scholarly journals A three-term recurrence relation for accurate evaluation of transition probabilities of the simple birth-and-death process

2021 ◽  
Author(s):  
Alberto Pessia ◽  
Jing Tang
Keyword(s):  
Hypergeometric Function ◽  
Recurrence Relation ◽  
Likelihood Function ◽  
Transition Probabilities ◽  
Transition Probability ◽  
Maximum Likelihood Estimates ◽  
Death Process ◽  
Birth And Death Process ◽  
Accurate Evaluation ◽  
Three Term Recurrence Relation

AbstractThe simple (linear) birth-and-death process is a widely used stochastic model for describing the dynamics of a population. When the process is observed discretely over time, despite the large amount of literature on the subject, little is known about formal estimator properties. Here we will show that its application to observed data is further complicated by the fact that numerical evaluation of the well-known transition probability is an ill-conditioned problem. To overcome this difficulty we will rewrite the transition probability in terms of a Gaussian hypergeometric function and subsequently obtain a three-term recurrence relation for its accurate evaluation. We will also study the properties of the hypergeometric function as a solution to the three-term recurrence relation. We will then provide formulas for the gradient and Hessian of the log-likelihood function and conclude the article by applying our methods for numerically computing maximum likelihood estimates in both simulated and real dataset.

On Sieved Orthogonal Polynomials II: Random Walk Polynomials

1986 ◽  
Vol 38 (2) ◽  
pp. 397-415 ◽  
Author(s):  
Jairo Charris ◽  
Mourad E. H. Ismail
Keyword(s):  
Random Walk ◽  
Markov Process ◽  
Orthogonal Polynomials ◽  
Transition Probabilities ◽  
Death Process ◽  
Birth And Death Process ◽  
Stationary Markov Process ◽  
Image Position

A birth and death process is a stationary Markov process whose states are the nonnegative integers and the transition probabilities(1.1)satisfy(1.2)as t → 0. Here we assume βn > 0, δn + 1 > 0, n = 0, 1, …, but δ0 ≦ 0. Karlin and McGregor [10], [11], [12], showed that each birth and death process gives rise to two sets of orthogonal polynomials. The first is the set of birth and death process polynomials {Qn(x)} generated by

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (3) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

Level phase independence for GI/M/1-type Markov chains

2000 ◽  
Vol 37 (4) ◽  
pp. 984-998 ◽  
Author(s):  
Guy Latouche ◽  
P. G. Taylor
Keyword(s):  
Phase Transitions ◽  
Markov Chains ◽  
State Space ◽  
Transition Probabilities ◽  
The Other ◽  
One Dimension ◽  
Death Process ◽  
Two Dimensional ◽  
Birth And Death Process ◽  
Special Case

GI/M/1-type Markov chains make up a class of two-dimensional Markov chains. One dimension is usually called the level, and the other is often called the phase. Transitions from states in level k are restricted to states in levels less than or equal to k+1. For given transition probabilities in the interior of the state space, we show that it is always possible to define the boundary transition probabilities in such a way that the level and phase are independent under the stationary distribution. We motivate our analysis by first considering the quasi-birth-and-death process special case in which transitions from any state are restricted to states in the same, or adjacent, levels.

Time-average optimal constrained semi-Markov decision processes

1986 ◽  
Vol 18 (2) ◽  
pp. 341-359 ◽  
Author(s):  
Frederick J. Beutler ◽  
Keith W. Ross
Keyword(s):  
Transition Probabilities ◽  
Decision Processes ◽  
The State ◽  
Death Process ◽  
Birth And Death Process ◽  
Average Reward ◽  
Acceptance Region ◽  
Dynamic Programming Equation ◽  
Markov Decision ◽  
Time Average

Optimal causal policies maximizing the time-average reward over a semi-Markov decision process (SMDP), subject to a hard constraint on a time-average cost, are considered. Rewards and costs depend on the state and action, and contain running as well as switching components. It is supposed that the state space of the SMDP is finite, and the action space compact metric. The policy determines an action at each transition point of the SMDP.Under an accessibility hypothesis, several notions of time average are equivalent. A Lagrange multiplier formulation involving a dynamic programming equation is utilized to relate the constrained optimization to an unconstrained optimization parametrized by the multiplier. This approach leads to a proof for the existence of a semi-simple optimal constrained policy. That is, there is at most one state for which the action is randomized between two possibilities; at all other states, an action is uniquely chosen for each state. Affine forms for the rewards, costs and transition probabilities further reduce the optimal constrained policy to ‘almost bang-bang’ form, in which the optimal policy is not randomized, and is bang-bang except perhaps at one state. Under the same assumptions, one can alternatively find an optimal constrained policy that is strictly bang-bang, but may be randomized at one state. Application is made to flow control of a birth-and-death process (e.g., an M/M/s queue); under certain monotonicity restrictions on the reward and cost structure the preceding results apply, and in addition there is a simple acceptance region.

