threshold effect with stochastic fluctuation in bacteria colony like proliferation dynamics as analyzed through a comparative study of reaction diffusion equations and cellular automata we report a comparative study on pattern formation between the methods of cellular automata ca and reaction diffusion equations rd applying to a morphology of bacterial colony formation to do so we began the study with setting an extremely simple model which was designed to realize autocatalytic proliferation of bacteria denoted as x fed with nutrition n and their inactive state prespore state p1 due to starvation x n 2x and x p1 respectively it was found numerically that while the ca could successfully generate rich patterns ranging from the circular fat structure to the viscous finger like complicated one the naive rd reproduced only the circular pattern but failed to give a finger structure augmenting the rd equations by adding two physical factors i a threshold effect in the dynamics of x n 2x breaking the continuity limit of rd and ii internal noise with onset threshold breaking the inherent symmetry of rd we have found that the viscous finger like realistic patterns are indeed recovered by thus modified rd this highlights the important difference between ca and rd and at the same time clarifies the necessary factors for the complicated patterns to emerge in such a surprisingly simple model system

the system point method in exponential queues a level crossing approach the purpose of this paper is to describe the new system point method for analyzing queues it considers the stationary probability distribution of the waiting time in variations of the m m r queue with first come first served discipline for a large class of service mechanisms it is shown that the stationary probability density function of the waiting time evaluated at w 0 can be interpreted as the long run average of the number of times that the virtual wait becomes less than w per unit time theorems are presented which establish this interpretation of the probability density function of the virtual waiting time in terms of point processes generated by level crossings in the state space these theorems in combination with the principle of stationary set balance generate a system of model equations that can be written directly in addition the forms of these equations are often linear volterra integral equations of the second kind with parameter which yield direct analytical solutions an analogous theorem is proved for a variant of m g 1 two illustrative examples are presented

on conditioning a random walk to stay nonnegative let s be a real valued random walk that does not drift to so p s k 0 for all k 0 we condition s to exceed n before hitting the negative halfline respectively to stay nonnegative up to time n we study under various hypotheses the convergence ofthese conditional laws as n first when s oscillates the two approximations converge to the same probability law this feature may be lost when s drifts to specifically under suitable assumptions on the upper tail of the step distribution the two approximations then converge to different probability laws

capturing the distributional behaviour of the maximum likelihood estimator of a changepoint we consider the problem of estimating the unknown changepoint in a sequence of time ordered observations upper and lower bounds are derived for the asymptotic distribution of the maximum likelihood estimator and methods of approximation are suggested a computationally efficient algorithm is presented for deriving the bounds and approximations for the asymptotic probabilities of the maximum likelihood estimator when the parameters before and after the changepoint are unknown we also show an essentially exponential rate of convergence of the probability distribution of the maximum likelihood estimator from finite samples to the case of infinite samples we apply the algorithm to the cases of normal and exponential distributions for the exponential distribution the lower and upper bounds for the right tail probabilities of the maximum likelihood estimator and the two approximations are identical this is not the case for the normal distribution finally we apply our changepoint analysis for the case of the exponential distribution to data on explosions in british coal mines keywords maximum likelihood estimator maximum of a random walk negative drift parameter change

spreading and shortest paths in systems with sparse long range connections spreading according to simple rules e g of fire or diseases and shortest path distances are studied on d dimensional systems with a small density p per site of long range connections small world lattices the volume v t covered by the spreading quantity on an infinite system is exactly calculated in all dimensions as a function of time t from this the average shortest path distance l r can be calculated as a function of euclidean distance r it is found that l r approximately r for r r c the characteristic length r c which governs the behavior of shortest path lengths diverges logarithmically with l for all p 0

a connection between a system of random walks and rumor transmission abstract we establish a relationship between the phenomenon of rumor transmission on a population and a probabilistic model of interacting particles on the complete graph more precisely we consider variations of the maki thompson epidemic model and the frog model of random walks which were introduced in the scientific literature independently and in different contexts we analyze the markov chains which describe these models and show a coupling between them our connection shows how the propagation of a rumor in a closed homogeneously mixing population can be described by a system of random walks on the complete graph additionally we discuss further applications of the random walk model which are relevant to the modeling of different biological dynamics

