Waiting-Time Asymptotics for the M/G/2 Queue with Heterogeneous Servers

This paper considers a heterogeneous M/G/2 queue. The service times at server 1 are exponentially distributed, and at server 2 they have a general distribution B(·). We present an exact analysis of the queue length and waiting time distribution in case B(·) has a rational Laplace–Stieltjes transform. When B(·) is regularly varying at infinity of index −ν, we determine the tail behaviour of the waiting time distribution. This tail is shown to be semi-exponential if the arrival rate is lower than the service rate of the exponential server, and regularly varying at infinity of index 1−ν if the arrival rate is higher than that service rate.