| Queuing networks occur in manufacturing systems, traffic flows and communication systems. Detailed methods exist to analyze the behavior of such networks. These networks can also be approximated by diffusion processes which are generally easier to handle than the original network itself. Our challenge is to describe classes of queuing networks for which non-linear diffusion approximations can be identified. We will look at some simple queuing networks with infinitely many servers under various service disciplines interpreting them as interacting particle systems following specified protocols of movement of particles. Then the network can eventually be described by an infinite system of stochastic differential equations. In the hyperbolic scaling1, this system of stochastic differential equations leads to non-linear partial differential equations which usually cannot be solved explicitly. To get a hold on the behavior of these approximations and thereby that of the original network, we propose to use Monte-Carlo and/or Fourier transform methods to approximate the solutions of such nonlinear partial differential equations. We explore solutions via Fourier methods and direct Monte Carlo simulation. In addition, a Monte Carlo approximation via interacting diffusion scheme has also been proposed. Using numerical descriptors and profile densities, we characterize and compare the properties of the solutions for special/general initial data in the case of certain networks. The networks that we consider are as follows: (1) Queuing networks with infinitely many servers in series 2; (2) Gossiping Secretaries queuing network with servers in series3; (3) A multi-server queue with state dependent rates3; (4) Procrastination queuing network with servers in series.; The last model has not been considered in the literature. The implementation of the Monte Carlo method using interacting diffusions for the last three networks is also new. Papers based on this work are in preparation.; 1Wojbor A. Woyczynski, Burgers-KPZ Turbulence---Gottingen Lectures, Lecture Notes in Math. 1700, Springer-Verlag, 1998. 2Srinivasan R., Queues in series via interacting particle systems, Math. of Operations Research, 18 (1993), 39--50. 3Margolius B. and Woyczynski, W. A. Nonlinear Diffusion Approximations of Queuing Networks, Stochastics in Finite and Infinite Dimensions, T. Hida et al Eds., Birkhauser 2000, 259--284. |
