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We shall use the Euler-Maruyama (EM) scheme with decreasing step size $\Lambda=(\eta_n)_{n\in \mathbb{N}}$ to approximate the steady state for piecewise $\alpha$-stable Ornstein-Uhlenbeck processes arising in queue networks. These processes do not have an explicit dissipation. We prove the EM scheme converges to the steady state with a rate $\eta^{1/\alpha}_n$ in Wasserstein-1 distance. In addition, we utilize the Sinkhorn--Knopp algorithm to compute the Wasserstein-1 distance and conduct the simulations for several examples., Comment: 30 pages,7 figures

We present an asymptotic analysis of a stochastic two-compartmental cell proliferation system with regulatory mechanisms. We model the system as a state-dependent birth and death process. Proliferation of hematopoietic stem cells (HSCs) is regulated by population density of HSC-derived clones and differentiation of HSC is regulated by population density of HSCs. By scaling up the initial population, we show the density of dynamics converges in distribution to the solution of a system of ordinary differential equations (ODEs). The system of ODEs has a unique non-trivial equilibrium that is globally stable. Furthermore, we show the scaled fluctuation of the population converges in law to a linear diffusion with time-dependent coefficients. With initial data being Gaussian, the limit is a Gauss-Markov process, and it behaves like the FCLT limit under equilibrium with constant coefficients at large times. This is proved by establishing exponential convergence in the 2-Wasserstein metric for the associated Gaussian measures in a $\mathcal{L}_2$ Hilbert space. We apply our results to analyze and compare two regulatory mechanisms in the hematopoietic system. Simulations are conducted to verify our large-scale and long-time approximation of the dynamics. We demonstrate some regulatory mechanisms are efficient (converge to steady state rapidly) but not effective (have large fluctuation around the steady state).

We study an individual-based stochastic spatial epidemic model where the number of locations and the number of individuals at each location both grow to infinity. Each individual is associated with a random infection-age dependent infectivity function. Individuals are infected through interactions across the locations with heterogeneous effects. The epidemic dynamics can be described using a time-space representation for the the total force of infection, the number of susceptible individuals, the number of infected individuals that are infected at each time and have been infected for a certain amount of time, as well as the number of recovered individuals. We prove a functional law of large numbers for these time-space processes, and in the limit, we obtain a set of time-space integral equations. We then derive the PDE models from the limiting time-space integral equations, in particular, the density (with respect to the infection age) of the time-age-space integral equation for the number of infected individuals tracking the age of infection satisfies a linear PDE in time and age with an integral boundary condition. These integral equation and PDE limits can be regarded as dynamics on graphon under certain conditions.

We study a stochastic epidemic model with multiple patches (locations), where individuals in each patch are categorized into three compartments, Susceptible, Infected and Recovered/Removed, and may migrate from one patch to another in any of the compartments. Each individual is associated with a random infectivity function which dictates the force of infectivity while the interactive infection process depends on the age of infection (elapsed time since infection). We prove a functional law of large number for the epidemic evolution dynamics including the aggregate infectivity process, the numbers of susceptible and recovered individuals as well as the number of infected individuals at each time that have been infected for a certain amount of time. From the limits, we derive a PDE model for the density of the number of infected individuals with respect to the infection age, which is a systems of linear PDE equations with a boundary condition that is determined by a set of integral equations.

We study an individual-based stochastic epidemic model in which infected individuals become susceptible again following each infection. In contrast to classical compartment models, after each infection, the infectivity is a random function of the time elapsed since one's infection. Similarly, recovered individuals become gradually susceptible after some time according to a random susceptibility function. We study the large population asymptotic behaviour of the model, by proving a functional law of large numbers (FLLN) and investigating the endemic equilibria properties of the limit. The limit depends on the law of the susceptibility random functions but only on the mean infectivity functions. The FLLN is proved by constructing a sequence of i.i.d. auxiliary processes and adapting the approach from the theory of propagation of chaos. The limit is a generalisation of a PDE model introduced by Kermack and McKendrick, and we show how this PDE model can be obtained as a special case of our FLLN limit.% for a particular set of infectivity and susceptibility random functions and initial conditions. For the endemic equilibria, if $ R_0 $ is lower than (or equal to) some threshold, the epidemic does not last forever and eventually disappears from the population, while if $ R_0 $ is larger than this threshold, the epidemic will not disappear and there exists an endemic equilibrium. The value of this threshold turns out to depend on the harmonic mean of the susceptibility a long time after an infection, a fact which was not previously known.

