The M/M\1 queue in a heavy-tailed random environment

We consider an M / M /1 queue with the special feature that the speed of the server alternates between two constant values s_L and s_H > s_L . The high-speed periods are exponentially distributed, and the low-speed periods have a regularly varying distribution. We obtain explicit asymptotics for the tail of the workload distribution. The two cases in which the offered traffic load is smaller respectively larger than the low service speed are shown to result in completely different asymptotics.