Two heavy traffic limit theorems are presented for a GI/M/s queue with server vacations. Server vacations are governed by an exogenous process which at any time specifies the number of servers who are on vacation. The number of vacationing servers can effect in the service rate of the remaining servers still functioning. In one case the number of servers going on vacation remains fixed as the queue approaches heavy traffic; in the other case the number grows at the rate of convergence to heavy traffic. The first limit is a diffusion process that is a mixture of an Ornstein-Uhlenbeck process and a Brownian motion with negative draft. The infinitesimal parameters of this limiting diffusion are governed by the exogenous vacation process. The second limit is a jump-diffusion process whose diffusion is a mixture of an Ornstein-Uhlenbeck process and a Brownian motion, while the jumps are from the vacation process. The parameters of this process also are controlled by the vacation process. The paper proposes diffusion approximations for such queueing models in heavy traffic and describes an application in Integrated Service Digital Newtowk communications links.
