The study of simple solar energy storage models leads to the question of analyzing the equilibrium distribution of Markov chains (Harris chains), for which the state at epoch (n + 1) (i.e. the temperature of the storage tank) depends on the state at epoch n and on a controlled input, acceptance of which entails a further decrease of the temperature level. Here we study the model where the input is exponentially distributed. For all values of the parameters involved an explicit expression for the equilibrium distribution of the Markov chain is derived, and from this we calculate, as one of the possible applications, the exact values of the mean of this equilibrium distribution.

The stochastic process {Xn} satisfying Xn+1 = max{Yn+1+ αβ Xn, βXn} where {Yn} is a stationary sequence of non-negative random variables and , 0<β <1, can be regarded as a simple thermal energy storage model with controlled input. Attention is mostly confined to the study of μ = EX where the random variable X has the stationary distribution for {Xn}. Even for special cases such as i.i.d. Yn or α = 0, little explicit information appears to be available on the distribution of X or μ . Accordingly, bounding techniques that have been exploited in queueing theory are used to study μ . The various bounds are illustrated numerically in a range of special cases.