Uniform stability of some large-scale parallel server networks

TL;DR

This paper establishes the uniform stability and exponential ergodicity of large-scale parallel server networks with multiple classes and tree topologies under certain staffing conditions, using a unified Lyapunov approach.

Contribution

It introduces a unified Lyapunov-based method to analyze stability of complex parallel server networks and defines system-wide work conserving policies for stability guarantees.

Findings

Positive spare capacity ensures exponential ergodicity.

Negative spare capacity leads to transience.

Zero spare capacity results in non-recurrence.

Abstract

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.

Equations

$$K_r:=\{x\in {\mathbb{R}}^m:\Vert x{\Vert }_1\le r\}.$$

$$\lVert\nu\rVert_{f}\,\coloneqq\,\sup_{\begin{subarray}{c}g\in{\mathcal{B}}({\mathbb{R}^{m}}),\;\lvert g\rvert\leq f\end{subarray}}\;\biggl{\lvert}\int_{{\mathbb{R}^{m}}}g(x)\,\nu(\mathrm{d}{x})\biggr{\rvert}\,,$$

$$\mathbb{R}^{{\mathcal{G}}}_{+}\,\coloneqq\,\bigl{\{}\xi=[\xi_{ij}]\in\mathbb{R}^{m\times J}_{+}\,\colon\xi_{ij}=0~{}~{}\text{for~{}}i\nsim j\bigr{\}}\,,$$

$$\frac{{\lambda }_i^n-n{\lambda }_i}{\sqrt{n}}\to {\hat{\lambda }}_i,\sqrt{n}({\mu }_{ij}^n-{\mu }_{ij})\to {\hat{\mu }}_{ij},\text{and}\frac{N_j^n-n{\nu }_j}{\sqrt{n}}\to {\hat{\nu }}_j.$$

$$\text{Minimize}\underset{j\in \mathcal{J}}{\max }\sum _{i\in \mathcal{I}(j)}{\xi }_{ij},\text{subject to}\sum _{j\in \mathcal{J}(i)}{\mu }_{ij}{\nu }_j{\xi }_{ij}={\lambda }_i\forall i\in \mathcal{I},$$

$$\sum _{i\in \mathcal{I}}{\xi }_{ij}^{\ast }=1,\forall j\in \mathcal{J},\text{and}{\xi }_{ij}^{\ast }>0\text{for all }i\sim j.$$

$$x_i^{\ast }=\sum _{j\in \mathcal{J}}{\xi }_{ij}^{\ast }{\nu }_j,\text{and }{z}_{ij}^{\ast }={\xi }_{ij}^{\ast }{\nu }_j.$$

$$Q_i^n(t)$$

$$Y_j^n(t)$$

$$q_i(x,z)$$

$$y_j(z)$$

$${\mathcal{Z}^{n}}(x)\,\coloneqq\,\bigl{\{}z\in\mathbb{Z}^{{\mathcal{G}}}_{+}\,\colon q_{i}(x,z)\wedge y_{j}^{n}(z)=0\,,~{}q_{i}(x,z)\geq 0\,,~{}y_{j}^{n}(z)\geq 0\quad\forall\,(i,j)\in{\mathcal{E}}\bigr{\}}\,.$$

$${\bar{z}}_{ij}^n:=\frac{1}{n}{\xi }_{ij}^{\ast }{N}_j^n,{\bar{x}}_i^n:=\sum _{j\in \mathcal{J}}{\bar{z}}_{ij}^n,$$

$$\begin{aligned} \hat{X}^{n}_{i}(t)&\,\coloneqq\,\frac{1}{\sqrt{n}}\bigl{(}X_{i}^{n}(t)-n\bar{x}^{n}_{i}\bigr{)}\,,\\[5.0pt] \hat{Q}^{n}_{i}(t)&\,\coloneqq\,\frac{1}{\sqrt{n}}Q_{i}^{n}(t)\,,\end{aligned}\qquad\begin{aligned} \hat{Z}^{n}_{ij}(t)&\,\coloneqq\,\frac{1}{\sqrt{n}}\bigl{(}Z_{ij}^{n}(t)-n\bar{z}^{n}_{ij}\bigr{)}\,,\\[5.0pt] \hat{Y}^{n}_{j}(t)&\,\coloneqq\,\frac{1}{\sqrt{n}}Y_{j}^{n}(t)\,.\end{aligned}$$

$${\hat{X}}_i^n(t)={\hat{Q}}_i^n(t)+\sum _{j\in \mathcal{J}(i)}{\hat{Z}}_{ij}^n(t)\forall i\in \mathcal{I},$$

$${\hat{Y}}_j^n(t)+\sum _{i\in \mathcal{I}(j)}{\hat{Z}}_{ij}^n(t)=0\forall j\in \mathcal{J}.$$

$${\hat{x}}^n:=\frac{x-n{\bar{x}}^n}{\sqrt{n}},{\hat{z}}^n:=\frac{z-n{\bar{z}}^n}{\sqrt{n}},$$

$${\mathscr{S}}^{n}\,\coloneqq\,\bigl{\{}\hat{x}\in{\mathbb{R}^{m}}\,\colon\sqrt{n}\hat{x}+n\bar{x}^{n}\in\mathbb{Z}^{m}_{+}\bigr{\}}\,.$$

