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CONDITIONAL SOJOURN TIMES OF PROCESSOR-SHARING QUEUES

Published online by Cambridge University Press:  20 December 2012

Wei-Yi Lee  and
Chia-Li Wang
Wei-Yi Lee
Affiliation:
Department of Applied Mathematics National Dong Hwa University, Hualien, Taiwan, ROC E-mail: [email protected]
Chia-Li Wang
Affiliation:
Department of Applied Mathematics National Dong Hwa University, Hualien, Taiwan, ROC E-mail: [email protected]
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Abstract

Queues operated by a processor-sharing mode have important applications in many modern systems. However, because of the simultaneous sharing of service capacity by all customers, the distribution function and moments of the sojourn time are difficult to derive, even with a given initial condition. In addition, when a limit on the number of customers in the system is enforced to ensure the quality of service, the sojourn time becomes more complicated. In recent literature, the distribution function is obtained via the Laplace–Stieltjes transform. In this paper, we take a pure algebraic approach to derive the moments of the sojourn time. We obtain an iterative formula and use it to investigate properties of the conditional sojourn time. The approach is simple and intuitive, and applies to queues with multiple class customers as well.

Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 27 , Issue 1 , January 2013 , pp. 99 - 114
Copyright
Copyright © Cambridge University Press 2013

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References

1.Altman, E., Avrachenkov, K., & Ayesta, U. (2006). A survey on discriminatory processor sharing. Queueing Systems 53: 53–63.CrossRefGoogle Scholar
2.Borst, S.C., Boxma, O.J., & Hegde, N. (2005). Sojourn times in finite-capacity processor-sharing queues. Proceedings of the 1st Euro-NGI Conference. Rome, 53–60.CrossRefGoogle Scholar
3.Boxma, O.J., Hegde, N., & Nunez-Queija, R. (2006). Exact and approximate analysis of sojourn times in finite discriminatory processor sharing queues. International Journal of Electronics and Communications 60: 109–115.CrossRefGoogle Scholar
4.http://www.cfdengineer.com/articles/BlockTriDiagonal.shtmlGoogle Scholar
5.Coffman, E.G., Muntz, R.R. Jr., & Trotter, H. (1970). Waiting time distributions for processor-sharing systems. Journal of the ACM 17: 123–130.CrossRefGoogle Scholar
6.Huang, Y. & McColl, W.F. (1997). Analytical Inversion of General Tridiagonal Matrices. Journal of Physics A: Mathematical and General 30: 7919–7933.CrossRefGoogle Scholar
7.Kleinrock, L. (1967). Time-shared systems: a theoretical treatment. Journal of the ACM 14: 242–261.CrossRefGoogle Scholar
8.Molinari, L.G. (2008). Determinants of block tridiagonal matrices. Linear Algebra and its Applications 429: 2221–2226.CrossRefGoogle Scholar
9.Naor, P. (1969). The regulation of queue size by levying tolls. Econometrica 37: 15–24.CrossRefGoogle Scholar
10.Sengupta, B. & Jagerman, D.L. (1985). A conditional response time of the M/M/1 processor-sharing queue. AT&T Technical Journal 64: 409–421.Google Scholar
11.Zhen, Q. & Knessl, C. (2009). On Sojourn Times in the M/M/1 − PS Model, Conditional on the Number of Other Users. Applied Mathematics Research eXpress 2: 142–167.Google Scholar