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American Journal of Mathematical and Management Sciences Volume 24, 2004 - Issue 1-2: Special State-of-the-Art Volume: Time-Dependent Birth and Death (BDP) Process Models With Applications
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Original Articles

Birth and Death (BDP) Process Models with Applications

Pages 1-212 | Published online: 14 Aug 2013

  • AbateJ. and WhittW. (1988) Approximations for the M/M/1 busy-period distribution. Queueing Systems, Theory and its Applications, (BoxmaO.J. and SyskiR., Eds.), North Holland, Amsterdam.[101, 103, 104].
  • AbateJ. and WhittW. (1999) Computing the Laplace Transforms for numerical inversion via continued fractions. INFORMS Jour Computing, 11, 394–405.[49].
  • AbramowitzM. and StegunI. (1965) Handbook of Mathematical Functions, Dover, New York.[75, 193].
  • AdanI.J.B.F. and ResingJ.A.C. (1996) Simple analysis of a fluid queue driven by an M/M/1 queue. Queueing Systems, 22, 171–174.[145].
  • Al-SalamW.A. (1990) Characterization theorems for orthogonal polynomials in Orthogonal Polynomials: Theory and Practice, (NevaiP.G., Ed.), 1–24, NATO ASI Series, Dordrecht:Kluwer.[73].
  • AokiM. (1996) New approaches to macroeconomic modeling, Cambridge University press, Cambridge.[7].
  • AndersonW.J. (1991) Continuous Time Markov Chains - An Application Oriented Approach, Springer-Verlag, New York.[6, 10, 80].
  • AndrewsG.E. (1976) The Theory of partitions, Encyclopedia of Mathematics, 2, Addison-Wesley, ReadingMA.[65, 193].
  • AndrewsG.E., AskeyR. and RoyR. (2000) Special Functions, Ency. Math. Appl. 71, Cambridge University Press, Cambridge.[6, 72].
  • AnickD., MitraD. and SondhiM.M. (1982) Stochastic theory of a data-handling system with multiple sources. The Bell System Technical Journal, 61, 1871–1894.[45, 122, 123, 127, 130, 130, 131, 132].
  • AronJ.L. and MayR.M. (1982) The population dynamics of malaria. Population Dynamics of infectious diseases, (Ed., AndersonR. M.), 139–179, Chapman and Hall, London.[181].
  • AskeyR. (1999) Continued fractions and orthogonal polynomials. Continued fractions: from analytic number theory to constructive approximation (Columbia, MO, 1998), 1–13, Contemp. Math., 236, Amer. Math. Soc., Providence.[6].
  • AskeyR. and IsmailM.E.H. (1984) Recurrence Relations, Continued Fractions and Orthogonal Polynomials, Memoirs of the Amer. Math. Soc., 49, Number 300.[6, 79].
  • BaileyN.T.J. (1964) The Elements of Stochastic Processes with Applications to the Natural Sciences, John Wiley, New York.[31, 179].
  • BaileyN.T.J. (1975) The Mathematical Theory of Infectious Diseases and its Applications, Griffin, London.[6, 179].
  • BanksR.B. (1994) Growth and Diffusion Phenomena, Springer-Verlag, New York.[31, 184, 184].
  • BarbotN. and SericolaB. (2002) Stationary solution to the fluid queue fed by an M/M/1 queue, J. Appl. Probab., 39, 359–369.[147, 148].
  • BarlowR.E. and ProschanF. (1996) Mathematical theory of reliability, SIAM, Philadelphia.[6].
  • BarberM.N. (1970) Random and Restricted Walks, Gordon and Breach, NY.[6].
  • BatherJ. (1963) Two non-linear birth and death processes. J. Austrl. Math. Soc., 3, 104–116.[47].
  • BerndtB.C. (1989) Ramanujan's Notebooks, Part II, Springer-Verlag, New York.[62, 65, 66, 66, 66, 70].
  • BertsekasD. and GallagerR. (1994) Data Networks, Prentice Hall of India.[6].
  • BhatU.N. (1984) Elements of Applied Stochastic Processes, John Wiley.[31].
  • BiroliniA. (1985) On the Use of Stochastic Processes in Modeling Reliability Problems: Lecture Notes in Economics and Mathematical Systems, Springer-Verlag, Berlin.[6].
  • BöhmW.M., KrinikA. and MohantyS.G. (1997) The combinatorics of birth-death processes and applications to queues. Queueing Systems Theory Appl., 26, 255–267.[27].
  • de BoorC. and GolubG.H. (1978) The numerically stable reconstruction of a Jacobi matrix from spectral data. Linear Algebra Appl., 21, 245–260.[96, 97].