$${\hat{\mathcal{Z}}^{n}}(\hat{x})\,\coloneqq\,\bigl{\{}\hat{z}\,\colon\sqrt{n}\hat{z}+n\bar{z}^{n}\in{\mathcal{Z}^{n}}(\sqrt{n}\hat{x}+n\bar{x}^{n})\bigr{\}}\,,\qquad\hat{x}\in{\mathscr{S}}^{n}\,.$$

$${\mathscr{L}}_{z}^{n}f(x)\,\coloneqq\,\sum_{i\in{\mathcal{I}}}\Biggl{(}\lambda_{i}^{n}\bigl{(}f(x+e_{i})-f(x)\bigr{)}+\sum_{j\in{\mathcal{J}}(i)}\mu_{ij}^{n}z_{ij}\bigl{(}f(x-e_{i})-f(x)\bigr{)}\Biggr{)}$$

$$\ell^{n}_{i}\,\coloneqq\,\frac{1}{\sqrt{n}}\biggl{(}\lambda^{n}_{i}-\sum_{j\in{\mathcal{J}}(i)}\mu_{ij}^{n}\xi^{*}_{ij}N_{j}^{n}\biggr{)}\,.$$

$${\ell }_i^n\underset{n\to \infty }{\to }{\ell }_i:={\hat{\lambda }}_i-\sum _{j\in \mathcal{J}(i)}{\hat{\mu }}_{ij}{z}_{ij}^{\ast }-\sum _{j\in \mathcal{J}(i)}{\mu }_{ij}{\xi }_{ij}^{\ast }{\hat{\nu }}_j,$$

$${\hat{\mathcal{L}}}_{\hat{z}}^{n}f(\hat{x})$$

$$\displaystyle\mspace{200.0mu}-\sum_{i\in{\mathcal{I}}}b^{n}_{i}(\hat{x},\hat{z})\,\frac{\mathfrak{d}f\bigl{(}\hat{x};-\tfrac{1}{\sqrt{n}}e_{i}\bigr{)}}{n^{-\nicefrac{{1}}{{2}}}}\,,\quad\hat{x}\in{\mathscr{S}}^{n}\,,\ \hat{z}\in{\hat{\mathcal{Z}}^{n}}(\hat{x})\,,$$

$$\mathfrak{d}f(x;y):=f(x+y)-f(x),x,y\in {\mathbb{R}}^m,$$

$$b_i^n(\hat{x},\hat{z}):={\ell }_i^n-\sum _{j\in \mathcal{J}(i)}{\mu }_{ij}^n{\hat{z}}_{ij},\hat{z}\in {\hat{\mathcal{Z}}}^n(\hat{x}),i\in \mathcal{I}.$$

$${\hat{q}}_i^n(\hat{x},\hat{z}):={\hat{x}}_i-\sum _{j\in \mathcal{J}(i)}{\hat{z}}_{ij},i\in \mathcal{I},{\hat{y}}_j^n(\hat{z}):=-\sum _{i\in \mathcal{I}(j)}{\hat{z}}_{ij},j\in \mathcal{J},$$

$$\hat{\vartheta}^{n}(\hat{x},\hat{z})\,\coloneqq\,\langle e,\hat{q}^{n}(\hat{x},\hat{z})\bigr{\rangle}\wedge\,\langle e,\hat{y}^{n}(\hat{z})\bigr{\rangle}\,.$$

$$\bigl{\langle}e,\hat{q}^{n}(\hat{x},\hat{z})\bigr{\rangle}\,=\,\hat{\vartheta}^{n}(\hat{x},\hat{z})+\langle e,\hat{x}\rangle^{+}\,,\quad\text{and}\quad\bigl{\langle}e,\hat{y}^{n}(\hat{z})\bigr{\rangle}\,=\,\hat{\vartheta}^{n}(\hat{x},\hat{z})+\langle e,\hat{x}\rangle^{-}$$

$${\Delta }_c:=\{u\in {\mathbb{R}}^m:u\ge 0,⟨e,u⟩=1\},\text{and }{\Delta }_s:=\{u\in {\mathbb{R}}^J:u\ge 0,⟨e,u⟩=1\},$$

$$\hat{q}^{n}(\hat{x},\hat{z})\,=\,\bigl{(}\hat{\vartheta}^{n}(\hat{x},\hat{z})+\langle e,\hat{x}\rangle^{+}\bigr{)}\,u^{c}\,,\quad\text{and}\quad\hat{y}^{n}(\hat{z})\,=\,\bigl{(}\hat{\vartheta}^{n}(\hat{x},\hat{z})+\langle e,\hat{x}\rangle^{-}\bigr{)}\,u^{s}\,.$$

$$\mathcal{D}\,\coloneqq\,\Bigl{\{}(\alpha,\beta)\in{\mathbb{R}^{m}}\times\mathbb{R}^{J}\,\colon\textstyle\sum_{i=1}^{m}\alpha_{i}=\sum_{j=1}^{J}\beta_{j}\Bigr{\}}\,.$$

Full text

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.

