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Infinite Networks with Varying Topology - A Mean-Field Approach

Presented by: 
AN Rybko Russian Academy of Sciences
Monday 12th August 2013 - 14:30 to 15:15
INI Seminar Room 1
We consider a model of a queuing network, containing infinite moving servers (nodes). The customers of different types are getting into the network in corresponding entrance nodes and each of the customer c has an exit node D(c) which it needs to reach. On the way to its destination the customer visits some intermediate nodes, where it stays in queues. Service times distributions depend on the customer and on the node types. After being served at any node v, the customer c is sent to the server v' which is the closest to the destination server D(c). Once it gets to D(c), it leaves the network. The main feature of the network is that its nodes are moving. So while the customer c is waiting in the queue in some node v, this node and its destination node D(c) can move. In such networks with moving nodes new effects take place, which are encountered in the situation with stationary nodes.

My talk is based on joint paper with Francois Baccelli and Senya Shlosman. Its main result can be described as follows: we start with the definition of the class of the networks with jumping nodes. The network can be finite or infinitely extended. In order to be able to treat them we consider the mean-field version of it, which consists of N copies of the network, interconnected in a mean-field manner. We show that as N increases, the limiting object becomes Non-Linear Markov Process in time. We establish the existence of the NLMP and the convergence to it.

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Presentation Material: 
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons