Abstract
We consider sojourn or response times in processor-shared queues that have a finite population of potential users. Computing the response time of a tagged customer involves solving a finite system of linear ODEs. Writing the system in matrix form, we study the eigenvectors and eigenvalues in the limit as the size of the matrix becomes large. This corresponds to finite population models where the total population is \(N\gg 1\). Using asymptotic methods we reduce the eigenvalue problem to that of a standard differential equation, such as the Hermite equation. The dominant eigenvalue leads to the tail of a customer’s sojourn time distribution.
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Acknowledgments
Q. Zhen was partly supported by a faculty scholarship grant from the University of North Florida. C. Knessl was partly supported by NSA grants H 98230-08-1-0102 and H 98230-11-1-0184.
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Zhen, Q., Knessl, C. On spectral properties of finite population processor shared queues. Math Meth Oper Res 77, 147–176 (2013). https://doi.org/10.1007/s00186-012-0421-6
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DOI: https://doi.org/10.1007/s00186-012-0421-6