On spectral properties of finite population processor shared queues |
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Authors: | Qiang Zhen Charles Knessl |
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Institution: | 1. Department of Mathematics and Statistics, University of North Florida, 1 UNF Dr, Jacksonville, FL, 32224, USA 2. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 South Morgan Street (M/C 249), Chicago, IL, 60607, USA
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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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