10,235 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
Critically loaded multi-server queues with abandonments, retrials, and time-varying parameters
In this paper, we consider modeling time-dependent multi-server queues that
include abandonments and retrials. For the performance analysis of those, fluid
and diffusion models called "strong approximations" have been widely used in
the literature. Although they are proven to be asymptotically exact, their
effectiveness as approximations in critically loaded regimes needs to be
investigated. To that end, we find that existing fluid and diffusion
approximations might be either inaccurate under simplifying assumptions or
computationally intractable. To address that concern, this paper focuses on
developing a methodology by adjusting the fluid and diffusion models so that
they significantly improve the estimation accuracy. We illustrate the accuracy
of our adjusted models by performing a number of numerical experiments
The snowball effect of customer slowdown in critical many-server systems
Customer slowdown describes the phenomenon that a customer's service
requirement increases with experienced delay. In healthcare settings, there is
substantial empirical evidence for slowdown, particularly when a patient's
delay exceeds a certain threshold. For such threshold slowdown situations, we
design and analyze a many-server system that leads to a two-dimensional Markov
process. Analysis of this system leads to insights into the potentially
detrimental effects of slowdown, especially in heavy-traffic conditions. We
quantify the consequences of underprovisioning due to neglecting slowdown,
demonstrate the presence of a subtle bistable system behavior, and discuss in
detail the snowball effect: A delayed customer has an increased service
requirement, causing longer delays for other customers, who in turn due to
slowdown might require longer service times.Comment: 23 pages, 8 figures -- version 3 fixes a typo in an equation. in
Stochastic Models, 201
On Asymptotic Optimality of Dual Scheduling Algorithm In A Generalized Switch
Generalized switch is a model of a queueing system where parallel servers are interdependent and have time-varying service capabilities. This paper considers the dual scheduling algorithm that uses rate control and queue-length based scheduling to allocate resources for a generalized switch. We consider a saturated system in which each user has infinite amount of data to be served. We prove the asymptotic optimality of the dual scheduling algorithm for such a system, which says that the vector of average service rates of the scheduling algorithm maximizes some aggregate concave utility functions. As the fairness objectives can be achieved by appropriately choosing utility functions, the asymptotic optimality establishes the fairness properties of the dual scheduling algorithm.
The dual scheduling algorithm motivates a new architecture for scheduling, in which an additional queue is introduced to interface the user data queue and the time-varying server and to modulate the scheduling process, so as to achieve different performance objectives. Further research would include scheduling with Quality of Service guarantees with the dual scheduler, and its application and implementation in various versions of the generalized switch model
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