Two Ergodic Sample-Path Properties of the Poisson Process

For increasing sequences of real numbers we consider two types of asymptotic behavior that remind of the defining property of a (homogeneous) Poisson process according to which the numbers of points in disjoint intervals are independent and follow Poisson distributions with specified parameters. We prove that almost all paths of a Poisson process show this asymptotic behavior, and characterize the Poisson process by these properties. Further we discuss the connection to equidistribution notions.     

On sets of integers with prescribed gaps

For a fixed setI of positive integers we consider the set of paths (p _o,...,p _k ) of arbitrary length satisfyingp _l –p _l–1I forl=2,...,k andp ₀=1,p _k =n. Equipping it with the uniform distribution, the random path lengthT _n is studied. Asymptotic expansions of the moments ofT _n are derived and its asymptotic normality is proved. The step lengthsp _l –p _l–1 are seen to follow asymptotically a restricted geometrical distribution. Analogous results are given for the free boundary case in which the values ofp ₀ andp _k are not specified. In the special caseI={m+1,m+2,...} (for some fixed m) we derive the exact distribution of a random m-gap subset of {1,...,n} and exhibit some connections to the theory of representations of natural numbers. A simple mechanism for generating a randomm-gap subset is also presented.     

Eine Erweiterung des Riemann-Lebesgue-Lemmas

It is proved that iff∈L ₁(?),f'∈L ₁(?) and ∫∣x∣ ⁱ ∣f(x)∣dx<∞ fori=1, ...,k?1 and ifA=(a _ij ) is a (k×k)-matrix with non-vanishing determinant, for $$\tilde f_A (\zeta ): = \smallint \exp (i\zeta _1 \sum\limits_{j = 1}^k {a_{1j} x^j } + ... + i\zeta _k \sum\limits_{j = 1}^k {a_{kj} x^j } )f(x)dx$$ the following relation holds: $$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$ .     

The (S−1,S) inventory model and its counterparts in queueing theory

We explore the relationship between the ($S?1,S$) inventory model and three well-known queueing models: the Erlang loss system, the machine-repair model and a two-node Jackson network. Exploiting this relationship allows us to obtain key performance measures of the ($S?1,S$) model, like the so-called virtual outdating time, the number of items on the shelf in steady state, the long-run rate of unsatisfied demands and the distribution of the empty shelf period.     

Non-stationary waiting times in a closed exponential tandem queue

A closed exponential tandem queue in which every customer has to be served by two servers is considered. Given the numbers of customers lined up at each of the two servers, we derive the probability distributions of the waiting time of any customer until his second service is completed, and of the total busy time of the system.     

Some exact expressions for the bulk-arrival queue mx/M/1

For a M/M/1 queueing system with group arrivals of random size the transition probabilities of the queue size process and the distribution of the maximal queue size during a time interval [0,t) are calculated. Simple formulae for the corresponding Laplace transforms are given.     

Asymptotic normality of discounted random series with applications in reliability and queueing

It is proved that a certain kind of randomly discounted random sums is asymptotically normal as the discount constant tends to zero. For replaceable systems with random lifetime, these sums represent the total discounted cost of policies of the age-replacement type; other applications to queueing and related areas are also indicated.     

The exact probability distribution of a two-dimensional random walk

A calculation is made of the exact probability distribution of the two-dimensional displacement of a particle at timet that starts at the origin, moves in straight-line paths at constant speed, and changes its direction after exponentially distributed time intervals, where the lengths of the straight-line paths and the turn angles are independent, the angles being uniformly distributed. This random walk is the simplest model for the locomotion of microorganisms on surfaces. Its weak convergence to a Wiener process is also shown.     

Gleichverteilungseigenschaften von Zufallsvariablen

We prove some equidistribution results for the expansion of random variables with respect to a given basis m.     

An optimal choice problem for a set of checking procedures

An item having a known initial failure probability is to be controlled by some out of a finite set of possible checks. Every check costs a certain amount of money, and budget constraints must be met. A check is characterized by the probabilities of three events: (i) letting a workable item pass, (ii) overlooking a failure if one is present, (iii) introducing a failure into a workable item. We show how any subset of checks to be employed is ordered optimally, and how the optimal subsequence of checks depends on the initial failure probability and oil the budget constraint.