Simple analytical solutions for the \(\mathrm{M}^b/\mathrm{E}_k/1/m, \mathrm{E}_k/\mathrm{M}^b/1/m\), and related queues. (English) Zbl 1498.60375
Summary: In this paper we revisit some classical queueing systems such as the \(\mathrm{M}^b/\mathrm{E}_k/1/m\) and \(\mathrm{E}_k/\mathrm{M}^b/1/m\) queues, for which fast numerical and recursive methods exist to study their main performance measures. We present simple explicit results for the loss probability and queue length distribution of these queueing systems as well as for some related queues such as the \(\mathrm{M}^b/\mathrm{D}/1/m\) queue, the \(\mathrm{D}/\mathrm{M}^b/1/m\) queue, and fluid versions thereof. In order to establish these results we first present a simple analytical solution for the invariant measure of the \(\mathrm{M}/\mathrm{E}_k /1\) queue that appears to be new.
MSC:
| 60K25 | Queueing theory (aspects of probability theory) |
| 68M20 | Performance evaluation, queueing, and scheduling in the context of computer systems |
Keywords:
queueing systems; analytical solution; loss probability; bulk service; bulk arrivals; Erlang-\(k\) distributionReferences:
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