On conditional passage time structure of birth-death processes
- 1 March 1984
- journal article
- Published by Cambridge University Press (CUP) in Journal of Applied Probability
- Vol. 21 (1) , 10-21
- https://doi.org/10.2307/3213660
Abstract
LetN(t) be a birth-death process onN= {0,1,2,· ··} governed by the transition rates λn> 0 (n≧ 0) andμη> 0 (n≧ 1). LetmTmbe the conditional first-passage time from r to n, given no visit tomwherem The downward conditional first-passage timenTmis defined similarly. It will be shown that, foranyλn> 0 andμη> 0. The limiting behavior ofis considerably different from that of the ordinary first-passage timewhere, under certain conditions, exponentiality sets in asn→∞. We will prove that, when λn→ λ > 0 andμη→μ> 0 asn→ ∞withρ= λ/μ <1, one hasasr→ ∞whereTBP(λ,μ)is the server busy period of anM/M/1 queueing system with arrival rate λand service rateμ.Keywords
This publication has 19 references indexed in Scilit:
- Markov Chain Models — Rarity and ExponentialityPublished by Springer Nature ,1979
- Exponential spectra as a tool for the study of server-systems with several classes of customersJournal of Applied Probability, 1978
- On exponential ergodicity and spectral structure for birth-death processes, IIStochastic Processes and their Applications, 1973
- On exponential ergodicity and spectral structure for birth-death processes IStochastic Processes and their Applications, 1973
- Taboo extinction, sojourn times, and asymptotic growth for the Markovian birth and death processJournal of Applied Probability, 1972
- Log-concavity and log-convexity in passage time densities of diffusion and birth-death processesJournal of Applied Probability, 1971
- A review of transient behavior in regular diffusion and birth-death processes. Part IIJournal of Applied Probability, 1965
- A review of transient behavior in regular diffusion and birth-death processesJournal of Applied Probability, 1964
- A CHARACTERIZATION OF BIRTH AND DEATH PROCESSESProceedings of the National Academy of Sciences, 1959
- The classification of birth and death processesTransactions of the American Mathematical Society, 1957