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.

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 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 &gt; 0, sup x∈B E x [τ B (δ)] &lt; ∞, 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.

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.

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 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.

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 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.

The stable M/G/1 queue in heavy traffic and its covariance function

1977 ◽  
Vol 9 (01) ◽  
pp. 169-186 ◽  
Author(s):  
Teunis J. Ott
Keyword(s):  
Brownian Motion ◽  
Waiting Time ◽  
Covariance Function ◽  
Busy Period ◽  
Stationary Process ◽  
Heavy Traffic ◽  
Time Process ◽  
Period Distribution ◽  
Virtual Waiting Time ◽  
Waiting Time Process

Let X(t) be the virtual waiting-time process of a stable M/G/1 queue. Let R(t) be the covariance function of the stationary process X(t), B(t) the busy-period distribution of X(t); and let E(t) = P{X(t) = 0|X(0) = 0}. For X(t) some heavy-traffic results are given, among which are limiting expressions for R(t) and its derivatives and for B(t) and E(t). These results are used to find the covariance function of stationary Brownian motion on [0, ∞).

scholarly journals Fractional Brownian Motion with H < 1/2 as a Limit of Scheduled Traffic

2012 ◽  
Vol 49 (03) ◽  
pp. 710-718 ◽  
Author(s):  
Victor F. Araman ◽  
Peter W. Glynn
Keyword(s):  
Brownian Motion ◽  
Limit Theorem ◽  
Fractional Brownian Motion ◽  
Heavy Traffic ◽  
Traffic Model ◽  
Traffic Models ◽  
Heavy Traffic Limit ◽  
Scheduled Traffic ◽  
Simple Class ◽  
Traffic Processes

In this paper we show that fractional Brownian motion with H &lt; ½ can arise as a limit of a simple class of traffic processes that we call ‘scheduled traffic models’. To our knowledge, this paper provides the first simple traffic model leading to fractional Brownnian motion with H &lt; ½. We also discuss some immediate implications of this result for queues fed by scheduled traffic, including a heavy-traffic limit theorem.

scholarly journals Abstract Functional Stochastic Evolution Equations Driven by Fractional Brownian Motion

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Mark A. McKibben ◽  
Micah Webster
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Evolution Equations ◽  
Separable Hilbert Space ◽  
Mild Solutions ◽  
Stochastic Evolution ◽  
Stochastic Evolution Equations ◽  
Convergence Results ◽  
Real Separable Hilbert Space ◽  
Wave Phenomena

We investigate a class of abstract functional stochastic evolution equations driven by a fractional Brownian motion in a real separable Hilbert space. Global existence results concerning mild solutions are formulated under various growth and compactness conditions. Continuous dependence estimates and convergence results are also established. Analysis of three stochastic partial differential equations, including a second-order stochastic evolution equation arising in the modeling of wave phenomena and a nonlinear diffusion equation, is provided to illustrate the applicability of the general theory.

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