Queues with slow servers and impatient customers

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 mu(0) (<= mu). 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. (C) 2009 Elsevier B. V. All rights reserved.