  In this paper we present a stability criterion for finite measure-valued
stochastic recursions, generalizing Loynes's Theorem to spaces of measures.
This result provides conditions for the reach of a "total stationary state" for
the queue with an infinity of servers and the single-server SRPT queue. Indeed,
we give in both cases a condition of existence of a stationary measure-valued
recursive sequence characterizing the queueing system exhaustively.
