The rate of protein synthesis in a cell is modulated by the number of RNA molecules present within the cell. RNA is synthesised by transcription from DNA in the chromosomes, and decays spontaneously after some time. Both synthesis and decay are stochastic, and lead to stochastic fluctuations in numbers of protein molecules. 

We model the system as an infinite-server queues with general service times and Cox process arrivals. We present functional large deviation principles for Cox processes, and apply them to study fluctuations of the queue size.