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 $n$ 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 $k_{max}$ 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 $k_{max}$ clique approximation coincides with a Kikuchi free energy approximation and prove that it is exact on chordal conflict graphs when $k_{max} = n$. 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 $d$ 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 $d$ 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 $N^2$ matrix exponential representation for their sojourn time distribution, where $N$ 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 $N$. 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