Sufficient Conditions For Ergodicity Research Articles

Stability of queueing-inventory systems with customers of different priorities

We study a production-inventory system with two customer classes with different priorities which are admitted to the system following a flexible admission control scheme. The inventory management is according to a base stock policy and arriving demand which finds the inventory depleted is lost (lost sales). We analyse the global balance equations of the associated Markov process and derive structural properties of the steady state distribution which provide insights into the equilibrium behaviour of the system. We derive a sufficient condition for ergodicity using the Foster-Lyapunov stability criterion. For a special case we show that the condition is necessary as well.

Ergodicity of Two Class Priority Queues with Preemptive Priority

Well-known results on the ergodicity of queues with preemptive priority were obtained under the assumption that jobs arrive according to the Poisson process. This assumption, however, does not always hold true in practice. In our work we find sufficient ergodicity conditions for queues with two priority classes with a single server, where interarrival times of high-priority jobs have either Erlang or hyperexponential distribution and interarrival times of low-priority jobs and service times of jobs of both classes have arbitrary continuous distributions. We present results in disciplines with loss of an interrupted job and with retrial of an interrupted job.

Threshold dynamics in a stochastic chemostat model under regime switching

A stochastic chemostat model in random environments that is driven by Brownian motions and subjected to Markov regime switching is considered. The new break-even concentration, i.e., critical value between persistence in mean and extinction is explored for the microorganism species. Moreover, sufficient conditions for ergodicity and positive recurrence is established by using stochastic Lyapunov analysis. Numerical simulations are accomplished to verify the analytical results.

Ergodicity of the probabilistic converter, a serial connection of two automata

Abstract The paper provides necessary and sufficient conditions for the the ergodicity of a serial connection of automata under which the output sequence of a substitution Mealy automaton is fed to the input of an output-free substitution automaton. It is shown that the condition of complete indecomposability of the state transition probability matrix of a Mealy automaton provides a sufficient condition for ergodicity of the probabilistic converter as a serial connection of automata. It is also shown that if the partial state transition functions of a Mealy automaton commute, then the condition of ergodicity of a serial connection is equivalent to that of both original probabilistic converters.

Ergodic functions over Zp

In this paper, we present ergodicity criteria for 1-Lipschitz functions on Zp, in terms of the van der Put coefficients as well as the inherent data associated with the function. These criteria are applied to provide sufficient conditions for ergodicity of the 1-Lipschitz p-adic functions with special features, such as everywhere/uniform differentiability with respect to the Mahler expansion. In particular, the ergodicity criteria are obtained for certain 1-Lipschitz functions on Z2 and Z3, which are known as B-functions, in terms of the Mahler and van der Put expansions. These functions are locally analytic functions of order 1 (and therefore contain polynomials). For arbitrary primes p≥5, an ergodicity criterion of B-functions on Zp is introduced, which leads to an efficient and practical method of constructing ergodic polynomials on Zp that realize a given unicyclic permutation modulo p. Thus, a complete description of ergodic polynomials modulo pμ, which are reduced from all ergodic B-functions on Zp, is provided where μ=μ(p)=3 for p∈{2,3} and μ=2 for p≥5.

The ergodic theorem for random walks on finite quantum groups

Necessary and sufficient conditions for a Markov chain to be ergodic are that the chain is irreducible and aperiodic. This result is manifest in the case of random walks on finite groups by a statement about the support of the driving probability: a random walk on a finite group is ergodic if and only if the support is not concentrated on a proper subgroup, nor on a coset of a proper normal subgroup. The study of random walks on finite groups extends naturally to the study of random walks on finite quantum groups, where a state on the algebra of functions plays the role of the driving probability. Necessary and sufficient conditions for ergodicity of a random walk on a finite quantum group are given on the support projection of the driving state.

Performance Analysis of Multi-processor Two-Stage Tandem Call Center Retrial Queues with Non-Reliable Processors

We analyze a multi-processor two-stage tandem call center retrial queueing network in which the processors are subject to active breakdowns and repairs at stage-I. A level-dependent quasi-birth-and-death (LDQBD) process is formulated and a sufficient condition for ergodicity of the system is discussed. Under the stability condition, the stationary distribution of the number of calls in the system, the mean number of calls in the orbit, the mean waiting time of calls in the orbit and the mean busy period of the system along with other descriptors of the system are determined by using the matrix-analytic techniques. Besides, the availability analysis of the processors is also studied. Further, we have also discussed the characteristics of the first-passage time to reach the orbit critical level, the number of calls served in the call center during this period and their corresponding moments. Finally, extensive numerical results are presented to highlight the impact of the system parameters on the performance measures of the two-stage tandem call center retrial queueing system under investigation.

Stochastic properties of solution for a chemostat model with a distributed delay and random disturbance

In this paper, stochastic properties of solution for a chemostat model with a distributed delay and random disturbance are studied, and we use distribution delay to simulate the delay in nutrient conversion. By the linear chain technique, we transform the stochastic chemostat model with weak kernel into an equivalent degenerate system which contains three equations. First, we state that this model has a unique global positive solution for any initial value, which is helpful to explore its stochastic properties. Furthermore, we prove the stochastic ultimate boundness of the solution of system. Then sufficient conditions for solution of the system tending toward the boundary equilibrium point at exponential rate are established, which means the microorganism will be extinct. Moreover, we also obtain some sufficient conditions for ergodicity of solution of this system by constructing some suitable stochastic Lyapunov functions. Finally, we provide some numerical examples to illustrate theoretical results, and some conclusions and analysis are given.

Ergodic Uniformly Differentiable Functions Modulo p on ℤp

We establish necessary and sufficient conditions for ergodicity of uniformly differentiable functions modulo p on ℤp. This result is applied to study ergodic rational functions on ℤ3.

