Skip to main content Accessibility help
×
Home
Hostname: page-component-559fc8cf4f-s65px Total loading time: 0.959 Render date: 2021-03-08T01:20:25.570Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": false, "newCiteModal": false, "newCitedByModal": true }

A queueing/inventory and an insurance risk model

Published online by Cambridge University Press:  11 January 2017

Onno Boxma
Affiliation:
Eindhoven University of Technology
Rim Essifi
Affiliation:
Eindhoven University of Technology
Augustus J. E. M. Janssen
Affiliation:
Eindhoven University of Technology
Corresponding
E-mail address:

Abstract

We study an M/G/1-type queueing model with the following additional feature. The server works continuously, at fixed speed, even if there are no service requirements. In the latter case, it is building up inventory, which can be interpreted as negative workload. At random times, with an intensity ω(x) when the inventory is at level x>0, the present inventory is removed, instantaneously reducing the inventory to 0. We study the steady-state distribution of the (positive and negative) workload levels for the cases ω(x) is constant and ω(x) = a x. The key tool is the Wiener–Hopf factorization technique. When ω(x) is constant, no specific assumptions will be made on the service requirement distribution. However, in the linear case, we need some algebraic hypotheses concerning the Laplace–Stieltjes transform of the service requirement distribution. Throughout the paper, we also study a closely related model arising from insurance risk theory.

Type
Research Article
Copyright
Copyright © Applied Probability Trust 2017 

Access options

Get access to the full version of this content by using one of the access options below.

References

[1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. U.S. Government Printing Office, Washington, DC.Google Scholar
[2] Albrecher, H. and Ivanovs, J. (2015). Strikingly simple identities relating exit problems for Lévy processes under continuous and Poisson observations. Preprint. Available at http://arxiv.org/abs/1507.03848.Google Scholar
[3] Albrecher, H. and Lautscham, V. (2013). From ruin to bankruptcy for compound Poisson surplus processes. ASTIN Bull. 43, 213243.CrossRefGoogle Scholar
[4] Albrecher, H., Gerber, H. U. and Shiu, E. S. W. (2011). The optimal dividend barrier in the gamma–omega model. Europ. Actuarial J. 1, 4355.CrossRefGoogle Scholar
[5] Albrecher, H., Boxma, O. J., Essifi, R. and Kuijstermans, R. A. C. (2016). A queueing model with randomized depletion of inventory. To appear in em Probab. Eng. Inform. Sc. Google Scholar
[6] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities, 2nd edn. World Scientific, Hackensack, NJ.CrossRefGoogle Scholar
[7] Berman, O., Parlar, M., Perry, D. and Posner, M. J. (2005). Production/clearing models under continuous and sporadic reviews. Methodol. Comput. Appl. Prob. 7, 203224.CrossRefGoogle Scholar
[8] Boxma, O., Essifi, R. and Janssen, A. J. E. M. (2015). A queueing/inventory and an insurance risk model. Preprint. Available at http://arxiv.org/abs/1510.07610.Google Scholar
[9] Brill, P. H. (2008). Level Crossing Methods in Stochastic Models. Springer, New York.CrossRefGoogle Scholar
[10] Cohen, J. W. (1975). The Wiener–Hopf technique in applied probability. In Perspectives in Probability and Statistics, Applied Probability Trust, Sheffield, pp.145156.Google Scholar
[11] Cohen, J. W. (YEAR). The Single Server Queue, 2nd edn. North-Holland, Amsterdam.Google Scholar
[12] Kashyap, B. R. K. (1966). The double-ended queue with bulk service and limited waiting space. Operat. Res. 14, 822834.CrossRefGoogle Scholar
[13] Liu, X., Gong, Q. and Kulkarni, V. G. (2015). Diffusion models for double-ended queues with renewal arrival processes. Stoch. Systems 5, 161.CrossRefGoogle Scholar
[14] Olver, F. W., Lozier, D. W., Boisvert, R. F. and Clark, C. W. (2010). The NIST Handbook of Mathematical Functions. Cambridge University Press.Google Scholar
[15] Perry, D., Stadje, W. and Zacks, S. (2005). Sporadic and continuous clearing policies for a production/inventory system under an M/G demand process. Math. Operat. Res. 30, 354368.CrossRefGoogle Scholar
[16] Polyanin, A. D. and Zaitsev, V. F. (2003). Handbook of Exact Solutions for Ordinary Differential Equations, 2nd edn. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[17] Srivastava, H. M. and Kashyap, B. R. K. (1982). Special Functions in Queuing Theory and Related Stochastic Processes. Academic Press, New York.Google Scholar
[18] Titchmarsh, E. C. (1939). Theory of Functions, 2nd edn. Oxford University Press.Google Scholar
[19] Welch, P. D. (1964). On a generalized M/G/1 queuing process in which the first customer of each busy period receives exceptional service. Operat. Res. 12, 736752.CrossRefGoogle Scholar
[20] Wolff, R. W. (1989). Stochastic Modeling and the Theory of Queues. Prentice Hall, Englewood Cliffs, NJ.Google Scholar

Full text views

Full text views reflects PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 94 *
View data table for this chart

* Views captured on Cambridge Core between 11th January 2017 - 8th March 2021. This data will be updated every 24 hours.

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

A queueing/inventory and an insurance risk model
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

A queueing/inventory and an insurance risk model
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

A queueing/inventory and an insurance risk model
Available formats
×
×

Reply to: Submit a response


Your details


Conflicting interests

Do you have any conflicting interests? *