A fluid model for a relay node in an ad-hoc network: the case of heavy-tailed input

Relay nodes in an ad hoc network can be modelled as fluid queues, in which the available service capacity is shared by the input and output. In this paper such a relay node is considered; jobs arrive according to a Poisson process and bring along a random amount of work. The total transmission capacity is fairly shared, meaning that, when n jobs are present, each job transmits traffic into the queue at rate 1/(n + 1) while the queue is drained at the same rate of 1/(n + 1). Where previous studies mainly concentrated on the case of exponentially distributed job sizes, the present paper addresses regularly varying jobs. The focus lies on the tail asymptotics of the sojourn time S. Using sample-path arguments, it is proven that P {S > x} behaves roughly as the residual job size, i.e., if the job sizes are regularly varying of index -nu, the tail of S is regularly varying of index 1 - nu. In addition, we address the tail asymptotics of other performance metrics, such as the workload in the queue, the flow transfer time and the queueing delay

Asymptotic Expansions for the Sojourn Time Distribution in the M/G/1M/G/1-PS Queue

We consider the $M/G/1$ queue with a processor sharing server. We study the conditional sojourn time distribution, conditioned on the customer's service requirement, as well as the unconditional distribution, in various asymptotic limits. These include large time and/or large service request, and heavy traffic, where the arrival rate is only slightly less than the service rate. Our results demonstrate the possible tail behaviors of the unconditional distribution, which was previously known in the cases $G=M$ and $G=D$ (where it is purely exponential). We assume that the service density decays at least exponentially fast. We use various methods for the asymptotic expansion of integrals, such as the Laplace and saddle point methods.Comment: 45 page

M/M/∞ transience : tail asymptotics of congestion periods

The c-congestion period, defined as a time interval in which the number of customers is larger than c all the time, is a key quantity in the design of communication networks. Particularly in the setting of M/M/infinity systems, the analysis of the duration of the congestion period, D_c, has attracted substantial attention; related quantities have been studied as well, such as the total area A_c above c, and the number of arrived customers N_c during a congestion period. Laplace transforms of these three random variables being known, as well as explicit formulae for their moments, this paper addresses the corresponding tail asymptotics. Our work addresses the following topics. In the so-called many-flows scaling, we show that the tail asymptotics are essentially exponential in the scaling parameter. The proof techniques stem from large deviations theory; we also identify the most likely way in which the event under consideration occurs. In the same scaling, we approximate the model by its Gaussian counterpart. Specializing to our specific model, we show that the (fairly abstract) sample-path largedeviations theorem for Gaussian processes, viz. the generalized version of Schilder’s theorem, can be written in a considerably more explicit way. Relying on this result, we derive the tail asymptotics for the Gaussian counterpart. Then we use change-of-measure arguments to find upper bounds, uniform in the model parameters, on the probabilities of interest. These change-of-measures are applied to devise of a number of importance-sampling schemes, for fast simulation of rare-event probabilities. They turn out to yield a substantial speed-up in simulation effort, compared to naïve, direct simulations

Flow-level models for multipath routing

In this paper we study coordinated multipath routing at the flow-level in networks with routes of length one. As a first step the static case is considered, in which the number of flows is fixed. A clustering pattern in the rate allocation is identified, and we describe a finite algorithm to find this rate allocation and the clustering explicitly. Then we consider the dynamic model, in which there are stochastic arrivals and departures; we do so for models with both streaming and elastic traffic, and where a peak-rate is imposed on the elastic flows (to be thought of as an access rate). Lacking explicit expressions for the equilibrium distribution of the Markov process under consideration, we study its fluid and diffusion limits; in particular, we prove the uniqueness of the equilibrium point. We demonstrate through a specific example how the diffusion limit can be identified; it also reveals structural results about the clustering pattern when the minimal rate is very small and the network grows large

Traffic generated by a semi-Markov additive process

We consider a semi-Markov additive process A(·)—that is, a Markov additive process for which the sojourn times in the various states have general (rather than exponential) distributions. Letting the Lévy processes Xi(·), which describe the evolution of A(·) while the background process is in state i, be increasing, it is shown how double transforms of the type (formule) can be computed. It turns out that these follow, for given nonnegative a and q, from a system of linear equations, which has a unique positive solution. Several extensions are considered as well

Open problems in Gaussian fluid queueing theory

We present three challenging open problems that originate from the analysis of the asymptotic behavior of Gaussian fluid queueing models. In particular, we address the problem of characterizing the correlation structure of the stationary buffer content process, the speed of convergence to stationarity, and analysis of an asymptotic constant associated with the stationary buffer content distribution (the so-called Pickands constant)

Mean sojourn times in two-queue fork-join systems : bounds and approximations

This paper considers a fork-join system (or: parallel queue), which is a two-queue network in which any arrival generates jobs at both queues and the jobs synchronize before they leave the system. The focus is on methods to quantify the mean value of the 'system's sojourn time' S: with Si denoting a job's sojourn time in queue i, S is defined as max{S1, S2}. Earlier work has revealed that this class of models is notoriously hard to analyze. In this paper, we focus on the homogeneous case, in which the jobs generated at both queues stem from the same distribution. We first evaluate various bounds developed in the literature, and observe that under fairly broad circumstances these can be rather inaccurate. We then present a number of approximations, that are extensively tested by simulation and turn out to perform remarkably well

Editorial (Special issue on Rare-event simulation for queues)

Without abstract

Estimation of the workload correlation in a Markov fluid queue

This paper considers a Markov fluid queue, focusing on the correlation function of the stationary workload process. A simulation-based computation technique is proposed, which relies on a coupling idea. Then an upper bound on the variance of the resulting estimator is given, which reveals how the coupling time and the busy period of the Markov fluid queue affect the performance of the computation procedure. A numerical assessment, in which we compare the proposed technique with naive simulation, gives an indication of the achievable efficiency gain in various scenarios

On the record process of time-reversible spectrally-negative Markov additive processes

We study the record process of a spectrally-negative Markov additive process (MAP). Assuming time-reversibility, a number of key quantities can be given explicitly. It is shown how these key quantities can be used when analyzing the distribution of the all-time maximum attained by MAPs with negative drift, or, equivalently, the stationary workload distribution of the associated storage system; the focus is on Markov-modulated Brownian mo- tion, spectrally-negative and spectrally-positive MAPs. It is also argued how our results are of great help in the numerical analysis of systems in which the driving MAP is a superposition of multiple time-reversible MAPs