Boolean Percolation on Doubling Graphs

We consider the discrete Boolean model of percolation on weighted graphs satisfying a doubling metric condition. We study sufficient conditions on the distribution of the radii of balls placed at the points of a Bernoulli point process for the absence of percolation, provided that the parameter of the underlying point process is small enough. We exhibit three families of interesting graphs where the main result of this work holds. Finally, we give sufficient conditions for ergodicity of the discrete Boolean model of percolation.

Necessary and sufficient conditions for ergodicity of CIR type SDEs with Markov switching

In this paper, we consider the ergodicity and transience of the Cox–Ingersoll–Ross (CIR) interest rate model with Markov switching. Using the theory of [Formula: see text]-matrices, we give some necessary and sufficient conditions for ergodicity of the CIR interest rate model with Markov switching. Besides, we show that the transition semigroup converges to the stationary distribution at an exponential rate in the Wasserstein distance. Finally, two examples are presented to illustrate our theory.

Sufficient ergodicity conditions for queueing systems with non-preemptive priority

Sufficient ergodicity conditions for queueing systems with non-preemptive priority

Ergodicity for population dynamics driven by stable processes with Markovian switching

In this paper, we consider ergodicity and transience for Lotka–Volterra model driven by α-stable processes () with Markovian switching. Sufficient conditions for ergodicity and extinction of such model are given. Two examples are given to illustrate our results.

A new look at a smart polling model

We analyze a Markovian smart polling model, which is a special case of the smart polling models studied in the work of Boon et al. (Queueing Syst 66:239–274, 2010), as well as a generalization of the gated M / M / 1 queue considered in Resing and Rietman (Stat Neerlandica 58:97–110, 2004). We first derive tractable expressions for the stationary distribution (when it exists) as well as the Laplace transforms of the transition functions of this polling model—while further assuming the system is empty at time zero—and we also present simple necessary and sufficient conditions for ergodicity of the smart polling model. Finally, we conclude the paper by briefly explaining how these techniques can be used to study other interesting variants of this smart polling model.

Necessary and sufficient conditions for ergodicity of CIR model driven by stable processes with Markov switching

In this paper, we consider long time behavior of the Cox-Ingersoll-Ross (CIR) interest rate model driven by stable processes with Markov switching. Under some assumptions, we prove an ergodicity-transience dichotomy, namely, the interest rate process is either ergodic or transient. The sufficient and necessary conditions for ergodicity and transience of such interest model are given under some assumptions. Finally, an application to interval estimation of the interest rate processes is presented to illustrate our results.

On a vector double autoregressive model

Motivated by the double autoregressive (DAR) model, in this paper, we study a vector double autoregressive model (VDAR). The model is a straightforward extension from univariate case to multivariate case. Sufficient ergodicity conditions are given for the model. Without existence of second moment conditions for observed time series, the quasi maximum likelihood estimator (QMLE) of the parameter in the model is shown to be asymptotically normal, which does not hold for classic vector autoregressive (VAR) model with i.i.d errors. Simulation results confirm that our estimators perform well. A given empirical study implies the proposed model has potential applications in practice.

Ergodic properties of generalized Ornstein–Uhlenbeck processes

We investigate ergodic properties of the solution of the SDE dVt=Vt−dUt+dLt, where (U,L) is a bivariate Lévy process. This class of processes includes the generalized Ornstein–Uhlenbeck processes. We provide sufficient conditions for ergodicity, and for subexponential and exponential convergence to the invariant probability measure. We use the Foster–Lyapunov method. The drift conditions are obtained using the explicit form of the generator of the continuous process. In some special cases the optimality of our results can be shown.

Ergodicity and transience of SDEs driven by -stable processes with Markovian switching

ABSTRACTIn this paper, we focus on ergodicity and transience of SDEs driven by symmetric -stable processes () with Markovian switching. Some sufficient conditions for ergodicity of several classes of stable processes with Markovian switching are given. Furthermore, an almost necessary and sufficient condition for ergodicity of Ornstein–Uhlenbeck type driven by symmetric -stable processes with Markovian switching is presented. As applications, some examples are provided to illustrate our results.

Steady-state analysis of a multiclass MAP/PH/cqueue with acyclic PH retrials

AbstractA multiclassc-server retrial queueing system in which customers arrive according to a class-dependent Markovian arrival process (MAP) is considered. Service and retrial times follow class-dependent phase-type (PH) distributions with the further assumption that PH distributions of retrial times are acyclic. A necessary and sufficient condition for ergodicity is obtained from criteria based on drifts. The infinite state space of the model is truncated with an appropriately chosen Lyapunov function. The truncated model is described as a multidimensional Markov chain, and a Kronecker representation of its generator matrix is numerically analyzed.

The Limit Behaviour of Imprecise Continuous-Time Markov Chains

We study the limit behaviour of a nonlinear differential equation whose solution is a superadditive generalisation of a stochastic matrix, prove convergence, and provide necessary and sufficient conditions for ergodicity. In the linear case, the solution of our differential equation is equal to the matrix exponential of an intensity matrix and can then be interpreted as the transition operator of a homogeneous continuous-time Markov chain. Similarly, in the generalised nonlinear case that we consider, the solution can be interpreted as the lower transition operator of a specific set of non-homogeneous continuous-time Markov chains, called an imprecise continuous-time Markov chain. In this context, our convergence result shows that for a fixed initial state, an imprecise continuous-time Markov chain always converges to a limiting distribution, and our ergodicity result provides a necessary and sufficient condition for this limiting distribution to be independent of the initial state.