Stochastic decomposition of the M/G/∞M/G/∞ queue in a random environment

We prove a stochastic decomposition formula for the number of customers in an M/G/∞M/G/∞ system with service speeds depending on an independent random environment. In addition, we present a new technique to analyze this kind of queues and apply it to solve some examples of M/M/∞M/M/∞ queues in random environment.