Abstract
A nonclassical queuing-theory problem with calls arising in a space is considered. Stations must be placed to minimize the service time for arising calls. The service time is an increasing function that depends on the distance between a call and a station. The time spent to overcome the same distance frequently depends on the direction of motion. In this case, a metric that considers the nonequivalence of coordinates of the space in order must be chosen to construct an adequate mathematical model. Optimum arrangements of stations can be found for problems of this kind only in exceptional situations. However, an asymptotical solution to the problem can be found that is acceptable from a practical viewpoint. An algorithm is given for constructing asymptotically optimum arrangements.
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References
L. V. Nazarov and S. N. Smirnov, “Service calls distributed in space,” Izv. Akad. Nauk SSSR, Tekh. Kibern., 1, 95–99 (1982).
T. V. Zakharova, “Optimization of the location of service stations on the plane,” Izv. Akad. Nauk SSSR, Tekh. Kibern., No. 6, 73–79 (1987).
T. V. Zakharova, “Optimization of the location of service stations in space,” Inform. Primen. 2 (2), 41–46 (2008).
T. V. Zakharova and A. A. Fisak, “Optimal positioning of service stations,” Mosc. Univ. Comput. Math. Cybern. No. 2, 40–47 (2018).
T. V. Zakharova, “Optimization of space-based queuing systems,” Cand. Sci. (Phys. Math.) Dissertation (Moscow State Univ., Moscow, 2008).
L. F. Toth, Lagerungen in der Ebene, auf der Kugel und im Raum (Spriger, Berlin, New York, 1953).
E. B. Saff and A. B. J. Kuijlaars, “Distributing many points on a sphere,” Math. Intell. 19 (1), 5–11 (1997).
A. Kashurin, “A problem of arrangement of service stations on the given territory,” Intelekt. Transp. Sist., No. 34, 111–115 (2010).
G. Caimi, L. Kroon, and C. Christian Liebchen, “Models for railway timetable optimization: applicability and applications in practice,” J. Rail Transp. Planning Manage. 6, 285–312 (2017).
A. Schobel, “An eigenmodel for iterative line planning, timetabling and vehicle scheduling in public transportation,” Transp. Res., Part C: Emerging Technol. 74, 348–365 (2017).
Funding
This work was supported by the Russian Foundation for Basic Research, project no. 19-07-00352.
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Russian Text © The Author(s), 2019, published in Vestnik Moskovskogo Universiteta, Seriya 15: Vychislitel’naya Matematika i Kibernetika, 2019, No. 3, pp. 6–10.
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Zakharova, T.V. Asymptotically Optimum Arrangements for a Special Class of Normed Spaces. MoscowUniv.Comput.Math.Cybern. 43, 89–94 (2019). https://doi.org/10.3103/S0278641919030075
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DOI: https://doi.org/10.3103/S0278641919030075