507 research outputs found
The Single Server Queue and the Storage Model: Large Deviations and Fixed Points
We consider the coupling of a single server queue and a storage model defined
as a Queue/Store model in Draief et al. 2004. We establish that if the input
variables, arrivals at the queue and store, satisfy large deviations principles
and are linked through an {\em exponential tilting} then the output variables
(departures from each system) satisfy large deviations principles with the same
rate function. This generalizes to the context of large deviations the
extension of Burke's Theorem derived in Draief et al. 2004.Comment: 20 page
Batch queues, reversibility and first-passage percolation
We consider a model of queues in discrete time, with batch services and
arrivals. The case where arrival and service batches both have Bernoulli
distributions corresponds to a discrete-time M/M/1 queue, and the case where
both have geometric distributions has also been previously studied. We describe
a common extension to a more general class where the batches are the product of
a Bernoulli and a geometric, and use reversibility arguments to prove versions
of Burke's theorem for these models. Extensions to models with continuous time
or continuous workload are also described. As an application, we show how these
results can be combined with methods of Seppalainen and O'Connell to provide
exact solutions for a new class of first-passage percolation problems.Comment: 16 pages. Mostly minor revisions; some new explanatory text added in
various places. Thanks to a referee for helpful comments and suggestion
Fixed points for multi-class queues
Burke's theorem can be seen as a fixed-point result for an exponential
single-server queue; when the arrival process is Poisson, the departure process
has the same distribution as the arrival process. We consider extensions of
this result to multi-type queues, in which different types of customer have
different levels of priority. We work with a model of a queueing server which
includes discrete-time and continuous-time M/M/1 queues as well as queues with
exponential or geometric service batches occurring in discrete time or at
points of a Poisson process. The fixed-point results are proved using
interchangeability properties for queues in tandem, which have previously been
established for one-type M/M/1 systems. Some of the fixed-point results have
previously been derived as a consequence of the construction of stationary
distributions for multi-type interacting particle systems, and we explain the
links between the two frameworks. The fixed points have interesting
"clustering" properties for lower-priority customers. An extreme case is an
example of a Brownian queue, in which lower-priority work only occurs at a set
of times of measure 0 (and corresponds to a local time process for the
queue-length process of higher priority work).Comment: 25 page
gym-saturation: Gymnasium environments for saturation provers (System description)
This work describes a new version of a previously published Python package -
gym-saturation: a collection of OpenAI Gym environments for guiding
saturation-style provers based on the given clause algorithm with reinforcement
learning. We contribute usage examples with two different provers: Vampire and
iProver. We also have decoupled the proof state representation from
reinforcement learning per se and provided examples of using a known ast2vec
Python code embedding model as a first-order logic representation. In addition,
we demonstrate how environment wrappers can transform a prover into a problem
similar to a multi-armed bandit. We applied two reinforcement learning
algorithms (Thompson sampling and Proximal policy optimisation) implemented in
Ray RLlib to show the ease of experimentation with the new release of our
package.Comment: 13 pages, 3 figures. This version of the contribution has been
accepted for publication, after peer review but is not the Version of Record
and does not reflect post-acceptance improvements, or any corrections. The
Version of Record is available online at:
https://doi.org/10.1007/978-3-031-43513-3_1
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