We consider a birth-death process whose birth and death rates are suggested by a chain sequence. We use an elegant transformation to find the transition probabilities in a simple closed form. We also find an explicit expression for time-dependent mean. We find parallel results in discrete time. Finally, we show that the processes under investigation are transient, and hence, the stationary distribution does not exist.

Panamalai Ramarao Parthasarathy

Binary sequences

Boundary conditions

Differential equations

Integral equations

Laplace transforms

Parallel processing systems

Polynomials

Probability distributions

Theorem proving

BIRTH-DEATH PROCESS

CHAIN SEQUENCE

Chebyshev polynomials

Transient behaviour

Markov processes

Fulltext

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IIT Madras

A birth-death process suggested by a chain sequence

Computers and Mathematics with Applications

In this work, the coefficients of orthogonal polynomials are obtained in closed form. Our formula works for all classes of orthogonal polynomials whose recurrence relation can be put in the form Rn (x) = x Rn - 1 (x) - αn - 2 Rn - 2 (x). We show that Chebyshev, Hermite and Laguerre polynomials are all members of the class of orthogonal polynomials with recurrence relations of this form. Our formula unifies the previously known formulas for the coefficients of these familiar polynomial families. © 2005 Elsevier Ltd. All rights reserved.

Applied Mathematics Letters

A formula for the coefficients of orthogonal polynomials from the three-term recurrence relations

Time-dependent system size probabilities of a birth and death process related to the Rogers-Ramanujan continued fraction are obtained. The range for the parameter in this continued fraction is obtained to ensure the positivity of the recursively defined birth and death rates. The general behavior of the birth and death rates is described and the asymptotic behavior of the transition probabilities and the (quasi)-stationary distribution is determined. For the transient case the birth and death process can be seen as a model in queuing theory where the length of the queue encourages customers to join the queue (the more the merrier). © 1998 Academic Press.

Journal of Mathematical Analysis and Applications

A Birth and Death Process Related to the Rogers-Ramanujan Continued Fraction

Springer Series in Statistics Continuous-Time Markov Chains

Examples of Continuous-Time Markov Chains

Proceedings of the American Mathematical Society

The parameters of a chain sequence

Orthogonal Polynomials

The Three Term Recurrence Relation and Spectral Properties of Orthogonal Polynomials

Journal of Applied Probability

Transient solution to the many-server Poisson queue: a simple approach

Birth-death process suggested by a chain sequence

Journal	Data powered by TypesetComputers and Mathematics with Applications
Publisher	Data powered by TypesetElsevier Science Ltd, Exeter
ISSN	08981221
Open Access	Yes