Abstract
We consider a two-node tandem queueing network in which the upstream queue is M/G/1 and each job reuses its upstream service requirement when moving to the downstream queue. Both servers employ the first-in-first-out policy. We investigate the amount of work in the second queue at certain embedded arrival time points, namely when the upstream queue has just emptied. We focus on the case of infinite-variance service times and obtain a heavy traffic process limit for the embedded Markov chain.
Similar content being viewed by others
References
Boxma, O.: Analysis of models for tandem queues. Ph.D. Thesis, University of Utrecht, Utrecht (1977)
Boxma, O.: On a tandem queueing model with identical service times at both counters, I. Adv. Appl. Probab 11, 616–643 (1979)
Karpelevich, F.I., Kreĭnin, A.Y.: Heavy Traffic Limits for Multiphase Queues, vol. 137 of Translations of Mathematical Monographs. American Mathematical Society, Providence (1994). (Translated from the Russian manuscript by Kreĭnin and A. Vainstein)
Karpelevitch, F.I., Kreinin, A.Y.: Asymptotic analysis of queueing systems with identical service. J. Appl. Probab. 33(1), 267–281 (1996)
Boxma, O.: On the longest service time in a busy period of the \(\text{ M }/\text{ G }/1\) queue. Stoch. Process. Appl. 8(1), 93–100 (1978)
Boxma, O.J., Deng, Q.: Asymptotic behaviour of the tandem queueing system with identical service times at both queues. Math. Methods Oper. Res. 52(2), 307–323 (2000)
Ethier, S.N., Kurtz, T.G.: Markov Processes: Characterization and Convergence, vol. 282. Wiley, Hoboken (2009)
Ballerini, R., Resnick, S.I.: Records in the presence of a linear trend. Adv. Appl. Probab. 19(4), 801–828 (1987)
Billingsley, P.: Convergence of Probability Measures. Tracts on Probability and Statistics. Wiley Series in Probability and Mathematical Statistics. Wiley, Hoboken (1968)
Bingham, N.H., Goldie, C.M., Teugels, J.L.: Regular Variation. Cambridge University Press, Cambridge (1987)
Resnick, S.I.: Heavy-tail Phenomena. Springer Series in Operations Research and Financial Engineering. Springer, New York (2007). (Probabilistic and statistical modeling)
Acknowledgements
Funding was provided by NWO grant number 639.033.413.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Gromoll, H.C., Terwilliger, B. & Zwart, B. Heavy traffic limit for a tandem queue with identical service times. Queueing Syst 89, 213–241 (2018). https://doi.org/10.1007/s11134-017-9560-z
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11134-017-9560-z