1. Introduction

Large-scale parallel server networks have been the subject of intense study, due to their use in modeling a variety of systems including telecommunications, data centers, customer services and manufacturing systems; see, e.g., [AAM, GKM03, brown-et-al, garnett-mandel-reiman, AIMMTYT, Shietal, van2017economies, van2018scalable]. In such networks, there are multiple classes of jobs and multiple server pools where each job class can be served by a subset of server pools while each server pool can serve a subset of job classes, thus requiring optimal routing and scheduling decisions. Many of these systems operate in the so-called the Halfin–Whitt regime (or Quality-and-Efficiency-Driven (QED) regime [halfin-whitt, W92, borst2004dimensioning]), where the arrival rates and the numbers of servers grow large as the scale of the system grows, while the service rates remain fixed in such a way that the system becomes critically loaded.

Ensuring stability of these systems through allocating available resources by means of adjusting controller parameters is of great importance. Existing work in the literature has addressed the following important questions:

1. (i)

Uniform stability of the multiclass single-pool “V” network. The study in [GS12] focused on the prelimit diffusion-scaled process and showed that, with square-root safety staffing in the single-pool of servers, the invariant probability distributions under all work-conserving scheduling policies are tight, and have a uniform exponential tail when the model has no abandonment (or a sub-Gaussian tail with abandonment). In [AHP18], a unified approach with a common Lyapunov function is developed to establish a Foster-Lyapunov equation for both the diffusion limit and the diffusion-scaled processes, which shows that the associated invariant probability measures have exponential tails, uniformly over the scale of the network, and over all stationary (work-conserving) Markov controls. 2. (ii)

Stability of the ‘N’ network under a static priority scheduling policy. With safety staffing in one server pool and no abandonment, Stolyar [Stolyar-15b] employed a integral type of Lyapunov function and established the tightness of stationary distributions of the diffusion-scaled process (there is no analysis of the rate of convergence though). 3. (iii)

Counterexamples for stability of multi-class multi-pool networks. Stolyar and Yudovina [Stolyar-Yudovina-13] showed that the stationary distributions of the diffusion-scaled processes may not be tight in these regimes under a natural load balancing scheduling policy, “Longest-queue freest-server” (LQFS-LB) (also true in the underloaded regime). 4. (iv)

Stability of multi-class multi-pool networks with pool-dependent service rates under the LQFS-LB policy [Stolyar-Yudovina-13]. We also refer the reader to [Stolyar-15, Stolyar-Yudovina-12], even though these concern the underloaded case. 5. (v)

Stability of multiclass multi-pool networks under a family of Markov policies. In [AP18, AP19], it is shown that a class of state-dependent policies, referred to as balanced saturation policies (BSP) are stabilizing for the prelimit diffusion-scaled queueing process, when at least one abandonment rate is strictly positive. 6. (vi)

Stability of the limiting controlled diffusions for multiclass multi-pool networks under a constant control. Arapostathis and Pang [AP16] developed a leaf elimination algorithm to derive an explicit expression of the drift, and, consequently, by using the structural properties of the drift, a static priority scheduling and routing control is identified which stabilizes the limiting diffusion, when at least one of the classes has a positive abandonment rate. 7. (vii)

Stabilizability of multiclass multi-pool networks of any tree topology without abandonment in the Halfin-Whitt regime. Hmedi, Arapostahis and Pang [HAP-OR] identified a system-wide safety staffing parameter and showed that that parameter being positive is a necessary and sufficient condition for the network to be stabilizable, that is, there exists a scheduling policy under which the stationary distributions of the controlled diffusion-scaled queueing processes are tight over the size of the network.

The stability results in (v) and (vi) are used in the aforementioned papers for the study of ergodic control problems for multiclass multi-pool networks. In [ABP15, AP16, AP18, AP19], due to the lack of the “uniform stability” (also called “blanket stability”) property, ergodic control problems were studied using a rather elaborate methodology. The uniform stability properties established in this paper render the ergodic control problem much simpler, and it can be studied by applying the methodology in [ABG12, Chapter 3.7].

Despite all the important results in (i)–(vi), the ergodic properties of multiclass multi-pool networks in the Halfin–Whitt regime are far from being well understood. The stability analysis of multiclass multi-pool networks in the Halfin–Whitt regime is considerably more challenging than the corresponding one for the ‘V’ network. The problem is particularly difficult when the system does not have abandonment.

Given the counterexamples in [Stolyar-Yudovina-13], uniform stability, that is, tightness of the invariant probability distributions, does not hold for multiclass multi-pool networks of any tree topology. In this paper we identify a large class of such networks that are indeed uniformly stable: (a) networks with one dominant server pool, that is, a single non-leaf server pool, which include the ‘N’, ‘M’ and generalized ‘N’, ‘M’ networks with diameters equal to three or four, and (b) networks with class-dependent service rates. It might appear to the reader that the topology in (a) is restrictive. One should note though that even for simple networks with two non-leaf server pools [Stolyar-Yudovina-13, Figure 2, p. 21] the parameters can be chosen so that uniform stability fails.

The classes of networks in (a)–(b) share an important structural property in their drift, that is, the matrix $B_{1}$ in Eq. 2.30 is diagonal. We establish a necessary and sufficient condition for uniform stability, via the so-called spare capacity (safety staffing) parameter defined in Eq. 2.35. For the networks under consideration, we show that if the spare capacity is negative, then the limiting diffusion is transient under any stationary Markov control, if it is zero, the diffusion cannot be positive recurrent, and if it is positive, the diffusion limit is uniformly stable over all stationary Markov controls (see Propositions 3.1 and 2.1). The analogous results for the diffusion-scaled processes are also established. Lastly, we provide a characterization of the spare capacity parameter for the limiting diffusion when the latter is positive recurrent. We show in Theorem 3.1 that the spare capacity is equal to an average ‘idleness’ weighted by the critical quantity in Eq. 3.1.

