1,725 research outputs found
Queue-length balance equations in multiclass multiserver queues and their generalizations
A classical result for the steady-state queue-length distribution of
single-class queueing systems is the following: the distribution of the queue
length just before an arrival epoch equals the distribution of the queue length
just after a departure epoch. The constraint for this result to be valid is
that arrivals, and also service completions, with probability one occur
individually, i.e., not in batches. We show that it is easy to write down
somewhat similar balance equations for {\em multidimensional} queue-length
processes for a quite general network of multiclass multiserver queues. We
formally derive those balance equations under a general framework. They are
called distributional relationships, and are obtained for any external arrival
process and state dependent routing as long as certain stationarity conditions
are satisfied and external arrivals and service completions do not
simultaneously occur. We demonstrate the use of these balance equations, in
combination with PASTA, by (i) providing very simple derivations of some known
results for polling systems, and (ii) obtaining new results for some queueing
systems with priorities. We also extend the distributional relationships for a
non-stationary framework
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
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
Analysis and Computation of the Joint Queue Length Distribution in a FIFO Single-Server Queue with Multiple Batch Markovian Arrival Streams
This paper considers a work-conserving FIFO single-server queue with multiple
batch Markovian arrival streams governed by a continuous-time finite-state
Markov chain. A particular feature of this queue is that service time
distributions of customers may be different for different arrival streams.
After briefly discussing the actual waiting time distributions of customers
from respective arrival streams, we derive a formula for the vector generating
function of the time-average joint queue length distribution in terms of the
virtual waiting time distribution. Further assuming the discrete phase-type
batch size distributions, we develop a numerically feasible procedure to
compute the joint queue length distribution. Some numerical examples are
provided also
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