On the time a Markov chain spends in a lumped state

1997 ◽  
Vol 34 (2) ◽  
pp. 340-345 ◽  
Author(s):  
Tommy Norberg
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Continuous Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Sojourn Time Distribution

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

On the time a Markov chain spends in a lumped state

1997 ◽  
Vol 34 (02) ◽  
pp. 340-345
Author(s):  
Tommy Norberg
Keyword(s):  
Markov Chain ◽  
State Space ◽  
Continuous Time ◽  
Sojourn Time ◽  
Time Distribution ◽  
Sojourn Time Distribution

The sojourn time that a Markov chain spends in a subset E of its state space has a distribution that depends on the hitting distribution on E and the probabilities (resp. rates in the continuous-time case) that govern the transitions within E. In this note we characterise the set of all hitting distributions for which the sojourn time distribution is geometric (resp. exponential).

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→∞.

Clustering of Bursts of Openings in Markov and Semi-Markov Models of Single Channel Gating

1997 ◽  
Vol 29 (01) ◽  
pp. 92-113 ◽  
Author(s):  
Frank Ball ◽  
Sue Davies
Keyword(s):  
Markov Chain ◽  
Ion Channel ◽  
State Space ◽  
Continuous Time ◽  
Markov Models ◽  
Numerical Study ◽  
Channel Gating ◽  
Markov Renewal Process ◽  
Time Interval ◽  
Continuous Time Markov Chain

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space. The state space is partitioned into two classes, termed ‘open’ and ‘closed’, and it is possible to observe only which class the process is in. In many experiments channel openings occur in bursts. This can be modelled by partitioning the closed states further into ‘short-lived’ and ‘long-lived’ closed states, and defining a burst of openings to be a succession of open sojourns separated by closed sojourns that are entirely within the short-lived closed states. There is also evidence that bursts of openings are themselves grouped together into clusters. This clustering of bursts can be described by the ratio of the variance Var (N(t)) to the mean[N(t)] of the number of bursts of openings commencing in (0, t]. In this paper two methods of determining Var (N(t))/[N(t)] and limt→∝Var (N(t))/[N(t)] are developed, the first via an embedded Markov renewal process and the second via an augmented continuous-time Markov chain. The theory is illustrated by a numerical study of a molecular stochastic model of the nicotinic acetylcholine receptor. Extensions to semi-Markov models of ion channel gating and the incorporation of time interval omission are briefly discussed.

A BMAP/SM/1 queueing system with Markovian arrival input of disasters

1999 ◽  
Vol 36 (03) ◽  
pp. 868-881
Author(s):  
Alexander Dudin ◽  
Shoichi Nishimura
Keyword(s):  
Sojourn Time ◽  
Queue Length ◽  
Time Distribution ◽  
Queueing System ◽  
Queuing System ◽  
Complicated System ◽  
Sojourn Time Distribution ◽  
Negative Arrivals

Disaster arrival in a queuing system with negative arrivals causes all customers to leave the system instantaneously. Here we obtain a queue-length and virtual waiting (sojourn) time distribution for the more complicated system BMAP/SM/1 with MAP input of disasters.

Quasi-ergodicity for non-homogeneous continuous-time Markov chains

1989 ◽  
Vol 26 (3) ◽  
pp. 643-648 ◽  
Author(s):  
A. I. Zeifman
Keyword(s):  
Differential Equation ◽  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Continuous Time ◽  
Sufficient Conditions ◽  
Continuous Time Markov Chain ◽  
Countable State Space ◽  
Continuous Time Markov Chains ◽  
Countable State

We consider a non-homogeneous continuous-time Markov chain X(t) with countable state space. Definitions of uniform and strong quasi-ergodicity are introduced. The forward Kolmogorov system for X(t) is considered as a differential equation in the space of sequences l1. Sufficient conditions for uniform quasi-ergodicity are deduced from this equation. We consider conditions of uniform and strong ergodicity in the case of proportional intensities.

scholarly journals Analysis of a Single-Sever Queue with Disasters and Repairs Under Bernoulli Vacation Schedule

2016 ◽  
Vol 4 (6) ◽  
pp. 547-559
Author(s):  
Jingjing Ye ◽  
Liwei Liu ◽  
Tao Jiang
Keyword(s):  
Performance Measures ◽  
Generating Functions ◽  
Sojourn Time ◽  
Time Distribution ◽  
Busy Period ◽  
Balance Equations ◽  
Sojourn Time Distribution ◽  
The Mean ◽  
System Length ◽  
Number Of Customers

AbstractThis paper studies a single-sever queue with disasters and repairs, in which after each service completion the server may take a vacation with probabilityq(0≤q≤1), or begin to serve the next customer, if any, with probabilityp(= 1− q). The disaster only affects the system when the server is in operation, and once it occurs, all customers present are eliminated from the system. We obtain the stationary probability generating functions (PGFs) of the number of customers in the system by solving the balance equations of the system. Some performance measures such as the mean system length, the probability that the server is in different states, the rate at which disasters occur and the rate of initiations of busy period are determined. We also derive the sojourn time distribution and the mean sojourn time. In addition, some numerical examples are presented to show the effect of the parameters on the mean system length.

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