On the distributions of infinite server queues with batch arrivals

Queues that feature multiple entities arriving simultaneously are among the oldest models in queueing theory, and are often referred to as “batch” (or, in some cases, “bulk”) arrival queueing systems. In this work, we study the effect of batch arrivals on infinite server queues. We assume that the arrival epochs occur according to a Poisson process, with treatment of both stationary and non-stationary arrival rates. We consider both exponentially and generally distributed service durations, and we analyze both fixed and random arrival batch sizes. In addition to deriving the transient mean, variance, and moment-generating function for time-varying arrival rates, we also find that the steady-state distribution of the queue is equivalent to the sum of scaled Poisson random variables with rates proportional to the order statistics of its service distribution. We do so through viewing the batch arrival system as a collection of correlated sub-queues. Furthermore, we investigate the limiting behavior of the process through a batch scaling of the queue and through fluid and diffusion limits of the arrival rate. In the course of our analysis, we make important connections between our model and the harmonic numbers, generalized Hermite distributions, and truncated polylogarithms.

     

Numerical study of mathematical models of micromechanics with periodic impulse action

We consider the results of the numerical study of the mathematical models of two microelectromechanical systems (MEMS). The models as initial boundary value problems describe the cylindrical flexure of an elastic beam as a movable electrode under action of a repetitive intensity impulse of the electrostatic field between the movable and fixed electrodes in a microgap. In the first problem, both ends of the beamare rigidly fixed, while in the second problemwe consider a cantilever beam. The range of the parameters is found where the model has two periodic solutions with periods of impulse action one of which is stable and the other is unstable.     

Exact travelling wave solutions of reaction-diffusion models of fractional order

Reaction-diffusion models are used in different areas of chemistry problems. Also, coupled reaction-diffusion systems describing the spatio- temporal dynamics of competition models have been widely applied in many real world problems. In this paper, we consider a coupled fractional system with diffusion and competition terms in ecology, and reaction-diffusion growth model of fractional order with Allee effect describing and analyzing the spread dynamic of a single population under different dispersal and growth rates. Finding the exact solutions of such models are very helpful in the theories and numerical studies. Exact traveling wave solutions of the above reaction-diffusion models are found by means of the $Q$-function method. Moreover, graphic illustrations in two and three dimensional plots of some of the obtained solutions are also given to predict their behaviours.     

Mathematical study of two-variable systems with adaptive numerical methods

In this paper, we consider reaction–diffusion systems arising from two-component predator–prey models with Smith growth functional response. The mathematical approach used here is in two folds since the time-dependent partial differential equations consist of both linear and nonlinear terms. We discretize the stiff or moderately stiff term with the fourth-order difference operator and advance the resulting nonlinear system of ordinary differential equations with the two competing families of the exponential time differencing (ETD) schemes, and we analyze them for stability. Numerical comparison between these two methods for solving various predator–prey population models with functional responses are also presented. Numerical results show that the techniques require less computational work. Also in the numerical results, some emerging spatial patterns are unveiled.     

Dynamics of a reaction–diffusion–ODE system with quiescence

We consider a reaction–diffusion–ODE quiescent model in which the species can switch between mobile and immobile categories. We assume that the population inhabits a bounded region and study how its dynamics depend on the parameters describing switching rates and local population dynamics. Our results suggest that the transfer displays a stabilizing effect and inhibits the generation of spatial periodic solutions. A new method to obtain global stability and dissipative structure is also explored by constructing Lyapunov functionals to overcome the loss of compactness.     

Evolution of dispersal in a spatially periodic integrodifference model

An integrodifference model describing the reproduction and dispersal of a population is introduced to investigate the evolution of dispersal in a spatially periodic habitat. The dispersal is determined by a kernel function, and the dispersal strategy is defined as the probability of population individuals’ moving to a different habitat. Both conditional and unconditional dispersal strategies are investigated, the distinction being whether dispersal depends on local environmental conditions. For competing unconditional dispersers, we prove that the population with the smaller dispersal probability always prevails. Alternatively, for conditional dispersers, it is shown that the strategy known as ideal free dispersal is both sufficient and necessary for evolutionary stability. These results extend those in the literature for discrete diffusion models in finite patchy landscapes and from reaction–diffusion models.     

