Abstract
Approximate formulas of the variance of the waiting-time (also called as delay-time variance) in a renewal-input general-service-time single-server (GI/GI/1) system play an important role in practical applications of the queueing theory. However, there exists almost no literature on the approximate formulas of the delay-time variance in the GI/GI/1 system. The goal of this paper is to present an approximate formula for the delay-time variance. Our approach is based on the combination of a higher-moment relationship between the unfinished work and the waiting time, and the diffusion process approximation for the unfinished work. To derive the former relationship, we apply Miyazawa’s rate conservation law for the stationary point process. Our approximate formula is shown to converge to the exact result for the Poisson-input system as traffic intensity goes to the unity. The accuracy of our approximation is validated by simulation results.
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References
Brumelle, S.L.: On the relationship between customer and time averages in queues. J. Appl. Prob. 8(3), 508–520 (1971)
Gelenbe, E.: Probabilistic models of computer systems, Part II, Diffusion approximations, waiting times and batch arrivals. Acta Informatica 12, 285–303 (1979)
Heyman, D.P.: A diffusion model approximation for the GI/G/1 queue in heavy traffic. Bell System Tech. J. 54, 1637–1646 (1976)
Hoshi, K., Takahashi, Y., Komatsu, N.: A further remark on diffusion approximations with elementary return and reflecting barrier boundaries. In: Proceedings of OR 2010 Munich, pp. I.6.TA-15-2:1–6. Springer (2010)
Miyazawa, M.: The derivation of invariance relations in complex queueing systems with stationary inputs. Adv. Appl. Prob. 15, 875–885 (1983)
Miyazawa, M.: Rate conservation laws: a survey. Queueing Systems 15(1-4), 1–58 (1994)
Takahashi, Y., Miyazawa, M.: Relationship between queue-length and waiting time distributions in a priority queue with batch arrivals. J. Operations Res. Soc. Japan 37(1), 48–63 (1994)
Takahashi, Y., Shikata, Y., Frey, A.: Diffusion approximation for a web-server system with proxy servers. In: Proceedings of OR 2010 Munich, pp. I.6.TC-15-3:1–6. Springer (2010)
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Takahashi, Y., Shikata, Y., Frey, A. (2012). Approximate Formula of Delay-Time Variance in Renewal-Input General-Service-Time Single-Server Queueing System. In: Klatte, D., Lüthi, HJ., Schmedders, K. (eds) Operations Research Proceedings 2011. Operations Research Proceedings. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-29210-1_80
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DOI: https://doi.org/10.1007/978-3-642-29210-1_80
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