In this paper, we develop a stochastic algorithm based on the Euler--Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of $G/Ph/n+GI$ queues in the Halfin-Whitt regime. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich (2014, 2022) plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of $G/Ph/n+GI$ queues and its Euler-Maruyama scheme. In particular, the variances in the CLT and MDP associated with the limiting diffusion are determined by Stein's equation and Malliavin calculus, in which properties of a mollified diffusion and an associated weighted occupation time play a crucial role., Comment: arXiv admin note: substantial text overlap with arXiv:2109.03623

This paper studies birth and death processes in interactive random environments where the birth and death rates and the dynamics of the state of the environment are dependent on each other. Two models of a random environment are considered: a continuous-time Markov chain (finite or countably infinite) and a reflected (jump) diffusion process. The background is determined by a joint Markov process carrying a specific interactive mechanism, with an explicit invariant measure whose structure is similar to a product form. We discuss a number of queueing and population-growth models and establish conditions under which the above-mentioned invariant measure can be derived. Next, an analysis of the rate of convergence to stationarity is performed for the models under consideration. We consider two settings leading to either an exponential or a polynomial convergence rate. In both cases we assume that the underlying environmental Markov process has an exponential rate of convergence, but the convergence rate of the joint Markov process is determined by certain conditions on the birth and death rates. To prove these results a coupling method turns out to be useful., Comment: 32 pages

This paper presents a law of large numbers result, as the size of the population tends to infinity, of SIR stochastic epidemic models, for a population distributed over $L$ distinct patches (with migrations between them) and $K$ distinct groups (possibly age groups). The limit is a set of Volterra-type integral equations, and the result shows the effects of both spatial and population heterogeneity. The novelty of the model is that the infectivity of an infected individual is infection age dependent. More precisely, to each infected individual is attached a random infection-age dependent infectivity function, such that the various random functions attached to distinct individuals are i.i.d. The proof involves a novel construction of a sequence of i.i.d. processes to invoke the law of large numbers for processes in $D$, by using the solution of a MacKean-Vlasov type Poisson-driven stochastic equation (as in the propagation of chaos theory). We also establish an identity using the Feynman-Kac formula for an adjoint backward ODE. The advantage of this approach is that it assumes much weaker conditions on the random infectivity functions than our earlier work for the homogeneous model in [20], where standard tightness criteria for convergence of stochastic processes were employed. To illustrate this new approach, we first explain the new proof under the weak assumptions for the homogeneous model, and then describe the multipatch-multigroup model and prove the law of large numbers for that model.

In this paper, we develop a stochastic algorithm based on Euler-Maruyama scheme to approximate the invariant measure of the limiting multidimensional diffusion of the $M/Ph/n+M$ queue. Specifically, we prove a non-asymptotic error bound between the invariant measures of the approximate model from the algorithm and the limiting diffusion of the queueing model. Our result also provides an approximation to the steady-state of the diffusion-scaled queueing processes in the Halfin-Whitt regime given the well established interchange of limits property. To establish the error bound, we employ the recently developed Stein's method for multi-dimensional diffusions, in which the regularity of Stein's equation developed by Gurvich \cite{Gur1} plays a crucial role. We further prove the central limit theorem (CLT) and the moderate deviation principle (MDP) for the occupation measures of the limiting diffusion of the $M/Ph/n+M$ queue and its Euler-Maruyama scheme. In particular, the variance of the CLT of the limiting queue is determined by using Stein's equation and Malliavin calculus., Comment: We updated the regularity estimate of Stein's equation and added CLT and LDP about the ergodic measures. The convergence rate of the approximation is improved

In this survey paper, we review the recent advances in individual based non--Markovian epidemic models. They include epidemic models with a constant infectivity rate, varying infectivity rate or infection-age dependent infectivity, infection-age recovery rate (or equivalently, general law of infectious period), as well as varying susceptibility/immunity. We focus on the scaling limits with a large population, functional law of large numbers (FLLN) and functional central limit theorems (FCLT), while the large and moderate deviations for some Markovian epidemic models are also reviewed. In the FLLN, the limits are a set of Volterra integral equations, and in the FCLT, the limits are stochastic Volterra integral equations driven by Gaussian processes. We relate our deterministic limits to the results in the seminal papers by Kermack and McKendrick published in 1927, 1932 and 1933, where the varying infectivity and susceptibility/immunity were already considered. We also discuss some extensions, including models with heterogeneous population, spatial models and control problems, as well as open problems.

