11 research outputs found
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
Rewriting History in Integrable Stochastic Particle Systems
Many integrable stochastic particle systems in one space dimension (such as
TASEP - Totally Asymmetric Simple Exclusion Process - and its -deformation,
the -TASEP) remain integrable if we equip each particle with its own speed
parameter. In this work, we present intertwining relations between Markov
transition operators of particle systems which differ by a permutation of the
speed parameters. These relations generalize our previous works
(arXiv:1907.09155, arXiv:1912.06067), but here we employ a novel approach based
on the Yang-Baxter equation for the higher spin stochastic six vertex model.
Our intertwiners are Markov transition operators, which leads to interesting
probabilistic consequences.
First, we obtain a new Lax-type differential equation for the Markov
transition semigroups of homogeneous, continuous-time versions of our particle
systems. Our Lax equation encodes the time evolution of multipoint observables
of the -TASEP and TASEP in a unified way, which may be of interest for the
asymptotic analysis of multipoint observables of these systems.
Second, we show that our intertwining relations lead to couplings between
probability measures on trajectories of particle systems which differ by a
permutation of the speed parameters. The conditional distribution for such a
coupling is realized as a "rewriting history" random walk which randomly
resamples the trajectory of a particle in a chamber determined by the
trajectories of the neighboring particles. As a byproduct, we construct a new
coupling for standard Poisson processes on the positive real half-line with
different rates.Comment: 76 pages, 24 figures. v2: minor typos correcte