Ah Andreas Löpker

The idle period of the finite G/M/1 queue with an interpretation in risk theory

We consider a G/M/1 queue with restricted accessibility in the sense that the maximal workload is bounded by 1. If the current workload Vt of the queue plus the service time of an arriving customer exceeds 1, only 1−Vt of the service requirement is accepted. We are interested in the distribution of the idle period, which can be interpreted as the deficit at ruin for a risk reserve process Rt in the compound Poisson risk model. For this risk process a special dividend strategy applies, where the insurance company pays out all the income whenever Rt reaches level 1. In the queueing context we further introduce a set-up time a∈[0,1]. At the end of every idle period, an arriving customer has to wait for a time units until the server is ready to serve it.

Connecting renewal age processes with M/D/1 and M/D/∞ queues through stick breaking

We consider the length of a busy period in the M/D/∞ queue and show that it coincides with the sojourn time of the first customer in an M/D/1 processor-sharing queue. We further show that the busy period is intimately related with the stationary waiting time in the M/D/1 first-come-first-served queue. We present three characterizations for the distribution function of the busy period and an asymptotic expression for its tail distribution. The latter involves complex-valued branches of the Lambert W function.

Perturbation analysis of inhomogeneous finite Markov chains

Abstract In this paper we provide a perturbation analysis of finite time-inhomogeneous Markov processes. We derive closed-form representations for the derivative of the transition probability at time t, with t > 0. Elaborating on this result, we derive simple gradient estimators for transient performance characteristics either taken at some fixed point in time t, or for the integrated performance over a time interval [0 , t]. Bounds for transient performance sensitivities are presented as well. Eventually, we identify a structural property of the derivative of the generator matrix of a Markov chain that leads to a significant simplification of the estimators.

A markov-modulated growth collapse model

We consider a growth collapse model in a random environment for which the input rates might depend on the state of an underlying irreducible Markov chain and at state change epochs there is a possible downward jump to a level that is a random fraction of the level just before the jump. The distributions of these jumps are allowed to depend on both the originating and target states. Under a very weak assumption we develop an explicit formula for the conditional moments (of all orders) of the time stationary distribution. We then consider special cases and show how to use this result to study a growth collapse process in which the times between collapses have a phase-type distribution.

On the overflow time of a fluid model

We consider a continuous time stochastic fluid model with alternating on- and off-periods. During on-periods the process increases linearly, during off periods there is an additional negative decrease rate, proportional to the content level. While the durations of the on-periods have an exponential distribution we allow for general distributions for the durations of the off-periods. We study the overflow time of the system and its behavior as the overflow level tends to infinity. Such systems can be related to queuing systems which switch between a one-server mode and phases with infinitely many available servers.

On a make-to-stock production/mountain modeln with hysteretic control

We consider a make-to-stock production-inventory model with one machine that produces stock in a buffer. The machine is subject to breakdowns. During up periods, the machine fills the buffer at a level-dependent rate

On the small-time behavior of subordinators

\alpha (x)>0

TRANSIENT MOMENTS OF THE WINDOW SIZE IN TCP

α(x)>0. During down periods, the production rate is zero, and the demand rate is either