This paper analyzes large deviation probabilities related to the number of customers in
a Markov modulated infinite-server queue, with state-dependent arrival and service rates.
Two specific scalings are studied: in the first, just the arrival rates are linearly scaled by N (for large N),
whereas in the second
in addition the Markovian background process is sped up by a factor N1+ϵ, for some ϵ>0.
In both regimes, (transient and stationary) tail probabilities decay essentially exponentially,
where the associated decay rate corresponds to that of the probability
that the sample mean of i.i.d.\ Poisson random variables
attains an atypical value