Abstract
We settle a conjecture of Kella et al. (J. Appl. Probab. 42:223–234, 2005): the distribution of the number of jobs in the system of a symmetric M/G/1 queue at a fixed time is independent of the service discipline if the system starts empty. Our derivations are based on a time-reversal argument for regenerative processes and a connection with a clearing model.
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References
Abate, J., Whitt, W.: Transient behavior of the M/M/1 queue via Laplace transforms. Adv. Appl. Probab. 20, 145–178 (1988)
Asmussen, S.: Applied Probability and Queues. Springer, New York (2003)
Bonald, T., Tran, M.-A.: On Kelly networks with shuffling. Queueing Syst. 59, 53–61 (2008)
Chao, X., Miyazawa, M., Pinedo, M.: Queueing Networks: Customers, Signals, and Product Form Solutions. Wiley, Chichester (1999)
Chen, H., Yao, D.D.: Fundamentals of Queueing Networks. Springer, New York (2001)
Cooper, R., Niu, L.: Benes’ formula for M/G/1-FIFO ‘explained’ by preemptive-resume LIFO. J. Appl. Probab. 23, 550–554 (1986)
Denisov, D., Sapozhnikov, A. On the distribution of the number of customers in the symmetric M/G/1 queue. Queueing Syst. 54, 237–241 (2006)
Jain, K., Sigman, K.: A Polleczek–Khintchine formulation for M/G/1 queues with disasters. J. Appl. Probab. 33, 1191–1200 (1996)
Kella, O., Zwart, A.P., Boxma, O.J.: Some time-dependent properties of symmetric M/G/1 queues. J. Appl. Probab. 42, 223–234 (2005)
Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)
Kitaev, M.Yu.: The M/G/1 Processor Sharing queue: transient behavior. Queueing Syst. 14, 239–273 (1993)
Kyprianou, A.: Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin (2006)
Rosenkrantz, W.: Calculation of the Laplace transform of the length of the busy period for the M/G/1 queue via martingales. Ann. Probab. 11, 817–818 (1983)
Serfozo, R.: Introduction to Stochastic Networks. Springer, Berlin (1999)
Virtamo, J.: (2003). Insensitivity of a network of symmetric queues with balanced service rates. http://netlab.hut.fi/tutkimus/fit/publ/insens_symm_qn.pdf.
Whitt, W.: The continuity of queues. Adv. Appl. Probab. 6, 175–183 (1974)
Whitt, W.: Stochastic-Process Limits. Springer, New York (2002)
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Bert Zwart is also affiliated with EURANDOM, VU University Amsterdam, and Georgia Institute of Technology. His research is partly supported by NSF Grants 0727400 and 0805979, an IBM faculty award, and a VIDI Grant from NWO. This research was initiated when the first author was affiliated with EURANDOM. We are grateful to Artem Sapozhnikov for useful comments.
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Fralix, B., Zwart, B. Time-dependent properties of symmetric queues. Queueing Syst 67, 33–45 (2011). https://doi.org/10.1007/s11134-010-9202-1
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DOI: https://doi.org/10.1007/s11134-010-9202-1