Buffer Content Process Research Articles

Large deviations for stochastic fluid networks with Weibullian tails

We consider a stochastic fluid network where the external input processes are compound Poisson with heavy-tailed Weibullian jumps. Our results comprise of large deviations estimates for the buffer content process in the vector-valued Skorokhod space which is endowed with the product \(J_1\) topology. To illustrate our framework, we provide explicit results for a tandem queue. At the heart of our proof is a recent sample-path large deviations result, and a novel continuity result for the Skorokhod reflection map in the product \(J_1\) topology.

Markov modulated fluid network process: Tail asymptotics of the stationary distribution

We consider a Markov modulated fluid network with a finite number of stations. We are interested in the tail asymptotics behavior of the stationary distribution of its buffer content process. Using two different approaches, we derive upper and lower bounds for the stationary tail decay rate in various directions. Both approaches are based on a well-known time-evolution formula of a Markov process, so-called Dynkin’s formula, where a key ingredient is a suitable choice of test functions. Those results show how multidimensional tail asymptotics can be studied for the more than two-dimensional case, which is known as a hard problem.

Efficient Rare-Event Simulation for Multiple Jump Events in Regularly Varying Random Walks and Compound Poisson Processes

We propose a class of strongly efficient rare-event simulation estimators for random walks and compound Poisson processes with a regularly varying increment/jump-size distribution in a general large deviations regime. Our estimator is based on an importance sampling strategy that hinges on a recently established heavy-tailed sample-path large deviations result. The new estimators are straightforward to implement and can be used to systematically evaluate the probability of a wide range of rare events with bounded relative error. They are “universal” in the sense that a single importance sampling scheme applies to a very general class of rare events that arise in heavy-tailed systems. In particular, our estimators can deal with rare events that are caused by multiple big jumps (therefore, beyond the usual principle of a single big jump) as well as multidimensional processes such as the buffer content process of a queueing network. We illustrate the versatility of our approach with several applications that arise in the context of mathematical finance, actuarial science, and queueing theory.

Gaussian queues in light and heavy traffic

In this paper we investigate Gaussian queues in the light-traffic and in the heavy-traffic regime. Let $Q^{(c)}_{X}\equiv\{Q^{(c)}_{X}(t):t\ge0\}$ denote a stationary buffer content process for a fluid queue fed by the centered Gaussian process X≡{X(t):t∈ℝ} with stationary increments, X(0)=0, continuous sample paths and variance function σ 2(⋅). The system is drained at a constant rate c>0, so that for any t≥0, We study $Q^{(c)}_{X}\equiv\{Q_{X}^{(c)}(t):t\ge0\}$ in the regimes c→0 (heavy traffic) and c→∞ (light traffic). We show for both limiting regimes that, under mild regularity conditions on σ, there exists a normalizing function δ(c) such that $Q^{(c)}_{X}(\delta(c)\cdot)/\sigma(\delta(c))$ converges to $Q^{(1)}_{B_{H}}(\cdot)$ in C[0,∞), where B H is a fractional Brownian motion with suitably chosen Hurst parameter H.

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).

Tail asymptotics for a Lévy-driven tandem queue with an intermediate input

We consider a Levy-driven tandem queue with an intermediate input assuming that its buffer content process obtained by a reflection mapping has the stationary distribution. For this queue, no closed form formula is known, not only for its distribution but also for the corresponding transform. In this paper, we consider only light-tailed inputs. For the Brownian input case, we derive exact tail asymptotics for the marginal stationary distribution of the second buffer content, while weaker asymptotic results are obtained for the general Levy input case. The results generalize those of Lieshout and Mandjes from the recent papers (Lieshout and Mandjes in Math. Methods Oper. Res. 66:275---298, 2007 and Queueing Syst. 60:203---226, 2008) for the corresponding tandem queue without an intermediate input.

Traffic with an fBm Limit: Convergence of the Stationary Workload Process

Highly-aggregated traffic in communication networks is often modeled as fractional Brownian motion (fBm). This is justified by the theoretical result that the sum of a large number of on–off inputs, with either on-times or off-times having a heavy-tailed distribution with infinite variance, converges to fBm, after rescaling time appropriately. For performance analysis purposes, the key question is whether this convergence carries over to the stationary buffer content process. In this paper it is shown that, in a heavy-traffic queueing environment, this property indeed holds.

A mountain process with state dependent input and output and a correlated dam

We consider a single machine, subject to breakdown, that produces items to inventory. While the machine is working (an ON period) there is a net flow, with general deterministic production rate, into the buffer. As a result, the production rate minus the demand rate generate a certain deterministic increase rate which is state dependent. There are two different types of OFF periods. One is the repair operation initiated after a breakdown; the second is a preventive maintenance operation. The length of an OFF period depends on the preceding ON period. During the OFF periods there is a deterministic outflow (whose release rule is general but state dependent), generated by the demand rate. The buffer content process can then be described as a fluid model called the mountain process. The main tool employed in determining its steady state law exploits an equivalence relationship between the mountain process and a special dam process with general release rule whose jumps are also state dependent. This study is a continuation and generalization of a previous work [Oper. Res. Lett. 26 (2000) 137–147], in which both the production rate and the demand rate are assumed to be constant.

A correlated M [formula omitted]G [formula omitted]1-type queue with randomized server repair and maintenance modes

We consider a single machine, subject to breakdown, that produces items to inventory, continuously and uniformly. While the machine is working (an ON period), there is a deterministic net flow into the buffer. There are also two different types of OFF periods; one is a repair operation initiated after a breakdown, and the other is a preventive maintenance operation. The length of an OFF period depends on the length of the preceding ON period. During OFF periods there is a uniform outflow from the inventory buffer. The buffer content process can then be described as a fluid model. The main tool employed in determining its steady-state law exploits an equivalence relationship between the buffer content process and the virtual waiting time process of a special single-server queue in which the interarrival times and the service requirements depend on each other. We present an appropriate cost function and provide an optimization scheme illustrated by some numerical results.