Level phase independence for GI/M/1-type Markov chains

2000 ◽  
Vol 37 (04) ◽  
pp. 984-998
Author(s):  
Guy Latouche ◽  
P. G. Taylor
Keyword(s):  
Phase Transitions ◽  
Markov Chains ◽  
State Space ◽  
Transition Probabilities ◽  
The Other ◽  
One Dimension ◽  
Death Process ◽  
Two Dimensional ◽  
Birth And Death Process ◽  
Special Case

GI/M/1-type Markov chains make up a class of two-dimensional Markov chains. One dimension is usually called the level, and the other is often called the phase. Transitions from states in level k are restricted to states in levels less than or equal to k+1. For given transition probabilities in the interior of the state space, we show that it is always possible to define the boundary transition probabilities in such a way that the level and phase are independent under the stationary distribution. We motivate our analysis by first considering the quasi-birth-and-death process special case in which transitions from any state are restricted to states in the same, or adjacent, levels.

Time-average optimal constrained semi-Markov decision processes

1986 ◽  
Vol 18 (02) ◽  
pp. 341-359 ◽  
Author(s):  
Frederick J. Beutler ◽  
Keith W. Ross
Keyword(s):  
Transition Probabilities ◽  
Decision Processes ◽  
The State ◽  
Death Process ◽  
Birth And Death Process ◽  
Average Reward ◽  
Acceptance Region ◽  
Dynamic Programming Equation ◽  
Markov Decision ◽  
Time Average

Optimal causal policies maximizing the time-average reward over a semi-Markov decision process (SMDP), subject to a hard constraint on a time-average cost, are considered. Rewards and costs depend on the state and action, and contain running as well as switching components. It is supposed that the state space of the SMDP is finite, and the action space compact metric. The policy determines an action at each transition point of the SMDP. Under an accessibility hypothesis, several notions of time average are equivalent. A Lagrange multiplier formulation involving a dynamic programming equation is utilized to relate the constrained optimization to an unconstrained optimization parametrized by the multiplier. This approach leads to a proof for the existence of a semi-simple optimal constrained policy. That is, there is at most one state for which the action is randomized between two possibilities; at all other states, an action is uniquely chosen for each state. Affine forms for the rewards, costs and transition probabilities further reduce the optimal constrained policy to ‘almost bang-bang’ form, in which the optimal policy is not randomized, and is bang-bang except perhaps at one state. Under the same assumptions, one can alternatively find an optimal constrained policy that is strictly bang-bang, but may be randomized at one state. Application is made to flow control of a birth-and-death process (e.g., an M/M/s queue); under certain monotonicity restrictions on the reward and cost structure the preceding results apply, and in addition there is a simple acceptance region.

Excursions of birth and death processes, orthogonal polynomials, and continued fractions

1999 ◽  
Vol 36 (03) ◽  
pp. 752-770 ◽  
Author(s):  
Fabrice Guillemin ◽  
Didier Pinchon
Keyword(s):  
Orthogonal Polynomial ◽  
Continued Fractions ◽  
Spectral Measure ◽  
Continued Fraction ◽  
Transition Probability ◽  
Polynomial System ◽  
Death Process ◽  
Probabilistic Interpretation ◽  
Birth And Death Process ◽  
Birth And Death Processes

On the basis of the Karlin and McGregor result, which states that the transition probability functions of a birth and death process can be expressed via the introduction of an orthogonal polynomial system and a spectral measure, we investigate in this paper how the Laplace transforms and the distributions of different transient characteristics related to excursions of a birth and death process can be expressed by means of the basic orthogonal polynomial system and the spectral measure. This allows us in particular to give a probabilistic interpretation of the series introduced by Stieltjes to study the convergence of the fundamental continued fraction associated with the system. Throughout the paper, we pay special attention to the case when the birth and death process is ergodic. Under the assumption that the spectrum of the spectral measure is discrete, we show how the distributions of different random variables associated with excursions depend on the fundamental continued fraction, the orthogonal polynomial system and the spectral measure.

scholarly journals The limiting behavior of transient birth and death processes conditioned on survival

1968 ◽  
Vol 8 (4) ◽  
pp. 716-722 ◽  
Author(s):  
Phillip Good
Keyword(s):  
Transition Probability ◽  
Transition Probability Matrix ◽  
Death Process ◽  
Initial Condition ◽  
Birth And Death Process ◽  
Birth And Death Processes ◽  
Limiting Behavior ◽  
Image Position ◽  
Number Of Individuals ◽  
Over Time

The development of a population over time can often be simulated by the behavior of a birth and death process, whose transition probability matrix P(t) = (Pij(t), where X(t) denotes the number of individuals at time t, satisfies the differential equations and the initial condition

scholarly journals On p-Cauchy numbers

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2731-2742 ◽  
Author(s):  
Mourad Rahmani
Keyword(s):  
Hypergeometric Function ◽  
Recurrence Relation ◽  
Confluent Hypergeometric Function ◽  
Bernoulli Numbers ◽  
Basic Properties ◽  
New Family ◽  
Cauchy Numbers ◽  
Three Term Recurrence Relation

In this paper we define a new family of p-Cauchy numbers by means of the confluent hypergeometric function. We establish some basic properties. As consequence, a number of algorithms based on three-term recurrence relation for computing Cauchy numbers of both kinds, Bernoulli numbers of the second kind are derived.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

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