We study M/M/c queues (c=1, 1<c<[infinity] and c=[infinity]) in a 2-phase (fast and slow) Markovian random environment, with impatient customers. The system resides in the fast phase (phase 1) an exponentially distributed random time with parameter [eta] and the arrival and service rates are [lambda] and [mu], respectively. The corresponding parameters for the slow phase (phase 0) are [gamma], [lambda]0, and . When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter [xi]. If the system does not change its environment from 0 to 1 before the customer's timer expires, the customer abandons the queue never to return. We concentrate on deriving analytic solutions to the queue-length distributions. We derive, for each case of c, the corresponding probability generating function, and calculate the mean queue size. Several extreme cases are investigated and numerical results are presented.
