Random Oper. and Stoch. Equ., Vol. 13, No. 4, pp. 399–405 (2005) c  VSP 2005 One-dimensional semi-Markov evolutions with general Erlang sojourn times A. A. POGORUI 1 and Ram´on M. RODR ´ IGUEZ-DAGNINO 2 1 Monterrey Institute of Technology (ITESM), M´ exico E-mail: pogorui@itesm.mx 2 Monterrey Institute of Technology (ITESM). Sucursal de correos “J” C.P. 64849, Monterrey, N.L., M´ exico E-mail: rmrodrig@itesm.mx Received for ROSE March 7, 2005 Abstract—In this paper we study a one-dimensional random motion by having a general Erlang distribution for the sojourn times and we obtain higher order hyperbolic equations for this case. We apply the methodology of random evolutions to find the partial differential equations governing the particle motion and we obtain a factorization of these equations. As a particular case we find the linear biwave equation for the symmetric motion case and 2-Erlang distributions for the sojourn times of a semi-Markov evolution. Key words and phrases: Random evolutions; semi-Markov processes; Erlang distributions; random velocities, telegrapher’s equation, biwave equation 1. INTRODUCTION In this paper we study a one-dimensional random motion performed with two alter- nating velocities, where the random times separating consecutive changes of velocities perform an alternating semi-Markov process. The sojourn times of this process are random variables with general Erlang distributions. Most of the papers on random motion are devoted to analysis of models in which motions are driven by a homogeneous Poisson process, so their processes are Markovian [1], [2], [3], and [4]. In paper [5], it is considered a non-Markovian generalization of the telegrapher’s random process where motion is driven by an alternating semi-Markov process with Erlang distributed interrenewal times. We assume that the particle moves on the line R in the following manner: At each instant it moves according to one of two velocities, namely v 1 > 0 or v 2 < 0. Starting at the position x 0 ∈ R the particle continues its motion with velocity v 1 > 0 during the random time τ 1 = ξ λ 1 + ··· + ξ λ n , where n ≥ 1 and ξ λ i is an exponential random variable (r.v.) with parameter λ i , then the particle moves with velocity v 2 < 0 during the random time τ 2 = ξ μ 1 + ··· + ξ μ m , where m ≥ 1 and ξ μ i is an exponential r.v. with parameter μ i . Moreover, the particle moves with velocity v 1 > 0 and so on. We study this one-dimensional random motion by assuming a general Erlang dis- tribution for the sojourn times. Meaning that τ 1 , respectively τ 2 , is general Erlang