0 A signal - recovery system : asymptotic properties , and construction of an infinite - volume limit

We consider a linear sequence of 'nodes', each of which can be in state 0 ('off') or 1 ('on'). Signals from outside are sent to the rightmost node and travel instantaneously as far as possible to the left along nodes which are 'on'. These nodes are immediately switched off, and become on again after a recovery time. The recovery times are independent exponentially distributed random variables. We present properties for finite systems and use some of these properties to construct an infinite-volume extension, with signals 'coming from infinity'. This construction is related to a question by D. Al-dous and we expect that it sheds some light on, and stimulates further investigation of, that question.