Covariance structure of interrupted Markov modulated poisson process

The paper considers the covariance structure of an interrupted Markov modulated Poisson process. In this process, periods of on-time and off-time alternate; the on-time interval has a phase-type distribution and the off-time interval has a general one. The off-time period represents the time during which no customer arrives. The on-time period on the other hand, represents the time during which customers do arrive. Here, the arrival rate depends on the phase condition of the on-time interval distribution. The Laplace-Stieltjes transform is derived for the inter-arrival time distribution. Using the results, the correlation structures of succeeding inter-arrival times are studied. When the on-time length distribution is hyperexponential, the covariance of succeeding inter-arrival lengths is positive, whereas it becomes negative for Erlangian on-time lengths.