  • BordesG. and RoehnerB. (1983) Applications of Stieltjes theory for S-fractions to birth and death processes. Adv. Appl. Prob., 15, 507–530.[82].
  • BoucherieR.J. and van DoornE.A. (1998) Uniformization for lambda-positive Markov chains. Comm. Statist. Stochastic Models, 14, 171–186.[20].
  • BowmanK.O. and ShentonL.R. (1989) Continued fractions in statistical applications. Marcel Dekker, New York.[49, 62, 63, 63].
  • BramsonM. and GriffeathD (1979) A note on the extinction rates of some birth-death processes. J. Appl. Probab., 16, 897–902.[27].
  • BrezinskiC. (1991) History of Continued Fractions and Páde Approximations, Springer, Berlin.[49].
  • BrownM. (1991) Spectral analysis without eigenvectors for Markov chains. Prob. Eng. Inf. Sci., 5, 131–144.[96].
  • BuzacottJ.A. and ShantikumarJ.G. (1993) Stochastic Models of Manufacturing Systems, Prentice Hall, New Jersey.[6].
  • CarmichaelD.G. (1987) Engineering Queues in Construction and Mining, Ellis-Horwood, Chichester.[6].
  • ChaudhuryM.L. and TempletonJ.G.C. (1973) A note on the distribution of a busy period for M/M/c queueing system. Math. Operationsforsch. U. Statist., 4, 75–79.[101].
  • ChenA. and RenshawE. (1997) The M/M/1 queue with mass exodus and mass arrivals when empty. Jour. Appl. Probab., 34, 192–207.[178].
  • ChengH. and YauStephen S.T. (1997) More explicit formulas for the matrix exponential. Linear Algebra Appl., 262, 131–163.[18].
  • ChiangC.L. (1968) Introduction to stochastic processes in biostatistics, Second edition, New York.[31].
  • ChiharaT.S. (1978) An Introduction to Orthogonal Polynomials, Gordon and Breach, New York.[61, 79, 81, 98, 128].
  • ChiharaT.S. (1987) On the spectra of certain birth and death processes. SIAM J. Appl. Math., 47, No. 3, 662–669.[82].
  • ChiharaT.S. (1990b) The three term recurrence relation and spectral properties of orthogonal polynomials. Orthogonal polynomials, 991–114, Kluwer Acad. Publ., Dordrecht.[6].
  • ChiharaT.S. and IsmailM.E.H. (1982) Orthogonal polynomials suggested by a queueing model. Adv. in Appl. Math., 3, 441–462.[80, 113].
  • ClancyD. and O'NeillP. (1998) Approximation of epidemics by inhomogeneous birth and death processes. Stoc. Proc. Appl., 73, 233–245.[6].
  • CohenJ.W. (1982) The Single Server Queue, North Holland, Amsterdam.[6, 166].
  • ConollyB.W. (1971) On randomized random walks. SIAM Rev., 13, 81–99.[175].
  • ConollyB.W. (1975) Lecture Notes on Queueing Systems, Ellis Horwood, Chichester.[31].
  • ConollyB.W. and LangarisC. (1992) Solution of the stochastic birth-emigration model using continued fractions. Math. Scientist, 17, 120–127.[49].
  • ConollyB.W., ParthasarathyP.R. and DharmarajaS. (1997) A chemical queue. The Math. Sci., 22, 83–91.[171, 175].
  • ConollyB.W., ParthasarathyP.R. and SelvarajuN. (2002) Double ended queues with impatience. Comp. Oper. Res. 29, 2053–2072.[163].
  • Di CrescenzoA. (1998) First-passage-time densities and avoiding probabilities for birth-and-death processes with symmetric sample paths. J. Appl. Prob., 35, 383–394.[30].
  • DetteH. (2001) First return probabilities of birth and death chains associated orthogonal polynomials. Proc. Amer. Math. Soc., 129, 1805–1815.[82].
  • DietzK. (1988) Mathematical models for transmission and control of malaria, Principles and Practice of Malariogy, WernsdorfW.H. and IMcGregor (Eds.), 1091–1133, Edinburgh Churchill, Livingstone.[179].
  • van DoornE.A. (1981a) The transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab., 18, 499–506.[113].
  • van DoornE.A. (1981b) Stochastic Monotonicity and Queueing Applications of Birth and Death Processes, Springer, Berlin.[6].
  • van DoornE.A. (1984) On the overflow process from a finite Markovian queue. Perf. Eval., 4, 233–240.[27].
