A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic

Delgado, Rosario

doi:10.1016/j.spa.2006.07.001

Cited by 38 publications

(62 citation statements)

References 7 publications

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“…This problem is overcome under a stronger assumption on R, which we quote below (cf. Proposition 4.2 of [22] and Section 2 of [4]). We will impose throughout the following assumption on the reflection matrix R.…”

Section: D Y I Is Continuous Nondecreasing and Can Increase Only Wmentioning

confidence: 99%

“…Under (HR), it can be shown (cf. [4] and [22]) that if B H is adapted to some filtration {F t : t ≥ 0} then (Z, Y ) is adapted to the filtration {G t : t ≥ 0}, with G t = F t ∨ N , where N denotes the collection of P-null sets in F . Henceforth, with an abuse of notation, we will assume that (Z, Y ) is adapted to the filtration {F t : t ≥ 0}.…”

Section: D Y I Is Continuous Nondecreasing and Can Increase Only Wmentioning

confidence: 99%

“…Let R d be the d-dimensional Euclidean space, and, for (0), H, , r 0 , R) will be called the data for an RFBM. The following definition is similar to that given in [4]. Definition 2.1.…”

mentioning

confidence: 99%

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A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Lee

2011

Journal of Applied Probability

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We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R d + , with drift r 0 ∈ R d and Hurst parameter H ∈ ( 1 2 , 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chainZ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact setfor an exponentially growing Lyapunov function V : S → [1, ∞). For a wide class of Markov processes, such a drift inequality is known as a necessary and sufficient condition for exponential ergodicity. Indeed, similar drift inequalities have been established for reflected processes driven by standard Brownian motions, and our result can be viewed as their fractional Brownian motion counterpart. We also establish that the return times to the set C itself are geometrically bounded. Motivation for this study is that RFBM appears as a limiting workload process for fluid queueing network models fed by a large number of heavy-tailed ON/OFF sources in heavy traffic.

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Section: D Y I Is Continuous Nondecreasing and Can Increase Only Wmentioning

confidence: 99%

Section: D Y I Is Continuous Nondecreasing and Can Increase Only Wmentioning

confidence: 99%

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A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Lee

2011

Journal of Applied Probability

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“…Reflection is also considered in Kushner (2001), for the case of a general model with short range dependence and light tails features. First results on heavy traffic with fBm in the limit can be found in Konstantopoulos and Lin (1996), Dȩbicki and Mandjes (2004), and Delgado (2007). The second relation in (1.4) viewed as a mapping of η appears in Reed and Ward (2004), Ward and Kumar (2008) where it is called a generalized regulator mapping.…”

Section: Introductionmentioning

confidence: 99%

Heavy Traffic Approximations of a Queue with Varying Service Rates and General Arrivals

2012

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Heavy traffic limit theorems are established for a class of single server queueing models including those with heavy-tailed or long-range dependent arrivals and time-varying service rates. The models are motivated by wireless queueing systems for which there is an increasing evidence of the presence of heavy-tailed or long range dependent arrivals, and where the service rates vary with the changes in the wireless medium. Since changes in the wireless medium occur slowly, the service rate is assumed to vary slower than that of the arrivals.The main focus of the paper is to obtain the different possible limit forms that can arise depending on the relationship between established scalings for both the arrival and departure processes. The limit forms obtained here are driven by either Brownian motion (when the contribution from the departure process dominates the limit) or the limits of properly scaled arrivals (when the contribution from the arrival process dominates the limit), typical examples being stable Lévy motion or fractional Brownian motion. In particular, for the case where arrival process is given by the infinite source Poisson process, this relationship, which determines the type of the limiting queue-length process, is a simple condition involving the heavy tail exponent, arrival rate and channel variation parameter in the wireless medium model. The limit forms are also affected by the assumption of a slower varying service rate.To establish these limit results, two approaches are studied. In one approach, when the limit is driven by Brownian motion, the perturbed test function method is extended to incorporate reflection. In contrast, the second approach allows for non-Markovian driving processes in the limit. Both approaches involve averaging in the drift term arising from random service rates at the departures. In the second approach, this averaging is carried out directly and pathwise, thus sidestepping the assumption of driving Brownian motion used in the perturbed test function method. In the case of the infinite source Poisson model, when the limit is stable Lévy motion, it is also argued how the limit form can be deduced using infinitesimal operators.

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“…Following this paradigm shift, there has been some recent work on heavy traffic modeling of queueing systems under LRD/HT assumptions on the data traffic in wireline systems (see [3], [5], [4]). In these papers reflected fractional Brownian motion (RFBM) models for the queue process are obtained.…”

Section: Introductionmentioning

confidence: 99%

Heavy traffic limits in a wireless queueing model with long range dependence

Buche

Ghosh

Pipiras

2007

2007 46th IEEE Conference on Decision and Control

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Abstract-High-speed wireless networks carrying multimedia applications are becoming a reality and the transmitted data exhibit long range dependence and heavy-tailed properties. We consider the heavy traffic approach in working towards queue models under these properties, extending the model in [2]. Our focus is on the scalings used in the heavy traffic approach which are determined by combinations of the source rate of an infinite source Poisson model of the arrival process, the tail distribution of data transmitted by these sources, and the rate of variation of the random process (channel process) modeling the wireless medium. A fundamental inequality between the exponent in the power tail distribution of the data from the source and the parameter specifying the rate of channel variations is obtain. This inequality is important in both the "fast growth" and "slow growth" regimes for the arrival process and along with the source rate is used to define the possible cases for obtaining limit models for the queueing process. Across the cases, the possible limit models include reflected Brownian motion, reflected stable Lévy motion, or reflected fractional Brownian motion.

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A reflected fBm limit for fluid models with ON/OFF sources under heavy traffic

Cited by 38 publications

References 7 publications

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

A Geometric Drift Inequality for a Reflected Fractional Brownian Motion Process on the Positive Orthant

Heavy Traffic Approximations of a Queue with Varying Service Rates and General Arrivals

Heavy traffic limits in a wireless queueing model with long range dependence

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