12,628 research outputs found
On the maximum queue length in the supermarket model
There are queues, each with a single server. Customers arrive in a
Poisson process at rate , where . Upon arrival each
customer selects servers uniformly at random, and joins the queue at a
least-loaded server among those chosen. Service times are independent
exponentially distributed random variables with mean 1. We show that the system
is rapidly mixing, and then investigate the maximum length of a queue in the
equilibrium distribution. We prove that with probability tending to 1 as
the maximum queue length takes at most two values, which are
.Comment: Published at http://dx.doi.org/10.1214/00911790500000710 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Large deviations sum-queue optimality of a radial sum-rate monotone opportunistic scheduler
A centralized wireless system is considered that is serving a fixed set of
users with time varying channel capacities. An opportunistic scheduling rule in
this context selects a user (or users) to serve based on the current channel
state and user queues. Unless the user traffic is symmetric and/or the
underlying capacity region a polymatroid, little is known concerning how
performance optimal schedulers should tradeoff "maximizing current service
rate" (being opportunistic) versus "balancing unequal queues" (enhancing
user-diversity to enable future high service rate opportunities). By contrast
with currently proposed opportunistic schedulers, e.g., MaxWeight and Exp Rule,
a radial sum-rate monotone (RSM) scheduler de-emphasizes queue-balancing in
favor of greedily maximizing the system service rate as the queue-lengths are
scaled up linearly. In this paper it is shown that an RSM opportunistic
scheduler, p-Log Rule, is not only throughput-optimal, but also maximizes the
asymptotic exponential decay rate of the sum-queue distribution for a two-queue
system. The result complements existing optimality results for opportunistic
scheduling and point to RSM schedulers as a good design choice given the need
for robustness in wireless systems with both heterogeneity and high degree of
uncertainty.Comment: Revised version. Major changes include addition of
details/intermediate steps in various proofs, a summary of technical steps in
Table 1, and correction of typos
Fast simulation of the leaky bucket algorithm
We use fast simulation methods, based on importance sampling, to efficiently estimate cell loss probability in queueing models of the Leaky Bucket algorithm. One of these models was introduced by Berger (1991), in which the rare event of a cell loss is related to the rare event of an empty finite buffer in an "overloaded" queue. In particular, we propose a heuristic change of measure for importance sampling to efficiently estimate the probability of the rare empty-buffer event in an asymptotically unstable GI/GI/1/k queue. This change of measure is, in a way, "dual" to that proposed by Parekh and Walrand (1989) to estimate the probability of a rare buffer overflow event. We present empirical results to demonstrate the effectiveness of our fast simulation method. Since we have not yet obtained a mathematical proof, we can only conjecture that our heuristic is asymptotically optimal, as k/spl rarr//spl infin/
Functional Large Deviations for Cox Processes and Queues, with a Biological Application
We consider an infinite-server queue into which customers arrive according to
a Cox process and have independent service times with a general distribution.
We prove a functional large deviations principle for the equilibrium queue
length process. The model is motivated by a linear feed-forward gene regulatory
network, in which the rate of protein synthesis is modulated by the number of
RNA molecules present in a cell. The system can be modelled as a tandem of
infinite-server queues, in which the number of customers present in a queue
modulates the arrival rate into the next queue in the tandem. We establish
large deviation principles for this queueing system in the asymptotic regime in
which the arrival process is sped up, while the service process is not scaled.Comment: 36 pages, 2 figures, to appear in Annals of Applied Probabilit
Sample-path large deviations for tandem and priority queues with Gaussian inputs
This paper considers Gaussian flows multiplexed in a queueing network. A
single node being a useful but often incomplete setting, we examine more
advanced models. We focus on a (two-node) tandem queue, fed by a large number
of Gaussian inputs. With service rates and buffer sizes at both nodes scaled
appropriately, Schilder's sample-path large-deviations theorem can be applied
to calculate the asymptotics of the overflow probability of the second queue.
More specifically, we derive a lower bound on the exponential decay rate of
this overflow probability and present an explicit condition for the lower bound
to match the exact decay rate. Examples show that this condition holds for a
broad range of frequently used Gaussian inputs. The last part of the paper
concentrates on a model for a single node, equipped with a priority scheduling
policy. We show that the analysis of the tandem queue directly carries over to
this priority queueing system.Comment: Published at http://dx.doi.org/10.1214/105051605000000133 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
First-Passage Time and Large-Deviation Analysis for Erasure Channels with Memory
This article considers the performance of digital communication systems
transmitting messages over finite-state erasure channels with memory.
Information bits are protected from channel erasures using error-correcting
codes; successful receptions of codewords are acknowledged at the source
through instantaneous feedback. The primary focus of this research is on
delay-sensitive applications, codes with finite block lengths and, necessarily,
non-vanishing probabilities of decoding failure. The contribution of this
article is twofold. A methodology to compute the distribution of the time
required to empty a buffer is introduced. Based on this distribution, the mean
hitting time to an empty queue and delay-violation probabilities for specific
thresholds can be computed explicitly. The proposed techniques apply to
situations where the transmit buffer contains a predetermined number of
information bits at the onset of the data transfer. Furthermore, as additional
performance criteria, large deviation principles are obtained for the empirical
mean service time and the average packet-transmission time associated with the
communication process. This rigorous framework yields a pragmatic methodology
to select code rate and block length for the communication unit as functions of
the service requirements. Examples motivated by practical systems are provided
to further illustrate the applicability of these techniques.Comment: To appear in IEEE Transactions on Information Theor
Rare event analysis of Markov-modulated infinite-service queues: A Poisson limit
This paper studies an infinite-server queue in a Markov environment, that is, an infinite-server
queue with arrival rates and service times depending on the state of a Markovian background
process. Scaling the arrival rates by a factor and the rates of the background process by N^{1+\vareps}
(for some \vareps > 0), the focus is on the tail probabilities of the number of customers in the system, in
the asymptotic regime that tends to . In particular, it is shown that the logarithmic asymptotics
correspond to those of a Poisson distribution with an appropriate mean
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