Moments of first passage times in general birth-death processes

Mathematical Methods of Operations Research

Volumn 68, Issue 1, 2008, Pages 49-76

  Jouini, Oualid ^a   Dallery, Yves ^a  

^a ECOLE CENTRALE PARIS   (France)

Author keywords

Birth death processes; Busy period; Conditional first passage times; First passage times; Transient analysis

Indexed keywords

CLUSTER ANALYSIS; ELECTRON TRANSITIONS; RANDOM VARIABLES; STANDARDS;

BIRTH-DEATH PROCESSES; BUSY PERIOD; CLOSED-FORM EXPRESSIONS; CONDITIONAL FIRST PASSAGE TIMES; CYCLE DURATION; ERGODIC; EXISTENCE CONDITIONS; FIRST PASSAGE TIMES; MARKOVIAN; QUEUEING SYSTEMS; SOME APPLICATIONS; STATE-DEPENDENT; TRANSIENT ANALYSIS; WAITING-TIME;

QUEUEING THEORY;

EID: 49149089425     PISSN: 14322994     EISSN: 14325217     Source Type: Journal    
DOI: 10.1007/s00186-007-0174-9     Document Type: Article

References (36)

1. Abate J, Whitt W (1988) Transient behavior of the M/M/1 queue via Laplace transforms. Adv Appl Probab 20:145-178
2. Abate J, Whitt W (1989) Calculating time-dependent performance measures for the M/M/1 queue. IEEE Trans Commun 37:1102-1104
3. Akhiezer NI (1965) The classical moment problem and some related questions in analysis. Hafner Publishing Co., New York (Translated by N. Kemmer)
4. Ancker CJ, Gafarian A (1962) Queueing with impatient customers who leave at random. J Ind Eng 13:84-90
5. Bailey NTJ (1954) A continuous time treatment of a single queue using generating functions. J R Stat Soc Ser 16:288-291
6. Ball F, Stefanov VT (2001) Further approaches to computing fundamental characteristics of birth-death processes. J Appl Probab 38:995-1005
7. Bertsimas D, Popescu I (2005) Optimal inequalities in probability theory: a convex optimization approach. SIAM J Optim 15:780-804
8. Champernowne DG (1956) An elementary method of solution of the queueing problem with a single server and a constant parameter. J R Stat Soc Ser 18:125-128
9. Coolen-Schrijner P, van Doorn EA (2002) The deviation matrix of a continuous-time Markov chain. Probabe Eng Info Sci 16:351-366
10. Flajolet P, Guillemin F (2000) The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv Appl Probab 32:750-778
11. Gross D, Harris CM (1985) Fundamentals of queueing theory. 2nd edn. Wiley series in probability and mathematical statistics
12. Guillemin F (2005) Spectral analysis of birth and death processes. Working paper, submitted to J Appl Probab
13. Guillemin F, Pinchon D (1999) Excursions of birth and death processes, orthogonal polynomials, and continued fractions. J Appl Probab 36:752-770
14. Joanes DN, Gill CA (1998) Comparing measures of sample skewness and kurtosis. J R Stat Soc (Series D) Stat 47:183-189
15. Jouini O, Dallery Y, Akšin OZ (2006) Modeling call centers with delay information. Working paper, Ecole Centrale Paris and Kot University
16. Karlin S, McGregor J (1957a) The classification of birth and death processes. Trans Am Math Soc 86: 366-401
17. Karlin S, McGregor J (1957b) The differential equation of birth and death processes, and the setieltjes moment problem. Trans Am Math Soc 85:489-546
18. Keilson J (1964a) A review of transient behavior in regular diffusion and birth-death processes. Part 2. J Appl Probab 1:247-266
19. Keilson J (1964b) A review of transient behavior in regular diffusion and birth-death processes. Part 1. J Appl Probab 1:247-266
20. Keilson J (1979) Markov chain models-rarity and exponentiality. Springer, New York
21. Keilson J (1981) On the unimodality of passage time densities in birth-death processes. Stat Neerl 35:49-55
22. Kijima M (1997) Markov processes for stochastic modeling, 1st edn. Chapman & Hall, London
23. Kleinrock L (1975) Queueing systems, theory, vol 1. Wiley, London
24. Koole G (2004) A formula for tail probabilities of Cox distributions. J Appl Probab 41:935-938
25. Krinik A, Sourouri Y (1990) Taylor series solutions of classical queueing systems. Abstracts American Mathematical Society, vol 11
26. Leguesdron P, Pellaumail J, Rubino G, Sericola B (1993) Transient analysis of the M/M/1 queue. Adv Appl Probab 25:702-713
27. Loève M (1977) Probability theory I, 4th edn. Springer, New York
28. Mao YH (2004) Ergodic degrees for continuous-time Markov chains. Sci China Ser A 47:161-174
29. Parthasarathy PR (1987) A transient solution to a M/M/1 queue: a new simple approach. Adv Appl Probab 19:997-998
30. Rosenlund SI (1977) Upwards passage times in the non-negative birth-death process. Scand J Stat 4:90-92
31. Shohat JA, Tamarkin JD (1943) The problem of moments. American Mathematical Society Mathematical surveys, Vol 2, American Mathematical Society, New York
32. Sumita U (1984) On conditional passage time structure of birth-death processes. J Appl Probab 21:10-21
33. Takâcs L (1960) Introduction to the theory of queues. Oxford University Press, New York
34. Tarabia AMK (2000) Transient analysis of M/M/1/N queue-an alternative approach. Tamkang J Sci Eng 3:263-266
35. Ward AR, Glynn PW (2003) A diffusion approximation for a Markovian queue with reneging. queueing systems: theory and applications QUESTA 43:103-128
36. Whitt W (1999) Predicting queueing delays. Manage Sci 45:870-888