An exponential queue with competition for service

We consider an M/M/c queue, where a single ‘special’ customer Cp arrives. If Cp must wait, then either Cp will eventually start service at some completion epoch with probability p, or Cp will ‘float’ into service at the end of a busy period. The model is interpreted as Cp competing for service against the regular customers. We study the waiting time of Cp. We analyze the effect of Cp on the waiting times of the regular customers. Examples and applications are discussed.