  Motivated by a real failure dataset in a two-dimensional context, this paper
presents an extension of the Markov modulated Poisson process (MMPP) to two
dimensions. The one-dimensional MMPP has been proposed for the modeling of
dependent and non-exponential inter-failure times (in contexts as queuing, risk
or reliability, among others). The novel two-dimensional MMPP allows for
dependence between the two sequences of inter-failure times, while at the same
time preserves the MMPP properties, marginally. The generalization is based on
the Marshall-Olkin exponential distribution. Inference is undertaken for the
new model through a method combining a matching moments approach with an
Approximate Bayesian Computation (ABC) algorithm. The performance of the method
is shown on simulated and real datasets representing times and distances
covered between consecutive failures in a public transport company. For the
real dataset, some quantities of importance associated with the reliability of
the system are estimated as the probabilities and expected number of failures
at different times and distances covered by trains until the occurrence of a
failure.
