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Time-dependent results in storage theory

Published online by Cambridge University Press:  14 July 2016

N. U. Prabhu
N. U. Prabhu*
Affiliation:
The University of Western Australia
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Extract

The probability theory of storage systems formulated by P. A. P. Moran in 1954 has now developed into an active branch of applied probability. An excellent account of the theory, describing results obtained up to 1958 is contained in Moran's (1959) monograph, Considerable progress has since been made in several directions-the study ofthe time-dependent behaviour ofstochastic processes underlying Moran's original model, modifications of this model, as well as the formulation and solution of new models. The aim of this paper is to give an expository account of these developments; a comprehensive treatment will be found in the author's forthcoming book [Prabhu (1964)].

Type
Review Paper
Information
Journal of Applied Probability , Volume 1 , Issue 1 , June 1964 , pp. 1 - 46
Copyright
Copyright © Applied Probability Trust 1964 

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References

Baxter, Glen and Donsker, M.D. (1957) On the distribution of the supremum functional for processes with stationary and independent increments. Trans. Amer. Math. Soc. 85, 99–124.Google Scholar
Doob, J. L. (1953) Stochastic Processes. John Wiley, New York.Google Scholar
Dvoretzky, A., Kiefer, J. and Wolfowitz, J. (1953) Sequential decision problems for processes with continuous time parameter: Testing hypotheses. Ann. Math. Statist. 24, 254–264.Google Scholar
Kemperman, J. H. B. (1961) The First Passage Problem for a Stationary Markov Chain. University of Chicago Press.Google Scholar
Levy, P. (1948) Processus Stochastiques et Mouvement Brownien. Gauthier Villars, Paris.Google Scholar
Reich, Edgar (1961) Some combinatorial theorems for continuous time parameter processes. Math. Scand. 2, 243–257.CrossRefGoogle Scholar
Spitzer, Frank (1956) A combinatorial lemma and its applications to probability theory. Trans. Amer. Math. Soc. 82, 323–339. b) Storage theory and related topics Google Scholar
Arrow, W. J., Karlin, S. and Scarf, H. (1958) Studies in the Mathematical Theory of Inventory and Production. Stanford University Press.Google Scholar
Avi-Itzhak, B. and Ben-Tuvia, S. (1963) A problem of optimizing a collecting reservoir system. Operat. Res. 11, 122–136.Google Scholar
Bather, J. A. (1962) Optimal regulation policies for finite dams. J. Soc. Indust. Appl. Math. 10, 395–423.Google Scholar
Bather, J. A. (1963) The optimal regulation of dams in continuous time. J. Soc. Indust. Appl. Math. 11, 33–63.Google Scholar
Beneş, V. E. (1957) On queues with Poisson arrivals. Ann. Math. Statist. 28, 670–677.Google Scholar
Bather, J. A. (1961) Theory of queues with one server. Trans. Amer. Math. Soc. 94, 282–294.Google Scholar
Bhat, B. R. and Gani, J. (1959) On the independence of yearly inputs in dams. ONR Report No. Non r-266 (259).Google Scholar
Downton, F. (1957) A note on Moran's theory of dams. Quart. J. Math. (2) 8, 282–286.Google Scholar
Gani, J. (1955) Some problems in the theory of provisioning and dams. Biometrika 42, 179–200.Google Scholar
Gani, J., Problems in the probability theory of storage systems. J. R. Statist. Soc. B 19, 181–206.Google Scholar
Gani, J., (1958) Elementary methods in an occupancy problem of storage. Math. Ann. 136, 454–465.CrossRefGoogle Scholar
