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  • 1 University of Delhi, Delhi-110007, India
  • | 2 University of Delhi, Delhi-110007, India
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For fixed integers n(= 0) and μ, the number of ways in which a moving particle taking a horizontal step with probability p and a vertical step with probability q, touches the line Y = n+μX for the first time, have been counted. The concept has been applied to obtain various probability distributions in independent and Markov dependent trials.

  • [1]

    Balakrishnan, N. (Editor), Advances in Combinatorial Methods and Applications to Probability and Statistics, Birkháuser, 1997.

  • [2]

    Bhat, U. N. and Lal, R., Number of successes in Markov trials, Journal of Applied Probability, 20 (1988), 677680.

  • [3]

    Consul, P. C., Geeta Distribution and its properties, Communications in Statistics-Theory and Methods, 19 (1990), 30513068.

  • [4]

    Consul, P. C., Two stochastic models for Geeta distribution, Communications in Statistics-Theory and Methods, 19 (1990), 36993706.

  • [5]

    Feller, W., An Introduction to Probability Theory and its Applications, Vol. I (Wiley, New York, 3rd edition), 1968.

  • [6]

    Gabriel, K. R., The distribution of the number of successes in a sequence of dependent trials, Biometrika, 46 (1959), 454460.

  • [7]

    Johnson, N. L., Kotz, S. and Balakrishnan, N., Univariate discrete distribu-tions, 2nd edition, New York: John Wiley & Sons, 1992.

  • [8]

    Krattenthaler, C., The enumeration of lattice paths with respect to their number of turns, Advances in Combinatorial Methods and Applications to Probability and Statistics, (ed. Balakrishnan, N.) Birkháuser, 29–58, 1997.

    • Search Google Scholar
    • Export Citation
  • [9]

    Mohanty, S. G., On a generalized two-coin tossing problem, Biometrische Zeit-schrift, 8 (1966), 266272.

  • [10]

    Mohanty, S. G., An r-coin tossing game and the associated partition of generalized Fibonacci number, Sankhya, Series A, 29 (1996), 207214.

    • Search Google Scholar
    • Export Citation
  • [11]

    Mohanty, S. G., Lattice Path Counting and Applications, New York, Academic Press, 1979.

  • [12]

    Mohanty, S. G., Success runs of length k in Markov dependent trials, Annals of the Institute of Statistical Mathematics, 46 (1994), 777796.

    • Search Google Scholar
    • Export Citation
  • [13]

    Narayana, T. V., A partial order and its applications in probability theory, Sankhya, 21, 91–98.

  • [14]

    Narayana, T. V. and Sathe, Y. S., Minimum variance unbiased estimation in coin tossing problems, Sankhya, Series A, 23 (1961), 183186.

    • Search Google Scholar
    • Export Citation
  • [15]

    Riordan, J., An Introduction to Combinatorial Analysis, Wiley, New York, 1958.

  • [16]

    Sen, K., On some combinatorial relations concerning the symmetric random walk, Publ. Math. Inst. Hungar. Acad. Sci., 9 (1964), 335357.

    • Search Google Scholar
    • Export Citation
  • [17]

    Sen, K., Lattice path Combinatorics in Probability and Statistics, Section Presi-dential address in eighty-sixth session of Indian Science Congress Association 1998–1999, 1999.

    • Search Google Scholar
    • Export Citation
  • [18]

    Sen, K., Lattice path approach to transient analysis of M/G/1/N-non Markovian queues using Cox distribution, Journal of Statistical Planning and Inference 101(1–2) (2001), 133147.

    • Search Google Scholar
    • Export Citation
  • [19]

    Sen, K. and Agarwal, M., Transient busy period analysis of initially nonempty M/G/1 queues –Lattice path approach, Advances in Combinatorial Methods and Application in Probability and Statistics (ed. Balakrishnan, N.) Birkháuser, 301–315, 1997.

    • Search Google Scholar
    • Export Citation
  • [20]

    Sen, K. and Agarwal, M., Lattice path combinatorics applied to transient queue length distribution of C2=M=1 queues and busy period analysis of bulk queues Cb 2=M=1, Journal of Statistical Planning and Inference, 100(2) (2002), 365397.

    • Search Google Scholar
    • Export Citation
  • [21]

    Sen, K. and Gupta, R., Transient behaviour of M/M/1 queues –A Lattice path approach, Assam Statistical Review, 2(7) (1993), 7188.

  • [22]

    Sen, K. and Gupta, R., Transient analysis of threshold T-Policy M/M/1 queues with server control, Sankhya, Series B, 56 (1994), 3951.

    • Search Google Scholar
    • Export Citation
  • [23]

    Sen, K. and Gupta, R., Transient solution of Mb=M=1 system under threshold control policy, Journal of Statistical Research, 2(30) (1996), 109120.

    • Search Google Scholar
    • Export Citation
  • [24]

    Sen, K. and Gupta, R., Transient solution of M/M/1 queues with batch arrival –A new approach, Statistica, L VI, 3 (1996), 333343.

  • [25]

    Sen, K. and Gupta, R, Time dependent analysis of T-Policy M/M/1 queues –A new approach, Stud.Sci.Math.Hung., 34 (1997), 453473.

  • [26]

    Sen, K. and Jain, J. L., Combinatorial approach to Markovian queuing models, Journal of Statistical Planning and Inference, 34(1) (1993), 269279.

    • Search Google Scholar
    • Export Citation
  • [27]

    Sen, K., Jain, J. L. and Gupta, J. M., Lattice path approach to transient solution to M/M/1 with (0; k) control policy, Journal of Statistical Planning and Inference, 34(2) (1993), 259267.

    • Search Google Scholar
    • Export Citation
  • [28]

    Takács, L., On the ballot theorems, Advances in Combinatorial Methods and Applications to Probability and Statistics, (ed. Balakrishnan, N.) Birkháuser, 677680, 1997.

    • Search Google Scholar
    • Export Citation