Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T14:27:12.940Z Has data issue: false hasContentIssue false
  • English
  • Français

STEADY-STATE OPTIMIZATION OF AN EXHAUSTIVE LÉVY STORAGE PROCESS WITH INTERMITTENT OUTPUT AND RANDOM OUTPUT RATE

Published online by Cambridge University Press:  17 April 2020

Royi Jacobovic Open the ORCID record for Royi Jacobovic [Opens in a new window]  and
Offer Kella
Royi Jacobovic
Affiliation:
Department of Statistics and Data Science, The Hebrew University of Jerusalem, Jerusalem 9190501, Israel E-mail: [email protected]; [email protected]
Offer Kella
Affiliation:
Department of Statistics and Data Science, The Hebrew University of Jerusalem, Jerusalem 9190501, Israel E-mail: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Consider a regenerative storage process with a nondecreasing Lévy input (subordinator) such that every cycle may be split into two periods. In the first (off), the output is shut off and the workload accumulates. This continues until some stopping time. In the second (on), the process evolves like a subordinator minus a positive drift (output rate) until it hits the origin. In addition, we assume that the output rate of every on period is a random variable, which is determined at the beginning of this period. For example, at each period, the output rate may depend on the workload level at the beginning of the corresponding busy period. We derive the Laplace–Stieltjes transform of the steady-state distribution of the workload process and then apply this result to solve a steady-state cost minimization problem with holding, setup and output capacity costs. It is shown that the optimal output rate is a nondecreasing deterministic function of the workload level at the beginning of the corresponding on period.

Keywords

Lévy driven queuesoptimization with partial informationoutput rate controlqueues with server vacations
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 35 , Issue 3 , July 2021 , pp. 730 - 744
Copyright
© Cambridge University Press 2020

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Asmussen, S. & Kella, O. (2000). A multi-dimensional martingale for Markov additive processes and its applications. Advances in Applied Probability 32(2): 376–393.CrossRefGoogle Scholar
2.Bekker, R. (2009). Queues with Lévy input and hysteretic control. Queueing Systems 63(1–4): 281.CrossRefGoogle Scholar
3.Bekker, R., Boxma, O.J., & Kella, O. (2008). Queues with delays in two-state strategies and Lévy input. Journal of Applied Probability 45(2): 314–332.CrossRefGoogle Scholar
4.Bekker, R., Boxma, O.J., & Resing, J.A.C. (2009). Lévy processes with adaptable exponent. Advances in Applied Probability 41(1): 177–205.CrossRefGoogle Scholar
5.Boxma, O., Mandjes, M., & Kella, O. (2008). On a queuing model with service interruptions. Probability in the Engineering and Informational Sciences 22(4): 537–555.CrossRefGoogle Scholar
6.Boxma, O., Kella, O., & Mandjes, M. (2010). On a generic class of Lévy-driven vacation models. Probability in the Engineering and Informational Sciences 24(1): 1–12.CrossRefGoogle Scholar
7.Debicki, K. & Mandjes, M (2015). Queues and Lévy fluctuation theory. Springer.Google Scholar
8.Jacobovic, R. & Kella, O. (2019). Minimizing a stochastic convex function subject to stochastic constraints and some applications. arXiv:1906.09604.Google Scholar
9.Ke, J.C., Wu, C.H., & Zhang, Z.G. (2010). Recent developments in vacation queueing models: A short survey. International Journal of Operations Research 7(4): 3–8.Google Scholar
10.Kella, O. (1998). An exhaustive Lévy storage process with intermittent output. Stochastic Models 14(4): 979–992.CrossRefGoogle Scholar
11.Kella, O. & Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. Journal of Applied Probability 29(2): 396–403.CrossRefGoogle Scholar
12.Kyprianou, A.E (2006). Introductory lectures on fluctuations of Lévy processes with applications. Springer Science and Business Media.Google Scholar
13.Palmowski, Z. & Vlasiou, M. (2011). A Lévy input model with additional state-dependent services. Stochastic Processes and Their Applications 121(7): 1546–1564.CrossRefGoogle Scholar
14.Protter, P.E. (2004). Stochastic integration and differential equations, 2nd ed. Springer.Google Scholar
15.Tian, N. & Zhang, Z.G (2006). Vacation queueing models: theory and applications, vol. 93. Springer Science & Business Media.Google Scholar