Stochastic theory of a fluid model of producers and consumers coupled by a buffer

Abstract

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.

Simple models of network access, with applications to the design of joint rate and admission control

“…As Q (r) is a generator, it has an eigenvalue 0, and hence one of the eigenvalues z (r) j is zero, say z (r) j * = 0, cf. [21]. With this in mind integration immediately yields that G (r) (x) equals…”

“…There are N + 1 such eigenvalues [21]. Notice that this also entails that the equilibrium distribution of Y (t) is given by w.…”

“…The fraction of fluid that is lost can be found as the ratio of the fluid loss rate (21) Maximization of system throughput. As a final application we show how the threshold level B * and the number of sources N may be jointly chosen such that the system throughput is maximized, with the fraction of lost fluid not to exceed = 10 −6 .…”

Performance of Telecommunication Systems: Selected Topics

“…2, which represents the overflow probability curve corresponding to the fluid-flow counterpart of the original traffic and which matches quite satisfactorily with the exact result over the burst level region. For further information on (primarily Markovian) fluid-flow models see: [1,40,43,49] for the basic theoretical foundation and analysis techniques, [3,37,42] for embelishments of the theory and efficient computational algorithms, and [38,44] for multiple-scale phenomena occurring when the traffic possesses burstlevel dynamics with a finer structure.…”

Exhaustive fluid vacation model with positive fluid rate during service

“…With the evolution of fluid queueing models [6,7,8,9] and their use in applied modeling [10] the question of the fluid counterpart of vacation and polling models has arisen. A first step toward this direction is by Czerniak and Yechiali [11] whose model is rather limited with respect to the stochastic evolution of the considered process: the load and the fluid service rate of stations are constant and the only stochastic ingredient of the model is the switchover time.…”