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Guodong Pang

Publications and source records attributed to Guodong Pang.

52 records · Page 3Linked to original sources
arXiv2019

Ergodic control of diffusions with compound Poisson jumps under a general structural hypothesis

We study the ergodic control problem for a class of controlled jump diffusions driven by a compound Poisson process. This extends the results of [SIAM J. Control Optim. 57 (2019), no. 2, 1516-1540] to running costs that are not near-monotone. This generality is needed in applications such as optimal scheduling of large-scale parallel server networks. We provide a full characterization of optimality via the Hamilton-Jacobi-Bellman (HJB) equation, for which we additionally exhibit regularity of solutions under mild hypotheses. In addition, we show that optimal stationary Markov controls are a.s. pathwise optimal. Lastly, we show that one can fix a stable control outside a compact set and obtain near-optimal solutions by solving the HJB on a sufficiently large bounded domain. This is useful for constructing asymptotically optimal scheduling policies for multiclass parallel server networks.

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arXiv2019

Uniform polynomial rates of convergence for a class of Lévy-driven controlled SDEs arising in multiclass many-server queues

We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $α$-stable processes which arise as the limiting equations of multiclass queueing models in the Halfin-Whitt regime that have heavy-tailed arrival processes. When the safety staffing parameter is positive, we show that the SDEs are uniformly ergodic and enjoy a polynomial rate of convergence to the invariant probability measure in total variation, which is uniform over all stationary Markov controls resulting in a locally Lipschitz continuous drift. We also derive a matching lower bound on the rate of convergence (under no abandonment). On the other hand, when all abandonment rates are positive, we show that the SDEs are exponentially ergodic uniformly over the above-mentioned class of controls. Analogous results are obtained for Lévy-driven SDEs arising from multiclass many-server queues under asymptotically negligible service interruptions. For these equations, we show that the aforementioned ergodic properties are uniform over all stationary Markov controls. We also extend a key functional central limit theorem concerning diffusion approximations so as to make it applicable to the models studied here.

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arXiv2018

Ergodicity of Lévy-driven SDEs arising from multiclass many-server queues

We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anisotropic Lévy process as in (2) and a pure-jump Lévy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $α$-stable Lévy process as a special case. We identify conditions on the parameters in the drift, the Lévy measure and/or covariance function which result in subexponential and/or exponential ergodicity. We show that these assumptions are sharp, and we identify some key necessary conditions for the process to be ergodic. In addition, we show that for the queueing models described above with no abandonment, the rate of convergence is polynomial, and we provide a sharp quantitative characterization of the rate via matching upper and lower bounds.

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arXiv2018

Optimal control of Markov-modulated multiclass many-server queues

We study multiclass many-server queues for which the arrival, service and abandonment rates are all modulated by a common finite-state Markov process. We assume that the system operates in the "averaged" Halfin-Whitt regime, which means that it is critically loaded in the average sense, although not necessarily in each state of the Markov process. We show that under any static priority policy, the Markov-modulated diffusion-scaled queueing process is geometrically ergodic. This is accomplished by employing a solution to an associated Poisson equation in order to construct a suitable Lyapunov function. We establish a functional central limit theorem for the diffusion-scaled queueing process and show that the limiting process is a controlled diffusion with piecewise linear drift and constant covariance matrix. We address the infinite-horizon discounted and long-run average (ergodic) optimal control problems and establish asymptotic optimality.

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arXiv2018

Ergodic control of a class of jump diffusions with finite Lévy measures and rough kernels

We study the ergodic control problem for a class of jump diffusions in $\mathbb{R}^d$, which are controlled through the drift with bounded controls. The Levy measure is finite, but has no particular structure; it can be anisotropic and singular. Moreover, there is no blanket ergodicity assumption for the controlled process. Unstable behavior is `discouraged' by the running cost which satisfies a mild coercive hypothesis (i.e., is near-monotone). We first study the problem in its weak formulation as an optimization problem on the space of infinitesimal ergodic occupation measures, and derive the Hamilton-Jacobi-Bellman equation under minimal assumptions on the parameters, including verification of optimality results, using only analytical arguments. We also examine the regularity of invariant measures. Then, we address the jump diffusion model, and obtain a complete characterization of optimality.

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arXiv2018

Infinite horizon asymptotic average optimality for large-scale parallel server networks

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.

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arXiv2017

Equivalence of Fluid Models for $G_t/GI/N+GI$ Queues

Four different fluid model formulations have been recently developed for $G_t/GI/N+GI$ queues, including a two-parameter fluid model in Whitt (2006) by tracking elapsed service and patience times of each customer, a measure-valued fluid model in Kang and Ramanan (2010) and its extension in Zu{ñ}iga (2014) by tracking elapsed service and patience times of each customer, and a measure-valued fluid model in Zhang (2013) by tracking residual service and patience times of each customer. We show that the two fluid models tracking elapsed times (Whitt's and Kang and Ramanan's fluid models) are equivalent formulations for the same $G_t/GI/N+GI$ queue, whereas Zu{ñ}iga's fluid model and Zhang's fluid model are not entirely equivalent under general initial conditions. We then identify necessary and sufficient conditions under which Zu{ñ}iga's fluid model and Zhang's fluid model can be derived from each other for the same system, in which certain measure-valued fluid processes tracking residual service and patience times of each customer derived from Kang-Ramanan and Zu{ñ}iga's fluid models play an important role.The equivalence properties discovered provide important implications for the understanding of the recent development for non-Markovian many-server queues.

