Queues with slow servers and impatient customers

We study M / M / c queues (c = 1, 1 < c < ∞ and c = ∞) 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 η and the arrival and service rates are λ and μ, respectively. The corresponding parameters for the slow phase (phase 0) are γ, λ₀, and μ₀ (≤ μ). When in the slow phase, customers become impatient. That is, each customer, upon arrival, activates an individual timer, exponentially distributed with parameter ξ. 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.