optimality of s s policies in inventory models with markovian demand this paper is concerned with a generalization of classical inventory models with fixed ordering costs that exhibit s s policies in our model the distribution of demands in successive periods is dependent on a markov chain the model includes the case of cyclic or seasonal demand the model is further extended to incorporate some other realistic features such as no ordering periods and storage and service level constraints both finite and infinite horizon nonstationary problems are considered we show that s s policies are also optimal for the generalized model as well as its extensions

epidemics and percolation in small world networks we study some simple models of disease transmission on small world networks in which either the probability of infection by a disease or the probability of its transmission is varied or both the resulting models display epidemic behavior when the infection or transmission probability rises above the threshold for site or bond percolation on the network and we give exact solutions for the position of this threshold in a variety of cases we confirm our analytic results by numerical simulation

confidence regions and tests for a change point in a sequence of exponential family random variables summary maximum likelihood methods are used to test for a change in a sequence of independent exponential family random variables with particular emphasis on the exponential distribution the exact null and alternative distributions of the test statistics are found and the power is compared with a test based on a linear trend statistic exact and approximate confidence regions for the change point are based on the values accepted by a level x likelihood ratio test and a modification of the method proposed by cox spj0tvoll 1982 the methods are applied to a classical data set on the time intervals between coal mine explosions and the change in variation of stock market returns in both cases the confidence regions for the change point cover historical events that may have caused the changes

critical behavior of propagation on small world networks we report numerical evidence that an epidemiclike model which can be interpreted as the propagation of a rumor exhibits critical behavior at a finite randomness of the underlying small world network the transition occurs between a regime where the rumor dies in a small neighborhood of its origin and a regime where it spreads over a finite fraction of the whole population critical exponents are evaluated through finite size scaling analysis and the dependence of the critical randomness with the network connectivity is studied the behavior of this system as a function of the small network randomness bears noticeable similarities with an epidemiological model reported recently m kuperman and g abramson phys rev lett 86 2909 2001 in spite of substantial differences in the respective dynamical rules

an improved upper bound for the critical probability of the frog model on homogeneous trees we study the frog model on homogeneous trees a discrete time system of simple symmetric random walks whose description is as follows there are active and inactive particles living on the vertices each active particle performs a simple symmetric random walk having a geometrically distributed random lifetime with parameter 1 p when an active particle hits an inactive particle the latter becomes active we obtain an improved upper bound for the critical parameter for having indefinite survival of active particles in the case of one particle per vertex initial configuration the main tool is to construct a class of branching processes which are dominated by the frog model and analyze their supercritical behavior this approach allows us also to present an upper bound for the critical probability in the case of random initial configuration

poisson arrivals see time averages in many stochastic models particularly in queueing theory poisson arrivals both observe see a stochastic process and interact with it in particular cases and or under restrictive assumptions it has been shown that the fraction of arrivals that see the process in some state is equal to the fraction of time the process is in that state in this paper we present a proof of this result under one basic assumption the process being observed cannot anticipate the future jumps of the poisson process

the maki thompson rumor model on infinite cayley trees in this paper we study the maki thompson rumor model on infinite cayley trees the basic version of the model is defined by assuming that a population represented by a graph is subdivided into three classes of individuals ignorants spreaders and stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate one at the same rate a spreader becomes a stifler after a contact with other nearest neighbor spreaders or stiflers in this work we study this model on infinite cayley trees which is formulated as a continuous times markov chain and we extend our analysis to the generalization in which each spreader ceases to propagate the rumor right after being involved in a given number of stifling experiences we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

dynamics of rumor propagation on small world networks we study the dynamics of an epidemiclike model for the spread of a rumor on a small world network it has been shown that this model exhibits a transition between regimes of localization and propagation at a finite value of the network randomness here by numerical means we perform a quantitative characterization of the evolution in the two regimes the variant of dynamic small worlds where the quenched disorder of small world networks is replaced by randomly changing connections between individuals is also analyzed in detail and compared with a mean field approximation

