2,035 research outputs found
Transient solution of the M/Ek/1 queueing system
In this thesis, the Erlang queueing model Af/i/l, where customers arrive at random mean rate A and service times have an Erlang distribution with parameter k and iro service rate u, has been considered from different perspectives. Firstly, an analytic metl of obtaining the time-dependent probabilities, pn,,(() for the M/Ek/l system have t> proposed in terms of a new generalisation of the modified Bessel function when initk there are no customers in the system. Results have been also generalised to the case wl initially there are a customers in the system. Secondly, a new generalisation of the modified Bessei function and its generating function have been presented with its main properties and relations to other special functii (generalised Wright function and Mittag-Leffler function) haw been noted. Thirdly, the mean waiting tune in the queue, H',(f), has been evaluated, using Lucha results. The double-exponential approximation of computing Yq(t) has been proposed different values of p. which gives results within about % of the 'exact1 values obtained fr numerical solution of the differential-difference equations. The advantage of this approximation is that it provides additional information, via its functional form of the characterisl of the transient solution. Fourthly, the inversion of the Laplace transform with the application to the queues 1 been studied and verified for A//A//1 and M/Ek/l models of computing Wq{t}. Finally, an application of the A//fi/l queue has been provided in the example of hour traffic flow for the Severn Bridge. One of the main reasons for studying queue models from a theoretical point of view is to develop ways of modelling real-life system. The analytic results have been confirmed with the simulation
Factorization identities for reflected processes, with applications
We derive factorization identities for a class of preemptive-resume queueing
systems, with batch arrivals and catastrophes that, whenever they occur,
eliminate multiple customers present in the system. These processes are quite
general, as they can be used to approximate Levy processes, diffusion
processes, and certain types of growth-collapse processes; thus, all of the
processes mentioned above also satisfy similar factorization identities. In the
Levy case, our identities simplify to both the well-known Wiener-Hopf
factorization, and another interesting factorization of reflected Levy
processes starting at an arbitrary initial state. We also show how the ideas
can be used to derive transforms for some well-known
state-dependent/inhomogeneous birth-death processes and diffusion processes
Analysis of Markov-modulated infinite-server queues in the central-limit regime
This paper focuses on an infinite-server queue modulated by an independently
evolving finite-state Markovian background process, with transition rate matrix
. Both arrival rates and service rates are depending
on the state of the background process. The main contribution concerns the
derivation of central limit theorems for the number of customers in the system
at time , in the asymptotic regime in which the arrival rates
are scaled by a factor , and the transition rates by a
factor , with . The specific value of
has a crucial impact on the result: (i) for the system
essentially behaves as an M/M/ queue, and in the central limit theorem
the centered process has to be normalized by ; (ii) for ,
the centered process has to be normalized by , with the
deviation matrix appearing in the expression for the variance
Time-Limited and k-Limited Polling Systems: A Matrix Analytic Solution
In this paper, we will develop a tool to analyze polling systems with the
autonomous-server, the time-limited, and the k-limited service discipline. It
is known that these disciplines do not satisfy the well-known branching
property in polling system, therefore, hardly any exact result exists in the
literature for them. Our strategy is to apply an iterative scheme that is based
on relating in closed-form the joint queue-length at the beginning and the end
of a server visit to a queue. These kernel relations are derived using the
theory of absorbing Markov chains. Finally, we will show that our tool works
also in the case of a tandem queueing network with a single server that can
serve one queue at a time
Coupled queues with customer impatience
Motivated by assembly processes, we consider a Markovian queueing system with multiple coupled queues and customer impatience. Coupling means that departures from all constituent queues are synchronised and that service is interrupted whenever any of the queues is empty and only resumes when all queues are non-empty again. Even under Markovian assumptions, the state space grows exponentially with the number of queues involved. To cope with this inherent state space explosion problem, we investigate performance by means of two numerical approximation techniques based on series expansions, as well as by deriving the fluid limit. In addition, we provide closed-form expressions for the first terms in the series expansion of the mean queue content for the symmetric coupled queueing system. By an extensive set of numerical experiments, we show that the approximation methods complement each other, each one being accurate in a particular subset of the parameter space. (C) 2017 Elsevier B.V. All rights reserved
Interference Queueing Networks on Grids
Consider a countably infinite collection of interacting queues, with a queue
located at each point of the -dimensional integer grid, having independent
Poisson arrivals, but dependent service rates. The service discipline is of the
processor sharing type,with the service rate in each queue slowed down, when
the neighboring queues have a larger workload. The interactions are translation
invariant in space and is neither of the Jackson Networks type, nor of the
mean-field type. Coupling and percolation techniques are first used to show
that this dynamics has well defined trajectories. Coupling from the past
techniques are then proposed to build its minimal stationary regime. The rate
conservation principle of Palm calculus is then used to identify the stability
condition of this system, where the notion of stability is appropriately
defined for an infinite dimensional process. We show that the identified
condition is also necessary in certain special cases and conjecture it to be
true in all cases. Remarkably, the rate conservation principle also provides a
closed form expression for the mean queue size. When the stability condition
holds, this minimal solution is the unique translation invariant stationary
regime. In addition, there exists a range of small initial conditions for which
the dynamics is attracted to the minimal regime. Nevertheless, there exists
another range of larger though finite initial conditions for which the dynamics
diverges, even though stability criterion holds.Comment: Minor Spell Change
On Markov Chains with Uncertain Data
In this paper, a general method is described to determine uncertainty intervals for performance measures of Markov chains given an uncertainty region for the parameters of the Markov chains. We investigate the effects of uncertainties in the transition probabilities on the limiting distributions, on the state probabilities after n steps, on mean sojourn times in transient states, and on absorption probabilities for absorbing states. We show that the uncertainty effects can be calculated by solving linear programming problems in the case of interval uncertainty for the transition probabilities, and by second order cone optimization in the case of ellipsoidal uncertainty. Many examples are given, especially Markovian queueing examples, to illustrate the theory.Markov chain;Interval uncertainty;Ellipsoidal uncertainty;Linear Programming;Second Order Cone Optimization
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