On Fluid Approximations for Conservative Networks

D.D. Botvich, A.A. Zamyatin

1995, v.1, №1, 113-140

ABSTRACT

In this paper we study the fluid dynamics of stable work-conserving networks. On the fluid level these networks behave in a similar manner to Jackson networks with Poisson arrivals and exponential service times. We develop the analysis of the fluid dynamics in [H. Chen and A. Mandelbaum, Discrete flow networks: bottleneck analysis and fluid approximations, Math. Oper. Res., 1991, v. 16, 408-446] in more detail. This paper describes the fluid dynamics in terms of the oblique reflection mapping. We use another approach and describe the dynamics in terms of the so-called second vector field in an orthant $R^N_+$, where $N$ is the number of nodes in the network [V.A. Malyshev, Networks and dynamical systems, Adv. Appl. Prob., 1993, v. 25, 140-175; V.A. Malyshev and M.V. Menshikov, Ergodicity, continuity and analyticity of countable Markov chains, Trans. Moscow Math. Soc., 1981, v. 1, 1-48]. Our goal here is to construct the fluid dynamics in explicit form and to provide an efficient algorithm for its calculation. In particular, we calculate the exact time it takes for all non-bottlenecks to empty or, for an ergodic network, the time for the system to empty. Other properties of the dynamics are also studied. We prove that the fluid dynamics of each stable work-conserving network is strictly acyclic in the sense of [V.A. Malyshev, Networks and dynamical systems, Adv. Appl. Prob., 1993, v. 25, 140-175].

Keywords: stable work-conserving networks,Jackson networks,fluid approximation,Euler queue length-time scaling limit,oblique reflection mapping,bottlneck and non-bottlneck nodes,random walks,second vector field,induced chains,ergodic and non-ergodic faces,induced vectors