We study epidemic models where the infectivity of each individual is a random function of the infection age (the elapsed time of infection). To describe the epidemic evolution dynamics, we use a stochastic process that tracks the number of individuals at each time that have been infected for less than or equal to a certain amount of time, together with the aggregate infectivity process. We establish the functional law of large numbers (FLLN) for the stochastic processes that describe the epidemic dynamics. The limits are described by a set of deterministic integral equations, which has a further characterization using PDEs under some regularity conditions. The solutions are characterized with boundary conditions that are given by a system of Volterra equations. We also characterize the equilibrium points for the PDEs in the SIS model with infection-age dependent infectivity. To establish the FLLNs, we employ a useful criterion for weak convergence for the two-parameter processes together with useful representations for the relevant processes via Poisson random measures.

We study the semimartingale properties for the generalized fractional Brownian motion (GFBM) introduced by Pang and Taqqu (2019) and discuss the applications of the GFBM and its mixtures to financial asset pricing. The GFBM is self-similar and has non-stationary increments, whose Hurst index $H \in (0,1)$ is determined by two parameters. We identify the regions of these two parameter values where the GFBM is a semimartingale. We next study the mixed process made up of an independent BM and a GFBM and identify the range of parameters for it to be a semimartingale, which leads to $H \in (1/2,1)$ for the GFBM. We also derive the associated equivalent Brownian measure. This result is in great contrast with the mixed FBM with $H \in \{1/2\}\cup(3/4,1]$ proved by Cheridito (2001) and shows the significance of the additional parameter introduced in the GFBM. We then study the semimartingale asset pricing theory with the mixed GFBM, in presence of long range dependence, and applications in option pricing and portfolio optimization. Finally we discuss the implications of using GFBM on arbitrage theory, in particular, providing an example of semimartingale asset pricing model of long range dependence without arbitrage.

In this paper, we prove functional central limit theorems (FCLTs) for a stochastic epidemic model with varying infectivity and general infectious periods recently introduced in Forien, Pang and Pardoux (2020).The infectivity process (total force of infection at each time) is composed of the independent infectivity random functions of each infectious individual, which starts at the time of infection. These infectivity random functions induce the infectious periods (as well as exposed, recovered or immune periods in full generality), whose probability distributions can be very general. The epidemic model includes the generalized non-Markovian SIR, SEIR, SIS, SIRS models with infection-age dependent infectivity. In the FCLTs for the generalized SIR and SEIR models, the limits of the diffusion-scaled fluctuations of the infectivity and susceptible processes are a unique solution to a two-dimensional Gaussian-driven stochastic Volterra integral equations, and then given these solutions, the limits for the infected (exposed/infectious) and recovered processes are Gaussian processes expressed in terms of the solutions to those stochastic Volterra integral equations. We also present the FCLTs for the generalized SIS and SIRS models.

The generalized fractional Brownian motion is a Gaussian self-similar process whose increments are not necessarily stationary. It appears in applications as the scaling limit of a shot noise process with a power law shape function and non-stationary noises with a power-law variance function. In this paper we study sample path properties of the generalized fractional Brownian motion, including Holder continuity, path differentiability/non-differentiability, and functional and local Law of the Iterated Logarithms.

We introduce a "system-wide safety staffing" (SWSS) parameter for multiclass multi-pool networks of any tree topology, Markovian or non-Markovian, in the Halfin-Whitt regime. This parameter can be regarded as the optimal reallocation of the capacity fluctuations (positive or negative) of order $\sqrt{n}$ when each server pool employs a square-root staffing rule. We provide an explicit form of the SWSS as a function of the system parameters, which is derived using a graph theoretic approach based on Gaussian elimination. For Markovian networks, we give an equivalent characterization of the SWSS parameter via the drift parameters of the limiting diffusion. We show that if the SWSS parameter is negative, the limiting diffusion and the diffusion-scaled queueing processes are transient under any Markov control, and cannot have a stationary distribution when this parameter is zero. If it is positive, we show that the diffusion-scaled queueing processes are uniformly stabilizable, that is, there exists a scheduling policy under which the stationary distributions of the controlled processes are tight over the size of the network. In addition, there exists a control under which the limiting controlled diffusion is exponentially ergodic. Thus we have identified a necessary and sufficient condition for the uniform stabilizability of such networks in the Halfin-Whitt regime. We use a constant control resulting from the leaf elimination algorithm to stabilize the limiting controlled diffusion, while a family of Markov scheduling policies which are easy to compute are used to stabilize the diffusion-scaled processes. Finally, we show that under these controls the processes are exponentially ergodic and the stationary distributions have exponential tails., Comment: 36 pages

