The Embedded Random Walk in the Stationary M/M/1 Queue

We consider the stationary queue length process of the standard M/M/1 queue just after its jump times (i.e., the times of arrivals or departures). Simple formulas for the distribution of this embedded random walk with immediate reflection at zero are derived. We study the monotonicity properties of the corresponding expected values and point out an asymmetry between the number of arrivals and the number of departures until the n-th jump time.     

Exact probability distributions for noncorrelated random walk models

A stochastic model for the idealized locomotion of cells is studied. The cell is assumed to cover a polygonal line in ⁿ , the times between turns are exponentially distributed and independent of the directions, and the density of thenth directione does not depend on the (n–1)th directione. The resulting Markov process (X(t), D(t)) for position and direction of the motion at timet is studied by using the integrodifferential equation for the transition function. For example, the joint distribution of (X(t), D(t)) is derived in closed form ifn=2 orn=3 and all chosen directions (including the initial one) are uniformly distributed. For higher dimensions the combined Fourier-Laplace transform ofX(t) is given. The case of a fixed initial direction is also considered.     

Contributions to the theory of first‐exit times of some compound processes in queueing theory

We consider compound processes that are linear with constant slope between i.i.d. jumps at time points forming a renewal process. These processes are basic in queueing, dam and risk theory. For positive and for negative slope we derive the distribution of the first crossing time of a prespecified level. The related problem of busy periods of single‐server queueing systems is also studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Two-stage queueing network models for quality control and testing

We study sojourn times in a two-node open queueing network with a processor sharing node and a delay node, with Poisson arrivals at the PS node. Motivated by quality control and blood testing applications, we consider a feedback mechanism in which customers may either leave the system after service at the PS node or move to the delay node; from the delay node, they always return to the PS node for new quality controls or blood tests. We propose various approximations for the distribution of the total sojourn time in the network; each of these approximations yields the exact mean sojourn time, and very accurate results for the variance. The best of the three approximations is used to tackle an optimization problem that is mainly inspired by a blood testing application.     

Asymptotic normality in a two-dimensional random walk model for cell motility

We prove the asymptotic normality of a two-dimensional random walk describing the locomotion of cells on planar surfaces and calculate the asymptotic covariance matrix. The trajectories of the walk are random broken lines covered with constant speed, where the time intervals between turns as well as the turn angles are random and stochastically independent.     

An inequality for Lp-norms with respect to the multivariate normal distribution

A partial converse of Jensen's inequality for integrals of norms on $\text{R}$^k is proved.     

On a loss system with a given number of customers

A one-server loss system with Poisson arrival stream and deterministic service times is considered conditional on the number of customers who appeared up to a givenT. This condition implies that the arrival times form a sample of the uniform distribution on (0,T]. We derive several characteristics of interest, such as the blocking probability at any given timet (0,T], the probability that exactlyi of the customers in (0,T] are served and, as a generalization, the distribution of the number of served customers arriving in any subinterval of (0,T].