A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (1) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

A storage model in which the net growth-rate is a Markov chain

1972 ◽  
Vol 9 (01) ◽  
pp. 129-139 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Rate Of Change ◽  
Continuous Time Markov Chain ◽  
Finite Capacity ◽  
Storage Model ◽  
Finite State ◽  
Net Growth Rate ◽  
The Times ◽  
Finite State Space

The distribution of the times to first emptiness and first overflow, together with the limiting distribution of content are determined for a dam of finite capacity. It is assumed that the rate of change of the level of the dam is a continuous-time Markov chain with finite state-space (suitably modified when the dam is full or empty).

scholarly journals Marked Continuous-Time Markov Chain Modelling of Burst Behaviour for Single Ion Channels

2007 ◽  
Vol 2007 ◽  
pp. 1-14 ◽  
Author(s):  
Frank G. Ball ◽  
Robin K. Milne ◽  
Geoffrey F. Yeo
Keyword(s):  
Markov Chain ◽  
Ion Channels ◽  
Continuous Time ◽  
Total Charge ◽  
Continuous Time Markov Chain ◽  
Open Time ◽  
Finite State ◽  
Gating Process ◽  
Finite State Space ◽  
Number Of Authors

Patch clamp recordings from ion channels often show bursting behaviour, that is, periods of repetitive activity, which are noticeably separated from each other by periods of inactivity. A number of authors have obtained results for important properties of theoretical and empirical bursts when channel gating is modelled by a continuous-time Markov chain with a finite-state space. We show how the use of marked continuous-time Markov chains can simplify the derivation of (i) the distributions of several burst properties, including the total open time, the total charge transfer, and the number of openings in a burst, and (ii) the form of these distributions when the underlying gating process is time reversible and in equilibrium.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (4) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φt)t≥0 be an additive functional defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level

2005 ◽  
Vol 37 (4) ◽  
pp. 1035-1055 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
High Level ◽  
The Relationship ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (Xt)t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v: E→ℝ\{0}, and let (φt)t≥0 be defined by φt=∫0tv(Xs)d s. We consider the case in which the process (φt)t≥0 is oscillating and that in which (φt)t≥0 has a negative drift. In each of these cases, we condition the process (Xt,φt)t≥0 on the event that (φt)t≥0 hits level y before hitting 0 and prove weak convergence of the conditioned process as y→∞. In addition, we show the relationship between the conditioning of the process (φt)t≥0 with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (φt)t≥0 with a negative drift to drift to ∞ and the conditioning of it to hit large levels before hitting 0.

scholarly journals Similar States in Continuous-Time Markov Chains

2009 ◽  
Vol 46 (02) ◽  
pp. 497-506 ◽  
Author(s):  
V. B. Yap
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Matrix ◽  
Base Substitution ◽  
Continuous Time Markov Chain ◽  
Rate Matrix ◽  
Dna Base ◽  
Substitution Models ◽  
Finite State ◽  
Finite State Space

In a homogeneous continuous-time Markov chain on a finite state space, two states that jump to every other state with the same rate are called similar. By partitioning states into similarity classes, the algebraic derivation of the transition matrix can be simplified, using hidden holding times and lumped Markov chains. When the rate matrix is reversible, the transition matrix is explicitly related in an intuitive way to that of the lumped chain. The theory provides a unified derivation for a whole range of useful DNA base substitution models, and a number of amino acid substitution models.

Aggregated Markov processes with negative exponential time interval omission

1990 ◽  
Vol 22 (04) ◽  
pp. 802-830 ◽  
Author(s):  
Frank Ball
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Single Channel ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Negative Exponential Distribution ◽  
Finite State ◽  
Negative Exponential ◽  
Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of f O(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. II. Hitting a high level

2005 ◽  
Vol 37 (04) ◽  
pp. 1035-1055 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
High Level ◽  
The Relationship ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v: E→ℝ\{0}, and let (φ t ) t≥0 be defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 hits level y before hitting 0 and prove weak convergence of the conditioned process as y→∞. In addition, we show the relationship between the conditioning of the process (φ t ) t≥0 with a negative drift to oscillate and the conditioning of it to stay nonnegative for a long time, and the relationship between the conditioning of (φ t ) t≥0 with a negative drift to drift to ∞ and the conditioning of it to hit large levels before hitting 0.

scholarly journals Conditioning an additive functional of a Markov chain to stay nonnegative. I. Survival for a long time

2005 ◽  
Vol 37 (04) ◽  
pp. 1015-1034 ◽  
Author(s):  
Saul D. Jacka ◽  
Zorana Lazic ◽  
Jon Warren
Keyword(s):  
Markov Chain ◽  
Weak Convergence ◽  
State Space ◽  
Continuous Time ◽  
Additive Functional ◽  
Finite State ◽  
Long Time ◽  
Negative Drift ◽  
Irreducible Markov Chain ◽  
Finite State Space

Let (X t ) t≥0 be a continuous-time irreducible Markov chain on a finite state space E, let v be a map v: E→ℝ\{0}, and let (φ t ) t≥0 be an additive functional defined by φ t =∫0 t v(X s )d s. We consider the case in which the process (φ t ) t≥0 is oscillating and that in which (φ t ) t≥0 has a negative drift. In each of these cases, we condition the process (X t ,φ t ) t≥0 on the event that (φ t ) t≥0 is nonnegative until time T and prove weak convergence of the conditioned process as T→∞.

scholarly journals Similar States in Continuous-Time Markov Chains

2009 ◽  
Vol 46 (2) ◽  
pp. 497-506 ◽  
Author(s):  
V. B. Yap
Keyword(s):  
Markov Chains ◽  
Continuous Time ◽  
Transition Matrix ◽  
Base Substitution ◽  
Continuous Time Markov Chain ◽  
Rate Matrix ◽  
Dna Base ◽  
Substitution Models ◽  
Finite State ◽  
Finite State Space

In a homogeneous continuous-time Markov chain on a finite state space, two states that jump to every other state with the same rate are called similar. By partitioning states into similarity classes, the algebraic derivation of the transition matrix can be simplified, using hidden holding times and lumped Markov chains. When the rate matrix is reversible, the transition matrix is explicitly related in an intuitive way to that of the lumped chain. The theory provides a unified derivation for a whole range of useful DNA base substitution models, and a number of amino acid substitution models.

Aggregated Markov processes with negative exponential time interval omission

1990 ◽  
Vol 22 (4) ◽  
pp. 802-830 ◽  
Author(s):  
Frank Ball
Keyword(s):  
State Space ◽  
Continuous Time ◽  
Single Channel ◽  
The State ◽  
Time Interval ◽  
Continuous Time Markov Chain ◽  
Negative Exponential Distribution ◽  
Finite State ◽  
Negative Exponential ◽  
Finite State Space

We consider a time reversible, continuous time Markov chain on a finite state space. The state space is partitioned into two sets, termed open and closed, and it is only possible to observe whether the process is in an open or a closed state. Further, short sojourns in either the open or closed states fail to be detected. We consider the situation when the length of minimal detectable sojourns follows a negative exponential distribution with mean μ–1. We show that the probability density function of observed open sojourns takes the form , where n is the size of the state space. We present a thorough asymptotic analysis of fO(t) as μ tends to infinity. We discuss the relevance of our results to the modelling of single channel records. We illustrate the theory with a numerical example.

Sign in / Sign up

Export Citation Format

Share Document