  • van DoornE.A. (1985) Conditions for exponential ergodicity and bounds for the decay parameter of a birth-death process. Adv. in Appl. Probab., 17, 514–530.[27].
  • van DoornE.A. (1987) The indeterminate rate problem for birth-death processes. Pacific J. Math., 130, 379–393.[80].
  • van DoornE.A., JagersA.A. and de SmitJ.S.J. (1988) A fluid reservoir regulated by a birth-death process. Stochastic Models, 3, 457–472.[127].
  • DshalalowJ.H., (Ed.) (1995) Advances in Queueing : Theory, Methods and Open Problems, CRC Press, Boca RatonFlorida.[6].
  • DshalalowJ.H., (Ed.) (1997) Frontiers in Queueing : Models and Applications in Science and Engineering, CRC Press, Boca RatonFlorida.[6].
  • ÉrdiP. and TóthJ. (1994) Mathematical Methods of Chemical Reactions. Manchester University Press, Manchester.[6].
  • FernandoK.V. (1997) On computing an eigenvector of a tridiagonal matrix, Part I : Basic results. SIAM J. Matrix Anal. Appl., 18, 1013–1034.[87, 87, 88].
  • FerrariD. (1996) Multimedia network protocols: Where are we?. Multimedia Syst., 299–304.[158].
  • FlajoletP. and GuilleminF. (2000) The formal theory of birth-and-death processes, lattice path combinatorics and continued fractions. Adv. Appl. Probab., 32, 750–778.[6].
  • GardinerC.W. (1985) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Second Edition, Springer-Verlag, Berlin.[6].
  • GautschiW., GolubG.H. and OpferG. (1999), (Eds.) Applications and computation of orthogonal polynomials. International Series of Numerical Mathematics 131. Birkhäuser, Basel.[72].
  • GaverD.P. and LehocskyJ.P. (1982) Channels that cooperatively service a data stream and voice messages. IEEE Trans. Comm., COM 28, 1153–1162.[123].
  • GillJ. (1999) A note on extending Euler connection between continued fractions and power series. J. Comp. Appl. Math., 106, 299–305.[52].
  • GilmoreE.C., NowakowskiR.S., CavinessV.S.Jr. and HerrupK. (2000) Cell birth, cell death, cell diversity and DNA breaks : how do they all fit together?. Trends in Neurosciences, 23, 100–105.[6].
  • GiornoV., NegriC. and NobileA.G. (1985) A solvable model for a finite capacity queueing system. J. Appl. Probab., 22, 903–911.[44].
  • GoelN.S. and Ritcher-DynR. (1974) Stochastic Models in Biology, Academic Press, New York.[45].
  • GoodI.J. (1958) Random motion and analytic continued fractions. Proc. Camb. Phil. Soc., 54, 43–47.[49].
  • GoyalS.K. and GiriB.C. (2001) Recent trends in modeling of deteriorating inventory. Euro. Jour. Oper. Res., 134, 1–16.[6].
  • GnedenkoB.W. and KönigD. (1984) Handbuch der Bedienungstheorie I, Akademic Verlag, Berlin.[6].
  • GrassmannW.K. (2000) (Ed.) Computional probability, Kulwer Academic Publisher, Boston.[85].
  • GuilleminF. and PinchonD. (1998) Continued fraction analysis of the duration of an excursion in an M/M/ ∞ system. J. Appl. Prob., 35, 165–183.[113].
  • GuilleminF. and PinchonD. (1999) Excursions of birth and death processes, orthogonal polynomials, and continued fractions. J. Appi. Prob., 36, 752–770.[53, 79].
  • HadidiN. (1975), A queueing model with variable arrival rates. Period. Math. Hung., 6, 39–47.[113].
  • HallamT.G. LevinS.A., (Eds.) (1984) Mathematical Ecology : An introduction. Biomathematics, 17, Springer, Berlin.[6].
  • HaninL.G. (2001) Iterated birth and death processes as a model of radiation cell survival. Math. Bios., 169, 89–107.[6].
  • HanschkeTh. (1992) Markov chains and generalized continued fractions, J. Appi. Prob., 29, 838–849.[49].
  • HansenE.R. (1975) A Table of Series and Products, Prentice Hall, New Jersey.[65, 188].
  • HenriciP. (1977) Applied and computational Complex Analysis, Vol. 2, Wiley, New York.[59].
  • HochstadtH. (1967) On some inverse problems in matrix theory. Arch. Math., 18, 201–207.[96].
  • HochstadtH. (1974) On the construction of a Jacobi Matrix from spectra data. Linear Algebra Appi, 8, 435–446.[98, 98].
