Heavy-Traffic Limit of theGI/GI/1 Stationary Departure Process and Its Variance Function

Abstract

Abstract. Heavy-traffic limits are established for the stationary departure process from a GI/GI/1 queue and its variance function. The limit process is a function of the Brownian motion limits of the arrival and service processes plus the stationary reflected Brownian motion (RBM) limit of the queue-length process. An explicit expression is given for the variance function, which depends only on the first two moments of the interarrival times and service times plus the previously determined correlation function of canonical (drift −1, diffusion coefficient 1) RBM. The limit for the variance function here is used to show that the approximation for the index of dispersion for counts of the departure process used in our new robust queueing network analyzer is asymptotically correct in the heavy-traffic limit.

A robust queueing network analyzer based on indices of dispersion

“…The present paper also relies heavily on heavy‐traffic limit theorems, but here we exploit our recent heavy‐traffic limits for the flows in Whitt and You (2018a, 2020).…”

“…With the aid of the HT limits established in Whitt and You (2018a, 2020), we now develop a network calculus to characterize the IDCs of the customer flows in an OQN. Similar nonparametric traffic descriptions have been studied in Jagerman et al (2004), Li and Hwang (1992, 1993), but they focused on single queues.…”

“…In Section 3, we introduce a set of linear equations, which we refer to as the IDC equations , to describe the combined effect of these three network operations. These IDC equations are derived from the HT limits in Whitt and You (2018a, 2020). We discuss the remaining technical details in the appendix.…”

“…In this section, we briefly describe the contribution of each of our previous papers (Whitt & You, 2018a, 2018b, 2019a, 2019b, 2020) and indicate how the present paper goes beyond them. To do so, it is helpful to classify OQN's according to structural complexity.…”

“…Even the network with two queues in series presents challenging new problems.The new problem presented by this class of OQNs is developing an effective approximation for the IDC of a departure process from a queue where the arrival process is partially characterized by its IDC. Significant progress was obtained by establishing a new heavy‐traffic limit theorem for the stationary departure process from a queue in Whitt and You (2018a). In addition, drawing on this limit theorem, an algorithm to approximate the IDC of a departure process was developed and tested in Whitt and You (2018a).…”