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

2011 ◽  
Vol 48 (03) ◽  
pp. 820-831
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Compact Set ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Heavy Tailed ◽  
Necessary And Sufficient

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 ∈ (½, 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 chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= E x [V(Ž̆(1))] − V(x) ≤ −βV(x) + b 1 C (x), x ∈ S, for 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.

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

2011 ◽  
Vol 48 (3) ◽  
pp. 820-831 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Compact Set ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Heavy Tailed ◽  
Necessary And Sufficient

We study a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = ℝ+d, with drift r0 ∈ ℝd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a geometric drift towards a compact set for the 1-skeleton chain Ž̆ of the RFBM process Z; that is, there exist β, b ∈ (0, ∞) and a compact set C ⊂ S such that ΔV(x):= Ex[V(Ž̆(1))] − V(x) ≤ −βV(x) + b1C(x), x ∈ S, for 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.

scholarly journals On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (01) ◽  
pp. 145-153 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Return Time ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Time Result ◽  
Heavy Tailed

We consider 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 ∈ (½, 1). Under a natural stability condition on the drift vector r 0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, sup x∈B E x [τ B (δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τ B (δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our 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.

scholarly journals On the Return Time for a Reflected Fractional Brownian Motion Process on the Positive Orthant

2011 ◽  
Vol 48 (1) ◽  
pp. 145-153 ◽  
Author(s):  
Chihoon Lee
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Return Time ◽  
Brownian Motion Process ◽  
Positive Orthant ◽  
Time Result ◽  
Heavy Tailed

We consider a d-dimensional reflected fractional Brownian motion (RFBM) process on the positive orthant S = R+d, with drift r0 ∈ Rd and Hurst parameter H ∈ (½, 1). Under a natural stability condition on the drift vector r0 and reflection directions, we establish a return time result for the RFBM process Z; that is, for some δ, κ > 0, supx∈BEx[τB(δ)] < ∞, where B = {x ∈ S : |x| ≤ κ} and τB(δ) = inf{t ≥ δ : Z(t) ∈ B}. Similar results are known for reflected processes driven by standard Brownian motions, and our result can be viewed as their FBM counterpart. Our 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.

scholarly journals Stationarity and control of a tandem fluid network with fractional Brownian motion input

2011 ◽  
Vol 43 (03) ◽  
pp. 847-874
Author(s):  
Chihoon Lee ◽  
Ananda Weerasinghe
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Fluid Network ◽  
State Process ◽  
Long Run ◽  
Convergence Results ◽  
Heavy Tailed ◽  
And Control

We consider a stochastic control model for a queueing system driven by a two-dimensional fractional Brownian motion with Hurst parameter 0 &lt; H &lt; 1. In particular, when H &gt; ½, this model serves to approximate a controlled two-station tandem queueing model with heavy-tailed ON/OFF sources in heavy traffic. We establish the weak convergence results for the distribution of the state process and construct an explicit stationary state process associated with given controls. Based on suitable coupling arguments, we show that each state process couples with its stationary counterpart and we use it to represent the long-run average cost functional in terms of the stationary process. Finally, we establish the existence result of an optimal control, which turns out to be independent of the initial data.

scholarly journals Stationarity and control of a tandem fluid network with fractional Brownian motion input

2011 ◽  
Vol 43 (3) ◽  
pp. 847-874 ◽  
Author(s):  
Chihoon Lee ◽  
Ananda Weerasinghe
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Fluid Network ◽  
State Process ◽  
Long Run ◽  
Convergence Results ◽  
Heavy Tailed ◽  
And Control

We consider a stochastic control model for a queueing system driven by a two-dimensional fractional Brownian motion with Hurst parameter 0 <H< 1. In particular, whenH> ½, this model serves to approximate a controlled two-station tandem queueing model with heavy-tailed ON/OFF sources in heavy traffic. We establish the weak convergence results for the distribution of the state process and construct an explicit stationary state process associated with given controls. Based on suitable coupling arguments, we show that each state process couples with its stationary counterpart and we use it to represent the long-run average cost functional in terms of the stationary process. Finally, we establish the existence result of an optimal control, which turns out to be independent of the initial data.

Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (02) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

Diffusion Limits for Open Networks of Finite-Buffer Queues

1996 ◽  
Vol 10 (3) ◽  
pp. 341-361
Author(s):  
Indrajit Bardhan
Keyword(s):  
Brownian Motion ◽  
Communication Systems ◽  
Heavy Traffic ◽  
Finite Buffer ◽  
Diffusion Approximations ◽  
Loss Networks ◽  
Positive Orthant ◽  
System Process ◽  
Open Networks ◽  
Diffusion Limits

This paper presents diffusion limits for congestion in networks of finite-buffer queues. We consider both loss networks, such as those in communication systems, and networks with manufacturing blocking. In both cases, the number in system process, under conditions of approximate balance under heavy traffic and appropriate scaling of buffers, is shown to behave like a multidimensional Brownian motion reflected to stay within a rectangle in the positive orthant. The two limits differ in directions of reflection off the faces representing full buffers. The limits suggest possible diffusion approximations for finitebuffer networks.

scholarly journals Heavy traffic analysis of a transportation network model

1996 ◽  
Vol 33 (03) ◽  
pp. 870-885
Author(s):  
William P. Peterson ◽  
Lawrence M. Wein
Keyword(s):  
Queueing Networks ◽  
Transportation Network ◽  
Heavy Traffic ◽  
Queueing Network ◽  
Network Models ◽  
Reflected Brownian Motion ◽  
Traffic Conditions ◽  
Heavy Traffic Analysis ◽  
Functional Central Limit ◽  
Heavy Traffic Conditions

We study a model of a stochastic transportation system introduced by Crane. By adapting constructions of multidimensional reflected Brownian motion (RBM) that have since been developed for feedforward queueing networks, we generalize Crane's original functional central limit theorem results to a full vector setting, giving an explicit development for the case in which all terminals in the model experience heavy traffic conditions. We investigate product form conditions for the stationary distribution of our resulting RBM limit, and contrast our results for transportation networks with those for traditional queueing network models.

scholarly journals Drift Parameter Estimation for a Reflected Fractional Brownian Motion Based on its Local Time

2013 ◽  
Vol 50 (2) ◽  
pp. 592-597 ◽  
Author(s):  
Yaozhong Hu ◽  
Chihoon Lee
Keyword(s):  
Parameter Estimation ◽  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Local Time ◽  
Heavy Traffic ◽  
Queueing Systems ◽  
Service Rate ◽  
Time Process ◽  
State Process ◽  
Drift Parameter

We consider a drift parameter estimation problem when the state process is a reflected fractional Brownian motion (RFBM) with a nonzero drift parameter and the observation is the associated local time process. The RFBM process arises as the key approximating process for queueing systems with long-range dependent and self-similar input processes, where the drift parameter carries the physical meaning of the surplus service rate and plays a central role in the heavy-traffic approximation theory for queueing systems. We study a statistical estimator based on the cumulative local time process and establish its strong consistency and asymptotic normality.

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