  • HockNg Chee (1996) Queueing Modelling Fundamentals. John Wiley, Chichester.[112].
  • HoffmanL.L. (1992) Characterizing linear birth and death processes. J. Amer. Statist. Assoc., 87, 1183–1187.[37].
  • HughesA.L. (2004) Birth-and-death evolution of protein-coding regions and concerted evolution of non-coding regions in the multi-component genomes of nanoviruses. Molecular Phylogenetics and Evolution, 30, Issue 2, 287–294.[6].
  • IosifescuM and TautuP. (1973), Stochastic Processes and Applications in Biology and Medicine, Part I: Theory k. Part II: Models, Springer, Berlin.[6].
  • IsmailM.E.H., LetessierJ. and ValentG. (1988) Linear birth and death models and associated Laguerre and Meixner polynomials. J. Approx. Theory, 55, 337–348.[81].
  • IsmailM.E.H., LetessierJ. and ValentG. (1989) Quadratic birth and death processes and associated continuous dual Hahn polynomials. SIAM J. Math. Anal., 20, 727–737.[47, 82].
  • IsmailM.E.H., LetessierJ. and ValentG. (1992) Birth and death processes with absorption. Int. J. Math. Sci., 15, 469–480.[82].
  • IsmailM.E.H., LetessierJ., MassonD.R. and ValentG. (1990) Birth and death processes and orthogonal polynomials. (P.Nevai, Ed.), Orthogonal Polynomials: Theory and Practice, 294, 229–255.[72].
  • IvanC. (1981) Weak convergence of the simple birth-and-death process. J. Appi Probab., 18, 245–252.[37].
  • JonesW.B. and SteinhardtA. (1982) Digital Filters and Continued Fractions, Led Notes Math. 932, Springer, Berlin.[50].
  • JonesW.B. and ThronW.J. (1980) Continued Fractions: Analytic Theory and Applications, Addison Wesley, New York.[6, 49, 52, 52, 61, 73].
  • KannanD. (1979) An Introduction to Stochastic Processes, North Holland, New York. [31].
  • van KämpenN.G. (1992) Stochastic Processes in Physics and Chemistry, 2nd Ed., North Holland, Amsterdam.[6].
  • KarlinS. and McGregorJ. (1957a) The differential equations of birth-and-death processes and the Stieltjes moment problem. Trans. Amer. Math. Soc., 85, 489–546. [79].
  • KarlinS. and McGregorJ. (1957b) The classification of birth and death processes. Trans. Amer. Math. Soc., 86, 366–400.[79, 81].
  • KaxlinS. and McGregorJ. (1958a) Many server queueing processes with Poisson input and exponential service times. Pacific J. Math., 8, 87–118.[60, 116].
  • KaxlinS. and McGregorJ. (1958b) Linear growth birth and death processes. J. Math. Mech., 7, 643–662.[30, 30].
  • KarlinS. and McGregorJ. (1959) Coincidence properties of birth and death processes. Pacific J. Math., 9, 1109–1140.[27].
  • KarlinS. and McGregorJ. (1961) The Hahn polynomials, formulas and an application. Scripta Math., 26, 33–46.[82].
  • KarlinS. and McGregorJ. (1965) Ehrenfest Urn Models, jour. Appl. Probab., 2, 352–376.[81].
  • KarlinS. and TaylorH.M. (1975) A First Course in Stochastic Processes, 2nd ed., Academic Press.[26, 31].
  • KeilsonJ. (1979) Markov Chain Models - Rarity and Exponentiality, Springer-Verlag, New York.[25].
  • Khovanskii (1963) The application of continued fractions and their generalizations to problems in approximation theory. P. Noordhoff N. V., Groningen.[67].
  • KijimaM. (1992), Evaluation of the decay parameter for some specialized birth-death processes. J. Appl. Probab., 29, 781–791.[27].
  • KijimaM. (1997) Markov Processes for Stochastic Modeling, Chapman and Hall, New York.[117].
  • KoekoekR. and SwarttouwR.F. (1998) The Askey-scheme of hypergeometric orthogonal polynomials and its q-analogue, Delft University of Technology, Report 98–17.[75].
  • Kopp-SchneiderA. (1992) Birth-death processes with piecewise constant rates. Statistics and Probability Letters, 13, 121–127.[37].
  • KouS.C. and KouS.G. (2003) Modeling growth stocks via birth-death processes. Adv. in Appl. Probab., 35, 641.[5].
  • KumarB.K., ParthasarathyP.R. and SharafaliM. (1993a) Transient solution of an M/M/1 queue with balking. Queueing Systems: Theory and Applications, 13, 441–448.[112].
