scholarly journals The quasi-stationary distribution of the subcritical contact process

2020 ◽  
Vol 148 (10) ◽  
pp. 4517-4525
Author(s):  
Franco Arrejoría ◽  
Pablo Groisman ◽  
Leonardo T. Rolla
Keyword(s):  
Stationary Distribution ◽  
Contact Process ◽  
Quasi Stationary Distribution

On the relationship between µ-invariant measures and quasi-stationary distributions for continuous-time Markov chains

1993 ◽  
Vol 25 (01) ◽  
pp. 82-102
Author(s):  
M. G. Nair ◽  
P. K. Pollett
Keyword(s):  
Invariant Measure ◽  
Stationary Distribution ◽  
Continuous Time ◽  
Sufficient Condition ◽  
Transition Rates ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Necessary And Sufficient ◽  
Forward Equations ◽  
Quasi Stationary Distribution

In a recent paper, van Doorn (1991) explained how quasi-stationary distributions for an absorbing birth-death process could be determined from the transition rates of the process, thus generalizing earlier work of Cavender (1978). In this paper we shall show that many of van Doorn's results can be extended to deal with an arbitrary continuous-time Markov chain over a countable state space, consisting of an irreducible class, C, and an absorbing state, 0, which is accessible from C. Some of our results are extensions of theorems proved for honest chains in Pollett and Vere-Jones (1992). In Section 3 we prove that a probability distribution on C is a quasi-stationary distribution if and only if it is a µ-invariant measure for the transition function, P. We shall also show that if m is a quasi-stationary distribution for P, then a necessary and sufficient condition for m to be µ-invariant for Q is that P satisfies the Kolmogorov forward equations over C. When the remaining forward equations hold, the quasi-stationary distribution must satisfy a set of ‘residual equations' involving the transition rates into the absorbing state. The residual equations allow us to determine the value of µ for which the quasi-stationary distribution is µ-invariant for P. We also prove some more general results giving bounds on the values of µ for which a convergent measure can be a µ-subinvariant and then µ-invariant measure for P. The remainder of the paper is devoted to the question of when a convergent µ-subinvariant measure, m, for Q is a quasi-stationary distribution. Section 4 establishes a necessary and sufficient condition for m to be a quasi-stationary distribution for the minimal chain. In Section 5 we consider ‘single-exit' chains. We derive a necessary and sufficient condition for there to exist a process for which m is a quasi-stationary distribution. Under this condition all such processes can be specified explicitly through their resolvents. The results proved here allow us to conclude that the bounds for µ obtained in Section 3 are, in fact, tight. Finally, in Section 6, we illustrate our results by way of two examples: regular birth-death processes and a pure-birth process with absorption.

A Markov chain associated with the minimal quasi-stationary distribution of birth–death chains

1995 ◽  
Vol 32 (01) ◽  
pp. 25-38
Author(s):  
Servet Martínez ◽  
Maria Eulália Vares
Keyword(s):  
Markov Chain ◽  
Stationary Distribution ◽  
Conditional Distribution ◽  
Transition Probabilities ◽  
The Matrix ◽  
Quasi Stationary Distribution ◽  
Birth Death

We show that if the limiting conditional distribution for an absorbed birth–death chain exists, then the chain conditioned to non-absorption converges to a Markov chain with transition probabilities given by the matrix associated with the minimal quasi-stationary distribution.

ON THE QUASI-STATIONARY DISTRIBUTION OF SIS MODELS

2016 ◽  
Vol 30 (4) ◽  
pp. 622-639 ◽  
Author(s):  
Gaofeng Da ◽  
Maochao Xu ◽  
Shouhuai Xu
Keyword(s):  
Stationary Distribution ◽  
Upper Bound ◽  
Hazard Rate ◽  
State Of The Art ◽  
Upper Bounds ◽  
Hazard Rate Order ◽  
Reversed Hazard Rate ◽  
Novel Method ◽  
Better Than ◽  
Quasi Stationary Distribution

In this paper, we propose a novel method for constructing upper bounds of the quasi-stationary distribution of SIS processes. Using this method, we obtain an upper bound that is better than the state-of-the-art upper bound. Moreover, we prove that the fixed point map Φ [7] actually preserves the equilibrium reversed hazard rate order under a certain condition. This allows us to further improve the upper bound. Some numerical results are presented to illustrate the results.

The quasi-stationary distributions of queues in heavy traffic

1972 ◽  
Vol 9 (04) ◽  
pp. 821-831 ◽  
Author(s):  
E. K. Kyprianou
Keyword(s):  
Stationary Distribution ◽  
Gamma Distribution ◽  
Arrival Time ◽  
Heavy Traffic ◽  
Waiting Times ◽  
Time Process ◽  
Queue Size ◽  
Virtual Waiting Time ◽  
Waiting Time Process ◽  
Quasi Stationary Distribution

This paper demonstrates that, when in heavy traffic, the quasi-stationary distribution of the virtual waiting time process of both the M/G/1 and GI/M/1 queues as well as the quasi-stationary distribution of the waiting times {Wn } of the M/G/1 queue can be approximated by the same gamma distribution. What characterises this approximating gamma distribution are the first two moments of the service time and inter-arrival time distributions only. A similar approximating behaviour is demonstrated for the queue size process.

Quasi-stationary distributions and one-dimensional circuit-switched networks

1987 ◽  
Vol 24 (04) ◽  
pp. 965-977 ◽  
Author(s):  
Ilze Ziedins
Keyword(s):  
Communication Networks ◽  
Stationary Distribution ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Distributions ◽  
Switched Networks ◽  
One Dimensional ◽  
Special Cases ◽  
Quasi Stationary Distribution

We discuss the quasi-stationary distribution obtained when a simple birth and death process is conditioned on never exceeding K. An application of this model to one-dimensional circuit-switched communication networks is described, and some special cases examined.

Evaluation of the decay parameter for some specialized birth-death processes

1992 ◽  
Vol 29 (4) ◽  
pp. 781-791 ◽  
Author(s):  
Masaaki Kijima
Keyword(s):  
State Space ◽  
Stationary Distribution ◽  
The State ◽  
Death Process ◽  
Decay Parameter ◽  
Speed Of Convergence ◽  
Tridiagonal Matrices ◽  
Decay Parameters ◽  
Quasi Stationary Distribution ◽  
Birth Death

Let N(t) be an exponentially ergodic birth-death process on the state space {0, 1, 2, ···} governed by the parameters {λn, μn}, where µ0 = 0, such that λn = λ and μn = μ for all n ≧ N, N ≧ 1, with λ < μ. In this paper, we develop an algorithm to determine the decay parameter of such a specialized exponentially ergodic birth-death process, based on van Doorn's representation (1987) of eigenvalues of sign-symmetric tridiagonal matrices. The decay parameter is important since it is indicative of the speed of convergence to ergodicity. Some comparability results for the decay parameters are given, followed by the discussion for the decay parameter of a birth-death process governed by the parameters such that limn→∞λn = λ and limn→∞µn = μ. The algorithm is also shown to be a useful tool to determine the quasi-stationary distribution, i.e. the limiting distribution conditioned to stay in {1, 2, ···}, of such specialized birth-death processes.

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