M/G/kM/G/k with staggered setup

We consider the M/G/k/staggeredM/G/k/staggered-setup, where idle servers are turned off to save cost, necessitating a setup time for turning a server back on; however, at most one server may be in setup mode at any time. We show that, for exponentially distributed setup times, the response time of an M/G/k/staggeredM/G/k/staggered-setup approximately decomposes into the sum of the response time for an M/G/kM/G/k and the setup time, where the approximation is nearly exact. This generalizes a prior decomposition result for an M/M/k/staggeredM/M/k/staggered-setup.