, despite its payment. We validated DTM^2 via a theoretical study using Markov chains and with a set of simulations in both urban and highway scenarios. Both theoretical and simulation results show that DTM^2 excludes from the network 100% of malicious nodes without causing any false-positive detection. Moreover, our solution guarantees a good ratio of reception, even in the presence of selfish nodes. [ABSTRACT FROM PUBLISHER]

We extend to Markov-modulated Brownian motion (MMBM) the renewal approach which has been successfully applied to the analysis of Markov-modulated fluid models. It has been shown recently that MMBM may be expressed as the limit of a parameterized family of Markov-modulated fluid models. We prove that the weak convergence also holds for systems with two reflecting boundaries, one at zero and one at $$b >0$$ , and that the stationary distributions of the approximating fluid models converge to the stationary distribution of the two-sided reflected MMBM. In so doing, we obtain a new representation for the stationary distribution. It is factorised into a vector determined by the phase behaviour when the fluid is either at the level 0 or the level $$b$$ , and a matrix expression characteristic of the process when the fluid is in the open interval $$(0,b)$$ . [ABSTRACT FROM AUTHOR]

Two sequentially Markov coalescent models (SMC and SMC9) are available as tractable approximations to the ancestral recombination graph (ARG). We present a Markov process describing coalescence at two fixed points along a pair of sequences evolving under the SMC9. Using our Markov process, we derive a number of new quantities related to the pairwise SMC9, thereby analytically quantifying for the first time the similarity between the SMC and the ARG. We use our process to show that the joint distribution of pairwise coalescence times at recombination sites under the SMC is the same as it is marginally under the ARG, which demonstrates that the SMC is, in a particular well-defined, intuitive sense, the most appropriate first-order sequentially Markov approximation to the ARG. Finally, we use these results to show that population size estimates under the pairwise SMC are asymptotically biased, while under the pairwise SMC they are approximately asymptotically unbiased. [ABSTRACT FROM AUTHOR]

This paper is concerned with the well-posedness of a problem for age-dependent delayed population equations describing the birth and death process with account of death caused by pregnancy. The proof of well-posedness is based on the authors' recent paper [Proceedings of the American Mathematical Society, 138(12)(2010) 4455-4468]. [ABSTRACT FROM AUTHOR]

Since fluctuation analysis was first introduced by Luria and Delbrück in 1943, it has been widely used to make inference about spontaneous mutation rates in cultured cells. Under certain model assumptions, the probability distribution of the number of mutants that appear in a fluctuation experiment can be derived explicitly, which provides the basis of mutation rate estimation. It has been shown that, among various existing estimators, the maximum likelihood estimator usually demonstrates some desirable properties such as consistency and lower mean squared error. However, its application in real experimental data is often hindered by slow computation of likelihood due to the recursive form of the mutant-count distribution. We propose a fast maximum likelihood estimator of mutation rates, MLE-BD, based on a birth–death process model with non-differential growth assumption. Simulation studies demonstrate that, compared with the conventional maximum likelihood estimator derived from the Luria–Delbrück distribution, MLE-BD achieves substantial improvement on computational speed and is applicable to arbitrarily large number of mutants. In addition, it still retains good accuracy on point estimation. [ABSTRACT FROM AUTHOR]

A multi-type continuous state and continuous time branching process with immigration satisfying some moment conditions is identified as a pathwise unique strong solution of certain stochastic differential equation with jumps. [ABSTRACT FROM AUTHOR]

Autocorrelated data arise in a variety of processes. To statistically monitor such processes, special statistical tools are needed to account for these correlations. The most common set of tools for such purpose is the autoregressive integrated moving average (ARIMA) models. Implementation of ARIMA models requires a fair amount of background understanding of how these models work, because the model selection step is essential. In this paper, we propose a new monitoring technique based on the use of hidden Markov models. The proposed monitoring method is powerful yet simple to use technique because it only requires a basic knowledge in statistics. Simulation results show that the proposed method performs similar to the ARIMA models in terms of average run length for detecting out of control processes and false alarms. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

