Approximation limits and occupation time problems for queueing networks

2005 Fiscal Year Final Research Report Summary

• Project/Area Number: 16540128
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: General mathematics (including Probability theory/Statistical mathematics)
• Research Institution: Kanagawa University
• Principal Investigator: YAMADA Keigo Kanagawa University, Department of Mathematics, Professor, 工学部, 教授 (90111369)
• Co-Investigator(Kenkyū-buntansha): NARITA Kiyomasa Kanagawa University, Department of Management Science, 工学部, 教授 (10211450)
• Project Period (FY): 2004 – 2005
• Keywords: queueing type network / approximation limit / occupation time problem / multivariate Bessel process / extended Ito formula / optimal control of queueing systems

Research Abstract

(1) It has been shown that a class of state dependent queueing systems can be approximated, under heavy traffic condition, by multivariate Bessel processes. We applied this result to the occupation time problems, and it turned out that their approximation 'limits are local times and principal values of multivariate Bessel processes.
(2) Using the above results, we have succeeded in decomposing multivariate Bessel process as the sum of two processes which correspond to arrival and departure processes respectively. This decomposition was applied to formulate optimal control problems of queueing systems under heavy traffic condition.
(3) An extended Ito formula for multivariate Bessel processes : Multivariate Bessel processes are not semimartingales in general, but belong to Directed processes. For such processes and for functions which are not smooth, we have succeeded in giving Ito formula using the above result, that is, a result of occupation time problems. This Ito formula made us possible to express multivariate Bessel processes as solutions of an extended Skorohod type equation.