Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates

Abstract

Motivated by the work of Whitt (Queueing Syst 81(4):341–378, 2015) and Ma and Whitt (in: Winter simulation conference, pp 2598–2609, 2015), that studied performance stabilization in a \(GI_t/GI_t/1\) queue, this research study investigates the stabilization of the mean virtual response time in a single-server processor sharing (PS) queueing system with a time-varying arrival rate and a service rate control (a \(GI_t/GI_t/1/PS\) queue). We propose and compare a modified square-root (SR) control and a difference-matching (DM) control to stabilize the mean virtual response time of a \(GI_t/GI_t/1/PS\) queue. Extensive simulation studies with various settings of arrival processes and service times show that the DM control outperforms the SR control for heavy-traffic conditions, and that the SR control performs better for light-traffic conditions.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

References

  1. Anselmi, J., & Verloop, M. (2011). Energy-aware capacity scaling in virtualized environments with performance guarantees. Performance Evaluation, 68(11), 1207–1221.

    Article  Google Scholar 

  2. CAIDA. (2016). CAIDA internet data: Realtime monitors. Retrieved August 24, 2018 from http://www.caida.org/data/realtime/passive/?monitor=equinix-chicago-dirB

  3. Chen, H., & Yao, D. D. (2001). Fundamentals of queueing networks. New York, NY: Springer.

    Book  Google Scholar 

  4. Feldman, Z., Mandelbaum, A., Massey, W. A., & Whitt, W. (2008). Staffing of time-varying queues to achieve time-stable performance. Management Science, 54(2), 324–338.

    Article  Google Scholar 

  5. Gautam, N. (2012). Analysis of queues: Methods and applications. Boca Raton, FL: Taylor and Francis.

    Book  Google Scholar 

  6. Gerhardt, I., & Nelson, B. L. (2009). Transforming renewal processes for simulation of nonstationary arrival processes. INFORMS Journal on Computing, 21(4), 630–640.

    Article  Google Scholar 

  7. Green, L., & Kolesar, P. (1991). The pointwise stationary approximation for queues with nonstationary arrivals. Management Science, 37(1), 84–97.

    Article  Google Scholar 

  8. Grishechkin, S. (1994). \(GI/G/1\) processor sharing queue in heavy traffic. Advances in Applied Probability, 26(2), 539–555.

    Article  Google Scholar 

  9. Gromoll, H. C. (2004). Diffusion approximation for a processor sharing queue in heavy traffic. Annals of Applied Probability, 14(2), 555–611.

    Article  Google Scholar 

  10. He, B., Liu, Y., & Whitt, W. (2016). Staffing a service system with non-Poisson non-stationary arrivals. Probability in the Engineering and Informational Sciences, 30(4), 593–621.

    Article  Google Scholar 

  11. Horvath, T., Abdelzaher, T., Skadron, K., & Liu, X. (2007). Dynamic voltage scaling in multitier web servers with end-to-end delay control. IEEE Transactions on Computers, 56(4), 444–458.

    Article  Google Scholar 

  12. Jennings, O. B., Mandelbaum, A., Massey, W. A., & Whitt, W. (1996). Server staffing to meet time-varying demand. Management Science, 42(10), 1383–1394.

    Article  Google Scholar 

  13. Ko, Y. M., & Cho, Y. (2014). A distributed speed scaling and load balancing algorithm for energy efficient data centers. Performance Evaluation, 79, 120–133.

    Article  Google Scholar 

  14. Kwon, S., & Gautam, N. (2016a). Guaranteeing performance based on time-stability for energy-efficient data centers. IIE Transactions, 48(9), 812–825.

    Article  Google Scholar 

  15. Kwon, S., & Gautam, N. (2016b). Time-stable performance in parallel queues with non-homogeneous and multi-class workloads. IEEE/ACM Transactions on Networking, 24(3), 1322–1335.

    Article  Google Scholar 

  16. Liao, D., Li, K., Sun, G., Anand, V., Gong, Y., & Tan, Z. (2015). Energy and performance management in large data centers: A queuing theory perspective. In 2015 international conference on computing, networking and communications (ICNC) (pp. 287–291).

  17. Liu, Y. (2018). Staffing to stabilize the tail probability of delay in service systems with time-varying demand. Operations Research, 66(2), 514–534.

    Article  Google Scholar 

  18. Liu, Y., & Whitt, W. (2012). Stabilizing customer abandonment in many-server queues with time-varying arrivals. Operations Research, 60(6), 1551–1564.

    Article  Google Scholar 

  19. Liu, Y., & Whitt, W. (2014). Stabilizing performance in networks of queues with time-varying arrival rates. Probability in the Engineering and Informational Sciences, 28(4), 419–449.

