Stochastic processes with orthogonal polynomial eigenfunctions

Markov processes which are reversible with either Gamma, Normal, Poisson or Negative Binomial stationary distributions in the Meixner class and have orthogonal polynomial eigenfunctions are characterized as being processes subordinated to well-known diffusion processes for the Gamma and Normal, and birth and death processes for the Poisson and Negative Binomial. A characterization of Markov processes with Beta stationary distributions and Jacobi polynomial eigenvalues is also discussed.     

Stationary and stationarizable regimes in normal stochastic differential systems

Narrow-sense stationary regimes are considered for multi-dimensional non-linear systems described by Ito stochastic differential equations with Wiener processes. The conditions for the existence of stationary and stationarizable one-dimensional distributions are derived. Exact expressions are obtained for stationary distributions in some mechanical systems.     

Stability and performance analysis of rate-based feedback flow controlled ATM networks

Motivated by the ABR class of service in ATM networks, we study a continuous-time queueing system with a feedback control of the arrival rate of some of the sources. The feedback regarding the queue length or the total workload is provided at regular intervals (variations on it, especially the EPRCA algorithm, are also considered). The propagation delays can be nonnegligible. For a general class of feedback algorithms, we obtain the stability of the system in the presence of one or more bottleneck nodes in the virtual circuit. We also obtain rates of convergence to the stationary distributions and finiteness of moments. For the single bottleneck case, we provide algorithms to compute the stationary distributions and the moments of the sojourn times in different sets of states. We also show analytically (by showing the continuity of stationary distributions and moments) that for small propagation delays, we can provide feedback algorithms which have higher mean throughput, lower probability of overflow, and lower delay jitter than any open-loop policy. Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Random locations of periodic stationary processes

We consider a family of random locations, called intrinsic location functionals, of periodic stationary processes. This family includes but is not limited to the location of the path supremum and first/last hitting times. We first show that the set of all possible distributions of intrinsic location functionals for periodic stationary processes is the convex hull generated by a specific group of distributions. We then focus on two special subclasses of these random locations. For the first subclass, the density has a uniform lower bound; for the second subclass, the possible distributions are closely related to the concept of joint mixability.     

Stability and performance analysis of rate-based feedback flow controlled ATM networks

Motivated by ABR class of service in ATM networks, we study a continuous time queueing system with a feedback control of the arrival rate of some of the sources. The feedback about the queue length or the total workload is provided at regular intervals (variations on it, especially the traffic management specification TM 4.0, are also considered). The propagation delays can be nonnegligible. For a general class of feedback algorithms, we obtain the stability of the system in the presence of one or more bottleneck nodes in the virtual circuit. Our system is general enough that it can be useful to study feedback control in other network protocols. We also obtain rates of convergence to the stationary distributions and finiteness of moments. For the single botterneck case, we provide algorithms to compute the stationary distributions and the moments of the sojourn times in different sets of states. We also show analytically (by showing continuity of stationary distributions and moments) that for small propagation delays, we can provide feedback algorithms which have higher mean throughput, lower probability of overflow and lower delay jitter than any open loop policy. Finally these results are supplemented by some computational results. This revised version was published online in June 2006 with corrections to the Cover Date.     

Strong Stationary Duality for Diffusion Processes

We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well-chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simulations of our primal and dual diffusion processes. Lastly, we show how our new definition of diffusion duality allows the spectral theory of cutoff phenomena to extend naturally from birth-and-death Markov chains to the present diffusion context.     

Convergence of integrated superpositions of Ornstein-Uhlenbeck processes to fractional Brownian motion

Superpositions of Ornstein-Uhlenbeck processes provide convenient ways to build stationary processes with given marginal distributions and long range dependence. After reviewing some of the basic features, we present several examples of processes with non-Gaussian marginal distributions. Our main results concern asymptotic properties of sums and partial sums of these processes and their polynomial functions. Further, we discuss some applications to estimation.     

EPSTA: The coincidence of time-stationary and customer-stationary distributions

In the first part of this paper we present an overview of relationships between time- and customer-stationary distributions of queueing processes. These have been proved by using the properties of random marked point processes, stochastic processes with embedded point processes, Palm distributions and an intensity conservation principle. In the second part a necessary and sufficient condition is established for the coincidence of the two types of stationary distributions, using conditional intensities. We also formulate the property of EPSTA that includes PASTA and ASTA as particular cases. A further result concerns the conditional EPSTA property. Applications to particular queueing systems are considered.     

Invariance of stationary probabilities of states for network of multiserver queues

A queueing network consisting of multiserver nodes with different sources of customers is considered. We investigate stationary probability distributions of network states. Their invariance with respect to the functional form of distributions of input and service processes is established under fixed expectations of these distributions and service discipline when any entering customer has a preemptive resume discipline.     

