The Reflection Map with Discontinuities

We study the multi-dimensional re ection map on the spaces D([0; T ];Rk ) and D([0;1);Rk ) of right-continuous Rk -valued functions on [0; T ] or [0;1) with left limits, endowed with variants of the Skorohod(1956) M1 topology. The re ection map was used with the continuous mapping theorem by Harrison and Reiman (1981) and Reiman (1984) to establish heavy-tra c limit theorems with re ected Brownian motion limit processes for vector-valued queue-length, waiting-time and workload stochastic processes in single-class open queueing networks. Since Brownian motion and re ected Brownian motion have continuous sample paths, the topology of uniform convergence over bounded intervals could be used for those results. Variants of the M1 topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes, as occurs in stochastic uid networks with bursty exogenous input processes, e.g., with on-o sources having heavy-tailed on periods or o -periods (having in nite variance). We show that the re ection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions, provided that the productM1 topology is used. As a consequence, the re ection map is continuous with the product M1 topology at all functions that have discontinuities in only one coordinate at a time. That continuity property also holds for more general re ection maps and is su cient to support limit theorems for stochastic processes in most applications. We apply the continuity of the re ection map to obtain limits for bu er-content stochastic processes in stochastic uid networks.