ABSTRACT We propose a new model for multivariate Markov chains of order one or higher on the basis of the mixture transition distribution (MTD) model. We call it the MTD-Probit. The proposed model presents two attractive features: it is completely free of constraints, thereby facilitating the estimation procedure, and it is more precise at estimating the transition probabilities of a multivariate or higher-order Markov chain than the standard MTD model. [ABSTRACT FROM AUTHOR]

We study the probability of extinction for single-type and multi-type continuous-time linear birth-and-death processes in a finite Markovian environment. The probability of extinction is equal to 1 almost surely if and only if the basic reproduction number $$R_0$$ is $$\le 1$$ , the key point being to identify a suitable definition of $$R_0$$ for such a result to hold. [ABSTRACT FROM AUTHOR]

The results of numerical modeling of the geomagnetic secular variation by the method of the Giant Gaussian Process (GGP) are presented and compared with the information derived from the presentday databases for paleointensity. The variances of the positions of the virtual geomagnetic pole (VGP) calculated from the synthetic and experimental data (Brunhes epoch, effusive rocks) are nearly similar, which supports the validity of the theoretical model. The average value of the virtual axial geomagnetic dipole (VADM) calculated from the PINT world database on paleointensity and the Sint-2000 model is lower than VADM calculated by the GGP model; at the same time, the estimates based on the archaeomagnetic data give the VADM value slightly above the model prediction. The largest difference is observed in the variances of VADM, which is for all the three databases noticeably higher than the value calculated from the GGP model. Most probably, this is due to the contribution of the neglected measurement errors of VADM. [ABSTRACT FROM AUTHOR]

We study the conditions for positive recurrence and transience of multi-dimensional birth-and-death processes describing the evolution of a large class of stochastic systems, a typical example being the randomly varying number of flow-level transfers in a telecommunication wire-line or wireless network. First, using an associated deterministic dynamical system, we provide a generic method to construct a Lyapunov function when the drift is a smooth function on RN. This approach gives an elementary and direct proof of ergodicity. We also provide instability conditions. Our main contribution consists of showing how discontinuous drifts change the nature of the stability conditions and of providing generic sufficient stability conditions having a simple geometric interpretation. These conditions turn out to be necessary (outside a negligible set of the parameter space) for piecewise constant drifts in dimension two. [ABSTRACT FROM AUTHOR]

For any branching process, we demonstrate that the typical total number rmp(vτ) of events triggered over all generations within any sufficiently large time window τ exhibits, at criticality, a superlinear dependence rmp(vτ) ~ (vτ)γ (with γ > 1) on the total number vτ of the immigrants arriving at the Poisson rate v. In branching processes in which immigrants (or sources) are characterized by fertilities distributed according to an asymptotic power-law tail with tail exponent 1 < γ < 2, the exponent of the superlinear law for rmp(vτ) is identical to the exponent y of the distribution of fertilities. For γ > 2 and for standard branching processes without power-law distribution of fertilities, rmp(vτ) ~ (vτ)². This scaling law replaces and tames the divergence vτ/(l -- n) of the mean total number Rt(τ) of events, as the branching ratio (defined as the average number of triggered events of first generation per source) tends to 1. The derivation uses the formalism of generating probability functions. The corresponding prediction is confirmed by numerical calculations, and an heuristic derivation enlightens its underlying mechanism. We also show that Rt(τ) is always linear in VT even at criticality (n = 1 ). Our results thus illustrate the fundamental difference between the mean total number, which is controlled by a few extremely rare realizations, and the typical behavior represented by rmp(vτ). [ABSTRACT FROM AUTHOR]

We consider a FIFO queue defined by a QBD process. When the number of phases of the QBD process is finite, it has been proved that the stationary distribution of sojourn times in that queue can be represented as a phase-type distribution. In this paper, we extend this result to the case where the number of phases of the QBD process is countably many and obtain several kinds of asymptotic formula for the steady-state tail probability of sojourn times in the queue when the tail probability decays in exact exponential form. [ABSTRACT FROM AUTHOR]

