An asymptotically exact decomposition of coupled Brownian systems

1993 ◽  
Vol 30 (4) ◽  
pp. 819-834
Author(s):  
Kerry W. Fendick
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Queue Length ◽  
Inventory Systems ◽  
Input Process ◽  
Simple Description ◽  
Flow Systems ◽  
Component Processes ◽  
Brownian Models ◽  
Brownian Flow Systems

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

An asymptotically exact decomposition of coupled Brownian systems

1993 ◽  
Vol 30 (04) ◽  
pp. 819-834
Author(s):  
Kerry W. Fendick
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Queue Length ◽  
Inventory Systems ◽  
Input Process ◽  
Simple Description ◽  
Flow Systems ◽  
Component Processes ◽  
Brownian Models ◽  
Brownian Flow Systems

Brownian flow systems, i.e. multidimensional Brownian motion with regulating barriers, can model queueing and inventory systems in which the behavior of different queues is correlated because of shared input processes. The behavior of such systems is typically difficult to describe exactly. We show how Brownian models of such systems, conditioned on one queue length exceeding a large value, decompose asymptotically into smaller subsystems. This conditioning induces a change in drift of the system's net input process and its components. The results here are analogous to results for jump-Markov queues recently obtained by Shwartz and Weiss. The Brownian setting leads to a simple description of the component processes' asymptotic behaviour, as well as to explicit distributional results.

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (03) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that thekth-moment c.d.f is thek-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k +1)th moment approaches its steady-state limit more slowly than thekthmoment. It is also easy to derive the asymptotic behavior ast→∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2)c.d.f.&s (mixtures of two exponentials). TheH2approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of theH2approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

Brownian Motion and Stochastic Flow Systems.

1987 ◽  
Vol 36 (1) ◽  
pp. 66
Author(s):  
J. F. C. Kingman ◽  
J. Michael Harrison
Keyword(s):  
Brownian Motion ◽  
Stochastic Flow ◽  
Flow Systems

Asymptotic stationarity of queueing processes

1997 ◽  
Vol 34 (4) ◽  
pp. 1041-1048
Author(s):  
Władysław Szczotka
Keyword(s):  
Waiting Time ◽  
Queue Length ◽  
Input Process ◽  
Natural Conditions ◽  
Queueing Processes

We show that if an input process ζ to a queue is asymptotic stationary in some sense, satisfies a condition AB and some other natural conditions, then the output processes (w, ζ) and (w, q,ζ) are asymptotic stationary in the same sense. Here, w and q are the waiting time and queue length processes, respectively.

Brownian Motion on a Symmetric Space of Non-Compact Type: Asymptotic Behaviour in Polar Coordinates

1991 ◽  
Vol 43 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
J. C. Taylor
Keyword(s):  
Brownian Motion ◽  
Symmetric Space ◽  
Asymptotic Behaviour ◽  
Polar Coordinates ◽  
Skew Product ◽  
Compact Type ◽  
Product Representation

AbstractThe results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.

Transient behavior of regulated Brownian motion, I: Starting at the origin

1987 ◽  
Vol 19 (3) ◽  
pp. 560-598 ◽  
Author(s):  
Joseph Abate ◽  
Ward Whitt
Keyword(s):  
Brownian Motion ◽  
Steady State ◽  
First Passage Time ◽  
Cumulative Distribution ◽  
Stochastic Flow ◽  
Completely Monotone ◽  
Constant Diffusion ◽  
Flow Systems ◽  
The Moment ◽  
First Moment

A natural model for stochastic flow systems is regulated or reflecting Brownian motion (RBM), which is Brownian motion on the positive real line with constant negative drift and constant diffusion coefficient, modified by an impenetrable reflecting barrier at the origin. As a basis for understanding how stochastic flow systems approach steady state, this paper provides relatively simple descriptions of the moments of RBM as functions of time. In Part I attention is restricted to the case in which RBM starts at the origin; then the moment functions are increasing. After normalization by the steady-state limits, these moment c.d.f.&s (cumulative distribution functions) coincide with gamma mixtures of inverse Gaussian c.d.f.&s. The first moment c.d.f. thus coincides with the first-passage time to the origin starting in steady state with the exponential stationary distribution. From this probabilistic characterization, it follows that the kth-moment c.d.f is the k-fold convolution of the first-moment c.d.f. As a consequence, it is easy to see that the (k + 1)th moment approaches its steady-state limit more slowly than the kth moment. It is also easy to derive the asymptotic behavior as t →∞. The first two moment c.d.f.&s have completely monotone densities, supporting approximation by hyperexponential (H2) c.d.f.&s (mixtures of two exponentials). The H2 approximations provide easily comprehensible descriptions of the first two moment c.d.f.&s suitable for practical purposes. The two exponential components of the H2 approximation yield simple exponential approximations in different regimes. On the other hand, numerical comparisons show that the limit related to the relaxation time does not predict the approach to steady state especially well in regions of primary interest. In Part II (Abate and Whitt (1987a)), moments of RBM with non-zero initial conditions are treated by representing them as the difference of two increasing functions, one of which is the moment function starting at the origin studied here.

ASYMPTOTIC BEHAVIOUR OF A CLASS OF RESOURCE COMPETITION BIOLOGY SPECIES SYSTEM BY THE FRACTIONAL BROWNIAN MOTION

2017 ◽  
Vol 58 (3-4) ◽  
pp. 491-499
Author(s):  
Q. ZHANG ◽  
M. YE ◽  
H. LEI ◽  
Q. JIN
Keyword(s):  
Brownian Motion ◽  
Asymptotic Behaviour ◽  
Fractional Brownian Motion ◽  
Lyapunov Functional ◽  
Resource Competition ◽  
Mortality Rates ◽  
Mean Square ◽  
Chemostat Model ◽  
Extrinsic Noise ◽  
The Mean

We analyse the asymptotic behaviour of a biological system described by a stochastic competition model with $n$ species and $k$ resources (chemostat model), in which the species mortality rates are influenced by the fractional Brownian motion of the extrinsic noise environment. By constructing a Lyapunov functional, the persistence and extinction criteria are derived in the mean square sense. Some examples are given to illustrate the effectiveness of the theoretical result.

scholarly journals On the Asymptotic Behaviour of a Diffusion Process with Singular Drift

1975 ◽  
Vol 57 ◽  
pp. 87-106
Author(s):  
Hitoshi Kaneta
Keyword(s):  
Brownian Motion ◽  
Markov Process ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Asymptotic Behaviour ◽  
Markov Processes ◽  
Smooth Boundary ◽  
Dimensional Euclidean Space ◽  
Singular Drift ◽  
Strong Markov

We discuss some peculiar features of the diffusion process whose characterization is given below. Let D be a bounded domain in the d-dimensional Euclidean space Ed with a smooth boundary ∂D. The domain D contains open balls (i = 1, · · ·, n) which are mutually disjoint. Our process is a diffusion process on the state space D ∪ ∂D which is locally equivalent to the Brownian motion except on the spheres ∂ and the boundary ∂D. By a diffusion process we mean a continuous strong Markov process. As to the terminology about Markov processes we refer to [2].

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