To prove the uniform exponential ergodicity for the limiting controlled diffusion, we use a common Lyapunov function given in Eq. 4.7. This Lyapunov function consists of two components that treat the positive and negative half spaces of the state space in a delicate manner. An important ‘tilting’ parameter must be carefully chosen to account for not only the different effects of queueing and idleness (positive and negative half state space), but also the second order derivatives of the extended generator of the diffusion. Note that these Lyapunov functions differ from the quadratic Lyapunov functions used in [AP16, AP18, AP19, DG13, APS19] for the study of stability under either constant controls or assuming abandonment, and also differ from that used in [AHP18] for the uniform stability of the ‘V’ network. In [DG13] for example, the stability analysis requires the existence of a common quadratic Lyapunov function which cannot be shown for the models under consideration. As it will be clear to the reader later in the paper, the uniform stability analysis without abandonment requires the use of the sum of two functions where each of them ‘dominates’ the other over a part of the state space. See for example the proofs of Lemmas 4.1 and 4.2.

The same Lyapunov function is used to prove the uniform exponential ergodicity for the prelimit diffusion-scaled processes. However, unlike the ‘V’ network studied in [AHP18], the Foster-Lyapunov equations for the limiting diffusion do not carry over to the analogous equations for the diffusion-scaled queueing processes over the entire state space. The reason lies in the jointly work conserving (JWC) condition (that is, all the queues have to be empty when there are idle servers) which is essential in establishing the weak convergence to the controlled limiting diffusion (see [Atar-05b, Atar-05a]). To tackle this difficulty, we first provide an explicit ‘drift’ representation of the diffusion-scaled processes which differs from the drift of the diffusion by an extra term that accounts for the deviation from the JWC condition in the $n^{\rm th}$ system, and which vanishes in the limit. A natural extension of the concept of work conservation for multiclass multi-pool networks is minimization of the idle servers at all times. This defines an action space which we call system-wide work conserving (SWC). Establishing the “uniform” geometric ergodicity over all SWC Markov policies when the spare capacity is positive, is accomplished by first proving a useful upper bound for the minimum of idle servers and cumulative queue size for the $n^{\mathrm{t}h}$ system, and then using this to derive the Foster–Lyapunov drift inequalities in the region of the state space where the drifts of the diffusion limit and the $n^{\mathrm{t}h}$ system do not match. This facilitates establishing the drift inequalities for the diffusion-scaled processes. As a consequence of the Foster-Lyapunov equations, the invariant probability measures of the diffusion-scaled queueing processes have uniform exponential tails.

The property of interchange of limits attests to the validity of the diffusion approximation for the queueing network. For stochastic networks in the conventional heavy traffic regime, we refer the readers to the papers [gamarnik-zeevi, Budhiraja-09, Gurvich-16, YY-16, YY-18, BDM-17] and references therein. For the ‘V’ network in the Halfin–Whitt regime, interchange of limits is established in [GS12, AHP18]. For the ‘N’ network, Stolyar [Stolyar-15b] has shown the interchange of limits under a specific static priority policy. This property also holds for networks with pool-dependent service rates under the LQFS-LB scheduling policy, as shown in [Stolyar-Yudovina-13, Section 7.2]. Stolyar and Yudovina [Stolyar-Yudovina-12] and Stolyar [Stolyar-15b] then proved tightness of the stationary distributions and interchange of limits of a leaf-activity priority policy in the sub-diffusion and diffusion scales, respectively, in the underloaded regime. This paper contributes to this literature by establishing that the limit of the diffusion-scaled invariant distributions is equal to the invariant distribution of the limiting diffusion process for the large classes of networks considered under any stationary Markov policy (see Remark 5.3).

1.1. Organization of the paper

In the next subsection, we summarize the notation used in the paper. In Subsection 2.1, we describe the model and state informally the assumptions used. We define the diffusion scaled processes, and characterize the corresponding controlled generator in Subsection 2.2. In Subsection 2.3, the notion of system-wide work conserving policies is introduced, and this is used in Subsection 2.4 to take limits and establish the diffusion approximation. In Section 3, we define the parameter of spare capacity ($\varrho$) for multiclass multi-pool networks and show that whenever $\varrho<0$, the process is transient under any stationary Markov control both for the diffusion limit and the $n^{\mathrm{th}}$ system for the models under consideration. In the same subsection, we establish the relation between the spare capacity and average idleness. In Section 4 we first provide equivalent characterizations of uniform exponential ergodicity of controlled diffusions, and then proceed to establish that the diffusion limits of the aforementioned classes of networks are uniformly exponentially ergodic and their invariant probability measures have uniform exponential tails. Finally, Section 5 is devoted to the study of uniform exponential ergodicity of the $n^{\mathrm{th}}$ system of networks under consideration.

