The transient state probabilities for a queueing model where potential customers are discouraged by queue length

1981 ◽  
Vol 18 (02) ◽  
pp. 499-506 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Queue Length ◽  
Transition Probabilities ◽  
Transient State ◽  
Death Process ◽  
Queueing Model ◽  
Potential Customers ◽  
Birth Death

Exact expressions are derived for the transition probabilities of the birth-death process with parametersandwhich serves as a queueing model where potential customers are discouraged by queue length.

The transient state probabilities for a queueing model where potential customers are discouraged by queue length

1981 ◽  
Vol 18 (2) ◽  
pp. 499-506 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
Queue Length ◽  
Transition Probabilities ◽  
Transient State ◽  
Death Process ◽  
Queueing Model ◽  
Potential Customers ◽  
Birth Death

Exact expressions are derived for the transition probabilities of the birth-death process with parameters and which serves as a queueing model where potential customers are discouraged by queue length.

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (01) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

scholarly journals On a Symmetric, Nonlinear Birth-Death Process with Bimodal Transition Probabilities

Symmetry ◽  
2009 ◽  
Vol 1 (2) ◽  
pp. 201-214 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Barbara Martinucci
Keyword(s):  
Transition Probabilities ◽  
Death Process ◽  
Birth Death

scholarly journals Extinction Probability in A Birth-Death Process with Killing

2005 ◽  
Vol 42 (1) ◽  
pp. 185-198 ◽  
Author(s):  
Erik A. Van Doorn ◽  
Alexander I. Zeifman
Keyword(s):  
Rate Of Convergence ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Logarithmic Norm ◽  
Absorbing State ◽  
The Common ◽  
Absorption Probabilities ◽  
Birth Death

We study birth-death processes on the nonnegative integers, where {1, 2,…} is an irreducible class and 0 an absorbing state, with the additional feature that a transition to state 0 may occur from any state. We give a condition for absorption (extinction) to be certain and obtain the eventual absorption probabilities when absorption is not certain. We also study the rate of convergence, as t → ∞, of the probability of absorption at time t, and relate it to the common rate of convergence of the transition probabilities that do not involve state 0. Finally, we derive upper and lower bounds for the probability of absorption at time t by applying a technique that involves the logarithmic norm of an appropriately defined operator.

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (1) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

On the transient state probabilities for a queueing model where potential customers are discouraged by queue length

1974 ◽  
Vol 11 (02) ◽  
pp. 345-354 ◽  
Author(s):  
Bent Natvig
Keyword(s):  
Queue Length ◽  
Transient State ◽  
Exact Expression ◽  
Single Server ◽  
Exact Formulation ◽  
Queueing Process ◽  
State Dependent ◽  
Contemporary Work ◽  
Potential Customers ◽  
Low Traffic

Earlier work by Hadidi and Conolly and contemporary work by the author point to the great operational advantages of state-dependent queueing models. Let pin (t) be the state probabilities and p∗ in the corresponding L.T.'s relative to the single server birth-and-death queueing process with parameters λn = λ/(n + 1), n ≥ 0, μn = μ, n ≥ 1. We have obtained an exact formulation of p ∗ i0 , p ∗ in (n ≥ 1) being determined recursively. An exact expression for p 10(t) is given in the case of low traffic intensities, and this has been approximated efficiently. Numerical evaluations show that the steady-state is reached very rapidly.

Quasi-stationary distributions and convergence to quasi-stationarity of birth-death processes

1991 ◽  
Vol 23 (4) ◽  
pp. 683-700 ◽  
Author(s):  
Erik A. Van Doorn
Keyword(s):  
State Space ◽  
Spectral Representation ◽  
Transition Probabilities ◽  
Initial Distribution ◽  
Death Process ◽  
Finite Support ◽  
Stationary Distributions ◽  
Absorbing State ◽  
Initial Distributions ◽  
Birth Death

For a birth–death process (X(t), ) on the state space {−1, 0, 1, ·· ·}, where −1 is an absorbing state which is reached with certainty and {0, 1, ·· ·} is an irreducible class, we address and solve three problems. First, we determine the set of quasi-stationary distributions of the process, that is, the set of initial distributions which are such that the distribution of X(t), conditioned on non-absorption up to time t, is independent of t. Secondly, we determine the quasi-limiting distribution of X(t), that is, the limit as t→∞ of the distribution of X(t), conditioned on non-absorption up to time t, for any initial distribution with finite support. Thirdly, we determine the rate of convergence of the transition probabilities of X(t), conditioned on non-absorption up to time t, to their limits. Some examples conclude the paper. Our main tools are the spectral representation for the transition probabilities of a birth–death process and a duality concept for birth–death processes.

scholarly journals Representations for the Decay Parameter of a Birth-Death Process Based on the Courant-Fischer Theorem

2015 ◽  
Vol 52 (01) ◽  
pp. 278-289 ◽  
Author(s):  
Erik A. van Doorn
Keyword(s):  
Orthogonal Polynomials ◽  
Lower Bounds ◽  
Transition Probabilities ◽  
Symmetric Matrix ◽  
Upper And Lower Bounds ◽  
Death Process ◽  
Decay Parameter ◽  
Absorbing State ◽  
State 1 ◽  
Birth Death

We study the decay parameter (the rate of convergence of the transition probabilities) of a birth-death process on {0, 1, …}, which we allow to evanesce by escape, via state 0, to an absorbing state -1. Our main results are representations for the decay parameter under four different scenarios, derived from a unified perspective involving the orthogonal polynomials appearing in Karlin and McGregor's representation for the transition probabilities of a birth-death process, and the Courant-Fischer theorem on eigenvalues of a symmetric matrix. We also show how the representations readily yield some upper and lower bounds that have appeared in the literature.

scholarly journals Weak convergence of conditioned birth-death processes in discrete time

1997 ◽  
Vol 34 (01) ◽  
pp. 46-53
Author(s):  
Pauline Schrijner ◽  
Erik A. Van Doorn
Keyword(s):  
Weak Convergence ◽  
Discrete Time ◽  
Transition Probabilities ◽  
Death Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Absorbing State ◽  
Limiting Behaviour ◽  
Necessary And Sufficient ◽  
Birth Death

We consider a discrete-time birth-death process on the non-negative integers with −1 as an absorbing state and study the limiting behaviour asn →∞ of the process conditioned on non-absorption until timen.By proving that a condition recently proposed by Martinez and Vares is vacuously true, we establish that the conditioned process is always weakly convergent when all self-transition probabilities are zero. In the aperiodic case we obtain a necessary and sufficient condition for weak convergence.

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