We introduce an epidemic model with varying infectivity and general exposed and infectious periods, where the infectivity of each individual is a random function of the elapsed time since infection, those function being i.i.d. for the various individuals in the population. This approach models infection-age dependent infectivity, and extends the classical SIR and SEIR models. We focus on the infectivity process (total force of infection at each time), and prove a functional law of large number (FLLN). In the deterministic limit of this LLN, the infectivity process and the susceptible process are determined by a two-dimensional deterministic integral equation. From its solutions, we then derive the exposed, infectious and recovered processes, again using integral equations. For the early phase, we study the stochastic model directly by using an approximate (non--Markovian) branching process, and show that the epidemic grows at an exponential rate on the event of non-extinction, which matches the rate of growth derived from the deterministic linearized equations. We also use these equations to derive the basic reproduction number $R_0$ during the early stage of an epidemic, in terms of the average individual infectivity function and the exponential rate of growth of the epidemic.

We study multi-patch epidemic models where individuals may migrate from one patch to another in either of the susceptible, exposed/latent, infectious and recovered states. We assume that infections occur both locally with a rate that depends on the patch as well as "from distance" from all the other patches. The exposed and infectious periods have general distributions, and are not affected by the possible migrations of the individuals. The migration processes in either of the three states are assumed to be Markovian, and independent of the exposed and infectious periods. We establish a functional law of large number (FLLN) and a function central limit theorem (FCLT) for the susceptible, exposed/latent, infectious and recovered processes. In the FLLN, the limit is determined by a set of Volterra integral equations. In the special case of deterministic exposed and infectious periods, the limit becomes a system of ODEs with delays. In the FCLT, the limit is given by a set of stochastic Volterra integral equations driven by a sum of independent Brownian motions and continuous Gaussian processes with an explicit covariance structure.

We study non-Markovian stochastic epidemic models (SIS, SIR, SIRS, and SEIR), in which the infectious (and latent/exposing, immune) periods have a general distribution. We provide a representation of the evolution dynamics using the time epochs of infection (and latency/exposure, immunity). Taking the limit as the size of the population tends to infinity, we prove both a functional law of large number (FLLN) and a functional central limit theorem (FCLT) for the processes of interest in these models. In the FLLN, the limits are a unique solution to a system of deterministic Volterra integral equations, while in the FCLT, the limit processes are multidimensional Gaussian solutions of linear Volterra stochastic integral equations. In the proof of the FCLT, we provide an important Poisson random measures representation of the diffusion-scaled processes converging to Gaussian components driving the limit process.

We study ergodic properties of a class of Markov-modulated general birth-death processes under fast regime switching. The first set of results concerns the ergodic properties of the properly scaled joint Markov process with a parameter that is taken large. Under very weak hypotheses, we show that if the averaged process is exponentially ergodic for large values of the parameter, then the same applies to the original joint Markov process. The second set of results concerns steady-state diffusion approximations, under the assumption that the 'averaged' fluid limit exists. Here, we establish convergence rates for the moments of the approximating diffusion process to those of the Markov modulated birth-death process. This is accomplished by comparing the generator of the approximating diffusion and that of the joint Markov process. We also provide several examples which demonstrate how the theory can be applied., Comment: 23 pages

