Discretization error for a two-sided reflected Lévy process

An obvious way to simulate a Lévy process X is to sample its increments over time 1 / n, thus constructing an approximating random walk \(X^{(n)}\). This paper considers the error of such approximation after the two-sided reflection map is applied, with focus on the value of the resulting process Y and regulators L, U at the lower and upper barriers at some fixed time. Under the weak assumption that \(X_\varepsilon /a_\varepsilon \) has a non-trivial weak limit for some scaling function \(a_\varepsilon \) as \(\varepsilon \downarrow 0\), it is proved in particular that \((Y_1-Y^{(n)}_n)/a_{1/n}\) converges weakly to \(\pm \, V\), where the sign depends on the last barrier visited. Here the limit V is the same as in the problem concerning approximation of the supremum as recently described by Ivanovs (Ann Appl Probab, 2018). Some further insight in the distribution of V is provided both theoretically and numerically.     

Stability and busy periods in a multiclass queue with state-dependent arrival rates

We introduce a multiclass single-server queueing system in which the arrival rates depend on the current job in service. The system is characterized by a matrix of arrival rates in lieu of a vector of arrival rates. Our proposed model departs from existing state-dependent queueing models in which the parameters depend primarily on the number of jobs in the system rather than on the job in service. We formulate the queueing model and its corresponding fluid model and proceed to obtain necessary and sufficient conditions for stability via fluid models. Utilizing the natural connection with the multitype Galton–Watson processes, the Laplace–Stieltjes transform of busy periods in the system is given. We conclude with tail asymptotics for the busy period for heavy-tailed service time distributions for the regularly varying case.     

Tail asymptotics for M/G/1 type queueing processes with subexponential increments

Bivariate regenerative Markov modulated queueing processes {I _n ,L _n } are described. {I _n } is the phase process, and {L _n } is the level process. Increments in the level process have subexponential distributions. A general boundary behavior at the level 0 is allowed. The asymptotic tail of the cycle maximum, , during a regenerative cycle, , and the asymptotic tail of the stationary random variable L _∞, respectively, of the level process are given and shown to be subexponential with L _∞ having the heavier tail. This revised version was published online in June 2006 with corrections to the Cover Date.     

Failure rates of regenerative systems with heavy tails

Let ξ_t be a regenerative process and assume that, at each state x, the process can fail with intensity α(x). If the inter-regeneration times have a finite exponential moment orinf _xα(x)>0, then α(ξ_t) tends to some limiting positive intensity as t→∞ (under mild additional restrictions). This fact is widely used in engineering because the limiting intensity can be employed in various calculations, say, in reliability theory. The paper contains a variety of examples showing that α(ξ_t) provided that inter-regeneration time has no exponential moment andinf _x α(x)=0. The speed of convergence depends, in general, on both the tail of inter-regeneration time and α(x). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part III.     

Branching random walks I

A general method is developed with which various theorems on the mean square convergence of functionals of branching random walks are proven. The results cover extensions and generalizations of classical central limit analogues as well as a result of a different type.     

Nuclear spin conversion of methane in pure and rare-gas mixed crystals

Spin species conversion has been observed by means of neutron scattering. Methane, matrixisolated in rare-gas crystals has been investigated by measuring the time dependence of the detailed balance factor of inelastic neutron spectra. The temperature dependence of conversion rates in the interval between 2 and 5 K turns out to be stronger than predicted by a single phonon process. In general, conversion times are of the order of hours in this temperature region. The influence of the matrix is well described by a coupled phonon density of states. Crystals of pure and krypton doped methane have been examined by measuring the time dependence of the total neutron scattering cross section. For pure methane, the temperature dependence of the conversion rates shows two regimes for both free rotator and ordered molecules. Below 5 K, the behavior is described by a single phonon process and above by an Arrhenius law. Rates are enhanced by krypton dotation, which is explained by changed level spacings due to orientational disorder.     

A new proof of convergence of MCMC via the ergodic theorem

A key result underlying the theory of MCMC is that any η-irreducible Markov chain having a transition density with respect to η and possessing a stationary distribution π is automatically positive Harris recurrent. This paper provides a short self-contained proof of this fact using the ergodic theorem in its standard form as the most advanced tool.     

Transient Properties of Many-Server Queues and Related QBDs

The time τ(n) of first passage from queue length x to queue length n>x in a many-server queue with both the arrival process and service intensities governed by a finite Markov process is considered. The mean and the Laplace transform are computed as solutions of systems of linear equations coming out by optional stopping of a martingale obtained as a stochastic integral of the exponential Wald martingale for Markov additive processes. Compared to existing techniques for QBD's, the approach has the advantage of being far more efficient for large n.