  We consider a general $k$ dimensional discounted infinite server queues
process (alternatively, an Incurred But Not Reported (IBNR) claim process)
where the multivariate inputs (claims) are given by a $k$ dimensional finite
state Markov Chain and the arrivals follow a renewal process. After deriving a
multidimensional integral equation for the moment generating function jointly
to the state of the input at time $t$ given the initial state of the input at
time 0, asymptotic results for the first and second (matrix) moments of the
process are provided. In particular, when the interarrival or service times are
exponentially distributed, transient expressions for the first two moments are
obtained. Also, the moment generating function for the process with
deterministic interarrival times is considered to provide more explicit
expressions. Finally, we demonstrate the potential of the present model by
showing how it allows us to study a semi-Markov modulated infinite server
queues where the customers (claims) arrival and service (reporting delay) times
depend on the state of the process immediately before and on the switching
times. AMS 2000 subject classications: Primary 60G50, 60K30, 62P05, 60K25.
