Phase-type models for cellular networks supporting voice, video and data traffic

This paper presents an analytical model for cellular networks supporting voice, video and data traffic. Self-similar and bursty nature of the incoming traffic causes correlation in inter-arrival times of the incoming traffic. Therefore, arrival of calls is modeled with Markovian arrival process as it allows for the correlation. Call holding times, cell residence times and retrial times are modeled as phase-type distributions. We consider that the cells in a cellular network are statistically homogeneous, so it is enough to investigate a single cell for the performance analysis of the entire networks. With appropriate assumptions, the stochastic process that describes the state of a cell is a Quasi-birth–death (QBD) process. We derive explicit expressions for the infinitesimal generator matrix of this QBD process. Also, expressions for performance measures are obtained. Further, complexity involved in computing the steady-state probabilities is discussed. Finally, queueing examples are provided that can be obtained as particular cases of the proposed analytical model.     

A Differentiation Theory for Itô's Calculus

Abstract

A peculiar feature of Itô's calculus is that it is an integral calculus that gives no explicit derivative with a systematic differentiation theory counterpart, as in elementary calculus. So, can we define a pathwise stochastic derivative of semimartingales with respect to Brownian motion that leads to a differentiation theory counterpart to Itô's integral calculus? From Itô's definition of his integral, such a derivative must be based on the quadratic variation process. We give such a derivative in this note and we show that it leads to a fundamental theorem of stochastic calculus, a generalized stochastic chain rule that includes the case of convex functions acting on continuous semimartingales, and the stochastic mean value and Rolle's theorems. In addition, it interacts with basic algebraic operations on semimartingales similarly to the way the deterministic derivative does on deterministic functions, making it natural for computations. Such a differentiation theory leads to many interesting applications, some of which we address in an upcoming article.     

A stochastic model for wound healing

Three separate activities of wound healing have been identified: migration, proliferation and differentiation. In this paper we present a mathematical model for the activities of migration and proliferation in an invitro system. The motion of a cell is modelled by a two-dimensional Brownian motion in the “unwounded” media. To reflect the proliferative activity in the wound area, we shall impose growth dynamics on the cells which are position dependent. From the resulting motile-growth stochastic model, we are able to estimate the expected number of cells in the wound at time t. From this, the expected time of wound closure can be predicted.     

Dynamically Active Compartments Coupled by a Stochastically Gated Gap Junction

We analyze a one-dimensional PDE-ODE system representing the diffusion of signaling molecules between two cells coupled by a stochastically gated gap junction. We assume that signaling molecules diffuse within the cytoplasm of each cell and then either bind to some active region of the cell’s membrane (treated as a well-mixed compartment) or pass through the gap junction to the interior of the other cell. We treat the gap junction as a randomly fluctuating gate that switches between an open and a closed state according to a two-state Markov process. This means that the resulting PDE-ODE is stochastic due to the presence of a randomly switching boundary in the interior of the domain. It is assumed that each membrane compartment acts as a conditional oscillator, that is, it sits below a supercritical Hopf bifurcation. In the ungated case (gap junction always open), the system supports diffusion-induced oscillations, in which the concentration of signaling molecules within the two compartments is either in-phase or anti-phase. The presence of a reflection symmetry (for identical cells) means that the stochastic gate only affects the existence of anti-phase oscillations. In particular, there exist parameter choices where the gated system supports oscillations, but the ungated system does not, and vice versa. The existence of oscillations is investigated by solving a spectral problem obtained by averaging over realizations of the stochastic gate.     

The optimal control of diffusions

Using a differentiation result of Blagovescenskii and Freidlin calculations of Bensoussan are simplified and the adjoint process identified in a stochastic control problem in which the control enters both the drift and diffusion coefficients. A martingale representation result of Elliott and Kohlmann is then used to obtain the integrand in a stochastic integral, and explicit forward and backward equations satisfied by the adjoint process are derived.This research was partially supported by NSERC under Grant A7964, the U.S. Air Force Office of Scientific Research under Contract AFOSR-86-0332, and the U.S. Army Research Office under Contract DAAL03-87-K-0102.     

Asymptotic behavior and stability switch for a mature–immature model of cell differentiation

We consider a nonlinear age-structured model, inspired by hematopoiesis modelling, describing the dynamics of a cell population divided into mature and immature cells. Immature cells, that can be either proliferating or non-proliferating, differentiate in mature cells, that in turn control the immature cell population through a negative feedback. We reduce the system to two delay differential equations, and we investigate the asymptotic stability of the trivial and the positive steady states. By constructing a Lyapunov function, the trivial steady state is proven to be globally asymptotically stable when it is the only equilibrium of the system. The asymptotic stability of the positive steady state is related to a delay-dependent characteristic equation. Existence of a Hopf bifurcation and stability switch for the positive steady state is established. Numerical simulations illustrate the stability results.     