  • KumarB.K. and ParthasarathyP.R. (1993) Density-dependent birth and death processes with controlled immigration. Stoch. Anal. Appl., 11, 181–196.[178].
  • LedermannW. and ReuterG.E.H. (1954) Spectral theory for the differential equations of simple birth and death processes. Phil. Trans. Royal Soć. London. A, 246, 321–369.[19].
  • LeninR.B., ParthasarathyP.R. ScheinhardtW.R.W. and van DoornE.A. (2000) Families of birth-death processes with similar time-dependent behaviour. J. Appl. Probab., 37, 835–849.[28].
  • LeninR.B. and ParthasarathyP.R. (2000a) A birth-death process suggested by a chain sequence. Comput. Math. Appl., 40, 239–247.[69, 69].
  • LeninR.B. and ParthasarathyP.R. (2000b) Fluid queues driven by an M/M/1/N queue. Math. Prob. Engg., 6, 439–460.[123, 133].
  • LeninR.B. and ParthasarathyP.R. (2000c) A computational approach to fluid queues driven by a truncated birth-death process. Methodol. Comput. Appl. Probab., 2, 373–392.[85].
  • LetesierJ. and ValentG. (1984) The generating function method for quadratic asymptotically symmetric birth and death processes, SIAM J. Appl. Math., 44, 773–783.[47, 82].
  • LettesierJ. and ValentG. (1986) Dual birth and death processes and orthogonal polynomials. SIAM J. Appl. Math., 46, 393–405.[82].
  • LetessierJ. and ValentG. (1988) Some exact solutions of the Kolmogrov boundary value problem. Appr. theory Appl., 4, 97–117.[47].
  • LevinS.A, HallamT.G. and GrossL.J. (1989), Applied Mathematical Ecology, Biomathematics Texts 18, Springer, Berlin.[6].
  • LiV.D.K and LaoW. (1997) Distributed multimedia systems. Proc. IEEE, 85, 1063–1108.[158].
  • LorentzenL. and WaadelandH. (1992) Continued fractions with applications, Studies in Computational Mathematics, 3, Elsevier Science.[6, 49].
  • LosoncziL. (1992) Eigenvalues and eigenvectors of some tridiagonal matrices. Acta Math. Hungar., 60, 309–322.[167].
  • MackenC.A. and PerelsonA.S. (1988) Stem cell proliferation and Differentiation, A Multitype branching Process Model. Lecture Notes in Biomath., 76, Springer, Berlin.[6].
  • MaruyamaT. (1980) Stochastic Problems in Population Genetics: Lecture Notes in Biomathematics, 17, Springer Verlag, Berlin.[6].
  • MassonD.R. (1988) Difference equations, continued fractions, Jacobi matrices and orthogonal polynomials. Nonlinear Numerical Methods and Rational Approximations, (Ed. CuytA.), Reidel, Dordrecht, 239–257.[6].
  • MatisJ.H. and KiffeT.R. (2000) Stochastic Population Models, A Compartmental Perspective. Lee. Notes in Stat., 145, Springer, Berlin.[6].
  • MatisJ.H., KiffeT.R. and ParthasarathyP.R. (1998) On the cumulant functions of the stochastic power logistic model. To appear in Theor. Pop. Biol., 53, 16–29. [184].
  • MencerO. (2000) Rational Arithmetic Units in Computer Systems, PhD dissertation, Stanford university. [50].
  • MeurantG. (1992) A review on the inverse of symmetric tridiagonal and block tridiagonal matrices. SIAM J. Matrix Anal. Appl., 13, 707–728. [96].
  • MitraD. (1988) Stochastic theory of fluid models of producers and consumers coupled by a buffer. Adv. Appl. Prob., 20, 646–676. [124].
  • MohantyS.G., Montazer-Haghighi and TruebloodR. (1993) On the transition behaviour of a finite birth-death process with application. Comput. Oper. Res., 20, 239–248. [23].
  • MolerC. and Van LoanC. (1978) Nineteen dubious ways to compute the exponential of a matrix. SIAM Rev., 114, 801–836. [18].
  • MorganB. J.T. (1978) Some recent applications of the linear birth and death process in biology. Math. Sci., bf 3, 103–116. [37].
  • MorganB.J.T. (1979) Four approaches to solving the linear birth-and-death (and similar) processes. Int. J. Math. Educ. Sci. Technol., 10, 51–64. [37].
  • MuirT. (1960) A Treatise on the Theory of Determinants, Dover Publications, New York. [188].
  • MuppalaJ.K. and TrivediK.S. (1992) Numerical transient solution of finite Markovian queueing systems, In Queueing and Related Models, Bhat and Basawa (Eds.), 262–284, Oxford University Press. [85].