Hsp70 molecular chaperones are ubiquitous. By preventing aggregation, promoting folding, and regulating degradation, Hsp70s are major factors in the ability of cells to maintain proteostasis. Despite a wealth of functional information, little is understood about the evolutionary dynamics of Hsp70s. We undertook an analysis of Hsp70s in the fungal clade Ascomycota. Using the well-characterized 14 Hsp70s of Saccharomyces cerevisiae, we identified 491 orthologs from 53 genomes. Saccharomyces cerevisiae Hsp70s fall into seven subfamilies: four canonical-type Hsp70 chaperones (SSA, SSB, KAR, and SSC) and three atypical Hsp70s (SSE, SSZ, and LHS) that play regulatory roles, modulating the activity of canonical Hsp70 partners. Each of the 53 surveyed genomes harbored at least one member of each subfamily, and thus establishing these seven Hsp70s as units of function and evolution. Genomes of some species contained only one member of each subfamily that is only seven Hsp70s. Overall, members of each subfamily formed a monophyletic group, suggesting that each diversified from their corresponding ancestral gene present in the common ancestor of all surveyed species. However, the pattern of evolution varied across subfamilies. At one extreme, members of the SSB subfamily evolved under concerted evolution. At the other extreme, SSA and SSC subfamilies exhibited a high degree of copy number dynamics, consistent with a birth–death mode of evolution. KAR, SSE, SSZ, and LHS subfamilies evolved in a simple divergent mode with little copy number dynamics. Together, our data revealed that the evolutionary history of this highly conserved and ubiquitous protein family was surprising complex and dynamic. [ABSTRACT FROM AUTHOR]

With the increase of product reliability, collecting time-to-failure data is becoming difficult, and degradation-based method has gained popularity. In this paper, a novel multi-hidden semi-Markov model is proposed to identify degradation and estimate remaining useful life of a system. Multiple fused features are used to describe the degradation process so as to improve the effectiveness and accuracy. The similarities of the features are depicted by a new variable combined with forward and backward variables to reduce computational effort. The degradation state is identified using modified Viterbi algorithm, in which linear function is adopted to describe the contribution of each feature to the state recognition. Subsequently, the remaining useful life can be forecasted by backward recursive equations. A case study is presented, and the results demonstrate the validity and effectiveness of the proposed method. Copyright © 2012 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

Duality relations between continuous-state and discrete-state stochastic processes with continuous time have already been studied and used in various research fields. We propose extended duality relations which enable us to derive discrete-state stochastic processes from arbitrary diffusion-type partial differential equations. The derivation is based on the Doi-Peliti formalism, and it will be clarified that additional states for the discrete-state stochastic processes must be considered in some cases. [ABSTRACT FROM AUTHOR]

Studies of wild vertebrates have provided evidence of substantial differences in lifetime reproduction among individuals and the sequences of life history 'states' during life (breeding, nonbreeding, etc.). Such differences may reflect 'fixed' differences in fitness components among individuals determined before, or at the onset of reproductive life. Many retrospective life history studies have translated this idea by assuming a 'latent' unobserved heterogeneity resulting in a fixed hierarchy among individuals in fitness components. Alternatively, fixed differences among individuals are not necessarily needed to account for observed levels of individual heterogeneity in life histories. Individuals with identical fitness traits may stochastically experience different outcomes for breeding and survival through life that lead to a diversity of 'state' sequences with some individuals living longer and being more productive than others, by chance alone. The question is whether individuals differ in their underlying fitness components in ways that cannot be explained by observable 'states' such as age, previous breeding success, etc. Here, we compare statistical models that represent these opposing hypotheses, and mixtures of them, using data from kittiwakes. We constructed models that accounted for observed covariates, individual random effects (unobserved heterogeneity), first-order Markovian transitions between observed states, or combinations of these features. We show that individual sequences of states are better accounted for by models incorporating unobserved heterogeneity than by models including first-order Markov processes alone, or a combination of both. If we had not considered individual heterogeneity, models including Markovian transitions would have been the best performing ones. We also show that inference about age-related changes in fitness components is sensitive to incorporation of underlying individual heterogeneity in models. Our approach provides insight into the sources of individual heterogeneity in life histories, and can be applied to other data sets to examine the ubiquity of our results across the tree of life. [ABSTRACT FROM AUTHOR]