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

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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.

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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).

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Correspondence to Young Myoung Ko.

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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 (\(s=10.0\))

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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 (2020) doi:10.1007/s10479-019-03511-9

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Keywords

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