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Weighted sums defined on a Markov chain (MC) are important in applications (e.g. to reservoir storage theory). The rather intractable theory of such sums simplifies to some extent when the transition p.d.f. of the chain {Xt } has a Laplace transform (LT) L(Xt +1; θ |Χ t=x) of the ‘exponential' form H(θ) exp{ – G(θ)x}. An algorithm is derived for the computation of the LT of Σat,Χ t for this class, and for a seasonal generalization of it. A special case of this desirable exponential type of transition LT for a continuous-state discrete-time MC is identified by comparison with the LT of the Bessel distribution. This is made the basis for a new derivation of a gamma-distributed MC proposed by Lampard (1968). A seasonal version of this process is developed, valid for any number of seasons. Reference is made to related chains with three-parameter gamma-like distributions (of the Kritskii–Menkel family) that may be generated from the above by a simple power transformation.

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The purpose of the work reported here is to derive the outflow distribution from a linear reservoir fed by a discrete‐time gamma‐distributed Markovian inflow. A satisfactory continuous time solution is available for the linear reservoir when the inflow distribution has independent increments. For the case of nonindependent increments discussed in the paper the treatment is in terms of a gamma‐distributed Markov chain inflow process in discrete time, for which the Laplace transform of the outflow rate distribution is derived. This is inverted, numerically, for a set of typical parameter values.