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Disasters in a Markovian inventory system for perishable items

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  01 July 2016

David Perry  and
Wolfgang Stadje
David Perry*
Affiliation:
University of Haifa
Wolfgang Stadje*
Affiliation:
Universität Osnabrück
*
∗ Postal address: Department of Statistics, University of Haifa, 31905 Haifa, Israel.
∗∗ Postal address: Universität Osnabrück, Fachbereich Mathematik/Informatik, 496069 Osnabrück, Germany. Email address: [email protected]
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Abstract

We study a Markovian model for a perishable inventory system with random input and an external source of obsolescence: at Poisson random times the whole current content of the system is spoilt and must be scrapped. The system can be described by its virtual death time process. We derive its stationary distribution in closed form and find an explicit formula for the Laplace transform of the cycle length, defined as the time between two consecutive item arrivals in an empty system. The results are used to compute several cost functionals. We also derive these functionals under the corresponding heavy traffic approximation, which is modeled using a Brownian motion in [0,1] reflected at 0 and 1 and restarted at 1 at the Poisson disaster times.

Keywords

inventory systemperishable itemsvirtual death processstationary distributioncost functionalsreflected Brownian motion

MSC classification

Primary: 90B05: Inventory, storage, reservoirs
Secondary: 60K30: Applications (congestion, allocation, storage, traffic, etc.) 60K25: Queueing theory
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 33 , Issue 1 , March 2001 , pp. 61 - 75
Copyright
Copyright © Applied Probability Trust 2001 

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References

Cohen, J. W. (1977). On up- and downcrossings. J. Appl. Prob. 14, 405–410.CrossRefGoogle Scholar
Doshi, B. (1992). Level-crossing analysis of queues. In Queueing and Related Models (Oxford Statist. Sci. Ser. 9), eds Bhat, U. N. and Basawa, I. V.. Oxford University Press, pp. 3–33.Google Scholar
Jain, G. and Sigman, K. (1996). A Pollaczek–Khintchine formula for M/G/1 queues with disasters. J. Appl. Prob. 33, 1191–1200.CrossRefGoogle Scholar
Kalpakam, S. and Sapna, K. P. (1994). Continuous review (s,S) inventory system with random lifetimes and positive lead-times. Operat. Res. Lett. 16, 115–119.CrossRefGoogle Scholar
Kalpakam, S. and Sapna, K. P. (1996a). A lost sales (S-1,S) perishable inventory system with renewal demand. Naval Res. Logist. 43, 129–142.3.0.CO;2-D>CrossRefGoogle Scholar
Kalpakam, S. and Sapna, K. P. (1996b). An (s,S) perishable system with arbitrarily distributed leadtimes. Opsearch 33, 1–19.Google Scholar
Kella, O. and Whitt, W. (1992). Useful martingales for stochastic storage processes with Lévy input. J. Appl. Prob. 29, 396–403.CrossRefGoogle Scholar
Kim, K. and Seila, A. F. (1993). A generalized cost model for stochastic clearing systems. Comput. Operat. Res. 20, 67–82.CrossRefGoogle Scholar
Liu, L. (1990). Continuous review models for inventory with random lifetimes. Operat. Res. Lett. 9, 161–167.CrossRefGoogle Scholar
Newell, G. F. (1982). Applications of Queueing Theory, 2nd edn. Chapman and Hall, London.CrossRefGoogle Scholar
Perry, D. (1999). Analysis of sampling control scheme for a perishable inventory system. Operat. Res. 47, 966–973.CrossRefGoogle Scholar
Perry, D. and Asmussen, S. (1995). Rejection rules in the M/G/1 type queue. Queueing Systems 19, 105–130.CrossRefGoogle Scholar
Perry, D. and Stadje, W. (1999a). Perishable inventory systems with impatient demands. Math. Meth. Operat. Res. 50, 77–90.CrossRefGoogle Scholar
Perry, D. and Stadje, W. (1999b). Heavy traffic analysis for a queueing system with bounded capacity for two types of customers. J. Appl. Prob. 36, 1155–1166.CrossRefGoogle Scholar
Ravichandran, N. (1995). Stochastic analysis of a continuous review perishable inventory system with positive lead time and Poisson demand. Eur. J. Operat. Res. 84, 444–457.CrossRefGoogle Scholar
Serfozo, R. and Stidham, S. (1978). Semi-stationary clearing processes. Stoch. Proc. Appl. 6, 165–178.CrossRefGoogle Scholar
Stidham, S. (1974). Stochastic clearing systems. Stoch. Proc. Appl. 2, 85–113.CrossRefGoogle Scholar
Stidham, S. (1977). Cost models for stochastic clearing systems. Operat. Res. 25, 100–127.CrossRefGoogle Scholar
Stidham, S. (1986). Clearing systems and (s,S) inventory systems with nonlinear costs and positive lead times. Operat. Res. 34, 276–280.CrossRefGoogle Scholar
Whitt, W. (1974). Heavy traffic limit theorems for queues: a survey. In Mathematical Methods in Queueing Theory (Lecture Notes Econ. Math. Systems 98), ed. Clarke, A. B.. Springer, Berlin, pp. 307–350.CrossRefGoogle Scholar