Stationarity of measure-valued stochastic recursions: Applications to the pure delay system and the SRPT queue (Q2381594)

Let \({\mathbf{M}}_f^+\) and \(C_b\) denote, respectively, the set of positive finite measures on \({\mathcal R}_+^ *\) and the set of bounded continuous functions from \({\mathcal R}\) to \({\mathcal R}\). Let \(C({\mathbf{M}}_f^+)\) be the space of continuous mappings from \({\mathbf{M}}_f^+\) into itself. The set \({\mathbf{M}}_f^+\) is endowed with the increasing partial order \(\prec\): for any two \(\mu\), \(\nu \in {\mathbf{M}}_f^+\), \(\mu \prec \nu\) if \(\mu ([a,\infty)) \leqslant \nu ([a,\infty))\) for any \(a \geqslant 0\). Let now \((\Omega ,F,{\mathbf{P}}^0 ,\theta)\) be a probability space furnished with a discrete bijective flow \(\theta\), under which \({\mathbf{P}}^0\) is stationary and ergodic. Let \(\Phi\) be a \(C({\mathbf{M}}_f^+)\)-valued random variable, \(\kappa \) be an \({\mathbf{M}}_f^+\)-valued random variable, and define the sequence of \({\mathbf{M}}_f^+\)-valued random variables: \(\mu_0^\kappa = \kappa\), \(\mu_n^\kappa = \Phi \circ \theta ^n (\mu_n^\kappa)\). The existence of a stationary version of \(\{ \mu_n^\kappa ,n \geqslant 0\}\) amounts to that of a solution of the following eqation: \(\kappa \circ \theta = \Phi (\kappa)\). Theorem 1.1 states that this equation admits a solution whenever \(\Phi\) is a.s. \(\prec\)-non-decreasing and the sequence \(\{ \mu_n^\varsigma \circ \theta ^{-n} ,n \geqslant 0\}\) is a.s. \(\prec\)-bounded (here \(\varsigma\) denotes the zero measure). This result is applied to \(G/G/\infty\) and the single-server SRPT queues.