1.2. Notation

We use $\mathbb{R}^{m}$ (and $\mathbb{R}^{m}_{+}$), $m\geq 1$, to denote real-valued $m$-dimensional (nonnegative) vectors, and write $\mathbb{R}$ for the real line. We use $z^{\mathsf{T}}$ to denote the transpose of a vector $z\in{\mathbb{R}^{m}}$. Throughout the paper $e\in{\mathbb{R}^{m}}$ stands for the vector whose elements are equal to $1$, that is, $e=(1,\dotsc,1)^{\mathsf{T}}$, and $e_{i}\in{\mathbb{R}^{m}}$ denotes the vector whose elements are all [math] except for the $i^{\mathrm{th}}$ element which is equal to $1$. For $x,y\in\mathbb{R}$, $x\vee y=\max\{x,y\}$, $x\wedge y=\min\{x,y\}$, $x^{+}=\max\{x,0\}$ and $x^{-}=\max\{-x,0\}$.

For a set $A\subseteq\mathbb{R}^{m}$, we use $A^{\mathsf{c}}$, $\partial A$, and $\mathds{1}_{A}$ to denote the complement, the boundary, and the indicator function of $A$, respectively. A ball of radius $r>0$ in $\mathbb{R}^{m}$ around a point $x$ is denoted by ${\mathscr{B}}_{r}(x)$, or simply as ${\mathscr{B}}_{r}$ if $x=0$. We also let ${\mathscr{B}}\equiv{\mathscr{B}}_{1}$. The Euclidean norm on $\mathbb{R}^{m}$ is denoted by $\lvert\,\cdot\,\rvert$, and $\langle\cdot\,,\,\cdot\rangle$ stands for the inner product. For $x\in\mathbb{R}^{m}$, we let $\lVert x\rVert_{1}\coloneqq\sum_{i}\lvert x_{i}\rvert$, and by $K_{r}$, or $K(r)$, for $r>0$, we denote the closed cube

[TABLE]

Also, we define $x_{\mathsf{max}}\coloneqq\max_{i}\,x_{i}$, and $x_{\mathsf{min}}\coloneqq\min_{i}\,x_{i}$, and x^{\pm}\coloneqq\bigl{(}x_{1}^{\pm},\dotsc,x_{m}^{\pm}\bigr{)}.

For a finite signed measure $\nu$ on ${\mathbb{R}^{m}}$, and a Borel measurable $f\colon{\mathbb{R}^{m}}\to[1,\infty)$, the $f$-norm of $\nu$ is defined by

[TABLE]

where ${\mathcal{B}}({\mathbb{R}^{m}})$ denotes the class of Borel measurable functions on ${\mathbb{R}^{m}}$.

2. The queueing network model and the diffusion limit

In this section, we consider a sequence of parallel server networks whose processes, parameters, and variables are indexed by $n$. We recall some of the definitions and notations used in [AP16, AP19].

2.1. Model and assumptions

Consider a general Markovian parallel server (multiclass multi-pool) network with $m$ classes of customers and $J$ server pools. Customer classes take values in ${\mathcal{I}}=\{1,\dotsc,m\}$ and server pools in ${\mathcal{J}}=\{1,\dotsc,J\}$. Forming their own queue, customers of each class are served according to a First-Come-First-Served (FCFS) service discipline. We assume throughout the paper that customers do not abandon. For all $i\in{\mathcal{I}}$, let ${\mathcal{J}}(i)$ denote the subset of server pools that can serve customer class $i$. On the other hand, for all $j\in{\mathcal{J}}$, let ${\mathcal{I}}(j)$ be the subset of customer classes that can be served by server pool $j$.

We form a bipartite undirected graph ${\mathcal{G}}=({\mathcal{I}}\cup{\mathcal{J}},{\mathcal{E}})$ with a set of edges defined by ${\mathcal{E}}=\{(i,j)\in{\mathcal{I}}\times{\mathcal{J}}\colon j\in{\mathcal{J}}(i)\}$, and use the notation $i\sim j$, if $(i,j)\in{\mathcal{E}}$, and $i\nsim j$, otherwise. We assume that the graph ${\mathcal{G}}$ is a tree. We define

[TABLE]

and analogously define $\mathbb{R}^{{\mathcal{G}}}$, $\mathbb{Z}^{{\mathcal{G}}}_{+}$, and $\mathbb{Z}^{{\mathcal{G}}}$.

In each server pool $j$, we let $N_{j}^{n}$ be the number of servers, and assume that the servers are statistically identical. For each $i\in{\mathcal{I}}$, class $i$ customer arrives according to a Poisson process with arrival rate $\lambda_{i}^{n}>0$. These customers are served at an exponential rate $\mu_{ij}^{n}>0$ at server pool $j$ if $j\in{\mathcal{J}}(i)$, and $\mu_{ij}^{n}=0$ otherwise. Finally, we assume that the arrival and service processes of all classes are mutually independent. We study these networks in the Halfin–Whitt regime, which involves the following assumption on the parameters. There exist positive constants $\lambda_{i}$ and $\nu_{j}$, nonnegative constants $\mu_{ij}$, with $\mu_{ij}>0$ for $i\sim j$ and $\mu_{ij}=0$ for $i\nsim j$, and constants $\hat{\lambda}_{i}$, $\hat{\mu}_{ij}$ and $\hat{\nu}_{j}$, such that the following limits exist as $n\to\infty$:

[TABLE]

The parameters $\lambda_{i}$ and $\mu_{ij}$ are the limiting arrival and service rates, $\nu_{j}$ is the limiting service capacity in pool $j$ in the fluid scale, while the parameters $\hat{\lambda}_{i}$, $\hat{\mu}_{ij}$ and $\hat{\nu}_{j}$ are the associated limits in the diffusion scale.

