The Stationary Behavior of Reflecting Coupled Brownian Motions with Applications to Shortest Remaining Processing Time Queues

Abstract

With the objective of characterizing the stationary behavior of the scaling limit for shortest remaining processing time (SRPT) queues with a heavy-tailed processing time distribution, as obtained in Banerjee, Budhiraja, and Puha (BBP, 2022), we study reflecting coupled Brownian motions (RCBM) $(W_t(a), a, t \geq 0)$. These RCBM arise by regulating coupled Brownian motions (CBM) $(\chi_t(a), a,t \geq 0)$ to remain nonnegative. Here, for $t\geq 0$, $\chi_t(0)=0$ and $\chi_t(a):=w(a)+\sigma B_t-\mu(a)t$ for $a>0$, $w(\cdot)$ is a suitable initial condition, $\sigma$ is a positive constant, $B$ is a standard Brownian motion, and $\mu(\cdot)$ is an unbounded, positive, strictly decreasing drift function. In the context of the BBP (2022) scaling limit, the drift function is determined by the model parameters, and, for each $a\geq 0$, $W_{\cdot}(a)$ represents the scaling limit of the amount of work in the system of size $a$ or less. Thus, for the BBP (2022) scaling limit, the time $t$ values of the RCBM describe the random distribution of the size of the remaining work in the system at time $t$. Our principal results characterize the stationary distribution of the RCBM in terms of a maximum process $M_*(\cdot)$ associated with CBM starting from zero. We obtain an explicit representation for the finite-dimensional distributions of $M_*(\cdot)$ and a simple formula for its covariance. We further show that the RCBM converge in distribution to $M_*(\cdot)$ as time $t$ approaches infinity. From this, we deduce the stationary behavior of the BBP (2022) scaling limit, including obtaining an integral expression for the stationary queue length in terms of the associated maximum process. While its distribution appears somewhat complex, we compute the mean and variance explicitly, and we connect with the work of Lin, Wierman, and Zwart (2011) to offer an illustration of Little's Law.