5,009 research outputs found
Heavy-traffic limits for waiting times in many-server queues with abandonment
We establish heavy-traffic stochastic-process limits for waiting times in
many-server queues with customer abandonment. If the system is asymptotically
critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a
bounding argument shows that the abandonment does not affect waiting-time
processes. If instead the system is overloaded, as in the efficiency-driven
(ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh
Annual Allerton Conference on Communication, Control and Computing (1999)
1095--1104], we treat customer abandonment by studying the limiting behavior of
the queueing models with arrivals turned off at some time . Then, the
waiting time of an infinitely patient customer arriving at time is the
additional time it takes for the queue to empty. To prove stochastic-process
limits for virtual waiting times, we establish a two-parameter version of
Puhalskii's invariance principle for first passage times. That, in turn,
involves proving that two-parameter versions of the composition and inverse
mappings appropriately preserve convergence.Comment: Published in at http://dx.doi.org/10.1214/09-AAP606 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
The morphing of fluid queues into Markov-modulated Brownian motion
Ramaswami showed recently that standard Brownian motion arises as the limit
of a family of Markov-modulated linear fluid processes. We pursue this analysis
with a fluid approximation for Markov-modulated Brownian motion. Furthermore,
we prove that the stationary distribution of a Markov-modulated Brownian motion
reflected at zero is the limit from the well-analyzed stationary distribution
of approximating linear fluid processes. Key matrices in the limiting
stationary distribution are shown to be solutions of a new quadratic equation,
and we describe how this equation can be efficiently solved. Our results open
the way to the analysis of more complex Markov-modulated processes.Comment: 20 page; the material on p7 (version 1) has been removed, and pp.8-9
replaced by Theorem 2.7 and its short proo
Random Fluid Limit of an Overloaded Polling Model
In the present paper, we study the evolution of an overloaded cyclic polling
model that starts empty. Exploiting a connection with multitype branching
processes, we derive fluid asymptotics for the joint queue length process.
Under passage to the fluid dynamics, the server switches between the queues
infinitely many times in any finite time interval causing frequent oscillatory
behavior of the fluid limit in the neighborhood of zero. Moreover, the fluid
limit is random. Additionally, we suggest a method that establishes finiteness
of moments of the busy period in an M/G/1 queue.Comment: 36 pages, 2 picture
Simple models of network access, with applications to the design of joint rate and admission control
At the access to networks, in contrast to the core, distances and feedback delays, as well as link capacities are small, which has network engineering implications that are investigated in this paper. We consider a single point in the access network which multiplexes several bursty users. The users adapt their sending rates based on feedback from the access multiplexer. Important parameters are the user's peak transmission rate p, which is the access line speed, the user's guaranteed minimum rate r, and the bound ε on the fraction of lost data. Two feedback schemes are proposed. In both schemes the users are allowed to send at rate p if the system is relatively lightly loaded, at rate r during periods of congestion, and at a rate between r and p, in an intermediate region. For both feedback schemes we present an exact analysis, under the assumption that the users' job sizes and think times have exponential distributions. We use our techniques to design the schemes jointly with admission control, i.e., the selection of the number of admissible users, to maximize throughput for given p, r, and ε. Next we consider the case in which the number of users is large. Under a specific scaling, we derive explicit large deviations asymptotics for both models. We discuss the extension to general distributions of user data and think times
Approximations for time-dependent distributions in Markovian fluid models
In this paper we study the distribution of the level at time of
Markovian fluid queues and Markovian continuous time random walks, the maximum
(and minimum) level over , and their joint distributions. We
approximate by a random variable with Erlang distribution and we
use an alternative way, with respect to the usual Laplace transform approach,
to compute the distributions. We present probabilistic interpretation of the
equations and provide a numerical illustration
Performance Modelling and Optimisation of Multi-hop Networks
A major challenge in the design of large-scale networks is to predict and optimise the
total time and energy consumption required to deliver a packet from a source node to a
destination node. Examples of such complex networks include wireless ad hoc and sensor
networks which need to deal with the effects of node mobility, routing inaccuracies, higher
packet loss rates, limited or time-varying effective bandwidth, energy constraints, and the
computational limitations of the nodes. They also include more reliable communication
environments, such as wired networks, that are susceptible to random failures, security
threats and malicious behaviours which compromise their quality of service (QoS) guarantees.
In such networks, packets traverse a number of hops that cannot be determined
in advance and encounter non-homogeneous network conditions that have been largely
ignored in the literature. This thesis examines analytical properties of packet travel in
large networks and investigates the implications of some packet coding techniques on both
QoS and resource utilisation.
Specifically, we use a mixed jump and diffusion model to represent packet traversal
through large networks. The model accounts for network non-homogeneity regarding
routing and the loss rate that a packet experiences as it passes successive segments of a
source to destination route. A mixed analytical-numerical method is developed to compute
the average packet travel time and the energy it consumes. The model is able to capture
the effects of increased loss rate in areas remote from the source and destination, variable
rate of advancement towards destination over the route, as well as of defending against
malicious packets within a certain distance from the destination. We then consider sending
multiple coded packets that follow independent paths to the destination node so as to
mitigate the effects of losses and routing inaccuracies. We study a homogeneous medium
and obtain the time-dependent properties of the packet’s travel process, allowing us to
compare the merits and limitations of coding, both in terms of delivery times and energy
efficiency. Finally, we propose models that can assist in the analysis and optimisation
of the performance of inter-flow network coding (NC). We analyse two queueing models
for a router that carries out NC, in addition to its standard packet routing function. The
approach is extended to the study of multiple hops, which leads to an optimisation problem
that characterises the optimal time that packets should be held back in a router, waiting
for coding opportunities to arise, so that the total packet end-to-end delay is minimised
Markov-modulated Brownian motion with two reflecting barriers
We consider a Markov-modulated Brownian motion reflected to stay in a strip
[0,B]. The stationary distribution of this process is known to have a simple
form under some assumptions. We provide a short probabilistic argument leading
to this result and explaining its simplicity. Moreover, this argument allows
for generalizations including the distribution of the reflected process at an
independent exponentially distributed epoch. Our second contribution concerns
transient behavior of the reflected system. We identify the joint law of the
processes t,X(t),J(t) at inverse local times.Comment: 13 pages, 1 figur
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