An additional standard assumption referred to as the complete resource pooling condition [williams-2000, Atar-05b] concerns the fluid scale equilibrium, and is stated as follows. The linear program (LP) given by

[TABLE]

has a unique solution $\xi^{*}=[\xi^{*}_{ij}]\in\mathbb{R}^{{\mathcal{G}}}_{+}$ satisfying

[TABLE]

We define $x^{*}\in\mathbb{R}^{m}$, and $z^{*}\in\mathbb{R}^{{\mathcal{G}}}_{+}$ by

[TABLE]

The quantity $\xi^{*}_{ij}$ represents the fraction of servers in pool $j$ allocated to class $i$ in the fluid equilibrium, $x^{*}_{i}$ represents the total number of class $i$ customers in the system, and $z^{*}_{ij}$ is the number of class $i$ customers in pool $j$. Note that the constraint in (2.3) is the rate balance equation for each class $i$ with allocations in each service pool $j$. Also, $\rho_{j}:=\sum_{i}\xi_{ij}$ can be interpreted as the traffic intensity in pool $j$, hence the condition in (2.4) implies that each pool is critically loaded.

For each $i\in{\mathcal{I}}$ and $j\in{\mathcal{J}}$, we let $X_{i}^{n}=\{X_{i}^{n}(t)\colon t\geq 0\}$ denote the total number of class $i$ customers in the system (both in service and in queue), $Z_{ij}^{n}=\{Z_{ij}^{n}(t),\,t\geq 0\}$ the number of class $i$ customers currently being served in pool $j$, $Q_{i}^{n}=\{Q_{i}^{n}(t),\,t\geq 0\}$ the number of class $i$ customers in the queue, and $Y_{j}^{n}=\{Y_{j}^{n}(t),\,t\geq 0\}$ the number of idle servers in server pool $j$. Let $X^{n}=(X_{i}^{n})_{i\in{\mathcal{I}}}$, $Y^{n}=(Y_{j}^{n})_{j\in{\mathcal{J}}}$, $Q^{n}=(Q_{i}^{n})_{i\in{\mathcal{I}}}$, and $Z^{n}=(Z_{ij}^{n})_{i\in{\mathcal{I}},\,j\in{\mathcal{J}}}$. The process $Z^{n}$ is the scheduling control. We have clearly the following balance equations

[TABLE]

Dropping the explicit dependence on $n$ for simplicity, let $(x,z)\in\mathbb{Z}_{+}^{m}\times\mathbb{Z}^{{\mathcal{G}}}_{+}$ denote a state-action pair. We rewrite Eq. 2.6 as

[TABLE]

and define the action space ${\mathcal{Z}^{n}}(x)$ by

[TABLE]

Note that this space consists of work-conserving actions only. It should be noted here that there is an abuse of notation in Eq. 2.7. The quantities $q_{i}(x,z)$ and $y_{j}(z)$ still represent the number of class $i$ customers in the queue and the number of idle servers in pool $j$ respectively. Equation Eq. 2.7 is used to show the dependence on $x$ and $z$ through the balance equations.

2.2. Diffusion scaling

With $\xi^{*}\in\mathbb{R}^{\mathcal{G}}_{+}$ the solution of the (LP), we define the centering quantities of the diffusion-scaled processes $\bar{z}^{n}\in\mathbb{R}^{{\mathcal{G}}}_{+}$ and $\bar{x}^{n}\in\mathbb{R}^{m}$ by

[TABLE]

and

[TABLE]

Using Eqs. 2.6 and 2.9, these obey the centered balance equations

[TABLE]

We introduce suitable notation in the diffusion scale as follows (see [AP19, Definition 2.3]).

For $x\in\mathbb{Z}^{m}_{+}$ and $z\in{\mathcal{Z}^{n}}(x)$, we define

[TABLE]

and let ${\mathscr{S}}^{n}$ denote the state space in the diffusion scale, that is,

[TABLE]

It is clear that the diffusion-scaled work-conserving action space ${\hat{\mathcal{Z}}^{n}}(\hat{x})$ takes the form

[TABLE]

Recall that a scheduling policy is called stationary Markov if $Z^{n}(t)=z(X^{n}(t))$ for some function $z\colon\mathbb{Z}^{m}_{+}\to\mathbb{Z}^{{\mathcal{G}}}_{+}$, in which case we identify the policy with the function $z$. Under a stationary Markov policy, $X^{n}$ is Markov with controlled generator

[TABLE]

for $f\in\mathcal{C}(\mathbb{R}^{m})$ and $x\in\mathbb{Z}^{m}_{+}$. Let $\ell^{n}=(\ell^{n}_{1},\dotsc,\ell^{n}_{m})^{\mathsf{T}}$ be defined by

[TABLE]