Internalization and end flux in morphogen gradient formation

Two simple reaction–diffusion systems of partial differential equations and auxiliary conditions governing the activities of diffusible ligands such as Dpp in anterior–posterior axis of Drosophila wing imaginal discs were previously formulated and investigated by numerical simulations in [Developmental Cell 2 (2002) 785–796]. System B focuses on diffusion, reversible binding with receptors and ligand-mediated degradation for a fixed receptor concentration uniform in time and space. System C extended this basic but meaningful model to allow for endocytosis, exocytosis and receptor synthesis and degradation. The present paper provides a mathematical underpinning for the computational studies of these two systems and some insight gained from our analysis. We will see for instance that the two boundary value problems governing the steady state for the two systems are identical in form. This result will enable us to avoid dealing with internalization explicitly when we investigate other complex morphogen activities such as the effects of (1) feedback and (2) diffusible and non-diffusible molecules competing for ligands and receptors to inhibit cell signaling and pattern formation. The principal contribution of the present work pertains to the extension of System C to allow for a ligand flux at the source end. The more general model has many significant consequences including the removal of a limitation of previous models on ligand synthesis rate for the existence of steady state behavior. Linear stability of the corresponding steady state behavior is established. While the actual decay rate of transients is less accessible in this new model, it is possible to obtain tight upper and lower bounds for the decay rate in terms of the (effective) degradation rate of the receptors and that of the ligand-receptor complexes.     

A parameter estimation scheme for a class of sequential hybrid systems

It may happen that the equations governing the response of dynamical systems have some parameters whose values may not be known a priori and have to be obtained using parameter estimation schemes. In this article, we present a parameter estimation scheme for a class of sequential hybrid systems. By hybrid systems, we refer to those systems whose response is described by different governing equations corresponding to various regimes/modes of operation along with some criteria to switch between the same. In a sequential hybrid system, the different modes are arranged in a specific sequence and the system can switch from a given mode to either the previous mode or the following mode in this sequence. Here, we consider those systems whose governing equations consist of ordinary differential equations and algebraic equations. The conditions for switching between the various modes (referred to as transition conditions) are in the form of linear inequalities involving the system output. We shall first consider the case where the transition conditions are known completely. We present a parameter update scheme along with sufficient conditions that will guarantee bounded parameter estimation errors. Then, we shall consider the case where the transition conditions are not known in the sense that some parameters in these conditions are not known. We present a parameter estimation scheme for this case. We illustrate the performance of the parameter estimation scheme in both cases with some examples.     

Competitive exclusion in a nonlocal reaction–diffusion–advection model of phytoplankton populations

We continue our study on the global dynamics of a nonlocal reaction–diffusion–advection system modeling the population dynamics of two competing phytoplankton species in a eutrophic environment, where both populations depend solely on light for their metabolism. In our previous work, we proved that system (1.1) is a strongly monotone dynamical system with respect to a non-standard cone related to the cumulative distribution functions, and further determined the global dynamics when the species have either identical diffusion rate or identical advection rate. In this paper, we study the trade-off of diffusion and advection and their joint influence on the outcome of competition. Two critical curves for the local stability of two semi-trivial equilibria are analyzed, and some new competitive exclusion results are obtained. Our main tools, besides the theory of monotone dynamical system, include some new monotonicity results for the principal eigenvalues of elliptic operators in one-dimensional domains.     

Effects of population dispersal and impulsive control tactics on pest management

In this paper, we firstly consider a Lotka–Volterra predator–prey model with impulsive constant releasing for natural enemies and a proportion of killing or catching pests at fixed moments, and we have proved that there exists a pest-eradication periodic solution which is globally asymptotically stable. Further, we extend the model for the population to move in a two-patch environment. The effects of population dispersal and impulsive control tactics are investigated, i.e. we chiefly address the question of whether population dispersal is beneficial or detrimental for pest persistence. To do this, some special cases are theoretically investigated and numerical investigations are done for general case. The results indicate that for some ranges of dispersal rates, population dispersal is beneficial to pest control, but for other ranges, it is harmful. These clarify that we can get some new effective pest control strategies by controlling the dispersal rates of pests and natural enemies.     

Adoption as a social marker: Innovation diffusion with outgroup aversion

Social identities are among the key factors driving behavior in complex societies. Signals of social identity are known to influence individual behaviors in the adoption of innovations. Yet the population-level consequences of identity signaling on the diffusion of innovations are largely unknown. Here we use both analytical and agent-based modeling to consider the spread of a beneficial innovation in a structured population in which there exist two groups who are averse to being mistaken for each other. We investigate the dynamics of adoption and consider the role of structural factors such as demographic skew and communication scale on population-level outcomes. We find that outgroup aversion can lead to adoption being delayed or suppressed in one group, and that population-wide underadoption is common. Comparing the two models, we find that differential adoption can arise due to structural constraints on information flow even in the absence of intrinsic between-group differences in adoption rates. Further, we find that patterns of polarization in adoption at both local and global scales depend on the details of demographic organization and the scale of communication. This research has particular relevance to widely beneficial but identity-relevant products and behaviors, such as green technologies, where overall levels of adoption determine the positive benefits that accrue to society at large.     

Constant-sign solutions of systems of higher order boundary value problems with integrable singularities

We consider some systems of boundary value problems where the nonlinearities may be singular in the independent variable and may also be singular in the dependent arguments. Using the Schauder fixed point theorem, we establish criteria such that the systems of boundary value problems have at least one constant-sign solution.     

