On the Temporal Order of First-Passage Times in One-Dimensional Lattice   Random Walks
  A random walk problem with particles on discrete double infinite linear grids is discussed. The model is based on the work of Montroll and others. A probability connected with the problem is given in the form of integrals containing modified Bessel functions of the first kind. By using several transformations simpler integrals are obtained from which for two and three particles asymptotic approximations are derived for large values of the parameters. Expressions of the probability for $n$ particles are also derived. 
J.B. Sanders, N.M.Temme