  • MurphyJ.A. and O'DonohoeM.R. (1975) Properties of continued fractions with applications in Markov processes. J. Inst. Maths. Appl., 16, 57–71. [49, 55, 56, 114].
  • NahmiasS. (1982) Perishable inventory theory, a review. Oper. Res., 30, 680–708. [6].
  • NasellI. (1985) Hybrid models of tropical infections, Springer, Heidelberg. [179].
  • NatvigB. (1974) On the transient state probabilities for a queueing model where potential customers are discouraged by queue length. J. Appl. Probab., 11, 345–354. [113].
  • NikiferovA.F., SuslovS.K. and UvarovV.B. (1991) Classical Orthogonal Polynomials of a discrete variable, Spriger, Berlin. [89].
  • NinhamB., NossalR. and ZwanzigR. (1969) Kinetics of a sequence of first-order reactions. J. Chem. Phy., 51, 5028–5033. [45, 165].
  • O'ReillyP. (1986) A fluid-flow approach to performance analysis of integrated voice-data systems with speech interpolation, in El ataAbu (Ed.), Modelling Techniques and Tools for Performance Analysis, 115–130. [137, 138, 140].
  • OppenheimI., ShulerK.K. and WeissG.H. (1977) Stochastic Processes in Chemical Physics : The Master Equation, The MIT Press, Cambridge. [6].
  • ParthasarathyP.R. (1987) A transient solution to an M/M/1 queue - A simple approach. Adv. Appl. Prob., 19, 997–998. [60, 100].
  • ParthasarathyP.R. and DharmarajaS. (1998) The transient solution of a local-jump heterogeneous chain of diatomic systems. J. Phys. A: Math. Gem., 31, 6579–6588. [172].
  • ParthasarathyP.R. and DharmarajaS. (1999) Exact transient solutions of kinetics of first-order reactions with end-effects. J. Math. Chem., 25, 281–295. [166].
  • ParthasarathyP.R. and DharmarajaS. (2001) Transient solution of a single link with finite capacity supporting multiple streams. Performance Evaluation, 43, 1–14. [154].
  • ParthasarathyP.R., DharmarajaS. and ManimaranG. (1999) Transient solution of multiprocessor systems with state-dependent arrivais. Intern. J. Computer Math., 73, 313–320. [157].
  • ParthasarathyP.R. and KumarB.K. (1991) Density-dependent birth and death processes with state-dependent immigration. Math. Comp. Mod., 15, 11–16. [178].
  • ParthasarathyP.R. and FeldmannS. (2000) On the inverse problem in Cauer networks. Inverse Problems, 16, 49–58. [68].
  • ParthasarathyP.R. and LeninR.B. (1997) On the exact transient solution of finite birth and death processes with specific quadratic rates. Math. Sci. 22, 92–105. [83].
  • ParthasarathyP.R. and LeninR.B. (1998) On the numerical solution of transient probabilities of quadratic birth and death process. J. Diff. Eqns., Appl., 4, 365–379. [92].
  • ParthasarathyP.R. and LeninR.B. (1999) An inverse problem in birth and death processes. Comput. Math. Appl., 38, 33–40. [97].
  • ParthasarathyP.R., LeninR.B., SchoutensW. and Van AsscheW. (1998) A birth and death process related to the Rogers-Ramanujan continued fraction. Journal of Mathematical Analysis and Application, 224, 297–315. [70].
  • ParthasarathyP.R. and SelvarajuN. (2001) Transient analysis of a queue where potential customers are discouraged by queue length. Math. Probl. Eng., 7, 433–454. [113].
  • ParthasarathyP.R., SelvarajuN. and LeninR.B. (2000) The numerical solution of a birth-death process arising in multimedia synchronization. Math. Comp. Mod., 34, 887–901. [158].
  • ParthasarathyP.R. and SharafaliM. (1989) Transient solution to many server Poisson queue - A simple approach. J. Appl. Prob., 26, 584–594. [112].
  • ParthasarathyP.R. and VijayalakshmiV. (1995) Transient solution of a state dependent many server Poisson queue with balking. J. Appl. Stat. Sci., 2, 207–220. [104].
  • ParthasarathyP.R. and VijayalakshmiV. (1996) Transient analysis of an (S – 1, S) inventory model - A numerical approach. Intern. J. Computer Math., 59, 177–185. [47].
  • ParthasarathyP.R. and VijayalakshmiV. (1998) On quadrature approximation for the busy period density function of an M/M/1 queueing system. Amer. J. Math. Mgmt. Sci., 18, 277–289. [101].