An ergodic theorem for Schlögl models with small migration

Summary We consider a class of reaction-diffusion processes with state space N^Zd. The reaction part is described by a birth and death process where the rates are given by certain polynomials. The diffusion part is an irreducible symmetric random walk. We prove ergodicity in the case of a sufficiently small migration rate. For the proof we couple two processes and show that the density of the discrepancies goes to zero.     

Filter‐based portfolio strategies in an HMM setting with varying correlation parametrizations

We consider portfolio optimization in a regime‐switching market. The assets of the portfolio are modeled through a hidden Markov model (HMM) in discrete time, where drift and volatility of the single assets are allowed to switch between different states. We consider different parametrizations of the involved asset covariances: statewise uncorrelated assets (though linked through the common Markov chain), assets correlated in a state‐independent way, and assets where the correlation varies from state to state. As a benchmark, we also consider a model without regime switches. We utilize a filter‐based expectation‐maximization (EM) algorithm to obtain optimal parameter estimates within this multivariate HMM and present parameter estimators in all three HMM settings. We discuss the impact of these different models on the performance of several portfolio strategies. Our findings show that for simulated returns, our strategies in many settings outperform naïve investment strategies, like the equal weights strategy. Information criteria can be used to detect the best model for estimation as well as for portfolio optimization. A second study using real data confirms these findings.     

Nonuniqueness for the vanishing viscosity solution with fixed initial condition in a nonstrictly hyperbolic system of conservation laws

We consider the Riemann problem for a system of two decoupled, nonstrictly hyperbolic, Burgers-like conservation equations with added artificial viscosity. We analytically establish two different vanishing viscosity limits for the solution of this system, which correspond to the two cases where one of the viscosities vanishes much faster than the other. This is done without altering the initial condition as is necessary with travelling wave methods. Numerical evidence is then provided to show that when the two viscosities vanish at the same rate, the solution converges to a limit that lies strictly between the two previously established limits. Finally, we use control theory to explain the mechanism behind this nonuniqueness behavior, which indicates other systems of nonstrictly hyperbolic conservation laws where nonuniqueness will occur.     

Bifurcation diagrams of population models with nonlinear,diffusion

We develop analytical and numerical tools for the equilibrium solutions of a class of reaction–diffusion models with nonlinear diffusion rates. Such equations arise from population biology and material sciences. We obtain global bifurcation diagrams for various nonlinear diffusion functions and several growth rate functions.     

Necessary and sufficient conditions for quadratic stabilizability of switched systems on non-uniform time domains

In this paper, we consider the quadratic stabilizability via state feedback for a particular class of switched systems that evolve on a non-uniform time domain by introducing time scales theory. The system considered switches between a continuous-time subsystem with variable lengths and a discrete-time subsystem with variable discrete step sizes. Necessary and sufficient conditions are derived to guarantee the quadratic stability of this class of switched systems via a switching state feedback law based on the existence of a common positive definite matrix satisfying the quadratic stabilizability condition by considering that the two subsystems are unstable. By state feedback, we mean that the switching among subsystems depends on the system states. Current results for this kind of state switching feedback control are derived only for switched systems evolving on a continuous time domain or a discrete time domain with fixed step’s size. These results are not applicable for the particular class of switched systems where there is a mixing between the continuous and discrete dynamics. This motivates the derivation of a new and more general state feedback control law for switched systems in this work. A numerical example illustrating the results is presented.     

Escape times for branching processes with random mutational fitness effects

We consider a large declining population of cells under an external selection pressure, modeled as a subcritical branching process. This population has genetic variation introduced at a low rate which leads to the production of exponentially expanding mutant populations, enabling population escape from extinction. Here we consider two possible settings for the effects of the mutation: Case (I) a deterministic mutational fitness advance and Case (II) a random mutational fitness advance. We first establish a functional central limit theorem for the renormalized and sped up version of the mutant cell process. We establish that in Case (I) the limiting process is a trivial constant stochastic process, while in Case (II) the limit process is a continuous Gaussian process for which we identify the covariance kernel. Lastly we apply the functional central limit theorem and some other auxiliary results to establish a central limit theorem (in the large initial population limit) of the first time at which the mutant cell population dominates the population. We find that the limiting distribution is Gaussian in both Cases (I) and (II), but a logarithmic correction is needed in the scaling for Case (II). This problem is motivated by the question of optimal timing for switching therapies to effectively control drug resistance in biomedical applications.     

Markov-modulated Brownian motions perturbed by catastrophes

ABSTRACT

We consider a one-sided Markov-modulated Brownian motion perturbed by catastrophes that occur at some rates depending on the modulating process. When a catastrophe occurs, the level drops to zero for a random recovery period. Then the process evolves normally until the next catastrophe. We use a semi-regenerative approach to obtain the stationary distribution of this perturbed MMBM. Next, we determine the stationary distribution of two extensions: we consider the case of a temporary change of regime after each recovery period and the case where the catastrophes can only happen above a fixed threshold. We provide some simple numerical illustrations.