Abstract
We consider a single server system consisting of n queues with different types of customers and k permanent customers. The permanent customers and those at the head of the queues are served in processor-sharing by the service facility (head-of-the-line processor-sharing). By means of Loynes’ monotonicity method a stationary work load process is constructed and using sample path analysis general stability conditions are derived. They allow to decide which queues are stable and, moreover, to compute the fraction of processor capacity devoted to the permanent customers. In case of a stable system the constructed stationary state process is the only one and for any initial state the system converges pathwise to the steady state.
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References
A. Brandt and M. Brandt, On the sojourn times for many-queue head-of-the-line processor-sharing systems with permanent customers, Math. Methods Oper. Res. 47 (1998) 181–220.
A. Brandt, P. Franken and B. Lisek, Stationary Stochastic Models (Akademie-Verlag, Berlin; Wiley, Chichester, 1990).
J.L. Doob, Stochastic Processes (Wiley, New York, 1990).
P. Franken, D. König, U. Arndt and V. Schmidt, Queues and Point Processes (Akademie-Verlag, Berlin; Wiley, Chichester, 1982).
R.M. Loynes, The stability of a queue with non-independent inter-arrival and service times, Proc. Cambridge Philos. Soc. 58 (1962) 497–520.
S.F. Yashkov, Processor-sharing queues: Some progress in analysis, Queueing Systems 2 (1987) 1–17.
S.F. Yashkov, Analysis of Queues in Computers (Radio Svyaz, Moscow, 1989) (in Russian).
S.F. Yashkov, Mathematical problems in the theory of shared-processor systems, in: Itogi Nauki i Tekhniki, Seriya Teoriya Veroyatnostei, Matematicheskaya Statistika, Teoreticheskaya Kibernetika 29 (1990) pp. 3–82.
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Brandt, A., Brandt, M. A note on the stability of the many-queue head-of-the-line processor-sharing system with permanent customers. Queueing Systems 32, 363–381 (1999). https://doi.org/10.1023/A:1019155524681
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DOI: https://doi.org/10.1023/A:1019155524681