The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions

Advances in Applied Probability

Volumn 32, Issue 3, 2000, Pages 750-778

  Flajolet, Philippe ^a,b   Guillemin, Fabrice ^a  

^a INRIA ROCQUENCOURT   (France)

^b ORANGE LABS   (France)

Author keywords

Birth and death process; Continued fractions; Excursions; First passage time; Lattice path combinatorics; Orthogonal polynomials; Transient characteristics

Indexed keywords

COMBINATORIAL MATHEMATICS; MARKOV PROCESSES; MATHEMATICAL MODELS; POLYNOMIALS;

CONTINUED FRACTIONS; LATTICE PATH COMBINATORICS;

PROBABILITY;

EID: 0034266828     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1017/S0001867800010247     Document Type: Article

References (41)

1. ABATE, J. AND WHITT, W. (1999). Computing Laplace transforms for numerical inversion via continued fractions. INFORMS J. Comput. 11, 394-405.
2. ABATE, J. AND WHITT, W. (1998). Laplace transforms of density probability density functions with series representation. AT&T Labs Res. Rept.
3. ABATE, J. AND WHITT, W. (1995). Numerical inversion of Laplace transforms of probability distributions. ORSA J. Comput. 7, 36-43.
4. ASKEY, R. AND ISMAIL, M. E. H. (1984). Recurrence Relations, Continued Fractions, and Orthogonal Polynomials. Mem. Amer. Math. Soc. 49, No. 300.
5. ASMUSSEN, S. (1987). Applied Probability and Queues. John Wiley, New York.
6. BHARUCHA-REID, A. T. (1997). Elements of the Theory of Markov Processes and neir Applications. Dover, Mineola, NY.
7. BORDES, G. AND ROEHNER, B. ( 1983). Applications of Stieltjes theory for S-fractions to birth and death processes. Adv. Appl. Prob. 15, 507-530.
8. CHIHARA, T. S. (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York.
9. COHEN, J. W. (1982). The Single Server Queue. North-Holland, Amsterdam.
10. ERDéLYI, A. (1953). Higher Transcendental Functions. McGraw-Hill, New York.
11. FLAJOLET, P. (1980). Combinatorial aspects of continued fractions. Discrete Math. 32, 125-161.
12. FLAJOLET, P. (1982). On congruences and continued fractions for some classical combinatorial quantities. Discrete Math. 41, 145-153.
13. FLAJOLET, P., FRANçON, J. AND VUILLEMIN, J. (1980). Sequence of operations analysis for dynamic data structures. J. Algorithms 1, 111-141.
14. FLAJOLET, P., PUECH, C. AND VUILLEMIN, J. (1986). The analysis of simple list structure. Information Sei. 38, 121-146.
15. FLAJOLET, P. AND SCHOTT, R. (1990). Non-overlapping partitions, continued fractions, Bessel functions and a divergent series. Eur. J. Combin. 11, 421-432.
16. GODSIL, C. D. (1993). Algebraic Combinatorics. Chapman and Hall, New York.
17. GOULDEN, I. P. AND JACKSON, D. M. (1983). Combinatorial Enumeration. John Wiley, New York.
18. GOULDEN, I. P. AND JACKSON, D. M. (1986). Path generating functions and continued fractions. J. Combin. Theory A, 41, 1-10.
19. GUILLEMIN, F. AND PiNCHON, D. (1998). Continued fraction analysis of the duration of an excursion in an M/M/∞ system. J. Appl. Prob. 35, 165-183.
20. GUILLEMIN, F. AND PINCHON, D. (1999). On a random variable associated with excursion in an M/M/oo system. Queueing Systems.
21. GUILLEMIN, F. AND PINCHON, D. (1998). On the area swept under the occupation process of an M/M/1 queue in a busy period. Queueing Systems 20, 383-398.
22. GUILLEMIN, F. AND PINCHON, D. (1999). Excursions of birth and death processes, orthogonal polynomials, and continued fractions. J. Appl. Prob. 36, 752-770.
23. GOOD, I. J. (1958). Random motion and analytic continued fractions. Proc. Cambr. Philos. Soc. 54, 43-47.
24. HENRICI, P. (1977). Applied and Computational Complex Analysis. Vol. 2. John Wiley, New York.
25. ISMAIL, M. E. H., LETESSIER, J., MASSON, D. AND VALENT, G. (1990). Birth and death processes and orthogonal polynomials. In Orthogonal Polynomials, ed. P. Nevai. Kluwer, Amsterdam, pp. 229-255.
26. ISMAIL, M. E. H., LETESSIER, D. AND VALENT, G. (1988). Linear birth and death models and associated Laguerre and Meixner polynomials. J. Approx. Theory 55, 337-348.
27. JONES, W. B. AND MAGNUS, A. (1977). Application of Stieltjes fractions to birth-death processes. In Fade and Rational Approximation, eds E. B. Saff and R. S. Varga. Academic Press, New York, 173-179.
28. JONES, W. B. AND THRON, W. J. (1990). Continued fractions: analytic theory and applications. In Encyclopedia of Mathematics and its Applications. Vol. 11. Addison-Wesley, New York.
29. KARLIN, S. AND MCGREGOR, J. L. (1957). The differential equation of birth and death processes, and the Stieltjes moment problem. Trans. Amer. Math. Soc. 85, 489-546.
30. KARLIN, S. AND MCGREGOR, J. L. (1957). The classification of birth and death processes. Trans. Amer. Math. Soc. 86, 366-401
31. KARLIN, S. AND MCGREGOR, J. L. (1958). Linear growth birth and death processes. J. Math. Mech. 7, 643-662.
32. KARLIN, S. AND MCGREGOR, J. L. (1958). Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87-118.
33. KARLIN, S. AND MCGREGOR, J. L. (1959). A characterization of birth and death processes. Proc. Nat. Acad. Sei. USA 45, 375-379.
34. KARLIN, S. AND TAYLOR, H. (1975). A First Course in Stochastic Processes. Academic Press, New York.
35. KLEINROCK, L. (1975). Queueing systems. Vol. I: Theory. John Wiley, New York.
36. LEBEDEV, N. (1965). Special Functions and their Applications. Prentice Hall, New York.
37. MURPHY, J. A. AND O'UONOHOE, J. L. (1975). Some properties of continued fractions with applications in Markov processes. J. Inst. Math. Appl. 16, 57-71.
38. NORRIS, J. R. (1997). Markov Chains. Cambridge University Press.
39. PREATER, J. (1998). A//A//CO transience revisited. J. Appl Prob. 34, 1061-1067.
40. SUMITA, U. (1984). On conditional passage time structure of birth-death processes. J. Appl. Prob. 21, 10-21.
41. WALL, H. S. (1961). Analytic Theory of Continued Fractions. Chelsea, New York.