Asymptotic behavior of posterior distributions for random processes under incorrect models

In this paper, the asymptotic behavior of posterior distributions on parameters contained in random processes is examined when the specified model for the densities is not necessarily correct. Uniform convergence of likelihood functions in some way is shown to be a sufficient condition for the posterior distributions to be asymptotically confined to a set (Theorem 1). For ergodic stationary Markov processes uniform convergence of likelihood functions is established by the ergodic theorem for Banach-valued stationary processes (Proposition 1). A sufficient condition for the uniform convergence is also shown for general random processes (Proposition 2). These results are used to analyze the asymptotic behavior of posterior distributions on parameters contained in linear systems under incorrect models (Example 1 and 2).     

Semi-Selfsimilar Processes

A notion of semi-selfsimilarity of R ^d -valued stochastic processes is introduced as a natural extension of the selfsimilarity. Several topics on semi-selfsimilar processes are studied: the existence of the exponent for semi-selfsimilar processes; characterization of semi-selfsimilar processes as scaling limits; relationship between semi-selfsimilar processes with independent increments and semi-selfdecomposable distributions, and examples; construction of semi-selfsimilar processes with stationary increments; and extension of the Lamperti transformation. Semi-stable processes where all joint distributions are multivariate semi-stable are also discussed in connection with semi-selfsimilar processes. A wide-sense semi-selfsimilarity is defined and shown to be reducible to semi-selfsimilarity.     

On the local times of stationary processes with conditional local limit theorems

We investigate the connection between conditional local limit theorems and the local time of integer-valued stationary processes. We show that a conditional local limit theorem (at 0) implies the convergence of local times to Mittag-Leffler distributions, both in the weak topology of distributions and a.s. in the space of distributions.     

排队系统的渐近泊松话务过程

在文中,我们首先给出由马氏过程的一些跳跃时刻形成的简单点过程的有限维分布族弱收敛到泊松过程的相应分布族的条件,并讨论了有限维分布族弱收敛到泊松过程相应分布族的平稳马氏排队系统的话务过程,其次,我们证明了ＧＩ／Ｍ／１排队系统的离去过程的有限维分布族在重话务的情况下弱收敛到泊松过程的相应分布族。     

Ito游程理论与一般Markov链的平稳分布

在本文中,我们利用Ito游程理论给出一般Markov链的平稳分布,该公式包含了极小过程中断和含瞬时态的情形。最后我们给出一个含瞬时态的Markov链的计算例子。     

Asymptotic Expansions for Stationary Distributions of Nonlinearly Perturbed Semi-Markov Processes. 1

Methodology and Computing in Applied Probability - New algorithms for construction of asymptotic expansions for stationary distributions of nonlinearly perturbed semi-Markov processes with finite...     

Recursive computation of invariant distributions of Feller processes

This paper provides a general and abstract approach to compute invariant distributions for Feller processes. More precisely, we show that the recursive algorithm presented in Lamberton and Pagès (2002) and based on simulation algorithms of stochastic schemes with decreasing steps can be used to build invariant measures for general Feller processes. We also propose various applications: Approximation of Markov Brownian diffusion stationary regimes with a Milstein or an Euler scheme and approximation of a Markov switching Brownian diffusion stationary regimes using an Euler scheme.     

M/M/1 Queueing systems with inventory

We derive stationary distributions of joint queue length and inventory processes in explicit product form for various M/M/1-systems with inventory under continuous review and different inventory management policies, and with lost sales. Demand is Poisson, service times and lead times are exponentially distributed. These distributions are used to calculate performance measures of the respective systems. In case of infinite waiting room the key result is that the limiting distributions of the queue length processes are the same as in the classical M/M/1/∞-system. All authors were supported by DAAD/KBN grant number D/02/32206.     

Poisson Traffic Processes in RGSMP and Generalized Networks

1.Introductiontrafficprocessesinqueueingnetworksareanimportantoperatingfacetofsuchmodels,aswellasvaluableinthestudyofvaliddecompositionsofnetworks.IfwefindsometrafficprocessesinanetworkPoisson,thenitoftenrendersthemathematicalanalysistractable.Generalized…     

一维紧邻的接触过程与选举模型的混合过程的光滑性与单调性

证明了一维紧邻的接触过程与选举模型的混合过程,其极点平稳分布除在临界点λc 处外总是光滑的,且该混合过程具有单调性.     

On stochastic processes associated with relativistic stable distributions

Lévy processes with marginal relativistic α-stable distributions are described. Strictly stationary Ornstein-Uhlenbeck type processes with one-dimentional relativistic α-stable distributions are constructed. The exponential family as Esscher transforms of distributions on D _[0,∞)(R ^d ) of relativistic α-stable Lévy processes is obtained and the corresponding mixed exponential processes are characterized.