Guodong Pang

Ari Arapostathis, Hassan Hmedi, Guodong Pang

We study ergodic properties of Markovian multiclass many-server queues which are uniform over scheduling policies, as well as the size n of the system. The system is heavily loaded in the Halfin-Whitt regime, and the scheduling policies are work-conserving and preemptive. We provide a unified approach via a Lyapunov function method that establishes Foster-Lyapunov equations for both the limiting diffusion and the prelimit diffusion-scaled queueing processes simultaneously. We first study the limiting controlled diffusion, and we show that if the spare capacity (safety staffing) parameter is positive, then the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp, since when there is no abandonment and the spare capacity parameter is negative, then the controlled diffusion is transient under any Markov control. In addition, we show that if all the abandonment rates are positive, the invariant probability measures have sub-Gaussian tails, regardless whether the spare capacity parameter is positive or negative. Using the above results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queueing processes. In addition to providing a simpler proof of the results in Gamarnik and Stolyar [Queueing Syst (2012) 71:25-51], we extend these results to the multiclass models with renewal arrival processes, albeit under the assumption that the mean residual life functions are bounded. For the Markovian model with Poisson arrivals, we obtain stronger results and show that the convergence to the stationary distribution is at an exponential rate uniformly over all work-conserving stationary Markov scheduling policies.

Ari Arapostathis, Guodong Pang

We study infinite-horizon asymptotic average optimality for parallel server network with multiple classes of jobs and multiple server pools in the Halfin-Whitt regime. Three control formulations are considered: 1) minimizing the queueing and idleness cost, 2) minimizing the queueing cost under a constraints on idleness at each server pool, and 3) fairly allocating the idle servers among different server pools. For the third problem, we consider a class of bounded-queue, bounded-state (BQBS) stable networks, in which any moment of the state is bounded by that of the queue only (for both the limiting diffusion and diffusion-scaled state processes). We show that the optimal values for the diffusion-scaled state processes converge to the corresponding values of the ergodic control problems for the limiting diffusion. We present a family of state-dependent Markov balanced saturation policies (BSPs) that stabilize the controlled diffusion-scaled state processes. It is shown that under these policies, the diffusion-scaled state process is exponentially ergodic, provided that at least one class of jobs has a positive abandonment rate. We also establish useful moment bounds, and study the ergodic properties of the diffusion-scaled state processes, which play a crucial role in proving the asymptotic optimality.

Ari Arapostathis, Guodong Pang

We study the infinite horizon optimal control problem for N-network queueing systems, which consist of two customer classes and two server pools, under average (ergodic) criteria in the Halfin-Whitt regime. We consider three control objectives: 1) minimizing the queueing (and idleness) cost, 2) minimizing the queueing cost while imposing a constraint on idleness at each server pool, and 3) minimizing the queueing cost while requiring fairness on idleness. The running costs can be any nonnegative convex functions having at most polynomial growth. For all three problems we establish asymptotic optimality, namely, the convergence of the value functions of the diffusion-scaled state process to the corresponding values of the controlled diffusion limit. We also present a simple state-dependent priority scheduling policy under which the diffusion-scaled state process is geometrically ergodic in the Halfin-Whitt regime, and some results on convergence of mean empirical measures which facilitate the proofs.

Ari Arapostathis, Anup Biswas, Guodong Pang

We study a dynamic scheduling problem for a multi-class queueing network with a large pool of statistically identical servers. The arrival processes are Poisson, and service times and patience times are assumed to be exponentially distributed and class dependent. The optimization criterion is the expected long time average (ergodic) of a general (nonlinear) running cost function of the queue lengths. We consider this control problem in the Halfin-Whitt (QED) regime, that is, the number of servers $n$ and the total offered load $\mathbf{r}$ scale like $n\approx\mathbf{r}+\hatρ\sqrt{\mathbf{r}}$ for some constant $\hatρ$. This problem was proposed in [Ann. Appl. Probab. 14 (2004) 1084-1134, Section 5.2]. The optimal solution of this control problem can be approximated by that of the corresponding ergodic diffusion control problem in the limit. We introduce a broad class of ergodic control problems for controlled diffusions, which includes a large class of queueing models in the diffusion approximation, and establish a complete characterization of optimality via the study of the associated HJB equation. We also prove the asymptotic convergence of the values for the multi-class queueing control problem to the value of the associated ergodic diffusion control problem. The proof relies on an approximation method by spatial truncation for the ergodic control of diffusion processes, where the Markov policies follow a fixed priority policy outside a fixed compact set.

Ari Arapostathis, Guodong Pang

We consider Markovian multiclass multi-pool networks with heterogeneous server pools, each consisting of many statistically identical parallel servers, where the bipartite graph of customer classes and server pools forms a tree. Customers form their own queue and are served in the first-come first-served discipline, and can abandon while waiting in queue. Service rates are both class and pool dependent. The objective is to study the limiting diffusion control problems under the long run average (ergodic) cost criteria in the Halfin--Whitt regime. Two formulations of ergodic diffusion control problems are considered: (i) both queueing and idleness costs are minimized, and (ii) only the queueing cost is minimized while a constraint is imposed upon the idleness of all server pools. We develop a recursive leaf elimination algorithm that enables us to obtain an explicit representation of the drift for the controlled diffusions. Consequently, we show that for the limiting controlled diffusions, there always exists a stationary Markov control under which the diffusion process is geometrically ergodic. The framework developed in our earlier work is extended to address a broad class of ergodic diffusion control problems with constraints. We show that that the unconstrained and constrained problems are well posed, and we characterize the optimal stationary Markov controls via HJB equations.