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 $N^{1+\epsilon}$, for some $\epsilon>0$. 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.

Queues, infinite-server systems, Markov modulation, large deviations
Queueing theory (msc 60K25)
Life Sciences (theme 5)
Springer
dx.doi.org/10.1007/s11134-014-9412-z
Queueing Systems
Coarse grained stochastic methods for biochemical reactions
Life Sciences and Health

Blom, J.G, de Turck, K, Kella, O, & Mandjes, M.R.H. (2014). Tail asymptotics of a Markov-modulated infinite-server queue. Queueing Systems, 78(4), 337–357. doi:10.1007/s11134-014-9412-z