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arXiv2017

Infinite Horizon Average Optimality of the N-network Queueing Model in the Halfin-Whitt Regime

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.

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arXiv2015

Ergodic control of multi-class $M/M/N+M$ queues in the Halfin-Whitt regime

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.

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arXiv2015

A service system with on-demand agent invitations

We consider a service system where agents are invited on-demand. Customers arrive exogenously as a Poisson process and join a customer queue upon arrival if no agent is available. Agents decide to accept or decline invitations after some exponentially distributed random time, and join an agent queue upon invitation acceptance if no customer is waiting. A customer and an agent are matched in the order of customer arrival and agent invitation acceptance under the non-idling condition, and will leave the system simultaneously once matched (service times are irrelevant here). We consider a feedback-based adaptive agent invitation scheme, which controls the number of pending agent invitations, depending on the customer and/or agent queue lengths and their changes. The system process has two components -- `the difference between agent and customer queues' and `the number of pending invitations', and is a countable continuous-time Markov chain. For the case when the customer arrival rate is constant, we establish fluid and diffusion limits, in the asymptotic regime where the customer arrival rate goes to infinity, while the agent response rate is fixed. We prove the process stability and fluid-scale limit interchange, which in particular imply that that both customer and agent waiting times in steady-state vanish in the asymptotic limit. To do this we develop a novel (multi-scale) Lyapunov drift argument; it is required because the process has non-trivial behavior on the state space boundary. When the customer arrival rate is time-varying, we present a fluid limit for the processes in the same asymptotic regime. Simulation experiments are conducted to show good performance of the invitation scheme and accuracy of fluid limit approximations.

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arXiv2015

Ergodic Diffusion Control of Multiclass Multi-Pool Networks in the Halfin-Whitt Regime

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.

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arXiv2015

Ergodicity and fluctuations of a fluid particle driven by diffusions with jumps

In this paper, we study the long-time behavior of a fluid particle immersed in a turbulent fluid driven by a diffusion with jumps, that is, a Feller process associated with a non-local operator. We derive the law of large numbers and central limit theorem for the evolution process of the tracked fluid particle in the cases when the driving process: (i) has periodic coefficients, (ii) is ergodic or (iii) is a class of Lévy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.

math.PR↗
arXiv2010

Two-Parameter Heavy-Traffic Limits for Infinite-Server Queues

In order to obtain Markov heavy-traffic approximations for infinite-server queues with general non-exponential service-time distributions and general arrival processes, possibly with time-varying arrival rates, we establish heavy-traffic limits for two-parameter stochastic processes. We consider the random variables $Q^e(t,y)$ and $Q^r(t,y)$ representing the number of customers in the system at time $t$ that have elapsed service times less than or equal to time $y$, or residual service times strictly greater than $y$. We also consider $W^r(t,y)$ representing the total amount of work in service time remaining to be done at time $t+y$ for customers in the system at time $t$. The two-parameter stochastic-process limits in the space $D([0,\infty),D)$ of $D$-valued functions in $D$ draw on, and extend, previous heavy-traffic limits by Glynn and Whitt (1991), where the case of discrete service-time distributions was treated, and Krichagina and Puhalskii (1997), where it was shown that the variability of service times is captured by the Kiefer process with second argument set equal to the service-time c.d.f.

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arXiv2010

Continuity of a queueing integral representation in the ${M}_{\mathbf{1}}$ topology

We establish continuity of the integral representation $y(t)=x(t)+\int_0^th(y(s)) ds$, $t\ge0$, mapping a function $x$ into a function $y$ when the underlying function space $D$ is endowed with the Skorohod $M_1$ topology. We apply this integral representation with the continuous mapping theorem to establish heavy-traffic stochastic-process limits for many-server queueing models when the limit process has jumps unmatched in the converging processes as can occur with bursty arrival processes or service interruptions. The proof of $M_1$-continuity is based on a new characterization of the $M_1$ convergence, in which the time portions of the parametric representations are absolutely continuous with respect to Lebesgue measure, and the derivatives are uniformly bounded and converge in $L_1$.

math.PR↗
arXiv2007

Martingale proofs of many-server heavy-traffic limits for Markovian queues

This is an expository review paper illustrating the ``martingale method'' for proving many-server heavy-traffic stochastic-process limits for queueing models, supporting diffusion-process approximations. Careful treatment is given to an elementary model -- the classical infinite-server model $M/M/\infty$, but models with finitely many servers and customer abandonment are also treated. The Markovian stochastic process representing the number of customers in the system is constructed in terms of rate-1 Poisson processes in two ways: (i) through random time changes and (ii) through random thinnings. Associated martingale representations are obtained for these constructions by applying, respectively: (i) optional stopping theorems where the random time changes are the stopping times and (ii) the integration theorem associated with random thinning of a counting process. Convergence to the diffusion process limit for the appropriate sequence of scaled queueing processes is obtained by applying the continuous mapping theorem. A key FCLT and a key FWLLN in this framework are established both with and without applying martingales.

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