The Reflection Map with Discontinuities

We study the multidimensional reflection map on the spaces D([0, T], ℝ^k) and D([0, ∞), ℝ^k) of right-continuous ℝ^k-valued functions on [0, T] or [0, ∞) with left limits, endowed with variants of the Skorohod (1956) M₁ 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 reflected 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 fluid networks with bursty exogenous input processes, e.g., with on-off sources having heavy-tailed on periods or off periods (having infinite variance). We show that the reflection map is continuous at limits without simultaneous jumps of opposite sign in the coordinate functions, provided that the product M₁ topology is used. As a consequence, the reflection 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 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.