Diffusion limits in the quarter plane and non-semimartingale reflected Brownian motion

Abstract

We consider a continuous-time random walk in the quarter plane for which the transition intensities are constant on each of the four faces $(0,\infty)^2$, $F_1=\{0\}\times(0,\infty)$, $F_2=(0,\infty)\times\{0\}$ and $\{(0,0)\}$. We show that when rescaled diffusively it converges in law to a Brownian motion with oblique reflection direction $d^{(i)}$ on face $F_i$, $i=1,2$, defined via the Varadhan-Williams submartingale problem. A parameter denoted by $α$ was introduced in \cite{vw}, measuring the extent to which $d^{(i)}$ are inclined toward the origin. In the case of the quarter plane, $α$ takes values in $(-2,2)$, and it is known that the reflected Brownian motion is a semimartingale if and only if $α\in(-2,1)$. Convergence results via both the Skorohod map and the invariance principle for semimartingale reflected Brownian motion are known to hold in various settings in arbitrary dimension. In the case of the quarter plane, the invariance principle was proved for $α\in (-2,1)$ whereas for tools based on the Skorohod map to be applicable it is necessary (but not sufficient) that $α\in [-1,1)$. Another tool that has been used to prove convergence in general dimension is the extended Skorohod map, which in the case of the quarter plane provides convergence for $α=1$. This paper focuses on the range $α\in (1,2)$, where the Skorohod problem and the extended Skorohod problem do not possess a unique solution, the limit process is not a semimartingale, and convergence to reflected Brownian motion has not been shown before. The result has implications on the asymptotic analysis of two Markovian queueing models: The {\it generalized processor sharing model with parallelization slowdown}, and the {\it coupled processor model}.

Figures (1)

• Figure 1: For $i=1,2$, $d^{(i)}$ is the direction of reflection at face $x_i=0$, and $\theta^{(i)}$ is the angle between $e^{(i)}$ and $d^{(i)}$, positive when $d^{(i)}$ is toward the origin.

Theorems & Definitions (17)

• Definition 2.1
• Theorem 2.2
• Theorem 2.3
• Corollary 2.4
• Corollary 2.5
• Proposition 3.1
• Lemma 3.2
• Proposition 4.1
• Lemma 4.2
• proof