Analysis of random walks with an absorbing barrier and chemical rule

Recently Tarabia and El-Baz [A.M.K. Tarabia, A.H. El-Baz, Transient solution of a random walk with chemical rule, Physica A 382 (2007) 430–438] have obtained the transient distribution for an infinite random walk moving on the integers − ∞ < k < ∞ of the real line. In this paper, a similar technique is used to derive new elegant explicit expressions for the first passage time and the transient state distributions of a semi-infinite random walk having “chemical” rule and in the presence of an absorbing barrier at state zero. The walker starting initially at any arbitrary positive integer position i , i > 0 . In random walk terminology, the busy period concerns the first passage time to zero. This relation of these walks to queuing problems is pointed out and the distributions of the queue length in the system and the first passage time (busy period) are derived. As special cases of our result, the Conolly et al. [B.W. Conolly, P.R. Parthasarathy, S. Dharmaraja, A chemical queue, Math. Sci. 22 (1997) 83–91] solution and the probability density function (PDF) of the busy period for the M / M / 1 / ∞ queue are easily obtained. Finally, numerical values are given to illustrate the efficiency and effectiveness of the proposed approach.