44 research outputs found
Loss systems in a random environment
We consider a single server system with infinite waiting room in a random
environment. The service system and the environment interact in both
directions. Whenever the environment enters a prespecified subset of its state
space the service process is completely blocked: Service is interrupted and
newly arriving customers are lost. We prove an if-and-only-if-condition for a
product form steady state distribution of the joint queueing-environment
process. A consequence is a strong insensitivity property for such systems.
We discuss several applications, e.g. from inventory theory and reliability
theory, and show that our result extends and generalizes several theorems found
in the literature, e.g. of queueing-inventory processes.
We investigate further classical loss systems, where due to finite waiting
room loss of customers occurs. In connection with loss of customers due to
blocking by the environment and service interruptions new phenomena arise.
We further investigate the embedded Markov chains at departure epochs and
show that the behaviour of the embedded Markov chain is often considerably
different from that of the continuous time Markov process. This is different
from the behaviour of the standard M/G/1, where the steady state of the
embedded Markov chain and the continuous time process coincide.
For exponential queueing systems we show that there is a product form
equilibrium of the embedded Markov chain under rather general conditions. For
systems with non-exponential service times more restrictive constraints are
needed, which we prove by a counter example where the environment represents an
inventory attached to an M/D/1 queue. Such integrated queueing-inventory
systems are dealt with in the literature previously, and are revisited here in
detail
Stability of queueing-inventory systems with different priorities
We study a production-inventory system with two customer classes with
different priorities which are admitted to the system following a flexible
admission control scheme. The inventory management is according to a base stock
policy and arriving demand which finds the inventory depleted is lost (lost
sales). We analyse the global balance equations of the associated Markov
process and derive structural properties of the steady state distribution which
provide insights into the equilibrium behaviour of the system. We derive a
sufficient condition for ergodicity using the Foster-Lyapunov stability
criterion. For a special case we show that the condition is necessary as well
Queues in a random environment
Exponential single server queues with state dependent arrival and service
rates are considered which evolve under influences of external environments.
The transitions of the queues are influenced by the environment's state and the
movements of the environment depend on the status of the queues (bi-directional
interaction). The structure of the environment is constructed in a way to
encompass various models from the recent Operation Research literature, where a
queue is coupled e.g. with an inventory or with reliability issues. With a
Markovian joint queueing-environment process we prove separability for a large
class of such interactive systems, i.e. the steady state distribution is of
product form and explicitly given: The queue and the environment processes
decouple asymptotically and in steady state.
For non-separable systems we develop ergodicity criteria via Lyapunov
functions. By examples we show principles for bounding throughputs of
non-separable systems by throughputs of two separable systems as upper and
lower bound
Assessing the risk of disregarding urgent maintenance interventions on waterways infrastructures
the consequences which are considered are: additional cost of repairing after failure, number of fatalities, and the rate of lost customers upon an interruption of the lock service. Since quantifying all the consequences in economic terms could be difficult if not impossible, another approach is applied, which allows the aggregation of the consequences expressed in different scales. The suggested procedure, which has its roots in the probability applied to engineering, aims at supporting the planning of maintenance interventions on waterways infrastructures when resources and investment are limited.We propose a method for the prioritization of maintenance interventions already classified as urgent on waterways infrastructures, and especially locks. The method is based on the risk of the realization of a failure scenario due to damage processes. The probability of failure is computed by modelling the damage evolution as a stochastic proces