2,484 research outputs found
Join-Idle-Queue with Service Elasticity: Large-Scale Asymptotics of a Non-monotone System
We consider the model of a token-based joint auto-scaling and load balancing
strategy, proposed in a recent paper by Mukherjee, Dhara, Borst, and van
Leeuwaarden (SIGMETRICS '17, arXiv:1703.08373), which offers an efficient
scalable implementation and yet achieves asymptotically optimal steady-state
delay performance and energy consumption as the number of servers .
In the above work, the asymptotic results are obtained under the assumption
that the queues have fixed-size finite buffers, and therefore the fundamental
question of stability of the proposed scheme with infinite buffers was left
open. In this paper, we address this fundamental stability question. The system
stability under the usual subcritical load assumption is not automatic.
Moreover, the stability may not even hold for all . The key challenge stems
from the fact that the process lacks monotonicity, which has been the powerful
primary tool for establishing stability in load balancing models. We develop a
novel method to prove that the subcritically loaded system is stable for large
enough , and establish convergence of steady-state distributions to the
optimal one, as . The method goes beyond the state of the art
techniques -- it uses an induction-based idea and a "weak monotonicity"
property of the model; this technique is of independent interest and may have
broader applicability.Comment: 30 page
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
Continuous feedback fluid queues
We investigate a fluid buffer which is modulated by a stochastic background process, while the momentary behavior of the background process depends on the current buffer level in a continuous way. Loosely speaking the feedback is such that the background process behaves `as a Markov process' with generator at times when the buffer level is , where the entries of are continuous functions of . Moreover, the flow rates for the buffer may also depend continuously on the current buffer level. Such models are interesting in the context of closed-loop telecommunication networks, in which sources interact with network buffers, but may also be deployed in the study of certain production systems. \u
Asymptotic analysis by the saddle point method of the Anick-Mitra-Sondhi model
We consider a fluid queue where the input process consists of N identical
sources that turn on and off at exponential waiting times. The server works at
the constant rate c and an on source generates fluid at unit rate. This model
was first formulated and analyzed by Anick, Mitra and Sondhi. We obtain an
alternate representation of the joint steady state distribution of the buffer
content and the number of on sources. This is given as a contour integral that
we then analyze for large N. We give detailed asymptotic results for the joint
distribution, as well as the associated marginal and conditional distributions.
In particular, simple conditional limits laws are obtained. These shows how the
buffer content behaves conditioned on the number of active sources and vice
versa. Numerical comparisons show that our asymptotic results are very accurate
even for N=20
On a generic class of two-node queueing systems
This paper analyzes a generic class of two-node queueing systems. A first queue is fed by an on–off Markov fluid source; the input of a second queue is a function of the state of the Markov fluid source as well, but now also of the first queue being empty or not. This model covers the classical two-node tandem queue and the two-class priority queue as special cases. Relying predominantly on probabilistic argumentation, the steady-state buffer content of both queues is determined (in terms of its Laplace transform). Interpreting the buffer content of the second queue in terms of busy periods of the first queue, the (exact) tail asymptotics of the distribution of the second queue are found. Two regimes can be distinguished: a first in which the state of the first queue (that is, being empty or not) hardly plays a role, and a second in which it explicitly does. This dichotomy can be understood by using large-deviations heuristics
Distribution-valued heavy-traffic limits for the queue
We study the queue in heavy-traffic using tempered
distribution-valued processes which track the age and residual service time of
each customer in the system. In both cases, we use the continuous mapping
theorem together with functional central limit theorem results in order to
obtain fluid and diffusion limits for these processes in the space of tempered
distribution-valued processes. We find that our diffusion limits are tempered
distribution-valued Ornstein-Uhlenbeck processes.Comment: Published at http://dx.doi.org/10.1214/14-AAP1027 in the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Supermarket Model on Graphs
We consider a variation of the supermarket model in which the servers can
communicate with their neighbors and where the neighborhood relationships are
described in terms of a suitable graph. Tasks with unit-exponential service
time distributions arrive at each vertex as independent Poisson processes with
rate , and each task is irrevocably assigned to the shortest queue
among the one it first appears and its randomly selected neighbors. This
model has been extensively studied when the underlying graph is a clique in
which case it reduces to the well known power-of- scheme. In particular,
results of Mitzenmacher (1996) and Vvedenskaya et al. (1996) show that as the
size of the clique gets large, the occupancy process associated with the
queue-lengths at the various servers converges to a deterministic limit
described by an infinite system of ordinary differential equations (ODE). In
this work, we consider settings where the underlying graph need not be a clique
and is allowed to be suitably sparse. We show that if the minimum degree
approaches infinity (however slowly) as the number of servers approaches
infinity, and the ratio between the maximum degree and the minimum degree in
each connected component approaches 1 uniformly, the occupancy process
converges to the same system of ODE as the classical supermarket model. In
particular, the asymptotic behavior of the occupancy process is insensitive to
the precise network topology. We also study the case where the graph sequence
is random, with the -th graph given as an Erd\H{o}s-R\'enyi random graph on
vertices with average degree . Annealed convergence of the occupancy
process to the same deterministic limit is established under the condition
, and under a stronger condition ,
convergence (in probability) is shown for almost every realization of the
random graph.Comment: 32 page
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
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