241 research outputs found
Proportional fairness and its relationship with multi-class queueing networks
We consider multi-class single-server queueing networks that have a product
form stationary distribution. A new limit result proves a sequence of such
networks converges weakly to a stochastic flow level model. The stochastic flow
level model found is insensitive. A large deviation principle for the
stationary distribution of these multi-class queueing networks is also found.
Its rate function has a dual form that coincides with proportional fairness. We
then give the first rigorous proof that the stationary throughput of a
multi-class single-server queueing network converges to a proportionally fair
allocation. This work combines classical queueing networks with more recent
work on stochastic flow level models and proportional fairness. One could view
these seemingly different models as the same system described at different
levels of granularity: a microscopic, queueing level description; a
macroscopic, flow level description and a teleological, optimization
description.Comment: Published in at http://dx.doi.org/10.1214/09-AAP612 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large deviations analysis for the queue in the Halfin-Whitt regime
We consider the FCFS queue in the Halfin-Whitt heavy traffic
regime. It is known that the normalized sequence of steady-state queue length
distributions is tight and converges weakly to a limiting random variable W.
However, those works only describe W implicitly as the invariant measure of a
complicated diffusion. Although it was proven by Gamarnik and Stolyar that the
tail of W is sub-Gaussian, the actual value of was left open. In subsequent work, Dai and He
conjectured an explicit form for this exponent, which was insensitive to the
higher moments of the service distribution.
We explicitly compute the true large deviations exponent for W when the
abandonment rate is less than the minimum service rate, the first such result
for non-Markovian queues with abandonments. Interestingly, our results resolve
the conjecture of Dai and He in the negative. Our main approach is to extend
the stochastic comparison framework of Gamarnik and Goldberg to the setting of
abandonments, requiring several novel and non-trivial contributions. Our
approach sheds light on several novel ways to think about multi-server queues
with abandonments in the Halfin-Whitt regime, which should hold in considerable
generality and provide new tools for analyzing these systems
A Stochastic Resource-Sharing Network for Electric Vehicle Charging
We consider a distribution grid used to charge electric vehicles such that
voltage drops stay bounded. We model this as a class of resource-sharing
networks, known as bandwidth-sharing networks in the communication network
literature. We focus on resource-sharing networks that are driven by a class of
greedy control rules that can be implemented in a decentralized fashion. For a
large number of such control rules, we can characterize the performance of the
system by a fluid approximation. This leads to a set of dynamic equations that
take into account the stochastic behavior of EVs. We show that the invariant
point of these equations is unique and can be computed by solving a specific
ACOPF problem, which admits an exact convex relaxation. We illustrate our
findings with a case study using the SCE 47-bus network and several special
cases that allow for explicit computations.Comment: 13 pages, 8 figure
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
Load Balancing in the Non-Degenerate Slowdown Regime
We analyse Join-the-Shortest-Queue in a contemporary scaling regime known as
the Non-Degenerate Slowdown regime. Join-the-Shortest-Queue (JSQ) is a
classical load balancing policy for queueing systems with multiple parallel
servers. Parallel server queueing systems are regularly analysed and
dimensioned by diffusion approximations achieved in the Halfin-Whitt scaling
regime. However, when jobs must be dispatched to a server upon arrival, we
advocate the Non-Degenerate Slowdown regime (NDS) to compare different
load-balancing rules.
In this paper we identify novel diffusion approximation and timescale
separation that provides insights into the performance of JSQ. We calculate the
price of irrevocably dispatching jobs to servers and prove this to within 15%
(in the NDS regime) of the rules that may manoeuvre jobs between servers. We
also compare ours results for the JSQ policy with the NDS approximations of
many modern load balancing policies such as Idle-Queue-First and
Power-of--choices policies which act as low information proxies for the JSQ
policy. Our analysis leads us to construct new rules that have identical
performance to JSQ but require less communication overhead than
power-of-2-choices.Comment: Revised journal submission versio
Store-Forward and its implications for Proportional Scheduling
The Proportional Scheduler was recently proposed as a scheduling algorithm
for multi-hop switch networks. For these networks, the BackPressure scheduler
is the classical benchmark. For networks with fixed routing, the Proportional
Scheduler is maximum stable, myopic and, furthermore, will alleviate certain
scaling issued found in BackPressure for large networks. Nonetheless, the
equilibrium and delay properties of the Proportional Scheduler has not been
fully characterized.
In this article, we postulate on the equilibrium behaviour of the
Proportional Scheduler though the analysis of an analogous rule called the
Store-Forward allocation. It has been shown that Store-Forward has
asymptotically allocates according to the Proportional Scheduler. Further, for
Store-Forward networks, numerous equilibrium quantities are explicitly
calculable. For FIFO networks under Store-Forward, we calculate the policies
stationary distribution and end-to-end route delay. We discuss network
topologies when the stationary distribution is product-form, a phenomenon which
we call \emph{product form resource pooling}. We extend this product form
notion to independent set scheduling on perfect graphs, where we show that
non-neighbouring queues are statistically independent. Finally, we analyse the
large deviations behaviour of the equilibrium distribution of Store-Forward
networks in order to construct Lyapunov functions for FIFO switch networks
Concave Switching in Single and Multihop Networks
Switched queueing networks model wireless networks, input queued switches and
numerous other networked communications systems. For single-hop networks, we
consider a {()-switch policy} which combines the MaxWeight policies
with bandwidth sharing networks -- a further well studied model of Internet
congestion. We prove the maximum stability property for this class of
randomized policies. Thus these policies have the same first order behavior as
the MaxWeight policies. However, for multihop networks some of these
generalized polices address a number of critical weakness of the
MaxWeight/BackPressure policies.
For multihop networks with fixed routing, we consider the Proportional
Scheduler (or (1,log)-policy). In this setting, the BackPressure policy is
maximum stable, but must maintain a queue for every route-destination, which
typically grows rapidly with a network's size. However, this proportionally
fair policy only needs to maintain a queue for each outgoing link, which is
typically bounded in number. As is common with Internet routing, by maintaining
per-link queueing each node only needs to know the next hop for each packet and
not its entire route. Further, in contrast to BackPressure, the Proportional
Scheduler does not compare downstream queue lengths to determine weights, only
local link information is required. This leads to greater potential for
decomposed implementations of the policy. Through a reduction argument and an
entropy argument, we demonstrate that, whilst maintaining substantially less
queueing overhead, the Proportional Scheduler achieves maximum throughput
stability.Comment: 28 page
State space collapse and diffusion approximation for a network operating under a fair bandwidth sharing policy
We consider a connection-level model of Internet congestion control,
introduced by Massouli\'{e} and Roberts [Telecommunication Systems 15 (2000)
185--201], that represents the randomly varying number of flows present in a
network. Here, bandwidth is shared fairly among elastic document transfers
according to a weighted -fair bandwidth sharing policy introduced by Mo
and Walrand [IEEE/ACM Transactions on Networking 8 (2000) 556--567] []. Assuming Poisson arrivals and exponentially distributed document
sizes, we focus on the heavy traffic regime in which the average load placed on
each resource is approximately equal to its capacity. A fluid model (or
functional law of large numbers approximation) for this stochastic model was
derived and analyzed in a prior work [Ann. Appl. Probab. 14 (2004) 1055--1083]
by two of the authors. Here, we use the long-time behavior of the solutions of
the fluid model established in that paper to derive a property called
multiplicative state space collapse, which, loosely speaking, shows that in
diffusion scale, the flow count process for the stochastic model can be
approximately recovered as a continuous lifting of the workload process.Comment: Published in at http://dx.doi.org/10.1214/08-AAP591 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
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