Characterization of stem cells using mathematical models of multistage cell lineages

Stem cells dynamics is an important field of research with promising clinical impacts. Due to the revolutionary new technologies of biological data collection, an enormous amount of information on specific factors and genes responsible for cell differentiation is available. However, the mechanisms controlling stem cell self-renewal, maintenance and differentiation are still poorly understood and there exists no general characterization of stem cells based on observable cell properties. We address these problems with the help of mathematical models. Stem cells are described as the cell type that is most responsive to certain environmental signals. This results in a dynamic characterization of stemness that depends on environmental conditions and is not necessarily linked to a unique cell population.     

On the stability of hematopoietic model with feedback control

We propose and analyze a mathematical model of the production and regulation of blood cell population in the bone marrow (hematopoiesis). This model includes the primitive hematopoietic stem cells (PHSC), the three lineages of their progenitors and the corresponding mature blood cells (red blood cells, white cells and platelets). The resulting mathematical model is a nonlinear system of differential equations with several delays corresponding to the cell cycle durations for each type of cells. We investigate the local asymptotic stability of the trivial steady state by analyzing the roots of the characteristic equation. We also prove by a Lyapunov function the global asymptotic stability of this steady state. This situation illustrates the extinction of the cell population in some pathological cases.     

A continuum-mechanical analysis of the influence of mechanical stimuli on biological tissue

Mechanical stimuli play a crucial role in the differentiation process of mesenchymal stem cells (MSC). The resulting mechanical signals are important in the regulation of various cell functions and maintenance of many tissues. The underlying molecular and biophysical mechanisms of the differentiation process are poorly understood. Present remodelling and growth models are purely phenomenological without linkage to cell mechanisms. The presented macroscopic model of MSC mechanics is based on a multiphasic-multicomponent formulation within the framework of Theory of Porous Media (TPM), where a single cell is considered as a mixture of interacting constituents. In particular, the constituents are the solid cytoskeleton saturated by a fluid phase (cytoplasm), which itself consists of a liquid solvent and mobile components, e. g., chemical messengers, proteins, etc. To demonstrate the capabilities of the developed model, first qualitative numerical simulations of the impact of external forces on MSC are presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Inference of Probabilities over a Stochastic IL-System by Quantifier Elimination

An algebraic approach based on quantifier elimination is proposed for the inference of probabilistic parameters over stochastic Lindenmayer systems with interaction, IL-systems. We are concerned with a multi-cellular organism as an instance of a stochastic IL system. The organism starts with one or a few cells, and develops different types of cells with distinct functions. We have constructed a simple model with cell-type order conservation and have assessed conditions for high cell-type diversity. This model is based on the stochastic IL-system for three types of cells. The cell-type order conservation corresponds to interaction terms in the IL-system. In our model, we have successfully inferred algebraic relations between the probabilities for cell-type diversity by using a symbolic method, quantifier elimination (QE). Surprisingly, three modes for the proliferation and transition rates emerged for various ratios of the initial cells to the developed cells. Furthermore, we have found that the high cell-type diversity pattern originates from the cell-type order conservation. Thus, QE has yielded analysis of the IL-system, which has revealed that, during the developing process of multi-cellular organisms, complex but explicit relations exist between cell-type diversity patterns and developmental rates.      

Analysis of a System Describing Proliferative-Quiescent Cell Dynamics

Systems describing the dynamics of proliferative and quiescent cells are commonly used as computational models, for instance for tumor growth and hematopoiesis. Focusing on the very earliest stages of hematopoiesis, stem cells and early progenitors, the authors introduce a new method, based on an energy/Lyapunov functional to analyze the long time behavior of solutions. Compared to existing works, the method in this paper has the advantage that it can be extended to more complex situations. The authors treat a system with space variable and diffusion, and then adapt the energy functional to models with three equations.     

The effect of variety seeking behaviour on optimal product positioning

In this paper we develop a stochastic (first order Markovian) consumer choice model that represents variety seeking behaviour and we investigate the practical implications of this model for optimal product positioning relative to a zero order model that does not incorporate variety seeking. We show that the optimal positioning implications of a variety seeking process is indeed different than those of a (no-variety-seeking) zero order process. Based on intuition, one might expect increased variety seeking to imply that firms should increase the distance between their products in an attribute space. In fact, we show that this effect does occur for relatively low share brands. But just the opposite effect holds for relatively high share brands. That is, variety seeking behaviour generates a desire to more differentiation among low share brands, and a desire for less differentiation among high share brands.     