a rumor spreading model with variable forgetting rate abstract a rumor spreading model with the consideration of forgetting rate changing over time is examined in small world networks the mean field equations are derived to describe the dynamics of rumor spreading in small world networks further numerical solutions are conducted on livejournal an online social blogging platform to better understand the performance of the model results show that the forgetting rate has a significant impact on the final size of rumor spreading the larger the initial forgetting rate or the faster the forgetting speed the smaller the final size of the rumor spreading numerical solutions also show that the final size of rumor spreading is much larger under a variable forgetting rate compared to that under a constant forgetting rate

river stream flows in the northern quebec labrador region a multivariate change point analysis via maximum likelihood 1 spring stream flows of six rivers that flow in the northern quebec labrador region are modeled in a multivariate gaussian framework and have been analyzed for possible change points in their average flows the multivariate formulation takes into account correlations among the flows of the six rivers that flow in the same region significant change was detected in the multivariate mean vector and the unknown change point is estimated by the method of maximum likelihood estimation we then establish the asymptotic distribution of the change point maximum likelihood estimate wherein we show that the multivariate case can be transformed into an equivalent univariate problem a simulation study is carried out to investigate the robustness of the asymptotic distribution to departures from normality and independence as well as its closeness to finite samples the asymptotic distribution allowed us to compute confidence interval estimates of the change point in the river flows a decrease in the mean river flows in 1984 identified by the methodology may have been a consequence of a sharp decline in the region s snow cover that occurred about a year or two ahead

a note on a production control model for a facility with limited storage capacity in a random environment abstract this note presents an extension of our earlier paper on a production inventory system with limited storage capacity in a multi state random environment i e e mohebbi a production inventory model with randomly changing environmental conditions eur j res 174 2006 539 552 more specifically it assumes that the production rate as well as the occurrence rate and batch sizes of a compound poisson demand stream are influenced by the state of a continuous time homogeneous markov chain representing the random environment we present an adaptation of the system point method of level crossings which yields the limiting distribution of inventory level stock on hand for a two parameter production control policy when stockouts are lost

a production inventory model with randomly changing environmental conditions we consider a limited capacity production inventory system with linear production rate and compound poisson demand in which both production and demand processes are subject to independently and randomly changing environmental conditions assuming that these conditions are represented by two continuous time homogeneous markov chains and shortages are lost we derive the limiting distribution of the inventory level and present some numerical results in terms of the ensuing performance measures

a spatial stochastic model for rumor transmission we consider an interacting particle system representing the spread of a rumor by agents on the d dimensional integer lattice each agent may be in any of the three states belonging to the set 0 1 2 here 0 stands for ignorants 1 for spreaders and 2 for stiflers a spreader tells the rumor to any of its nearest ignorant neighbors at rate at rate a spreader becomes a stifler due to the action of other nearest neighbor spreaders finally spreaders and stiflers forget the rumor at rate one we study sufficient conditions under which the rumor either becomes extinct or survives with positive probability

mean field solution of the small world network model the small world network model is a simple model of the structure of social networks which possesses characteristics of both regular lattices and random graphs the model consists of a one dimensional lattice with a low density of shortcuts added between randomly selected pairs of points these shortcuts greatly reduce the typical path length between any two points on the lattice we present a mean field solution for the average path length and for the distribution of path lengths in the model this solution is exact in the limit of large system size and either a large or small number of shortcuts

laws of large numbers for the frog model on the complete graph the frog model is a stochastic model for the spreading of an epidemic on a graph in which a dormant particle starts to perform a simple random walk on the graph and to awaken other particles once i

