DOI QR코드

DOI QR Code

Conditional sojourn time distributions in M/G/1 and G/M/1 queues under PMλ-service policy

  • Kim, Sunggon (Department of Statistics, University of Seoul)
  • 투고 : 2018.05.15
  • 심사 : 2018.07.03
  • 발행 : 2018.07.31

초록

$P^M_{\lambda}$-service policy is a workload dependent hysteretic policy. The policy has two service states comprised of the ordinary stage and the fast stage. An ordinary service stage is initiated by the arrival of a customer in an idle state. When the workload of the server surpasses threshold ${\lambda}$, the ordinary service stage changes to the fast service state, and it continues until the system is empty. These service stages alternate in this manner. When the cost of changing service stages is high, the hysteretic policy is more efficient than the threshold policy, where a service stage changes immediately into the other service stage at either case of the workload's surpassing or crossing down a threshold. $P^M_{\lambda}$-service policy is a modification of $P^M_{\lambda}$-policy proposed to control finite dams, and also an extension of the well-known D-policy. The distributions of the stationary workload of $P^M_{\lambda}$-service policy and its variants are studied well. However, there is no known result on the sojourn time distribution. We prove that there is a relation between the sojourn time of a customer and the first up-crossing time of the workload process over the threshold ${\lambda}$ after the arrival of the customer. Using the relation and the duality of M/G/1 and G/M/1 queues, we obtain conditional sojourn time distributions in M/G/1 and G/M/1 queues under the policy.

키워드

참고문헌

  1. Abdel-Hameed M (2000). Optimal control of a dam using $P^M_{{\lambda},{\tau}}$ policies and penalty cost when the input process is a compound Poisson process with positive drift, Journal of Applied Probability, 37, 408-416. https://doi.org/10.1239/jap/1014842546
  2. Abdel-Hameed M (2011). Control of dams using $P^M_{{\lambda},{\tau}}$ policies when the input process is a nonnegative Levy process, International Journal of Stochastic Analysis, 2011, Article ID 916952.
  3. Abdel-Hameed MS and Nakhi YA (2006). Optimal control of a finite dam with diffusion input and state dependent release rates, Computers & Mathematics with Applications, 51, 317-324. https://doi.org/10.1016/j.camwa.2005.11.006
  4. Asmussen S (2000). Ruin Probabilities, World Scientific, Singapore.
  5. Bae J, Kim S, and Lee EY (2002). A $P^M_{\lambda}$-policy for an M/G/1 queueing system, Applied Mathematical Modelling, 26, 929-939. https://doi.org/10.1016/S0307-904X(02)00045-8
  6. Balachandran, KR (1973). Control policies for a single server system, Management Science, 19, 1013-1018. https://doi.org/10.1287/mnsc.19.9.1013
  7. Bekker R (2009). Queues with Levy input and hysteretic control, Queueing Systems, 63, 281-299. https://doi.org/10.1007/s11134-009-9123-z
  8. Cohen JW (1969). The Single Server Queue, North-Holland Amsterdam, Amsterdam.
  9. Faddy MJ (1974). Optimal control of finite dams: discrete (2-stage) output procedure, Journal of Applied Probability, 11, 111-121. https://doi.org/10.2307/3212588
  10. Hur S, Lee H, and Kim S (2006). A simple approximation method for workload analyses in some queueing systems with control policies, Computers & Industrial Engineering, 51, 183-195. https://doi.org/10.1016/j.cie.2006.07.010
  11. Kim J, Bae J, and Lee EY (2006). An optimal $P^M_{\lambda}$-service policy for an M/G/1 queueing system, Applied Mathematical Modelling, 30, 38-48. https://doi.org/10.1016/j.apm.2005.03.007
  12. Kim S and Bae J (2008). A G/M/1 queueing system with $P^M_{\lambda}$-service policy, Operations Research Letters, 36, 201-204. https://doi.org/10.1016/j.orl.2007.09.001
  13. Lee J and Kim J (2006). A workload-dependent M/G/1 queue under a two-stage service policy, Operations Research Letters, 34, 531-538. https://doi.org/10.1016/j.orl.2005.08.002
  14. Lee J and Kim J (2007). Workload analysis of an M/G/1 queue under the $P^M_{\lambda}$ policy with a set-up time, Applied Mathematical Modelling, 31, 236-244. https://doi.org/10.1016/j.apm.2005.09.003
  15. Prabhu NU (1997). Stochastic Storage Processes: Queues, Insurance Risk, Dams, and Data Communication (2nd ed), Springer, New York.
  16. Stadje W and Zacks S (2003). Upper first-exit times of compound Poisson processes revisited, Probability in the Engineering and Informational Sciences, 17, 459-465.
  17. Yeh L (1985). Optimal control of a finite dam: average-cost case, Journal of Applied Probability, 22, 480-484. https://doi.org/10.2307/3213793