1,856 research outputs found
Reversibility in Queueing Models
In stochastic models for queues and their networks, random events evolve in
time. A process for their backward evolution is referred to as a time reversed
process. It is often greatly helpful to view a stochastic model from two
different time directions. In particular, if some property is unchanged under
time reversal, we may better understand that property. A concept of
reversibility is invented for this invariance. Local balance for a stationary
Markov chain has been used for a weaker version of the reversibility. However,
it is still too strong for queueing applications.
We are concerned with a continuous time Markov chain, but dose not assume it
has the stationary distribution. We define reversibility in structure as an
invariant property of a family of the set of models under certain operation.
The member of this set is a pair of transition rate function and its supporting
measure, and each set represents dynamics of queueing systems such as arrivals
and departures. We use a permutation {\Gamma} of the family menmbers, that is,
the sets themselves, to describe the change of the dynamics under time
reversal. This reversibility is is called {\Gamma}-reversibility in structure.
To apply these definitions, we introduce new classes of models, called
reacting systems and self-reacting systems. Using those definitions and models,
we give a unified view for queues and their networks which have reversibility
in structure, and show how their stationary distributions can be obtained. They
include symmetric service, batch movements and state dependent routing.Comment: Submitted for publicatio
Arrival first queueing networks with applications in kanban production systems
In this paper we introduce a new class of queueing networks called {\it arrival first networks}. We characterise its transition rates and derive the relationship between arrival rules, linear partial balance equations, and product form stationary distributions. This model is motivated by production systems operating under a kanban protocol. In contrast with the conventional {\em departure first networks}, where a transition is initiated by service completion of items at the originating nodes that are subsequently routed to the destination nodes (push system), in an arrival first network a transition is initiated by the destination nodes of the items and subsequently those items are processed at and removed from the originating nodes (pull system). These are similar to the push and pull systems in manufacturing systems
Bose--Einstein Condensation in the Large Deviations Regime with Applications to Information System Models
We study the large deviations behavior of systems that admit a certain form
of a product distribution, which is frequently encountered both in Physics and
in various information system models. First, to fix ideas, we demonstrate a
simple calculation of the large deviations rate function for a single
constraint (event). Under certain conditions, the behavior of this function is
shown to exhibit an analogue of Bose--Einstein condensation (BEC). More
interestingly, we also study the large deviations rate function associated with
two constraints (and the extension to any number of constraints is conceptually
straightforward). The phase diagram of this rate function is shown to exhibit
as many as seven phases, and it suggests a two--dimensional generalization of
the notion of BEC (or more generally, a multi--dimensional BEC). While the
results are illustrated for a simple model, the underlying principles are
actually rather general. We also discuss several applications and implications
pertaining to information system models
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