Abstract
We show in this paper that the computation of the distribution of the sojourn time of an arbitrary customer in a M/M/1 with the processor sharing discipline (abbreviated to M/M/1 PS queue) can be formulated as a spectral problem for a self-adjoint operator. This approach allows us to improve the existing results for this queue in two directions. First, the orthogonal structure underlying the M/M/1 PS queue is revealed. Second, an integral representation of the distribution of the sojourn time of a customer entering the system while there are n customers in service is obtained.
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Guillemin, F., Boyer, J. Analysis of the M/M/1 Queue with Processor Sharing via Spectral Theory. Queueing Systems 39, 377–397 (2001). https://doi.org/10.1023/A:1013913827667
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DOI: https://doi.org/10.1023/A:1013913827667