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• W2004687618 abstract "Previous article Next article On Randomized Random WalksB. W. ConollyB. W. Conollyhttps://doi.org/10.1137/1013005PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] William Feller, An introduction to probability theory and its applications. Vol. I, John Wiley and Sons, Inc., New York, 1957xv+461, 2nd ed. MR0088081 0077.12201 Google Scholar[2] A. E. Gibson, Masters Thesis, Some aspects of time-dependent one dimensional random walks, Doctoral thesis, Virginia Polytechnic Institute, Blacksburg, Va., 1968 Google Scholar[3] E. Sparre-Andersen, On the fluctuations of sums of random variables, Math. Scand., 1 (1953), 263–285 MR0058893 0053.09701 CrossrefGoogle Scholar[4] E. Sparre-Andersen, On the fluctuations of sums of random variables. II, Math. Scand., 2 (1954), 195–223 MR0068154 0058.12102 Google Scholar[5] Frank Spitzer, Principles of random walk, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1964xi+406 MR0171290 0119.34304 CrossrefGoogle Scholar[6] William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons Inc., New York, 1966xviii+636 MR0210154 0138.10207 Google Scholar[7] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, 1944vi+804 MR0010746 0063.08184 Google Scholar[8A] A. Erdélyi, Tables of integral transforms. Vol. I, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954xx+391 MR0061695 0055.36401 Google Scholar[8B] A. Erdélyi, Tables of integral transforms. Vol. II, McGraw-Hill Book Company, Inc., New York-Toronto-London, 1954xvi+451 MR0065685 0058.34103 Google Scholar[9] William Feller, Infinitely divisible distributions and Bessel functions associated with random walks, SIAM J. Appl. Math., 14 (1966), 864–875 10.1137/0114071 MR0207060 0161.37401 LinkISIGoogle Scholar[10] S. Goldstein, On the mathematics of exchange processes in fixed columns. I. Mathematical solutions and asymptotic expansions, Proc. Roy. Soc. London. Ser. A., 219 (1953), 151–171 MR0058824 0053.46106 CrossrefISIGoogle Scholar[11] B. W. Conolly, Some applications of the theory of infinite capacity service systems to a single server system with linearly state dependent service, J. Appl. Probability, 8 (1971), 202–207 MR0279912 0216.47301 CrossrefISIGoogle Scholar[12] Nasser Hadidi, On the service time distribution and the waiting time process of a potentially infinite capacity queueing system, J. Appl. Probability, 6 (1969), 594–603 MR0258150 0191.50401 CrossrefISIGoogle Scholar[13] Frank Spitzer, A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 82 (1956), 323–339 MR0079851 0071.13003 CrossrefGoogle Scholar[14] Lajos Takács, Combinatorial methods in the theory of stochastic processes, John Wiley & Sons Inc., New York, 1967xi+262 MR0217858 0162.21303 Google Scholar[15] L. Takács, On certain sojourn time problems in the theory of stochastic processes, Acta Math. Acad. Sci. Hungar., 8 (1957), 169–191 MR0088092 0081.13303 CrossrefGoogle Scholar[16] L. Takács, On limiting distributions conncerning a sojourn time problem, Acta Math. Acad. Sci. Hungar., 8 (1957), 279–294 MR0096315 0208.44303 CrossrefGoogle Scholar[17] I. J. Good, The frequency count of a Markov chain and the transition to continuous time, Ann. Math. Statist., 32 (1961), 41–48 MR0126948 0099.12801 CrossrefISIGoogle Scholar[18] Irwin Greenberg, The distribution of busy time for a simple queue, Operations Res., 12 (1964), 503–504 MR0166851 0122.13901 CrossrefISIGoogle Scholar[19] A. E. Gibson and , B. W. Conolly, On a three-state sojourn time problem, J. Appl. Probability, 8 (1971), 716–723 MR0292177 0231.60055 CrossrefISIGoogle Scholar[20] B. W. Conolly, Marche aléatoire dont la répartition de la longueur des étapes suit une loi exponentielle négative, Ann. Inst. H. Poincaré Sect. B (N.S.), 2 (1965/1966), 173–184 MR0196816 0142.14203 ISIGoogle Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Spectral analysis of bilateral birth–death processes: some new explicit examples15 June 2022 | Advances in Applied Probability, Vol. 379 Cross Ref Upper bound on the rate of convergence and truncation bound for non-homogeneous birth and death processes on ZApplied Mathematics and Computation, Vol. 423 Cross Ref On a class of birth-death processes with time-varying intensity functionsApplied Mathematics and Computation, Vol. 379 Cross Ref Continuous-Time Birth-Death Chains Generate by the Composition Method15 April 2020 Cross Ref First-passage times and related moments for continuous-time birth–death chains18 December 2018 | Ricerche di Matematica, Vol. 68, No. 2 Cross Ref Asymptotic Results for Random Walks in Continuous Time with Alternating Rates8 February 2014 | Journal of Statistical Physics, Vol. 154, No. 5 Cross Ref On a Bilateral Linear Birth and Death Process in the Presence of Catastrophes Cross Ref A Double-ended Queue with Catastrophes and Repairs, and a Jump-diffusion Approximation11 February 2011 | Methodology and Computing in Applied Probability, Vol. 14, No. 4 Cross Ref On a bilateral birth-death process with alternating rates4 November 2011 | Ricerche di Matematica, Vol. 61, No. 1 Cross Ref Cellular Statistical Models of Broken Cloud Fields. 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