Stratonovich Calculus with Respect to Fractional Brownian Sheet

Abstract

We introduce two types of Stratonovich stochastic integrals for two-parameter process. The relationship of Stratonovich integrals to Skorohod integrals will be investigated. By using this relationship, we prove that a differentiation formula for fractional Brownian sheet in Stratonovich form can be expressed as the sum of Stratonovich integrals of two types introduced in this article.     

Solution of the stochastic transport equation of neutral particles with anisotropic scattering using RVT technique

Much of the literatures are directed toward the development of a mathematical formalism for a rigorous estimation of the ensemble average of the solution process of a stochastic differential equation (SDE). The Random Variable Transformation technique (RVT) is a powerful technique to get the complete solution for the SDE represented by the probability-density function of the solution process. In this paper, the RVT technique together with a simple integral transformation to the input stochastic process are implemented to get the complete solution of the one-speed transport equation for neutral particles in a semi-infinite stochastic medium with linear anisotropic scattering. The extinction function of the medium (input stochastic process) is assumed to be a continuous random function of position. The probability-density function and hence, the higher order statistical moments of the solution process are presented. Numerical results are given for different distributions of the input stochastic process.     

Modeling and simulation of cell populations interaction

Regenerative medicine and cell therapy provide great hopes for the use of adult and stem cells. The latter are far less present in tissue than the former and must be expanded using cell culture. Stem cells culture requires the conservation of their proliferation and self-renewal capabilities. Still, the complex interaction between cell populations, for example in primary cell cultures, are not well-known and may account for part of the variability of such cultures. In order to represent and understand the evolution of cultured stem cells, we present here a mathematical model of cell proliferation and differentiation. Based on the formalism of cellular automata, this model simulates the evolution of several cell classes (which may represent either different levels of differentiation or different cell types) in an environment modeling the growth medium. We model the cell cycle as on the one hand a quiescence phase during which a cell rests, and on the other hand a division phase during which the cell starts the division process. In order to represent cell–cell interaction, the transition probability between those phases depends on the local composition of the growth medium depending itself on neighboring cells. An interaction between cellular populations is represented by a quantitative parameter which has a direct impact on cellular proliferation. Differentiation results in a change of the cell class and depends on the biological model studied : it may result from an asymmetric division or be a consequence of the local composition of the growth medium. This mathematical model aims at a better understanding of the interactions between cell populations in a culture. By defining constraints on the potential or the type of the cells at the end of a culture, it will then be possible to find optimal experimental conditions for cell production.     

Population growth: Stochastic treatment in compartments

The growth of microbial cultures is studied by means of stochastic processes. A compartment structure is used to take into account cell maturation stages, having an inflow of cells from one compartment to the next. Dilution and cell division are also considered. A differential equation, first proposed by Kiefer, is studied and the expected number of cells in each compartment is found for several interesting cases.     

A multi-agent model describing self-renewal of differentiation effects on the blood cell population

In this work, a new multi-agent model is used to describe blood cell population dynamics. More particularly, we focus our simulations here on differentiation and self-renewal process based on cell communication. We consider the different cases where progenitor cells are able to self-renew or not in the bone marrow. As a consequence of this study, we give some possible explanations of the mechanism for recovery of the system under important blood loss or blood diseases such as anemia.     

Effect of bounded noise on chaotic motions of stochastically perturbed slowly varying oscillator

The effect of bounded noise on the chaotic behavior of a class of slowly varying oscillators is investigated. The stochastic Melnikov method is employed and then the criteria in both mean and mean-square sense are derived. The threshold amplitude of bounded noise given by stochastic Melnikov process is in good comparison with one determined by the numerical simulation of top Lyapunov exponents. The presence of noise scatters the chaotic domain in parameter space and the larger noise intensity results in a sparser and more irregular region. Both the simple cell mapping method and the generalized cell mapping method are applied to demonstrate the effects of noises on the attractors. Results show that the attractors are diffused and smeared by bounded noise and if the noise intensity increases, the diffusion is exacerbated.     

Curved exponential families of stochastic processes and their envelope families

Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.     

Existence for dynamic contact of a stochastic viscoelastic Gao Beam

This work presents and analyzes a model for the vibrations of a viscoelastic Gao Beam, which may come in contact with a deformable random foundation and allows for stochastic inputs. The body force involves a stochastic integral that includes Brownian motion. In addition, the gap between the beam and the foundation is a stochastic process, which is one of the novelties in the paper, and contact is described with the normal compliance condition. The existence and uniqueness of strong solutions to the model is established and it is shown that the solutions are adapted to the filtration determined by a given Wiener process for the stochastic force noise term.