Gani, J., (1961) First emptiness of two dams in parallel. Ann. Math. Statist. 32, 219–229.CrossRefGoogle Scholar
Gani, J. (1962 a) A stochastic dam process with non-homogeneous Pois, on inputs. Studia Math. 21, 307–315. (Corrigenda, 22 (1963) 371).Google Scholar
Gani, J., (1962 b) The time-dependent solution for a dam with ordered Poisson inputs. Studies in Applied Probability and Management Science (Stanford University Press).Google Scholar
Gani, J. and Moran, P. A. P. (1955) A solution of dam equations by Monte Carlo methods. Aust. J. Appl. Sci. 6, 267–273.Google Scholar
Gani, J. and Prabhu, N. U. (1957) Stationary distributions of the negative exponential type for the infinite dam. J. R. Statist. Soc. B 19, 342–351.Google Scholar
Gani, J. and Prabhu, N. U., (1958) Continuous time treatment of a storage problem. Nature 182, 39–40.Google Scholar
Gani, J. and Prabhu, N. U., (1959 a) Remarks on the dam with Poisson type inputs. Aust. J. Appl. Sci. 10. 113–122.Google Scholar
Gani, J. and Prabhu, N. U., (1959 b) The time-dependent solution for a storage model with Poisson input J. Math. and Mech. 8, 653–664.Google Scholar
Gani, J. and Prabhu, N. U., (1963) A storage model with continuous infinitely divisible inputs. Proc. Camb. Phil. Soc. 59, 417–429.Google Scholar
Gani, J. and Pyke, R. (1960 a) The content of a dam as the supremum of an infinitely divisible process. J. Math. and Mech. 2, 639–652.Google Scholar
Gani, J. and PYKE, R., (1960 b) Inequalities for first emptiness probabilities of a dam with ordered inputs. Tech. Report No. 1, National Sci. Found. Grant G-9670, Appl. Math. and Stat. Lab., Stanford University (see also J. R. Statist. Soc. B 24 (1962) 102–106).Google Scholar
Gaver, D. P. and Miller, R. G. Jr. (1962) Limiting distributions for some storage problems. Studies in Applied Probability and Management Science. Stanford University Press.Google Scholar
Ghosal, A. (1959) On the continuous analogue of Holdaway's problem for the finite dam. Aust. J. Appl. Sci. 10, 365–370.Google Scholar
Ghosal, A., (1960 a) Problem of emptiness in Holdaway's finite dam. Bull. Calcutta Statist. Ass. 2, 111–116.Google Scholar
Ghosal, A., (1960 b) Emptiness in the finite dam. Ann. Math. Statist. 31, 803–808.Google Scholar
Ghosal, A., (1962) Finite dam with negative binomial input. Aust. J. Appl. Sci. 13, 71–74.Google Scholar
Goldstein, S. (1951) On diffusion by discontinuous movements and on the telegraph equation. Quart. J. Mech. and Appl. Math. 4, 129–156.Google Scholar
Jarvis, C. L. (1963) An application of Moran's theory of dams to the Ord River Project. M.Sc. Thesis, The University of Western Australia.Google Scholar
Kendall, , (1957) Some problems in the theory of dams. J. R. Statist. Soc. B 19, 207–212.Google Scholar
Kingman, J. F. C. (1963) On continuous time models in the theory of dams. J. Aust. Math. Soc. 3, 480–487.Google Scholar
Langbein, W. B. (1958) Queueing theory and water storage. J. Hydrol. Div. 84, Paper 1811.Google Scholar
Langbein, W. B., (1961) Reservoir storage—general solution of a queueing model. Geol. Survey Res. Article 298.Google Scholar
Lindley, D. V. (1952) The theory of queues with a single server. Proc. Camb. Phil. Soc. 48 277–289.Google Scholar
Lloyd, E. H. (1963 a) Reservoirs with serially correlated inputs. Technometrics 5, 85–93.Google Scholar
Lloyd, E. H., (1963 b) The epochs of emptiness of a semi-infinite discrete reservoir. J. R. Statist. Soc. B 25, 131–136.Google Scholar
Loynes, R. M. (1962) The stability of a queue with non-independent inter-arrival and service times. Proc. Camb. Phil. Soc. 58, 497–520.Google Scholar
