458 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
Continuity theorems for the queueing system
In this paper continuity theorems are established for the number of losses
during a busy period of the queue. We consider an queueing
system where the service time probability distribution, slightly different in a
certain sense from the exponential distribution, is approximated by that
exponential distribution. Continuity theorems are obtained in the form of one
or two-sided stochastic inequalities. The paper shows how the bounds of these
inequalities are changed if further assumptions, associated with specific
properties of the service time distribution (precisely described in the paper),
are made. Specifically, some parametric families of service time distributions
are discussed, and the paper establishes uniform estimates (given for all
possible values of the parameter) and local estimates (where the parameter is
fixed and takes only the given value). The analysis of the paper is based on
the level crossing approach and some characterization properties of the
exponential distribution.Comment: Final revision; will be published as i
Fluid flow models in performance analysis
We review several developments in fluid flow models: feedback fluid models, linear stochastic fluid networks and bandwidth sharing networks. We also mention some promising new research directions
A Stochastic Fluid Model Approach to the Stationary Distribution of the Maximum Priority Process
In traditional priority queues, we assume that every customer upon arrival
has a fixed, class-dependent priority, and that a customer may not commence
service if a customer with a higher priority is present in the queue. However,
in situations where a performance target in terms of the tails of the
class-dependent waiting time distributions has to be met, such models of
priority queueing may not be satisfactory. In fact, there could be situations
where high priority classes easily meet their performance target for the
maximum waiting time, while lower classes do not.
Here, we are interested in the stationary distribution at the times of
commencement of service of this maximum priority process. Until now, there has
been no explicit expression for this distribution. We construct a mapping of
the maximum priority process to a tandem fluid queue, which enables us to find
expressions for this stationary distribution. We derive the results for the
stationary distribution of the maximum priority process at the times of the
commencement of service.Comment: The Eleventh International Conference on Matrix-Analytic Methods in
Stochastic Models (MAM11), 2022, Seoul, Republic of Kore
A fluid queue with a finite buffer and subexponential input
We consider a fluid model similar to that of Kella and Whitt [33], but with a buffer having finite capacity K. The connections between the infinite buffer fluid model and the G/G/1 queue established in [33] are extended to the finite buffer case. It is shown that the stationary distribution of the buffer content is related to the stationary distribution of the finite dam. We also derive a number of new results for the latter model. In particular, an asymptotic expansion for the loss fraction is given for the case of subexponential service times. The stationary buffer content distribution of the fluid model is also related to that of the corresponding model with infinite buffer size, by showing that the two corresponding probability measures are proportional on [0,K) if the silence periods are exponentially distributed. These results are applied to obtain large buffer asymptotics for the loss fraction and the mean buffer content when the fluid queue is fed by N on-off sources with subexponential on-periods. The asymptotic results show a significant influence of heavy-tailed input characteristics on the performance of the fluid queue
Occupation densities in solving exit problems for Markov additive processes and their reflections
This paper solves exit problems for spectrally negative Markov additive
processes and their reflections. A so-called scale matrix, which is a
generalization of the scale function of a spectrally negative \levy process,
plays a central role in the study of exit problems. Existence of the scale
matrix was shown in Thm. 3 of Kyprianou and Palmowski (2008). We provide a
probabilistic construction of the scale matrix, and identify the transform. In
addition, we generalize to the MAP setting the relation between the scale
function and the excursion (height) measure. The main technique is based on the
occupation density formula and even in the context of fluctuations of
spectrally negative L\'{e}vy processes this idea seems to be new. Our
representation of the scale matrix W(x)=e^{-\Lambda x}\eL(x) in terms of nice
probabilistic objects opens up possibilities for further investigation of its
properties
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