214 research outputs found
Free Energy Approximations for CSMA networks
In this paper we study how to estimate the back-off rates in an idealized
CSMA network consisting of links to achieve a given throughput vector using
free energy approximations. More specifically, we introduce the class of
region-based free energy approximations with clique belief and present a closed
form expression for the back-off rates based on the zero gradient points of the
free energy approximation (in terms of the conflict graph, target throughput
vector and counting numbers). Next we introduce the size clique free
energy approximation as a special case and derive an explicit expression for
the counting numbers, as well as a recursion to compute the back-off rates. We
subsequently show that the size clique approximation coincides with a
Kikuchi free energy approximation and prove that it is exact on chordal
conflict graphs when . As a by-product these results provide us
with an explicit expression of a fixed point of the inverse generalized belief
propagation algorithm for CSMA networks. Using numerical experiments we compare
the accuracy of the novel approximation method with existing methods
Global attraction of ODE-based mean field models with hyperexponential job sizes
Mean field modeling is a popular approach to assess the performance of large
scale computer systems. The evolution of many mean field models is
characterized by a set of ordinary differential equations that have a unique
fixed point. In order to prove that this unique fixed point corresponds to the
limit of the stationary measures of the finite systems, the unique fixed point
must be a global attractor. While global attraction was established for various
systems in case of exponential job sizes, it is often unclear whether these
proof techniques can be generalized to non-exponential job sizes. In this paper
we show how simple monotonicity arguments can be used to prove global
attraction for a broad class of ordinary differential equations that capture
the evolution of mean field models with hyperexponential job sizes. This class
includes both existing as well as previously unstudied load balancing schemes
and can be used for systems with either finite or infinite buffers. The main
novelty of the approach exists in using a Coxian representation for the
hyperexponential job sizes and a partial order that is stronger than the
componentwise partial order used in the exponential case.Comment: This paper was accepted at ACM Sigmetrics 201
Improved Load Balancing in Large Scale Systems using Attained Service Time Reporting
Our interest lies in load balancing jobs in large scale systems consisting of
multiple dispatchers and FCFS servers. In the absence of any information on job
sizes, dispatchers typically use queue length information reported by the
servers to assign incoming jobs. When job sizes are highly variable, using only
queue length information is clearly suboptimal and performance can be improved
if some indication can be provided to the dispatcher about the size of an
ongoing job. In a FCFS server measuring the attained service time of the
ongoing job is easy and servers can therefore report this attained service time
together with the queue length when queried by a dispatcher.
In this paper we propose and analyse a variety of load balancing policies
that exploit both the queue length and attained service time to assign jobs, as
well as policies for which only the attained service time of the job in service
is used. We present a unified analysis for all these policies in a large scale
system under the usual asymptotic independence assumptions. The accuracy of the
proposed analysis is illustrated using simulation.
We present extensive numerical experiments which clearly indicate that a
significant improvement in waiting (and thus also in response) time may be
achieved by using the attained service time information on top of the queue
length of a server. Moreover, the policies which do not make use of the queue
length still provide an improved waiting time for moderately loaded systems
Performance Analysis of Load Balancing Policies with Memory
Joining the shortest or least loaded queue among randomly selected queues
are two fundamental load balancing policies. Under both policies the dispatcher
does not maintain any information on the queue length or load of the servers.
In this paper we analyze the performance of these policies when the dispatcher
has some memory available to store the ids of some of the idle servers. We
consider methods where the dispatcher discovers idle servers as well as methods
where idle servers inform the dispatcher about their state.
We focus on large-scale systems and our analysis uses the cavity method. The
main insight provided is that the performance measures obtained via the cavity
method for a load balancing policy {\it with} memory reduce to the performance
measures for the same policy {\it without} memory provided that the arrival
rate is properly scaled. Thus, we can study the performance of load balancers
with memory in the same manner as load balancers without memory. In particular
this entails closed form solutions for joining the shortest or least loaded
queue among randomly selected queues with memory in case of exponential job
sizes. Moreover, we obtain a simple closed form expression for the (scaled)
expected waiting time as the system tends towards instability.
We present simulation results that support our belief that the approximation
obtained by the cavity method becomes exact as the number of servers tends to
infinity.Comment: 30 pages, 3 figure
Commuting matrices in the sojourn time analysis of MAP/MAP/1 queues
Queues with Markovian arrival and service processes, i.e., MAP/MAP/1 queues, have been useful in the analysis of computer and communication systems and different representations for their sojourn time distribution have been derived. More specifically, the class of MAP/MAP/1 queues lies at the intersection of the class of QBD queues and the class of semi-Markovian queues. While QBD queues have an order matrix exponential representation for their sojourn time distribution, where is the size of the background continuous time Markov chain, the sojourn time distribution of the latter class allows for a more compact representation of order .
In this paper we unify these two results and show that the key step exists in establishing the commutativity of some fundamental matrices involved in the analysis of the MAP/MAP/1 queue. We prove, using two different approaches, that the required matrices do commute and identify several other sets of commuting matrices. Finally, we generalize some of the results to queueing systems with batch arrivals and services
A Refined Mean Field Approximation
International audienceMean field models are a popular means to approximate large and complex stochastic models that can be represented as N interacting objects. Recently it was shown that under very general conditions the steady-state expectation of any performance functional converges at rate O(1/N) to its mean field approximation. In this paper we establish a result that expresses the constant associated with this 1/N term. This constant can be computed easily as it is expressed in terms of the Jacobian and Hessian of the drift in the fixed point and the solution of a single Lyapunov equation. This allows us to propose a refined mean field approximation. By considering a variety of applications, that include coupon collector, load balancing and bin packing problems, we illustrate that the proposed refined mean field approximation is significantly more accurate that the classic mean field approximation for small and moderate values of N: relative errors are often below 1% for systems with N=10
TTL Approximations of the Cache Replacement Algorithms LRU(m) and h-LRU
International audienceComputer system and network performance can be significantly improved by caching frequently used information. When the cache size is limited, the cache replacement algorithm has an important impact on the effectiveness of caching. In this paper we introduce time-to-live (TTL) approximations to determine the cache hit probability of two classes of cache replacement algorithms: h-LRU and LRU(m). These approximations only require the requests to be generated according to a general Markovian arrival process (MAP). This includes phase-type renewal processes and the IRM model as special cases. We provide both numerical and theoretical support for the claim that the proposed TTL approximations are asymptotically exact. In particular, we show that the transient hit probability converges to the solution of a set of ODEs (under the IRM model), where the fixed point of the set of ODEs corresponds to the TTL approximation. We use this approximation and trace-based simulation to compare the performance of h-LRU and LRU(m). First, we show that they perform alike, while the latter requires less work when a hit/miss occurs. Second, we show that as opposed to LRU, h-LRU and LRU(m) are sensitive to the correlation between consecutive inter-request times. Last, we study cache partitioning. In all tested cases, the hit probability improved by partitioning the cache into different parts—each being dedicated to a particular content provider. However, the gain is limited and the optimal partition sizes are very sensitive to the problem's parameters
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