  • ParthasarathyP.R. and VijayashreeK.V. (2003a), Fluid queues driven by birth and death process with quadratic rates. International Journal of Computers and Mathematics, 80, 1385–1395. [138].
  • ParthasarathyP.R. and VijayashreeK.V. (2003b), Fluid queues driven by a discouraged arrivals queue. International Journal of Mathematics and Mathematical Sciences, 2003, 1509–1528. [145].
  • ParthasarathyP.R., VijayashreeK.V. and LeninR.B. (2002) An M/M/1 driven fluid queue - continued fraction approach. Queueing Systems, 42, 189–199. [142].
  • PearceC.E.M. (1989) Extended Continued fractions, recurrence relations and two-dimensional Markov processes. Adv. Appl, Prob., 21, 357–375. [49].
  • PengN.F. (1996) Spectral representation of the transient probability matrices for continuous time finite Markov chains. J. Appl. Probab., 33, 28–33. [96].
  • PerronO. (1950) Die Lehre Von Den Kertenbsüchen, Chelsea Publ. Co., NY. [6].
  • PrabhuN.U. (1997) Foundations of Queueing Theory, Kluwer Academic Publisher, Boston. [6].
  • PrabhuN.U. (1998), Stochastic storage processes: queues, insurance risk, dams, and data communication, Springer : New York. [6].
  • RaghuP. (2001) Explicit form of impedance of Cauer networks. Int. J. Comp. Math., 77, 523–533. [68].
  • RajuS.N. and BhatU.N. (1982) A computationally oriented analysis of the G/M/1 queue. Opsearch, 19, 67–83. [106, 106].
  • RenshawE. (1991) Modelling Biological Populations in Space and Time, Cambridge University Press, Cambridge. [6, 184, 184, 185, 185].
  • RenshawE. and ChenA. (1997) Birth-death processes with mass annihilation and state-dependent immigration. Comm. Stat.: Stochastic Models, 13, 239–253. [38].
  • RiskenH. (1989) The Fokker-Plank equation, Springer, Berlin. [6].
  • RoehnerB. and ValentG. (1982) Solving the birth and death processes with quadratic symmetric transition rates. SIAM J. Appl. Math., 42, 1020–1046.[47, 82].
  • RosenlundS.I. (1978) Transition probabilities for a truncated birth-death process. Scan J. Statist., 5, 119–122. [15, 16].
  • RossS.M. (1996) Stochastic Processes, 2nd Ed., Wiley, New York. [31, 179].
  • SchoutensW. (2000a) Stochastic Processes and Orthogonal Polynomials, Springer, New York. [6].
  • SchoutensW. (2000b) Birth and death processes, orthogonal polynomials and limiting conditional distributions. Math. Sci., 25, 87–93. [27, 83].
  • Schweiger (2000) Multidimensional continued fractions, Oxford University press, Oxford. [49].
  • SerfozoR.F. (1981) Optimal control of random walks, birth and death processes, and queues. Adv. in Appl. Probab., 13, 61–83. [27].
  • SerfozoR.F. (1988) Extreme values of birth-death processes and queues. Stoc. Proc. Appl, 27, 291–306. [27].
  • SericolaB. (1998) Transient analysis of stochastic fluid models, Performance Evaluation, 32, 245–263. [151].
  • SericolaB., ParthasarathyP.R. and VijayashreeK.V. (2003) Exact transient solution of an M/M/1 driven fluid queues. To appear in International Journal of Computers and Mathematics. [149].
  • SericolaB. and TuffinB. (1999) A fluid queue driven by a Markovian queue. Queueing Systems, 31, 253–264. [122, 145].
  • SharafaliM. and ParthasarathyP.R. (1989) On the distribution busy period for the many server Poisson queue. Opsearch, 26, 125–132. [101].
  • SharmaO.P. (1990) Markovian Queues, Ellis Horwood Ltd., England. [6].
  • SharmaO.P. and NairN.S.K. (1996) Transient analysis of finite state birth and death process with absorbing boundary states. Stoc. Anal. Appl., 14, 565–589. [26].
  • SmithP.J. (1997) Sequences of linear birth, death and immigration processes with applications to cascades of optical amplifiers. Prob. Engg. Inf. Sci., 11, 387–394. [37].
  • SrivastavaH.M. and KashyapB.R.K. (1982) Special Functions in Queueing Theory and Related Stochastic Processes, Academic Press, New York. [6].
  • StadjeW. and ParthasarathyP.R. (1999) On the convergence to stationarity of the many-server Poisson queue. J. Appl. Probab., 36, 546–557. [117].
