Review of the methods used for the estimation of sojourn time for semi-Markov process

Semi-Markov process is a stochastic process which can be applied in many areas such as survival analysis, queueing theory, inventory problems, seismological efforts or reliability analysis. The process makes transitions between states in accordance with the Markovian property while spending a random time in the states of the system. The random time that the system spends in any state is called sojourn time. However, according to the process, the sojourn times of the system can follow different distributions. When sojourn times of the process follow different distributions, the stochastic process is called semi-Markov process. In this study, semi-Markov process which is a stochastic process have been investigated in detail. The methods that exists in the literature for estimation of distribution parameters of sojourn times for semi-Markov process are reviewed. Official match results and dates of a football team are characterized by semi-Markov process and EM-algorithm is used for the estimation of distribution parameters of sojourn times. After the estimation of parameters, stationary distributions and mean sojourn times for semi-Markov process are estimated by both empirical estimators and the way which is shown in the literature. Finally, a comparison is made between the results that obtained with given methods.