scaling and percolation in the small world network model in this paper we study the small world network model of watts and strogatz which mimics some aspects of the structure of networks of social interactions we argue that there is one nontrivial length scale in the model analogous to the correlation length in other systems which is well defined in the limit of infinite system size and which diverges continuously as the randomness in the network tends to zero giving a normal critical point in this limit this length scale governs the crossover from large to small world behavior in the model as well as the number of vertices in a neighborhood of given radius on the network we derive the value of the single critical exponent controlling behavior in the critical region and the finite size scaling form for the average vertex vertex distance on the network and using series expansion and pad e approximants find an approximate analytic form for the scaling function we calculate the effective dimension of small world graphs and show that this dimension varies as a function of the length scale on which it is measured in a manner reminiscent of multifractals we also study the problem of site percolation on small world networks as a simple model of disease propagation and derive an approximate expression for the percolation probability at which a giant component of connected vertices first forms in epidemiological terms the point at which an epidemic occurs the typical cluster radius satisfies the expected finite size scaling form with a cluster size exponent close to that for a random graph all our analytic results are confirmed by extensive numerical simulations of the model

theory of rumour spreading in complex social networks we introduce a general stochastic model for the spread of rumours and derive mean field equations that describe the dynamics of the model on complex social networks in particular those mediated by the internet we use analytical and numerical solutions of these equations to examine the threshold behaviour and dynamics of the model on several models of such networks random graphs uncorrelated scale free networks and scale free networks with assortative degree correlations we show that in both homogeneous networks and random graphs the model exhibits a critical threshold in the rumour spreading rate below which a rumour cannot propagate in the system in the case of scale free networks on the other hand this threshold becomes vanishingly small in the limit of infinite system size we find that the initial rate at which a rumour spreads is much higher in scale free networks than in random graphs and that the rate at which the spreading proceeds on scale free networks is further increased when assortative degree correlations are introduced the impact of degree correlations on the final fraction of nodes that ever hears a rumour however depends on the interplay between network topology and the rumour spreading rate our results show that scale free social networks are prone to the spreading of rumours just as they are to the spreading of infections they are relevant to the spreading dynamics of chain emails viral advertising and large scale information dissemination algorithms on the internet

a fluid model for systems with random disruptions motivated by modeling manufacturing systems in which job arrivals and processing times are essentially deterministic but the environment is typically random we develop a fluid model with random disruptions the quality and relevance of such a model are supported by the following facts which we establish in this study the fluid model is more susceptible to analysis its dynamical sample paths are continuous and piecewise linear and its stationary behavior can be studied using standard approaches in random walk the model is a limit of corresponding queueing systems with random disruptions that are usually difficult to analyze there exist pathwise bounds between the fluid model and d d 1 queues with random disruptions and the bounds are simple and tight

a general regression changepoint test for time series data abstractthis article develops a test for a single changepoint in a general setting that allows for correlated time series regression errors a seasonal cycle time varying regression factors and covariate information within a changepoint statistic is constructed from likelihood ratio principles and its asymptotic distribution is derived the asymptotic distribution of the changepoint statistic is shown to be invariant of the seasonal cycle and the covariates should the latter obey some simple limit laws however the limit distribution depends on any time varying factors a new test based on arma residuals is developed and is shown to have favorable properties with finite samples driving our work is a changepoint analysis of the mauna loa record of monthly carbon dioxide concentrations this series has a pronounced seasonal cycle a nonlinear trend heavily correlated regression errors and covariate information in the form of climate oscillations in the end we find a prominent changepoint in the ea

stability analysis of i2s2r rumor spreading model in complex networks abstract in this paper we propose i 2 s 2 r rumor spreading model with general incidence function in homogeneous networks and i 2 s 2 r rumor spreading model in heterogeneous network in the first model we establish a dynamic model for rumor propagation called i 2 s 2 r in which we consider two general incidence functions then the basic reproduction number the locally stability and the global stability of rumor free equilibrium are discussed in the second model we mainly consider another network structure i e heterogeneous network thus we propose a new dynamic model for rumor spreading by deriving the mean field equation the basic reproduction number is obtained moreover the global stability of rumor free equilibrium is established by comparison principle by using a monotone iterative algorithm the global attractivity of the positive equilibrium is proved finally we obtain some important conclusions with numerical simulations