  • StadjeW. and ParthasarathyP.R. (2000) Generating function analysis of some joint distributions for Poisson loss systems. Queueing Systems, 34, 183–197. [117, 121].
  • StefanovV.T. (1995), Mean passage times for tridiagonal transition matrices. J. Appl. Probab., 32, 846–849. [27, 85].
  • SternT.E. and ElwalidA.I. (1991) Analysis of separable Markov-modulated rate models for information-handling systems. Adv. Appl. Prob., 23, 105–139. [124].
  • StewartW.J. (1994) Introduction to the Numerical Solution of Markov Chains, Princeton University Press, Princeton. [85].
  • StewartJ. (Ed.,) (1995) Computations with Markov Chains, Kluwer, Boston. [27].
  • StockmayerW.H., GobushW. and NorvichR. (1971) Local-jump models for chain dynamics. Pure and Appl. Chem., 26, 537–543. [171, 173].
  • SumitaU. (1984) On conditional passage time structure of birth-death processes. J. Appl. Probab., 21, 10–21. [25].
  • SwiftR.J. (2001), A logistic birth-death-immigration-emigration process. Math. Sci., 26, 25–33. [38].
  • SzegöG. (1959) Orthogonal polynomials, Amer. Math. Soc. Colloquium Publications, 23, rev. ed., Amer. Math. Soc., New York. [72, 73, 77].
  • SzegöG. (1975) Orthogonal Polynomials, 4th ed., ProvidenceRI: Amer.Math.Soc. [49].
  • TakácsL. (1962) Introduction to the Theory of Queues, Oxford Uni. Press, New York. [6, 44, 89, 117, 168, 170].
  • TakashimaM. (1956) Note on evolutionary processes. Bull. Math. Stat., 7, 18–24. [46].
  • TanW.Y. (1976) On the absorption probabilities and absorption times of finite homogeneous birth-death processes. Biometrics, 32, 745–752. [25].
  • TanW.Y. (1986) A stochastic Gompertz birth-death process. Statist. Probab. Lett., 4, 25–28. [42].
  • TanW.Y. (1991) Stochastic Models of Carcinogenesis, Marcel Dekker, New York. [6].
  • TaylorH.M. and KarlinS. (1998) An Introduction to Stochastic Modeling, 3rd edition, Academic Press, New York. [31].
  • ValentG. (1996) Exact solutions of some quadratic and quartic birth and death processes and related orthogonal polynomials, J. Comp. Appl. Mat., 67, 103–127. [47].
  • Van AsscheW. (1987) Asymptotics for Orthogonal Polynomials, Lecture Notes in Math., 1265, Springer-Verlag, Berlin. [6, 6, 73].
  • Van AsscheW. (1999) Multiple orthogonal polynomials, irrationality and transcendence. Contemp. Math., 236, 325–341. [74, 75].
  • Van AsscheW., ParthasarathyP.R. and LeninR.B. (1999) Spectral representation of four finite birth and death processes. Math. Sci., 24, 105–112. [83].
  • WallH.S. (1948) Analytical Theory of Continued Fractions, Chelsea Publ. Co., NY. [6, 50, 59, 61, 64, 64, 65, 66, 67, 67, 67, 73].
  • WangZ. and YangX. (1992) Birth and Death processes and Markov Chains, Springer, Berlin. [6].
  • WatsonG.N. (1995) A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press. [100].
  • WaughW.A.O'N (1972) Taboo extinction, sojourn times and asymptotic growth for the Markovian birth and death process. J. Appl. Probability, 9, 486–506. [37].
  • WeidlichW. and HaagG. (1983) Concepts and models of quantitative sociology, Springer, Berlin. [7].
  • WeinbergL. (1962) Network Analysis and Synthesis, McGraw Hill, New York. [68].
  • WhittW. (1983) Untold horrors of the waiting room : What the equilibrium distribution will never tell about the queue-length process. Management Science, 29, 395–408. [6].
  • WhittW. (2001) Stochastic Process Limits, An introduction to stochastic process limits and their application to queues, Springer, Heidelberg. [6].
  • WilkinsonJ.H. (1965) The Algebraic Eigenvalue Problem, Prentice Hall, New York. [55, 87].
  • YcartB. (1988) A charateristic property of linear growth birth and death processes. Sankhy Ser., A. 50, 184–189. [40].
  • ZhengQ. (1997) A unified approach to a class of stochastic carcinogenesis models. Risk Anal., 17, 617–624. [6].
  • ZhengQ. (1998) Note on the nonhomogeneous Prendville process. Math. Biosci., 148, 1–5. [46].

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