817 research outputs found
Some aspects of queueing and storage processes : a thesis in partial fulfilment of the requirements for the degree of Master of Science in Statistics at Massey University
In this study the nature of systems consisting of a single queue are first considered. Attention is then drawn to an analogy between such systems and storage systems.
A development of the single queue viz queues with feedback is considered after first considering feedback processes in general. The behaviour of queues, some with feedback loops, combined into networks is then considered. Finally, the application of such networks to the analysis of interconnected reservoir systems is considered and the conclusion drawn that such analytic methods complement the more recently developed mathematical programming methods by providing analytic solutions for
sub systems behaviour and thus guiding the development of a system model
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
Stationary distributions of the multi-type ASEP
We give a recursive construction of the stationary distribution of multi-type
asymmetric simple exclusion processes on a finite ring or on the infinite line
. The construction can be interpreted in terms of "multi-line diagrams" or
systems of queues in tandem. Let be the asymmetry parameter of the system.
The queueing construction generalises the one previously known for the totally
asymmetric () case, by introducing queues in which each potential service
is unused with probability when the queue-length is . The analysis is
based on the matrix product representation of Prolhac, Evans and Mallick.
Consequences of the construction include: a simple method for sampling exactly
from the stationary distribution for the system on a ring; results on common
denominators of the stationary probabilities, expressed as rational functions
of with non-negative integer coefficients; and probabilistic descriptions
of "convoy formation" phenomena in large systems.Comment: 54 pages, 4 figure
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
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
Gordon and Newell queueing networks and copulas
In this paper we have found an analytical formula for a copula that connects the numbers Ni of customers in the nodes of a Gordon and Newell queueing network. We have considered two cases: the first one is the case of the network with 2 nodes, and the second one is the case of the network with at least 3 nodes. The analytical formula for the second case has been found for the most general case (none of the constants from a list is equal to a given value), and the other particular cases have been obtained by limit.Gordon and Newell queueing networks, copulas.
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