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Asymptotics of waiting time distributions in the accumulating priority queue

Abstract

We analyze the asymptotics of waiting time distributions in the two-class accumulating priority queue with general service times. The accumulating priority queue was suggested by Kleinrock in the 60s—he coined it time-dependent priority—to diversify waiting time objectives of different classes in a paramaterized way. It also avoids the typical starvation problem of regular priority queues. All customers build up priority linearly while waiting in the queue but at a class-dependent rate. At a service opportunity epoch, the customer with highest priority present is served. Stanford and colleagues recently calculated the Laplace–Stieltjes Transform (LST) of the waiting time distributions of the different classes, but only invert these LSTs numerically. In this paper, we analytically calculate the asymptotics of the corresponding distributions from these LSTs. We show that different singularities of the LST can play a role in the asymptotics, depending on the magnitude of service differentiation between both classes.

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Notes

  1. In this paper, we denote the LST corresponding to a density function f(t) by \({\tilde{F}}(s)\).

  2. Not to be confused with class-1 customers.

  3. Other distribution types can be treated as well, but make the (writing down of the) analysis more cumbersome. We refer to [1] for more details.

  4. We follow notation of Stanford et al. [14] as closely as possible.

  5. For exponential service times of class 1, an explicit expression is possible.

  6. \(f(t) \sim g(t), t\rightarrow t_0 \Leftrightarrow \lim _{t\rightarrow t_0}f(t)/g(t)=1\).

  7. Note that our formulas are not entirely identical as in Abate and Whitt [1], since they assumed \(\mu _1=1\).

References

  1. Abate, J., Whitt, W.: Asymptotics for M/G/1 low-priority waiting-time tail probabilities. Queue. Syst. 25(1–4), 173–233 (1997)

    Article  Google Scholar 

  2. Altman, E., Avrachenkov, K., Ayesta, U.: A survey on discriminatory processor sharing. Queue. Syst. 53, 53–63 (2006)

    Article  Google Scholar 

  3. Banchs, A., Perez, X.: Distributed weighted fair queuing in 802.11 wireless LAN. In Proceedings of the 2002 IEEE International Conference on Communications (ICC), pp 3121–3127, New York, (2002)

  4. De Vuyst, S., Wittevrongel, S., Bruneel, H.: Place reservation: Delay analysis of a novel scheduling mechanism. Comput. Operat. Res. 35(8), 2447–2462 (2008)

    Article  Google Scholar 

  5. Flajolet, P., Sedgewick, R.: Analytic combinatorics. Cambridge University Press, (2009)

  6. Kleinrockl, L.: Queueing systems volume II: Computer applications. John Wiley & Sons, New York, (1976)

  7. Knessl, C., Tier, C., Choi, D.: A dynamic priority queue model for simultaneous service of two traffic types. SIAM J. Appl. Math. 63(2), 398–422 (2003)

    Article  Google Scholar 

  8. Li, N., Stanford, D., Sharif, A., Caron, R., Pardhan, A.: Optimising key performance indicator adherence with application to emergency department congestion. Eur. J. Oper. Res. 272, 313–323 (2019)

    Article  Google Scholar 

  9. Li, N., Stanford, D., Taylor, P., Ziedins, I.: Nonlinear accumulating priority queues with equivalent linear proxies. Oper. Res. 65(6), 1712–1721 (2017)

    Article  Google Scholar 

  10. Lim, Y., Kobza, J.: Analysis of a delay-dependent priority discipline in an integrated multiclass traffic fast packet switch. IEEE Trans. Commun. 38(5), 659–685 (1990)

    Article  Google Scholar 

  11. Maertens, T., Walraevens, J., Bruneel, H.: On priority queues with priority jumps. Perform. Eval. 63(12), 1235–1252 (2006)

    Article  Google Scholar 

  12. Maheswaran, M., Ali, S., Siegel, H., Hensgen, D., Freund, R.: Dynamic mapping of a class of independent tasks onto heterogeneous computing systems. J. Parallel Distrib. Comput. 59, 107–131 (1999)

    Article  Google Scholar 

  13. Parekh, A.L., Gallager, R.G.: A generalized processor sharing approach to flow control in integrated services networks: the multiple node case. IEEE/ACM Trans. Netw. 2(2), 137–150 (1994)

    Article  Google Scholar 

  14. Stanford, D., Taylor, P., Ziedins, I.: Waiting time distributions in the accumulating priority queue. Queue. Syst. 77, 297–330 (2014)

    Article  Google Scholar 

  15. Stolyar, A., Ramanan, K.: Largest weighted delay first scheduling: large deviations and optimality. Ann. Appl. Probab. 11(1), 1–48 (2001)

    Article  Google Scholar 

  16. Takagi, H.: Queueing analysis: a foundation of performance evaluation, volume 1: vacation and priority systems, part 1. North-Holland, (1991)

  17. Takine, T.: The nonpreemptive priority MAP/G/1 queue. Oper. Res. 47(6), 917–927 (1999)

    Article  Google Scholar 

  18. Van Mieghem, J.: Dynamic scheduling with convex delay costs: the generalized \(c\mu \) rule. Ann. Appl. Probab. 5(3), 809–833 (1995)

    Google Scholar 

  19. Vasiliadis, D., Rizos, G., Vassilakis, C.: Class-based weighted fair queuing scheduling on quad-priority delta networks. Int. J. Parallel Emerg. Distrib. Syst. 27(5), 435–457 (2012)

    Article  Google Scholar 

  20. Walraevens, J., Steyaert, B., Bruneel, H.: Delay characteristics in discrete-time GI-G-1 queues with non-preemptive priority queueing discipline. Perform. Eval. 50(1), 53–75 (2002)

    Article  Google Scholar 

  21. Walraevens, J., van Leeuwaarden, J.S.H., Boxma, O.J.: Power series approximations for two-class generalized processor sharing systems. Queue. Syst. 66(2), 107–130 (2010)

    Article  Google Scholar 

  22. Wang, Y., Huang, K., Wang, F.: Scheduling online mixed-parallel workflows of rigid tasks in heterogeneous multi-cluster environments. Futur. Gener. Comput. Syst. 60, 35–47 (2016)

    Article  Google Scholar 

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Acknowledgements

The authors wish to thank the two anonymous referees and associate editor for valuable comments that improved the paper.

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Correspondence to Joris Walraevens.

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Walraevens, J., Van Giel, T., De Vuyst, S. et al. Asymptotics of waiting time distributions in the accumulating priority queue. Queueing Syst 101, 221–244 (2022). https://doi.org/10.1007/s11134-022-09839-7

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Keywords

  • Accumulating priority
  • Dominant singularity analysis

Mathematics Subject Classification

  • 60K25
  • 90B22