On Polling Systems where Servers Wait for Customers

#### L.G. Afanassieva, F. Delcoigne, G. Fayolle

1997, v.3, №4, 527-545

ABSTRACT

In this paper, a particular polling system with $N$ queues and $V$ servers is analyzed. Whenever a server visits an empty queue, it waits for the next customer to come to this queue. A customer chooses his destination according to a routing matrix $P$. The model originates from specific problems arising in transportation networks. A global classification of the process describing the system is given under broad assumptions. It is shown that only transience or null recurrence can take place. A detailed classification of each node, together with limit laws after proper time-scaling, are obtained. The method of analysis relies on the Central Limit Theorem and a coupling with a reference system in which transportation times are identically zero. It is shown that a tight time-varying injection of servers can empty the system. Finally a system where servers may become impatient is briefly considered. Ergodicity conditions are given, showing that the system can stabilize, contrary to the case without impatience.

Keywords: network,polling,random walk,recurrence,transience,Central Limit Theorem