A Novel Birth and Death Chain Formulation and Solution to a Spatial Queuing Problem

2019 ◽  
Author(s):  
Cheng Hua ◽  
Arthur Swersey
Keyword(s):  
Birth And Death Chain ◽  
Spatial Queuing

Existence of non-trivial quasi-stationary distributions in the birth-death chain

1992 ◽  
Vol 24 (04) ◽  
pp. 795-813 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Servet Martínez ◽  
Pierre Picco
Keyword(s):  
Continued Fractions ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Birth And Death Chain ◽  
Birth Death

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

scholarly journals A Stochastic Model for Phylogenetic Trees

2009 ◽  
Vol 46 (2) ◽  
pp. 601-607 ◽  
Author(s):  
Thomas M. Liggett ◽  
Rinaldo B. Schinazi
Keyword(s):  
Stochastic Model ◽  
Phylogenetic Tree ◽  
Mutation Rate ◽  
Phylogenetic Trees ◽  
Birth Rate ◽  
Long Time ◽  
New Type ◽  
Birth And Death Chain ◽  
Fixed Distribution ◽  
Simple Stochastic Model

We propose the following simple stochastic model for phylogenetic trees. New types are born and die according to a birth and death chain. At each birth we associate a fitness to the new type sampled from a fixed distribution. At each death the type with the smallest fitness is killed. We show that if the birth (i.e. mutation) rate is subcritical, we obtain a phylogenetic tree consistent with an influenza tree (few types at any given time and one dominating type lasting a long time). When the birth rate is supercritical, we obtain a phylogenetic tree consistent with an HIV tree (many types at any given time, none lasting very long).

Calculating extinction probabilities for the birth and death chain in a random environment

1979 ◽  
Vol 16 (04) ◽  
pp. 709-720 ◽  
Author(s):  
William C. Torrez
Keyword(s):  
State Space ◽  
Random Environment ◽  
Previous Investigation ◽  
Transition Probabilities ◽  
Finite Difference Methods ◽  
Fixed Sequence ◽  
Marginal Process ◽  
Difference Methods ◽  
Birth And Death Chain ◽  
Irreducible Markov Chain

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn ) evolving in a random environment controlled by an irreducible Markov chain (Yn ) with state space 𝒴 The process (Yn, Zn ) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn ) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn ). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn )-conditioned process (Zn ) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

Calculating extinction probabilities for the birth and death chain in a random environment

1979 ◽  
Vol 16 (4) ◽  
pp. 709-720 ◽  
Author(s):  
William C. Torrez
Keyword(s):  
State Space ◽  
Random Environment ◽  
Previous Investigation ◽  
Transition Probabilities ◽  
Finite Difference Methods ◽  
Fixed Sequence ◽  
Marginal Process ◽  
Difference Methods ◽  
Birth And Death Chain ◽  
Irreducible Markov Chain

In a previous investigation (Torrez (1978)) conditions were given for extinction and instability of a stochastic process (Zn) evolving in a random environment controlled by an irreducible Markov chain (Yn) with state space 𝒴 The process (Yn, Zn) is Markovian with state space 𝒴 × {0,1, ···, N} where 𝒴 = {1,· ··,m} and the marginal process (Zn) is a birth and death chain on {0,1,· ··,N}, with 0 and N made absorbing, when conditioned on a fixed sequence of environmental states of (Yn). This paper provides bivariate finite difference methods for calculating (i) P(Zn → 0) when this probability is not one; and (ii) the expected duration of the process Zn. For (i), the cases when the transition probabilities of the (Yn)-conditioned process (Zn) are non-homogeneous and homogeneous are considered separately. Examples are given to illustrate these methods.

scholarly journals The Birth and Death Chain in a Random Environment: Instability and Extinction Theorems

1978 ◽  
Vol 6 (6) ◽  
pp. 1026-1043 ◽  
Author(s):  
William C. Torrez
Keyword(s):  
Random Environment ◽  
Birth And Death Chain

Existence of non-trivial quasi-stationary distributions in the birth-death chain

1992 ◽  
Vol 24 (4) ◽  
pp. 795-813 ◽  
Author(s):  
Pablo A. Ferrari ◽  
Servet Martínez ◽  
Pierre Picco
Keyword(s):  
Continued Fractions ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Birth And Death Chain ◽  
Birth Death

We study conditions for the existence of non-trivial quasi-stationary distributions for the birth-and-death chain with 0 as absorbing state. We reduce our problem to a continued fractions one that can be solved by using extensions of classical results of this theory. We also prove that there exist normalized quasi-stationary distributions if and only if 0 is geometrically absorbing.

A spatial queuing model for the emergency vehicle districting and location problem

2009 ◽  
Vol 43 (7) ◽  
pp. 798-811 ◽  
Author(s):  
Nikolas Geroliminis ◽  
Matthew G. Karlaftis ◽  
Alexander Skabardonis
Keyword(s):  
Location Problem ◽  
Emergency Vehicle ◽  
Queuing Model ◽  
Spatial Queuing
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