## Upper bounds on performance measures of heterogeneous $$M/M/c$$ queues.(English)Zbl 1235.90042

Summary: In many real-life queueing systems, the servers are often heterogeneous, namely they work at different rates. This paper provides a simple method to compute tight upper bounds on two important performance measures of single-class heterogeneous multi-server Markovian queueing systems, namely the average number in queue and the average waiting time in queue. This method is based on an expansion of the state space that is followed by an approximate reduction of the state space, only considering the most probable states. In most cases tested, we were able to approximate the actual behavior of the system with smaller errors than those obtained from traditional homogeneous multiserver Markovian queues, as shown by GPSS simulations. In addition, we have correlated the quality of the approximation with the degree of heterogeneity of the system, which was evaluated using its Gini index. Finally, we have shown that the bounds are robust and still useful, even considering quite different allocation strategies. A large number of simulation results show the accuracy of the proposed method that is better than that of classical homogeneous multiserver Markovian formulae in many situations.

### MSC:

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

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