Ergodic properties of Poisson processes with almost periodic intensity

Summary We discuss in this paper a non-homogeneous Poisson process A driven by an almost periodic intensity function. We give the stationary version A ^* and the Palm version A ⁰ corresponding to A ^*. Let (T _i ,i) be the inter-point distance sequence in A and (T _i ⁰ ,i) in A ⁰. We prove that forj, the sequence (T _i+j,i) converges in distribution to (T _i ⁰ ,i). If the intensity function is periodic then the convergence is in variation.     

Relationships between characteristics in periodic Poisson queues

We study single server periodic queues in the day equilibrium conditions. The following characteristics of interest are considered at time of dayt: V^p(t)-the work load, L^p(t)-the number of customers and u^p(t)-the departure rate. We give relationships between E[V^p(t)], E[L^p(t)] and u^p(t). We also prove that E[V^p(t)] < and E[L^p(t)] < provided the second moment of the service time is finite.     

The exact asymptotic of the collision time tail distribution for independent Brownian particles with different drifts

In this note we consider the time of the collision τ for n independent Brownian motions X ¹ _t ,...,X _t ⁿ with drifts a ₁,...,a _n , each starting from x = (x ₁,...,x _n ), where x ₁ < ... < x _n . We show the exact asymptotics of ${\mathbb{P}}_{\bf x}(\tau > t) = Ch({\bf x})t^{-\alpha} {\rm e}^{-\gamma t}(1 + o(1))$ as t → ∞ and identify C, h(x), α, γ in terms of the drifts.     

Micro- and ultramicrodetermination of the formyl and isonitrile groups in the presence of other acyl groups

Gravimetric and colorimetric micromethods for determination of formyl and isonitrile groups are reported. The gravimetric method is based on a quantitative cleavage of the formyl group and decomposition of formed formic acid to carbon monoxide by means of 60% sulfuric acid. Carbon monoxide is oxidized and the formed carbon dioxide is determined gravimetrically.The colorimetric ultramicromethod is based on the basic hydrolysis of the formyl group. After acidifying the hydrolysate, the quantitatively formed formic acid is distilled under special conditions. The acid is then reduced to formaldehyde. Its quantity is determined spectrophotometrically with chromotropic acid.     

Conditions for finite moments of waiting times inG/G/1 queues

We find conditions for E(W ) to be finite whereW is the stationary waiting time random variable in a stableG/G/1 queue with dependent service and inter-arrival times.Supported in part by KBN under grant 640/2/9, and at the Center for Stochastic Processes, Department of Statistics at the University of North Carolina Chapel Hill by the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Office Grant No. DAAL09-92-G-0008.     

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

We consider a Lévy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Lévy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275–298, 2007 and Queueing Syst. 60:203–226, 2008) for the corresponding tandem queue without an intermediate input.     

A Note on Transient Gaussian Fluid Models

In this paper we study the distribution of the supremum over interval [0,T] of a centered Gaussian process with stationary increments with a general negative drift function. This problem is related to the distribution of the buffer content in a transient Gaussian fluid queue Q(T) at time T, provided that at time 0 the buffer is empty. The general theory is illustrated by detailed considerations of different cases for the integrated Gaussian process and the fractional Brownian motion. We give asymptotic results for P(Q(T)>x) and P(sup _0tT Q(t)>x) as x.