**6. Conclusions**

The queueing systems with heterogeneous servers have many real applications. The optimal control policy which minimizes the mean number of customers in the system without preemption under certain assumptions belongs to a threshold policy. Classical methods, such as the solution of difference equations, matrix-analytic and dynamic-programming approach, have significant restrictions due to the dimension of the random processes involved. A heuristic solution is obtained for the optimal threshold levels in a system with an arbitrary number of servers. The simple lower and upper bounds for the minimal mean number of customers in the system are derived using one dimensional processes for the equivalent heterogeneous queues with a preemption. The gap between the bounds increases with increasing of the servers' heterogeneity and the number of servers in the system. We have further conducted simulation to provide sensitivity analysis of the obtained HS to changes in inter-arrival and service time distributions. Simulation results showed that the optimal thresholds are likely to depend on the mean inter-arrival and service times and hence the proposed heuristic solution can be used as a quasi-optimal in systems with arbitrary distributions.

**Author Contributions:** Conceptualization, D.E.; Formal analysis, D.E., N.S.; Investigation, D.E., N.S., A.P.; Methodology, D.E., J.S.; Software, D.E., N.S.; Writing—original draft, D.E., N.S.; Writing—review & editing, D.E., J.S. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by RUND University Program gran<sup>t</sup> number 5-100.

**Acknowledgments:** The authors are very grateful to the reviewers for their valuable comments and suggestions which improved the quality and the presentation of the paper.

**Conflicts of Interest:** The authors declare no conflict of interest.