In this paper, we study optimal control problems for multiclass GI/M/n+M queues in an alternating renewal (up-down) random environment in the Halfin-Whitt regime. Assuming that the downtimes are asymptotically negligible and only the service processes are affected, we show that the limits of the diffusion-scaled state processes under non-anticipative, preemptive, work-conserving scheduling policies, are controlled jump diffusions driven by a compound Poisson jump process. We establish the asymptotic optimality of the infinite-horizon discounted and long-run average (ergodic) problems for the queueing dynamics. Since the process counting the number of customers in each class is not Markov, the usual martingale arguments for convergence of mean empirical measures cannot be applied. We surmount this obstacle by demonstrating the convergence of the generators of an augmented Markovian model which incorporates the age processes of the renewal interarrival times and downtimes. We also establish long-run average moment bounds of the diffusion-scaled queueing processes under some (modified) priority scheduling policies. This is accomplished via Foster-Lyapunov equations for the augmented Markovian model., Comment: 32 pages

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., Comment: 20 pages

In this paper we study the uniform stability properties of two classes of parallel server networks with multiple classes of jobs and multiple server pools of a tree topology. These include a class of networks with a single non-leaf server pool, such as the 'N' and 'M' models, and networks of any tree topology with class-dependent service rates. We show that with $\sqrt{n}$ safety staffing, and no abandonment, in the Halfin--Whitt regime, the diffusion-scaled controlled queueing processes are exponentially ergodic and their invariant probability distributions are tight, uniformly over all stationary Markov controls. We use a unified approach in which the same Lyapunov function is used in the study of the prelimit and diffusion limit. A parameter called the spare capacity (safety staffing) of the network plays a central role in characterizing the stability results: the parameter being positive is necessary and sufficient that the limiting diffusion is uniformly exponentially ergodic over all stationary Markov controls. We introduce the concept of "system-wide work conserving policies", which are defined as policies that minimize the number of idle servers at all times. This is stronger than the so-called joint work conservation. We show that, provided the spare capacity parameter is positive, the diffusion-scaled processes are geometrically ergodic and the invariant distributions are tight, uniformly over all "system-wide work conserving policies". In addition, when the spare capacity is negative we show that the diffusion-scaled processes are transient under any stationary Markov control, and when it is zero, they cannot be positive recurrent., Comment: 32 pages

In this article, relying on Foster-Lyapunov drift conditions, we establish subexponential upper and lower bounds on the rate of convergence in the $\mathrm{L}^p$-Wasserstein distance for a class of irreducible and aperiodic Markov processes. We further discuss these results in the context of Markov L\'evy-type processes. In the lack of irreducibility and/or aperiodicity properties, we obtain exponential ergodicity in the $\mathrm{L}^p$-Wasserstein distance for a class of It\^{o} processes under an asymptotic flatness (uniform dissipativity) assumption. Lastly, applications of these results to specific processes are presented, including Langevin tempered diffusion processes, piecewise Ornstein-Uhlenbeck processes with jumps under constant and stationary Markov controls, and backward recurrence time chains, for which we provide a sharp characterization of the rate of convergence via matching upper and lower bounds., Comment: 32 pages

A Markovian single-server queue is studied in an interactive random environment. The arrival and service rates of the queue depend on the environment, while the transition dynamics of the random environment depends on the queue length. We consider in detail two types of Markov random environments: a pure jump process and a reflected jump-diffusion. In both cases, the joint dynamics is constructed so that the stationary distribution can be explicitly found in a simple form (weighted geometric). We also derive an explicit estimate for exponential rate of convergence to the stationary distribution via coupling., Comment: 27 pages. Keywords: queues, random environment, stationary distribution, invariant measure, rate of convergence

We study the ergodic properties of a class of controlled stochastic differential equations (SDEs) driven by $\alpha$-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 L\'evy-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.

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., Comment: 28 pages

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., Comment: 27 pages

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., Comment: 21 pages

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 It\^o equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump L\'evy process, or (2) an anisotropic L\'evy process with independent one-dimensional symmetric $\alpha$-stable components, or (3) an anisotropic L\'evy process as in (2) and a pure-jump L\'evy process. We also study the class of models driven by a subordinate Brownian motion, which contains an isotropic (or rotationally invariant) $\alpha$-stable L\'evy process as a special case. We identify conditions on the parameters in the drift, the L\'evy 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., Comment: 42 pages

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., Comment: 35 pages. arXiv admin note: text overlap with arXiv:1602.03275

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., Comment: 35 pages

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., Comment: 32 pages

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 L\'evy processes. The presented results generalize the classical and well-known results for fluid flows driven by elliptic diffusion processes.

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{\~n}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{\~n}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{\~n}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{\~n}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., Comment: 33 pages