×

zbMATH — the first resource for mathematics

On the slow server problem. (English. Russian original) Zbl 1180.90078
Autom. Remote Control 70, No. 12, 2013-2023 (2009); translation from Avtom. Telemekh. 2009, No. 12, 81-91 (2009).
Summary: The problem of optimal control over a Markov queueing system with heterogeneous servers and a joint queue is considered, which is also known in the literature as “the slow server problem.” The classical model is generalized here to the case with delay and call serving penalties. It is proved here that the optimal control policy for servers’ activating is of monotonic and threshold nature.

MSC:
90B22 Queues and service in operations research
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kitaev, M.Yu. and Rykov, V.V., Controlled Queueing Systems, New York: CRC Press, 1995. · Zbl 0876.60077
[2] Rykov, V.V., On Monotonicity Conditions for Optimal Policies for Controlling Systems, Autom. Remote Control, 1999, no. 9, pp. 1290–1302. · Zbl 1058.60080
[3] Topkis, D., Minimizing a Submodular Function on a Lattice, Oper. Res. 1978, vol. 26, no. 2, pp. 305–321. · Zbl 0379.90089
[4] Vishnevskii, V.M. and Semenova, O.V., Sistemy pollinga: teoriya i primenenie v shirokopolosnykh besprovodnykh setyakh (Polling Systems: Theory and Application in the Broadband Networks), Moscow: Technosphera, 2007.
[5] Pedro, T., One Size Does Not Fit All: A Case for Heterogeneous Multiprocessor Systems, IADIS Int. Conf. Appl. Comput., 2005.
[6] Krishnamoorthy, B., On Poisson Queue with Two Heterogeneous Servers, Oper. Res., 1963, vol. 11, pp. 321–330. · Zbl 0173.20101
[7] Hajek, B., Optimal Control of Two Interacting Service Stations, IEEE Trans. Automat. Control, 1984, vol. 29, pp. 491–499. · Zbl 0555.90047
[8] Lin, W. and Kumar, P.R., Optimal Control of a Queueing System with Two Heterogeneous Servers, IEEE Trans. Automat. Control, 1984, vol. 29, pp. 696–703. · Zbl 0546.90035
[9] Koole, G., A Simple Proof of the Optimality of a Threshold Policy in Two-server Queueing System, Syst. & Control Lett., 1995, vol. 26, pp. 301–303. · Zbl 0876.90052
[10] Weber, R., On a Conjecture about Assigning Jobs to Processors of Different Speeds, IEEE Trans. Automat. Control, 1993, vol. 38, no. 1, pp. 166–170.
[11] Rykov, V.V., Monotone Control of Queueing System with Heterogeneous Servers, QUESTA, 2001, vol. 37, pp. 391–403. · Zbl 1017.90026
[12] Vericourt, F. and Zhou, Y.P., On the Incomplete Results for the Heterogeneous Server Problem, Queueing Syst., 2006, vol. 52, pp. 189–191. · Zbl 1142.60405
[13] Efrosinin, D.V. and Rykov, V.V., Numerical Study of the Optimal Control of a System with Heterogeneous Servers, Autom. Remote Control, 2003, no. 2, pp. 302–309. · Zbl 1071.60086
[14] Efrosinin, D., Controlled Queueing Systems with Heterogeneous Servers. Dynamic Optimization and Monotonicity Properties, Saarbr├╝cken: VDM Verlag, 2008.
[15] Efrosinin, D. and Breuer, L., Threshold Policies for Controlled Retrial Queues with Heterogeneous Servers, Ann. Oper. Res., 2006, vol. 141, pp. 139–162. · Zbl 1114.90020
[16] Ghoneim, H.A. and Stidham, S., Control of Arrivals to Two Queues in Series, Eur. J. Oper. Res., 1985, vol. 21, pp. 399–409. · Zbl 0569.60091
[17] Sennott, L.I., Average Cost Optimal Stationary Policies in Infinite State Markov Decision Processes with Unbounded Costs, Oper. Res., 1989, vol. 37, pp. 626–633. · Zbl 0675.90091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.