## Waiting-time asymptotics for the M/G/2 queue with heterogeneous servers.(English)Zbl 0993.60098

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 $$\lambda>\mu$$.

### MSC:

 60K25 Queueing theory (aspects of probability theory) 90B22 Queues and service in operations research
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