    Article  Google Scholar 

  20. Liu, Y., & Whitt, W. (2017). Stabilizing performance in a service system with time-varying arrivals and customer feedback. European Journal of Operational Research, 256(2), 473–486.

    Article  Google Scholar 

  21. Ma, N., & Whitt, W. (2015). Using simulation to study service-rate controls to stabilize performance in a single-server queue with time-varying arrival rate. In 2015 winter simulation conference (WSC) (pp. 2598–2609).

  22. Ma, N., & Whitt, W. (2016). Efficient simulation of non-Poisson non-stationary point processes to study queueing approximations. Statistics and Probability Letters, 109, 202–207.

    Article  Google Scholar 

  23. Ma, N., & Whitt, W. (2018). A rare-event simulation algorithm for periodic single-server queues. INFORMS Journal on Computing, 30(1), 71–89.

    Article  Google Scholar 

  24. Ma, N., & Whitt, W. (2019). Minimizing the maximum expected waiting time in a periodic single-server queue with a service-rate control. Stochastic Systems, 9(3), 261–290. https://doi.org/10.1287/stsy.2018.0027.

    Article  Google Scholar 

  25. Whitt, W. (1991). The pointwise stationary approximation for \({M}_t/{M}_t/s\) queues is asymptotically correct as the rates increase. Management Science, 37(3), 307–314.

    Article  Google Scholar 

  26. Whitt, W. (2015). Stabilizing performance in a single-server queue with time-varying arrival rate. Queueing Systems, 81(4), 341–378.

    Article  Google Scholar 

  27. Zhang, J., & Zwart, B. (2008). Steady state approximations of limited processor sharing queues in heavy traffic. Queueing Systems, 60(3), 227–246.

    Article  Google Scholar 

Download references

Acknowledgements

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2016R1D1A1B04933453).

Author information

Affiliations

Authors

Corresponding author

Correspondence to Young Myoung Ko.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendices

Appendix A: Inversion method to simulate a nonstationary non-Poisson process

For all the simulation experiments conducted in this study, we generate samples from the arrival process \(A(\cdot )\) using the inversion method described in Gerhardt and Nelson (2009). The following Algorithm 1 summarizes the procedure.

figurec

Appendix B: Performances of the suggested controls

B.1 Numerical data

See Tables 4, 5, 6, 7 and 8.

Table 4 Control performance of \(\mu _{SR}\) and \(\mu _{DM}\): \(M_t/M_t/1/PS\)
Table 5 Control performance of \(\mu _{SR}\) and \(\mu _{DM}\): \(ER_t/ER_t/1/PS\)
Table 6 Control performance of \(\mu _{SR}\) and \(\mu _{DM}\): \(LN_t/LN_t/1/PS\)
Table 7 Control performance of \(\mu _{SR}\) and \(\mu _{DM}\): \(ER_t/LN_t/1/PS\)
Table 8 Control performance of \(\mu _{SR}\) and \(\mu _{DM}\): \(LN_t/ER_t/1/PS\)

B.2 Plots

See Figs. 8, 9, 10, 11, 12, 13, 14, 15, 16 and 17.

Fig. 8
figure8

General performance measures of \(M_t/M_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 0.1 (\(s=0.1\))

Fig. 9
figure9

General performance measures of \(M_t/M_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 10.0 (\(s=10.0\))

Fig. 10
figure10

General performance measures of \(ER_t/ER_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 0.1 (\(s=0.1\))

Fig. 11
figure11

General performance measures of \(ER_t/ER_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 10.0 (\(s=10.0\))

Fig. 12
figure12

General performance measures of \(LN_t/LN_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 0.1 (\(s=0.1\))

Fig. 13
figure13

General performance measures of \(LN_t/LN_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 10.0 (\(s=10.0\))

Fig. 14
figure14

General performance measures of \(ER_t/LN_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 0.1 (\(s=0.1\))

Fig. 15
figure15

General performance measures of \(ER_t/LN_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 10.0 (\(s=10.0\))

Fig. 16
figure16

General performance measures of \(LN_t/ER_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 0.1 (\(s=0.1\))

Fig. 17
figure17

General performance measures of \(LN_t/ER_t/1/PS\) queues under \(\mu _{SR}\) and \(\mu _{DM}\) with target response time 10.0 (\(s=10.0\))

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cho, Y., Ko, Y.M. Stabilizing the virtual response time in single-server processor sharing queues with slowly time-varying arrival rates. Ann Oper Res 293, 27–55 (2020). https://doi.org/10.1007/s10479-019-03511-9

Download citation

Keywords

  • Stabilizing performance
  • Nonstationary queues
  • Processor sharing
  • Service rate control
  • Queueing simulation