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Ergodicity of age-dependent inventory control systems

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  24 October 2016

Fredrik Olsson  and
Tatyana S. Turova
Fredrik Olsson*
Affiliation:
Lund University
Tatyana S. Turova*
Affiliation:
Lund University
*
*Postal address: Department of Industrial Management and Logistics,Lund University, Box 118, SE-221 00 Lund, Sweden. Email address: [email protected]
**Postal address: Centre for Mathematical Sciences, Lund University, Box 118, SE-221 00 Lund, Sweden. Email address: [email protected]
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Abstract

We consider continuous review inventory systems with general doubly stochastic Poisson demand. In this specific case the demand rate, experienced by the system, varies as a function of the age of the oldest unit in the system. It is known that the stationary distributions of the ages in such models often have a strikingly simple form. In particular, they exhibit a typical feature of a Poisson process: given the age of the oldest unit the remaining ages are uniform. The model we treat here generalizes some known inventory models dealing with partial backorders, perishable items, and emergency replenishment. We derive the limiting joint density of the ages of the units in the system by solving partial differential equations. We also answer the question of the uniqueness of the stationary distributions which was not addressed in the related literature.

Keywords

Inventorybase-stock policyergodicitydoubly stochastic Poisson process

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 60K25: Queueing theory 90B05: Inventory, storage, reservoirs
Type
Research Papers
Information
Journal of Applied Probability , Volume 53 , Issue 3 , September 2016 , pp. 688 - 699
Copyright
Copyright © Applied Probability Trust 2016 

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References

[1] Asmussen, S. (2003).Applied Probability and Queues, 2nd edn.Springer,New York.Google Scholar
[2] Cox, D. R. (1955).The analysis of non-Markovian stochastic processes by the inclusion of supplementary variables.Proc. Camb. Philos. Soc. 51,433–441.CrossRefGoogle Scholar
[3] Decreusefond, L. and Moyal, P. (2008).A functional central limit theorem for the M/GI/∞ queue..Ann. Appl. Prob. 18,2156–2178.CrossRefGoogle Scholar
[4] Decreusefond, L. and Moyal, P. (2009).Fluid limit of a heavily loaded EDF queue with impatient customers.Markov Process. Relat. Fields 14,131–158.Google Scholar
[5] Gaukler, G. M. and Seifert, R. W. (2007).Item-level RFID in the retail supply chain..Production Operat. Manag. 16,65–76.CrossRefGoogle Scholar
[6] Gnedenko, B. V. and Kovalenko, I. N. (1968).Introduction to Queueing Theory,Israel Program for Scientific Translations,Jerusalem.Google Scholar
[7] Kallenberg, O. (2002).Foundations of Modern Probability, 2nd edn.Springer,New York.CrossRefGoogle Scholar
[8] Kaspi, H. and Perry, D. (1984).Inventory systems for perishable commodities with renewal input and Poisson output.Adv. Appl. Prob. 16,402–421.CrossRefGoogle Scholar
[9] Kingman, J. F. C. (1964).On the doubly stochastic Poisson processes.Proc. Camb. Philos. Soc. 60,923–930.CrossRefGoogle Scholar
[10] Kingman, J. F. C. (2009).The first Erlang century—and the next.Queueing Systems 63,3–12.CrossRefGoogle Scholar
[11] Moinzadeh, K. (1989).Operating characteristics of the (S−1,S) inventory system with partial backorders and constant resupply times.Manag. Sci. 35,472–477.CrossRefGoogle Scholar
[12] Moinzadeh, K. and Schmidt, C. P. (1991).An (S−1,S) inventory system with emergency orders.Operat. Res. 39,308–321.CrossRefGoogle Scholar
[13] Nahmias, S.,Perry, D. and Stadje, W. (2004).Perishable inventory systems with variable input and demand rates.Math. Meth. Operat. Res. 60,155–162.CrossRefGoogle Scholar
[14] Perry, D. (1999).Analysis of a sampling control scheme for a perishable inventory system.Operat. Res. 47,966–973.CrossRefGoogle Scholar
[15] Perry, D. and Posner, M. J. M. (1998).An (S−1, S) inventory system with fixed shelf life and constant lead times..Operat. Res. 46,S65–S71.CrossRefGoogle Scholar
[16] Perry, D. and Stadje, W. (2001).Disasters in a Markovian inventory system for perishable items.Adv. Appl. Prob. 33,61–75.CrossRefGoogle Scholar
[17] Rolski, T.,Serfozo, R. and Stoyan, D. (2015).Service-time ages, residuals, and lengths in an M/GI/∞ service system.Queueing Systems 79,173–181.CrossRefGoogle Scholar
[18] Schmidt, C. P. and Nahmias, S. (1985).(S−1,S) policies for perishable inventory.Manag. Sci. 31,719–728.CrossRefGoogle Scholar
[19] Shiryaev, A. N. (1996).Probability, 2nd edn.Springer,New York.CrossRefGoogle Scholar
[20] Takács, L. (1962).Introduction to the Theory of Queues}.,Oxford University Press.Google Scholar
[21] Zhang, G. P.,Patuwo, B. E. and Chu, C.-W. (2003).A hybrid inventory system with a time limit on backorders.IIE Trans. 35,679–687.CrossRefGoogle Scholar