By Eq. 2.8, the assumptions on the parameters in Eqs. 2.2 and 2.3, we have

[TABLE]

with $z^{*}$ as in Eq. 2.5. Let $\ell\coloneqq(\ell_{1},\dotsc,\ell_{m})^{\mathsf{T}}$. Note that $\ell_{i}^{n}$ and $\ell_{i}$ can be regarded as the deficit or surplus in the number of servers (of order ${\mathscr{O}}(\sqrt{n}$) allocated to class $i$ in the diffusion scale. Note also that $\ell_{i}^{n}$ and $\ell_{i}$ appears as constants in the drift of the diffusion-scaled and diffusion limit processes; see Eqs. 2.25 and 2.30

We drop the dependence on $n$ in the diffusion-scaled variables in order to simplify the notation. A work-conserving stationary Markov policy $z$, that is a map $z\colon\mathbb{Z}^{m}_{+}\to\mathbb{Z}^{{\mathcal{G}}}_{+}$ such that $z(x)\in{\mathcal{Z}^{n}}(x)$ for all $x\in\mathbb{Z}_{+}^{m}$, gives rise to a policy $\hat{z}\colon{\mathscr{S}}^{n}\to\mathbb{R}^{\mathcal{G}}$, with $\hat{z}(\hat{x})\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$ for all $\hat{x}\in{\mathscr{S}}^{n}$, via Eq. 2.11 (and vice-versa). Using Eqs. 2.13, 2.9, and 2.14 and rearranging terms, the controlled generator of the corresponding diffusion-scaled process can be written as

[TABLE]

where $\mathfrak{d}f$ is given by

[TABLE]

and the ‘drift’ $b^{n}=(b_{1}^{n},\cdots,b_{m}^{n})^{\mathsf{T}}$ is given by

[TABLE]

Abusing the notation for $\hat{x}\in{\mathscr{S}}^{n}$ and $\hat{z}\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$, we define (compare with Eq. 2.10)

[TABLE]

and

[TABLE]

Recall Eq. 2.9. The parameter $\hat{\vartheta}^{n}$ can therefore be regarded as the scaled minimum of the total number of customers in the queues and the total number of idle servers.

By Eq. 2.17, we have

[TABLE]

for all $\hat{x}\in{\mathscr{S}}^{n}$ and $\hat{z}\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$. Define the $(m-1)$ and $(J-1)$ simplexes

[TABLE]

and let $\varDelta\coloneqq\varDelta_{c}\times\varDelta_{s}$. By Eq. 2.19, there exists $u=(u^{c},u^{s})\in\varDelta$ such that

[TABLE]

Let

[TABLE]

As shown in [Atar-05a, Proposition A.2], there exists a unique linear map $\Phi=[\Phi_{ij}]\colon\mathcal{D}\to\mathbb{R}^{{\mathcal{G}}}$ solving

[TABLE]

The solutions $\Phi_{ij}$ correspond to the resource allocations $\hat{z}_{ij}$ in the diffusion scale, see (2.23). Since \bigl{(}\hat{x}-\hat{q}^{n}(\hat{x},\hat{z}),-\hat{y}^{n}(\hat{z})\bigr{)}\in\mathcal{D} by Eqs. 2.7 and 2.17, using the linearity of the map $\Phi$ and Eqs. 2.21 and 2.22, it follows that

[TABLE]

We describe an important property of the linear map $\Phi$ which we need later. Consider the matrices $B_{1}^{n}\in\mathbb{R}^{m\times m}$ and $B_{2}^{n}\in\mathbb{R}^{m\times J}$ defined by

[TABLE]

It is clear that for $B_{1}^{n}$ to be a nonsingular matrix the basis used in the representation of the linear map $\Phi$ should be of the form \mathfrak{D}=\bigl{(}\alpha,(\beta)_{-j}\bigr{)}, $j\in{\mathcal{J}}$, where $(\beta)_{-j}=\{\beta_{\ell}\,,\,\ell\neq j\}$. Since $\Phi$ has a unique representation in terms of such a basis, and since $B_{i}^{n}$, $i=1,2$, are determined uniquely from $\Phi$ by Eq. 2.24, abusing the terminology, we refer to such an $\mathfrak{D}$ as a basis for $B_{i}^{n}$, $i=1,2$. In [AP16, Lemma 4.3], the following property is asserted: Given any $\hat{\imath}\in{\mathcal{I}}$, there exists an ordering of $\{\alpha_{i}\,,\;i\in{\mathcal{I}}\}$ with $\alpha_{\hat{\imath}}$ the last element, and $\hat{\jmath}\in{\mathcal{J}}$, such that the matrix $B_{1}^{n}$ is lower diagonal with positive diagonal elements with respect to this ordered basis \bigl{(}\alpha,(\beta)_{-\hat{\jmath}}\bigr{)}. For more details, we refer the reader to [AP16, Section 4.1].

In view of Eqs. 2.23 and 2.24, for any $\hat{z}\in{\widetilde{\mathcal{Z}}^{n}}(\hat{x})$ with $\hat{x}\in{\mathscr{S}}^{n}$, there exists $u=u(\hat{x},\hat{z})\in\varDelta$ such that the drift $b^{n}$ in Eq. 2.16 takes the form

[TABLE]

2.3. Joint and system-wide work conservation

We start with the following definition.

