In this paper we consider a physical model in which a buffer receives messages from a finite number of statistically independent and identical information sources that asynchronously alternate between exponentially distributed periods in the ‘on’ and ‘off’ states. While on, a source transmits at a uniform rate. The buffer depletes through an output channel with a given maximum rate of transmission. This model is useful for a data‐handling switch in a computer network. The equilibrium buffer distribution is described by a set of differential equations, which are analyzed herein. The mathematical results render trivial the computation of the distribution and its moments and thus also the waiting time moments. The main result explicitly gives all the system's eigenvalues. While the insertion of boundary conditions requires the solution of a matrix equation, even this step is eliminated since the matrix inverse is given in closed form. Finally, the simple expression given here for the asymptotic behavior of buffer content is insightful, for purposes of design, and numerically useful. Numerical results for a broad range of system parameters are presented graphically.

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This paper analyzes, derives efficient computational procedures and numerically investigates the following fluid model which is of interest in manufacturing and communications: m producing machines supply a buffer, n consuming machines feed off it. Each machine independently alternates between exponentially distributed random periods in the ‘in service' and ‘failed' states. Producers/consumers have their own failure/repair rates and working capacities. When the buffer is either full or empty some of the machines in service are not utilized to capacity; otherwise they are fully utilized. Our main result is for the state distribution of the Markovian system in equilibrium which is the solution of a system of differential equations. The spectral expansion for its solution is obtained. Two important decompositions are obtained: the eigenvectors have the Kronecker-product form in lower-dimensional vectors; the characteristic polynomial is factored with each factor an explicitly given polynomial of degree at most 4. All eigenvalues are real. For each of various cases of the model, a system of linear equations is derived from the boundary conditions; their solution complete the spectral expansion. The count in operations of the entire procedure is O(m 3 n 3): independence from buffer size exemplifies an important attraction of fluid models. Computations have revealed several interesting features, such as the benefit of small machines and the inelasticity of production rate to inventory. We also give results on the eigenvalues of a more general fluid model, reversible Markov drift processes.

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