Miller, R. G. Jr. (1963) Continuous time stochastic storage processes with random linear inputs and outputs. J. Math. and Mech. 12, 275–291.Google Scholar
Moran, P. A. P. (1954) A probability theory of dams and storage systems. Aust. J. Appl. Sci. 5, 116–124.Google Scholar
Moran, P. A. P., (1955) A probability theory of dams and storage systems: modifications of the release rules. Aust. J. Appl. Sci. 6, 117–130.Google Scholar
Moran, P. A. P., (1956) A probability theory of dams with a continuous release. Quart. J. Math. (2) 7, 130–137.Google Scholar
Moran, P. A. P., (1957) The statistical treatment of flood flows. Trans. Amer. Geophys. Union 38, 519–523.Google Scholar
Moran, P. A. P., (1959) The Theory of Storage. Methuen, London.Google Scholar
Mott, J. L. (1963) The distribution of the time-to-emptiness of a discrete dam under steady demand. J. R. Statist. Soc. B 25, 137–139.Google Scholar
Phatarfod, R. M. (1963) Application of methods in sequential analysis to dam theory. Ann. Math. Statist. 34, 1588–1592.Google Scholar
Prabhu, N. U. (1958 a) Some exact results for the finite dam. Ann. Math. Statist. 29, 1234–1243.Google Scholar
Prabhu, N. U., (1958 b) On the integral equation for the finite dam. Quart. J. Math. (2) 2, 183–188.Google Scholar
Prabhu, N. U., (1959) Application of generating function to a problem in finite dam theory. J. Aust. Math. Soc. 1, 116–120.Google Scholar
Prabhu, N. U., (1960 a) A problem in optimum storage. Bull. Calcutta Statist. Ass. 10, 35–40.Google Scholar
Prabhu, N. U., (1960 b) Application of a storage theory to queues with Poisson arrivals. Ann. Math. Statist. 31, 475–482.Google Scholar
Prabhu, N. U., (1960 c) Some results for the queue with Poisson arrivals. J. R. Statist. Soc. B 22, 104–107.Google Scholar
Prabhu, N. U., (1961) On the ruin problem of collective risk theory. Ann. Math. Statist. 32, 757–764.Google Scholar
Prabhu, N. U., (1964) Queues and Inventories. Forthcoming.Google Scholar
Prabhu, N. U. and Bhat, , Narayan, U. (1963) Some first passage problems and their application to queues. Sankhya Series A 25, 281–292.Google Scholar
Reich, Edgar (1958) On the integro-differential equation of Takács I. Ann. Math. Statist. 29, 563–570.Google Scholar
Reich, Edgar, (1959) On the integro-differential equation of Takács II. Ann. Math. Statist. 30, 143–148.Google Scholar
Takács, Lajos (1955) Investigation of waiting time problems by reduction to Markov processes. Acta Math. Acad. Sci. Hung. 6, 101–129.CrossRefGoogle Scholar
Takács, Lajos, (1964) Combinatorial methods in the theory of dams. J. Appl. Prob. 1, 69–76.Google Scholar
Weesakul, B. (1961 a) The explicit time-dependent solution for a finite dam with geometric inputs. Aust. Math. Soc. Summer Research Institute Report (see also Ann. Math. Statist. 32, (1961) 765–769).Google Scholar
Weesakul, B. (1961 a) The explicit time-dependent solution for a finite dam with geometric inputs. Aust. Math. Soc. Summer Research Institute Report (see also Ann. Math. Statist. 32, (1961 b) First emptiness in a finite dam. J. R. Statist. Soc. B 23, 343–351.Google Scholar
Yeo, G. F. (1960) The time-dependent solution for a dam with geometric inputs. Aust. J. Appl. Sci. 11, 434–442.Google Scholar
Yeo, G. F., (1961 a) The time-dependent solution for an infinite dam with discrete additive inputs. J. R. Statist. Soc. B 23, 173–179.Google Scholar
Yeo, G. F., (1961 b) A discrete dam with ordered inputs. Aust. Math. Soc. Summer Research Institute Report.Google Scholar