Waiting-time asymptotics for the M/G/2 queue with heterogeneous servers (Q5961046)

A heterogeneous M/G/2 queue with FCFS service discipline and Poisson distribution of the arrivals with rate \(\lambda\) is considered. It is supposed that the service times at the server 1 are exponentially distributed with rate \(\mu\), but at server 2 they have general distribution \(B(\cdot)\) with mean \(\beta\). The stability condition \(\lambda< \mu+1/\beta\) is assumed to be satisfied. If \(B(\cdot)\) is regularly varying at infinity of index \(-\nu\), i.e. \(1-B(t)= t^{-\nu}L(t)\), \(t\to\infty\), with a slowly varying function \(L(\cdot)\), then it is proved that the waiting time tail is semi-exponential if \(\lambda< \mu\) and the waiting time tail is regularly varying of index \(1-\nu\) if \(\lambda>\mu\).