Validity of heavy traffic steady-state approximations in generalized Jackson networks

Abstract

We consider a single class open queueing network, also known as a generalized Jackson network (GJN). A classical result in heavytraffic theory asserts that the sequence of normalized queue length processes of the GJN converge weakly to a reflected Brownian motion (RBM) in the orthant, as the traffic intensity approaches unity. However, barring simple instances, it is still not known whether the stationary distribution of RBM provides a valid approximation for the steady-state of the original network. In this paper we resolve this open problem by proving that the re-scaled stationary distribution of the GJN converges to the stationary distribution of the RBM, thus validating a so-called "interchange-of-limits" for this class of networks. Our method of proof involves a combination of Lyapunov function techniques, strong approximations and tail probability bounds that yield tightness of the sequence of stationary distributions of the GJN.

Cross-Selling in a Call Center with a Heterogeneous Customer Population

“…For the deterministic version, and with arguments reminiscent of Lyapunov function tools used for stability proofs, we show that the system is in some sense attracted back into a certain domain. Finally, we remove the conditioning and apply some arguments from [5] to obtain the steady state bounds. Some of the arguments are common to several proofs.…”

“…We will then use Lemma 8 from §E and adapt the argument used in the proof of Theorem 5.1 in [5] to obtain a bound for the steady-state queue length process. The bound for the steady-state waiting time will then follow from an application of Little's law.…”

“…We now adapt arguments from the proof of Theorem 5.1 in Gamarnik and Zeevi [5], to establish bounds for (Q Λ (∞) − η Λ k ) + . Specifically, by the definition of stationarity we have that…”

“…Proof: We use a Lyapunov type of argument along the lines of Gamarnik and Zeevi [5]. First, assume that Q Λ (0) > 2M Λ for some constant M > 0 and define the random time…”

“…We now use equation (A78) to establish a bounds for the steady state queue length using a result from [5]. Towards that end, note first that since Q Λ (t) ≤ Q Λ (t) + A Λ (t),…”