On the integral equation of the continuous-state stochastic reservoir with seasonally varying autocorrelated inflows |
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| Authors: | E H Lloyd |
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| Institution: | Redbank House, Yealand Redmayne, Carnforth LA5 9TA, UK |
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| Abstract: | The paper deals with continuous-state reservoirs in discrete time, in terms of the (continuous) Moran transform Ut = Zt + Xt of the (discontinuous) storage levels Zt. In the non-seasonal case, if the inflows Xt have a first-order Markov structure, the modified storage levels Ut have a second-order Markov structure. It is shown that, for each value of u, the limit p(u,v) as t → ∞ of the joint density pt + 1 (u, v) of Ut+ 1 at u and Ut at v satisfies a certain non-standard first-order linear integral equation of the form or, equivalently, a standard second-order equation of the form In the seasonal case, with two seasons, the corresponding bivariate density p(u,v) for consecutive seasons satisfies a second-order integral equation of the form the kernel being a function of the season-to-season inflow transition densities. |
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