An inverse problem in birth and death processes. (English) Zbl 0938.60089
Summary: An inverse problem of constructing birth and death processes \(\{X(t)\}\) on finite state space \(\{0,1,2,\dots, N\}\) is considered. Given a set of \(2N+1\) distinct, nonnegative real numbers one of which is zero, say \(0= s_0< z_1< s_2<\cdots< z_N< s_N\), a procedure is established to obtain the birth and death rates of a birth and death process so that
\[
P(X(t)= 0)= \sum^N_{j= 0} \Biggl(\prod^N_{i=1} (z_i- s_j)\Biggl/ \prod^N_{t= 0,i\neq j}(s_i- s_j)\Biggr) e^{-s_jt}
\]
and other transient system size probabilities. This technique is illustrated numerically.
MSC:
| 60J80 | Branching processes (Galton-Watson, birth-and-death, etc.) |
Keywords:
tridiagonal matrices; tridiagonal determinants; persymmetric matrix; eigenvalues; orthogonal polynomialsReferences:
| [1] | Anderson, W. J.: Continuous time Markov chains–an application oriented approach. (1991) · Zbl 0731.60067 |
| [2] | Askey, R.; Ismail, M. E. H.: Recurrence relations, continued fractions and orthogonal polynomials. Memoirs of the amer. Math. soc. 300, 108 (1984) · Zbl 0548.33001 |
| [3] | Jones, W. B.; Thron, W. J.: Continued fractions: analytic theory and applications encyclopedia of mathematics and yts applications. 2 (1980) · Zbl 0445.30003 |
| [4] | Hald, O. H.: On discrete and numerical inverse Sturm-Liouville problems. Uppsala univ. Dept. of computer sci., rep. No. 42 (1972) |
| [5] | Hochstadt, H.: On some inverse problems in matrix theory. Arch. math. 18, 201-207 (1967) · Zbl 0147.27701 |
| [6] | Meurant, G.: A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix anal. Appl. 13, 707-728 (1992) · Zbl 0754.65029 |
| [7] | Bertsekas, D.; Gallager, R.: Data networks. (1994) |
| [8] | Conolly, B. W.; Parthasarathy, P. R.; Dharmaraja, S.: A chemical queue. The mathematical sci. 22, 83-91 (1997) · Zbl 0898.60087 |
| [9] | Van Doorn, E. A.: Stochastic monotonicity and queueing applications of birth and death processes. (1981) · Zbl 0454.60069 |
| [10] | Oppenheim, I.; Shuler, K. K.; Weiss, G. H.: Stochastic processes in chemical physics: the master equations. (1977) |
| [11] | Parthasarathy, P. R.: The effect of superinfection on the distribution of the infectious period–A continued fraction approximation. IMA J. Math. appl. In med. And biol. 14, 113-123 (1997) · Zbl 0871.92028 |
| [12] | Renshaw, E.: Modelling biological populations in space and time. (1991) · Zbl 0754.92018 |
| [13] | Karlin, S.; Mcgregor, J.: Linear growth, birth and death processes. Journal of mathematics and mechanics 7 (1958) · Zbl 0091.13804 |
| [14] | Parthasarathy, P. R.; Lenin, R. B.: On the exact transient solution of finite birth and death processes with specific quadratic rates. The mathematical scientist 22, 92-105 (1997) · Zbl 0898.60084 |
| [15] | Parthasarathy, P. R.; Sharafali, M.: Transient solution to many server Poisson queue–A simple approach. J. appl. Prob. 26, 584-594 (1989) · Zbl 0693.60076 |
| [16] | Takács, L.: Introduction to the theory of queues. (1962) · Zbl 0106.33502 |
| [17] | Buzacott, J. A.; Shantikumar, J. G.: Stochastic models of manufacturing systems. (1993) · Zbl 1094.90518 |
| [18] | Perros, H. G.; Altiok, T.: Queueing networks with blocking. (1989) · Zbl 0699.68010 |
| [19] | P.R. Parthasarathy, R.B. Lenin and J.H. Matis, On the numerical solution of transient probabilities of the stochastic power law logistic model, In Nonlinear Analysis (to appear). · Zbl 0947.60084 |
| [20] | Brown, M.: Spectral analysis, without eigenvectors, for Markov chains. Prob. eng. Inf. sci. 5, 131-144 (1991) · Zbl 1134.60373 |
| [21] | Peng, N. Fu: Spectral representations of the transition probability matrices for continuous time finite Markov chains. J. appl. Prob. 33, 28-33 (1996) · Zbl 0847.60060 |
| [22] | Harris, C. M.; Marchal, W. G.; Botta, R. F.: A note on generalized hyperexponential distributions. Commun. statist.–stochastic models 8, 179-191 (1992) · Zbl 0749.60009 |
| [23] | Neuts, M. F.: Matrix-geometric solutions in stochastic model. (1981) · Zbl 0469.60002 |
| [24] | De Boor, C.; Golub, G. H.: The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear algebra appl. 21, 245-260 (1978) · Zbl 0388.15010 |
| [25] | Gnedenko, B. W.; König, D.: Handbuch der bedienungstheorie I. (1984) |
| [26] | Murphy, J. A.; O’donohoe, M. R.: Properties of continued fractions with applications in Markov processes. J. inst. Maths. appl. 16, 57-71 (1975) · Zbl 0314.65057 |
| [27] | Wilkinson, J. H.: The algebraic eigenvalue problem. (1965) · Zbl 0258.65037 |
| [28] | Kijima, M.: Markov processes for stochastic modelling. (1997) · Zbl 0866.60056 |
| [29] | Hochstadt, H.: On the construction of a Jacobi matrix from spectra data. Linear algebra and its applications 8, 435-446 (1974) · Zbl 0288.15029 |
| [30] | Chihara, T. S.: An introduction to orthogonal polynomials. (1978) · Zbl 0389.33008 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
