Cyclic-type polling models with preparation times

We consider a system consisting of a server serving in sequence a fixed number of stations. At each station there is an infinite queue of customers that have to undergo a preparation phase before being served. This model is connected to layered queuing networks, to an extension of polling systems, and surprisingly to random graphs. We are interested in the waiting time of the server. The waiting time of the server satisfies a Lindley-type equation of a non-standard form. We give a sufficient condition for the existence of a limiting waiting time distribution in the general case, and assuming preparation times are exponentially distributed, we describe in depth the resulting Markov chain. We provide detailed computations for a special case and extensive numerical results investigating the effect of the system's parameters to the performance of the server.