BOUNDS FOR FLUID MODELS DRIVEN BY SEMI-MARKOV INPUTS

In this paper we consider an infinite buffer fluid model whose input is driven by independent semi-Markov processes. The output capacity of the buffer is a constant. We derive upper and lower bounds for the limiting distribution of the stationary buffer content process. We discuss examples and applications where the results can be used to determine bounds on the loss probability in telecommunication networks.

On–off fluid models in heavy traffic environment

We consider fluid models with infinite buffer size. Let \{Z_N(t)\} be the net input rate to the buffer, where \{Z_N(t)\} is a superposition of N homogeneous alternating on–off flows. Under heavy traffic environment \{Z_N (t)\} converges in distribution to a centred Gaussian process with covariance function of a single flow. The aim of this paper is to prove the convergence of the stationary buffer content process \{X^*_N(t)\} in the Nth model to the buffer content process \{X^*(t)\} in the limiting Gaussian model.

Large deviation properties of data streams that share a buffer

Using large deviation techniques, we analyze the tail behavior of the stationary distribution of the buffer content process for a two-station communication network. We also show how the associated rate function can be expressed as the solution to a finite-dimensional variational problem. Along the way, we develop a number of results and techniques that are of independent interest, including continuity results for the input-output mapping for certain multiclass fluid models and a new technique for obtaining large deviation principles for invariant distributions from sample path large deviation results.

The superposition of alternating on-off flows and a fluid model

An on-off process is a 0-1 process $\xi_t$ in which consecutive 0-periods ${T_{0, n}}$ alternate with 1-periods ${T_{1, n}}(n = 1, 2,\dots)$. The on and off time sequences are independent, each consisting of i.i.d. r.v.s. By the superposed flow, we mean the process $Z_t = \Sigma_{\ell=1}^N r^{\ell}\xi_t^{\ell}$, where $r^{\ell} > 0$ and ${\xi_t^1}, \dots,{\xi_t^N}$ are independent on-off flows. The process $\xi_t^{\ell}$ is not Markovian; however, with the age component $\eta_t^{\ell}$, the process $w_t^{\ell} = (\xi_t^{\ell}, \eta_t^{\ell})$ is a piecewise deterministic Markov process. In this paper we study the buffer content process for which the tail of its steady-state distribution $\Psi (b)$ fulfills inequality $C_- e^{\gammab} \leq \Psi (b) \leq C_+ e^{-\gammab}, where $\gamma > 0$ is the solution of some basic nonlinear system of equations.

An approximation method for complete solutions of Markov-modulated fluid models

This paper presents an approximation method for numerically solving general Markov-modulated fluid models which are widely used in modelling communications and computer systems. We show how the superposition of a group of heterogeneous sources (normally modeled by a multidimensional Markov process) can be approximated by a one-dimensional Markov process, which is then used as the modulating process of the buffer content process. The method effectively reduces the computation that is usually required to find exact (or asymptotic) solutions of fluid models. While this method is general, we focus our discussion on the models with only ON/OFF traffic sources. Numerous numerical results are provided to show the accuracy of the approximation.

The impact of preventive maintenance on system cost and buffer size

In this paper we consider a system with two production operations (typically machines) and a buffer between them. Our objective is to determine the optimal buffer inventory level, b ∗ , that triggers preventive maintenance on the first production operation. We assume that periodic preventive maintenance will, in the long run, decrease the first production operation's failure rate. A cost model for the system is specified that includes the costs of preventive maintenance and unscheduled repairs, starving the second production operation, and inventory. The cost model is dependent on the values of performance measures that are a function of b ∗ , which defines the preventive maintenance policy. We model the stochastic buffer content process of this system and obtain the performance measures with an embedded absorbing Markov chain analysis. Under certain scenarios, the implementation of the preventive maintenance program is shown to decrease the system cost and the output process variation. Furthermore, our model can be used to indicate oversized buffers in such an environment.

Second-Order Fluid Flow Models: Reflected Brownian Motion in a Random Environment

This paper considers a stochastic fluid model of a buffer content process {X(t), t ≥ 0} that depends on a finite-state, continuous-time Markov process {Z(t), t ≥ 0} as follows: During the time-intervals when Z(t) is in state i, X(t) is a Brownian motion with drift μi, variance parameter σi2and a reflecting boundary at zero. This paper studies the steady-state analysis of the bivariate process {(X(t), Z(t)), t ≥ 0} in terms of the eigenvalues and eigenvectors of a nonlinear matrix system. Algorithms are developed to compute the steady-state distributions as well as moments. Numerical work is reported to show that the variance parameter has a dramatic effect on the buffer content process.

State-Dependent Benes Buffer Model with Fast Loading and Output Rates

We consider a state-dependent generalization of the exponential Benes model of single-source buffer system in which the source process consists of alternating transmission and idle periods. Martingale methods are applied for analyzing limit non-stationary behavior of the buffer content process, when the buffer is loaded and depleted, with rates proportional to a large parameter N. Depending on traffic conditions, defined by parameters of the model, different types of approximations are established for the buffer content. We show that in heavy traffic the buffer content grows linearly in N , while the deviations of the order √ N from the deterministic limit are approximated by the Gaussian diffusion process. In the moderate traffic the buffer content grows as √ N , and the normalized buffer content is approximated by diffusion process with reflection at zero. In the case of normal traffic we show that the buffer utilization tends to the ratio of ”the input-to-output-rate”. Moreover, we show that the main contribution to the utilization comes from arbitrary small buffer content.