This paper considers models of large-scale networks where nodes are characterized by a set of states describing their local behavior , and by an explicit mobility model over a two-dimensional lattice. A stochastic model is given as a Markov population process that is, in general, infeasible to analyze due to the massive state space sizes involved. Building on recent results on fluid approximation, such a process admits a limit behavior as a system of ordinary differential equations , whose size is, unfortunately, dependent on the number of points in the lattice . Assuming an unbiased random walk model of nodes mobility, we prove convergence of the stochastic process to the solution of a system of partial differential equations of reaction-diffusion type . This provides a macroscopic view of the model which becomes independent of the lattice granularity, by approximating inherently discrete stochastic movements with continuous , deterministic diffusions. We illustrate the practical applicability of this result by modeling a network of mobile nodes with on/off behavior performing file transfers with connectivity to 802.11 access points. A numerical validation shows high quality of the  approximation even for low populations and coarse lattices , and excellent speed of convergence with increasing system sizes .
We consider Markov models of large-scale networks where nodes are characterized by their local behavior and by a mobility model over a two-dimensional lattice. By assuming random walk, we prove convergence to a system of partial differential equations (PDEs) whose size depends neither on the lattice size nor on the population of nodes . This provides a macroscopic view of the model which approximates discrete stochastic movements with continuous deterministic diffusions. We illustrate the practical applicability of this result by modeling a network of mobile nodes with on/off behavior performing file transfers with connectivity to 802.11 access points. By means of an empirical validation against discrete-event simulation we show high quality of the PDE approximation even for low populations and coarse lattices . In addition, we confirm the computational advantage in using the PDE limit over a traditional ordinary differential equation limit where the lattice is modeled discretely, yielding speed-ups of up to two orders of magnitude .
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A stochastic model is given as a Markov population process that is, in general, infeasible to analyze due to the massive state space sizes involved. Building on recent results on fluid approximation, such a process admits a limit behavior as a system of ordinary differential equations , whose size is, unfortunately, dependent on the number of points in the lattice . Assuming an unbiased random walk model of nodes mobility, we prove convergence of the stochastic process to the solution of a system of partial differential equations of reaction-diffusion type
By assuming random walk, we prove convergence to a system of partial differential equations (PDEs) whose size depends neither on the lattice size nor on the population of nodes
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By means of an empirical validation against discrete-event simulation we show
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