13 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
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
Lost-customers approximation of semi-open queueing networks with backordering -- An application to minimise the number of robots in robotic mobile fulfilment systems
We consider a semi-open queueing network (SOQN), where a customer requires
exactly one resource from the resource pool for service. If there is a resource
available, the customer is immediately served and the resource enters an inner
network. If there is no resource available, the new customer has to wait in an
external queue until one becomes available ("backordering"). When a resource
exits the inner network, it is returned to the resource pool and waits for
another customer. In this paper, we present a new solution approach. To
approximate the inner network with the resource pool of the SOQN, we consider a
modification, where newly arriving customers will decide not to join the
external queue and are lost if the resource pool is empty "lost customers". We
prove that we can adjust the arrival rate of the modified system so that the
throughputs in each node are pairwise identical to those in the original
network. We also prove that the probabilities that the nodes with constant
service rates are idling are pairwise identical too. Moreover, we provide a
closed-form expression for these throughputs and probabilities of idle nodes.
To approximate the external queue of the SOQN with backordering, we construct a
reduced SOQN with backordering, where the inner network consists only of one
node, by using Norton's theorem and results from the lost-customers
modification. In a final step, we use the closed-form solution of this reduced
SOQN, to estimate the performance of the original SOQN. We apply our results to
robotic mobile fulfilment systems (RMFSs). Instead of sending pickers to the
storage area to search for the ordered items and pick them, robots carry
shelves with ordered items from the storage area to picking stations. We model
the RMFS as an SOQN, analyse its stability and determine the minimal number of
robots for such systems using the results from the first part