On the limit behavior of a multicompartment storage model with an underlying Markov chain

1988 ◽  
Vol 20 (1) ◽  
pp. 208-227
Author(s):  
Eric S. Tollar
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Limit Behavior ◽  
Moment Conditions ◽  
Storage Model ◽  
Second Moments ◽  
First Moment ◽  
Inputs And Outputs

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

On the limit behavior of a multicompartment storage model with an underlying Markov chain

1988 ◽  
Vol 20 (01) ◽  
pp. 208-227
Author(s):  
Eric S. Tollar
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Limit Behavior ◽  
Moment Conditions ◽  
Storage Model ◽  
Second Moments ◽  
First Moment ◽  
Inputs And Outputs

The present paper considers a multicompartment storage model with one-way flow. The inputs and outputs for each compartment are controlled by a denumerable-state Markov chain. Assuming finite first and second moments, it is shown that the amounts of material in certain compartments converge in distribution while for others they diverge, based on appropriate first-moment conditions on the inputs and outputs. It is also shown that the diverging compartments under suitable normalization converge to functionals of Brownian motion, independent of those compartments which converge without normalization.

A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain

1971 ◽  
Vol 8 (02) ◽  
pp. 276-284 ◽  
Author(s):  
J. P. Lehoczky
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Sample Path ◽  
Exact Distribution ◽  
Total Delay ◽  
Traffic Flow Theory ◽  
Second Moments ◽  
Close Relationship ◽  
First Moment ◽  
Infinite Reservoir

Summary The first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized. The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.

A note on the first emptiness time of an infinite reservoir with inputs forming a Markov chain

1971 ◽  
Vol 8 (2) ◽  
pp. 276-284 ◽  
Author(s):  
J. P. Lehoczky
Keyword(s):  
Markov Chain ◽  
Generating Functions ◽  
Sample Path ◽  
Exact Distribution ◽  
Total Delay ◽  
Traffic Flow Theory ◽  
Second Moments ◽  
Close Relationship ◽  
First Moment ◽  
Infinite Reservoir

SummaryThe first emptiness time of an infinite reservoir with unit release and discrete input forming a stationary. Markov chain is investigated. The exact distribution of the first emptiness time is derived without the use of moment generating functions. The first and second moments of this distribution are given explicity. The close relationship between the process with stationary independent input and Markov chain input is emphasized.The first moment of the area beneath the sample path up to first emptiness is computed. This area is often used as a measure of total delay in traffic flow theory.

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.

A Brownian motion with two reflecting barriers and Markov-modulated speed

2004 ◽  
Vol 41 (04) ◽  
pp. 1237-1242 ◽  
Author(s):  
Offer Kella ◽  
Wolfgang Stadje
Keyword(s):  
Brownian Motion ◽  
Markov Chain ◽  
Linear Algebra ◽  
Stationary Distribution ◽  
The State ◽  
Long Run ◽  
Markov Modulated ◽  
Two Barriers ◽  
Reflecting Barriers

We consider a Brownian motion with time-reversible Markov-modulated speed and two reflecting barriers. A methodology depending on a certain multidimensional martingale together with some linear algebra is applied in order to explicitly compute the stationary distribution of the joint process of the content level and the state of the underlying Markov chain. It is shown that the stationary distribution is such that the two quantities are independent. The long-run average push at the two barriers at each of the states is also computed.

scholarly journals DIFFUSION LIMITS FOR A MARKOV MODULATED BINOMIAL COUNTING PROCESS

2019 ◽  
Vol 34 (2) ◽  
pp. 235-257
Author(s):  
Peter Spreij ◽  
Jaap Storm
Keyword(s):  
Markov Chain ◽  
Markov Process ◽  
Regime Switching ◽  
Limit Theorems ◽  
Counting Process ◽  
Limit Behavior ◽  
Diffusion Approximations ◽  
Default Rate ◽  
Markov Modulated ◽  
Rapid Switching

In this paper, we study limit behavior for a Markov-modulated binomial counting process, also called a binomial counting process under regime switching. Such a process naturally appears in the context of credit risk when multiple obligors are present. Markov-modulation takes place when the failure/default rate of each individual obligor depends on an underlying Markov chain. The limit behavior under consideration occurs when the number of obligors increases unboundedly, and/or by accelerating the modulating Markov process, called rapid switching. We establish diffusion approximations, obtained by application of (semi)martingale central limit theorems. Depending on the specific circumstances, different approximations are found.

FRACTIONAL BROWNIAN MOTION WITH STOCHASTIC VARIANCE: MODELING ABSOLUTE RETURNS IN STOCK MARKETS

2008 ◽  
Vol 19 (08) ◽  
pp. 1221-1242 ◽  
Author(s):  
H. E. ROMAN ◽  
M. PORTO
Keyword(s):  
Brownian Motion ◽  
Fractional Brownian Motion ◽  
Stock Markets ◽  
Distribution Functions ◽  
Conditional Heteroskedasticity ◽  
Arch Models ◽  
Variance Modeling ◽  
Probability Distribution Functions ◽  
Second Moments ◽  
Long Time

We discuss a model for simulating a long-time memory in time series characterized in addition by a stochastic variance. The model is based on a combination of fractional Brownian motion (FBM) concepts, for dealing with the long-time memory, with an autoregressive scheme with conditional heteroskedasticity (ARCH), responsible for the stochastic variance of the series, and is denoted as FBMARCH. Unlike well-known fractionally integrated autoregressive models, FBMARCH admits finite second moments. The resulting probability distribution functions have power-law tails with exponents similar to ARCH models. This idea is applied to the description of long-time autocorrelations of absolute returns ubiquitously observed in stock markets.

scholarly journals Multitype Branching Brownian Motion and Traveling Waves

2014 ◽  
Vol 46 (01) ◽  
pp. 217-240
Author(s):  
Yan-Xia Ren ◽  
Ting Yang
Keyword(s):  
Brownian Motion ◽  
Traveling Waves ◽  
Traveling Wave ◽  
Traveling Wave Solutions ◽  
Critical Value ◽  
System Of Equations ◽  
Moment Conditions ◽  
Asymptotic Behaviors ◽  
Branching Brownian Motion ◽  
Parabolic System Of Equations

In this article we study the parabolic system of equations which is closely related to a multitype branching Brownian motion. Particular attention is paid to the monotone traveling wave solutions of this system. Provided with some moment conditions, we show the existence, uniqueness, and asymptotic behaviors of such waves with speed greater than or equal to a critical value c̲ and nonexistence of such waves with speed smaller than c̲.

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