The reflection map with discontinuities

We study the multidimensional reflection map on the spaces D([0, T], R-k) a nd D([0, infinity), R-k) of right-continuous R-k-valued functions on [0, T] or [0, infinity) with left limits, endowed with variants of the Skorohod ( 1956) M-l topology. The reflection map was used with the continuous mapping theorem by Harrison and Reiman (1981) and Reiman (1984) to establish heavy -traffic limit theorems with reflected 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 reflecte d Brownian motion have continuous sample paths, the topology of uniform con vergence over bounded intervals could be used for those results. Variants o f the M-l topologies are needed to obtain alternative discontinuous limits approached gradually by the converging processes, as occurs in stochastic f luid networks with bursty exogenous input processes, e.g., with on-off sour ces having heavy-tailed on periods or off periods (having infinite variance ). We show that the reflection map is continuous at limits without simultan eous jumps of opposite sign in the coordinate functions, provided that the product M-l topology is used. As a consequence, the reflection map is conti nuous with the product M-l topology at all functions that have discontinuit ies in only one coordinate at a time. That continuity property also holds f or more general reflection maps and is sufficient to support limit theorems for stochastic processes in most applications. We apply the continuity of the reflection map to obtain limits for buffer-content stochastic processes in stochastic fluid networks.