Definition 2.1**.**

We say that an action $\hat{z}\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$ is jointly work conserving (JWC), if $\hat{\vartheta}^{n}(\hat{x},\hat{z})=0$. Recall that a work conserving policy refers to an action in which a server is idle if and only if there is no customer waiting in the queue that this server can serve. A jointly work conserving action keeps all servers busy unless all queues are empty. Let

[TABLE]

and

[TABLE]

We refer to ${\widetilde{\mathcal{Z}}^{n}}(\hat{x})$ as the system-wide work conserving (SWC) action set at $\hat{x}$. A stationary Markov scheduling policy $\hat{z}$ is called SWC if $\hat{z}(\hat{x})\in{\widetilde{\mathcal{Z}}^{n}}(\hat{x})$ for all $\hat{x}\in{\mathscr{S}}^{n}$. We let ${\widetilde{\mathfrak{Z}}^{n}}$ denote the class of all such policies. Since $z$ and $\hat{z}$ are related by Eq. 2.11, abusing this terminology, we also refer to a Markov policy $z\colon\mathbb{Z}_{+}^{m}\to\mathbb{Z}_{+}^{\mathcal{G}}$ as SWC, if it satisfies

[TABLE]

and we write $z\in{\widetilde{\mathfrak{Z}}^{n}}$.

We recall [Atar-05b, Lemma 3] which states that there exists $M_{0}>0$ such that the collection of sets $\breve{{\mathscr{X}}}^{n}$ defined by

[TABLE]

has the following property. If $\hat{x}\in\breve{{\mathscr{X}}}^{n}$, then for any pair $(\hat{q},\hat{y})$ such that $\sqrt{n}\hat{q}\in\mathbb{Z}^{m}_{+}$, $\sqrt{n}\hat{y}\in\mathbb{Z}^{J}_{+}$, and satisfying

[TABLE]

it holds that $\Phi(\hat{x}-\hat{q},-\hat{y})\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$. It follows from this lemma and Definition 2.1 that if $\hat{x}\in\breve{{\mathscr{X}}}^{n}$, then the actions in ${\widetilde{\mathcal{Z}}^{n}}(\hat{x})$ are JWC.

Remark 2.1*.*

Using Eq. 2.19, we know that under the JWC condition

[TABLE]

Rewriting Eq. 2.21 under the JWC condition, we therefore have the following

[TABLE]

and

[TABLE]

Hence, one can see that the control $u^{c}_{i}$ represents the proportion of the total queue length in the network at queue $i$, while $u^{s}_{j}$ represents the proportion of the total number of idle servers in the network at pool $j$.

Recall from Definition 2.1 that a JWC action keeps all servers busy unless all queues are empty. It is clear that this cannot be always enforced in multiclass multi-pool networks over the entire state space. A SWC control enforces the complementarity between overall queue length and service.

2.4. The diffusion limit

The diffusion approximation or diffusion limit of the queueing model described above is an $m$-dimensional stochastic differential equation (SDE) of the form (see [Atar-05a] and [Atar-05b, Section 2.5])

[TABLE]

Here, ${\{W_{t}\}_{t\geq 0}}$ is a standard $m$-dimensional Brownian motion, and the control $U_{t}$ takes values in the set $\varDelta=\varDelta_{c}\times\varDelta_{s}$ defined in Eq. 2.20. The drift $b$ can be derived as follows. Recall $\mathbb{R}^{{\mathcal{G}}}$ in (2.1). For $u=(u^{c},u^{s})\in\varDelta$, let $\widehat{\Phi}[u]\colon{\mathbb{R}^{m}}\to\mathbb{R}^{{\mathcal{G}}}$ be defined by

[TABLE]

with $\Phi$ as defined in Eq. 2.22. Then the drift $b$ takes the form

[TABLE]

By [AP16, Lemma 4.3], we also know that Eq. 2.29 can be expressed as

[TABLE]

where $B_{1}\in\mathbb{R}^{m\times m}$ is a lower diagonal matrix with positive diagonal elements, and $B_{2}\in\mathbb{R}^{m\times J}$. Of course $B_{i}$ in Eq. 2.30 and $B_{i}^{n}$ in Eq. 2.25, $i=1,2$, have the same functional form with respect to $\{\mu_{ij}\}$ and $\{\mu^{n}_{ij}\}$, respectively.

The diffusion matrix $\upsigma\in\mathbb{R}^{m\times m}$ is constant, and

[TABLE]

In addition, for $f\in\mathcal{C}^{2}(\mathbb{R}^{m})$, we define

[TABLE]

with $\nabla^{2}f$ denoting the Hessian of $f$.

In the following, we describe the drift of the diffusion limit of the networks under consideration while showing some examples in LABEL:{fig-networks}.

2.4.1. Networks with a dominant server pool

This network has one non-leaf server node, which, without loss of generality, we label as $j=1$. As in Subsection 2.1, the customer nodes are denoted by ${\mathcal{I}}=\{1,2,\dotsc,m\}$, and the server nodes by ${\mathcal{J}}=\{1,2,\dotsc,J\}$. Recall that ${\mathcal{J}}(i)$ is the collection of sever nodes connected to customer $i$. Owing to the tree structure of the network, server $1\in{\mathcal{J}}(i)$ for all $i\in\{1,2,\dotsc,m\}$. Let ${\mathcal{J}}_{1}(i)\coloneqq{\mathcal{J}}(i)\setminus\{1\}$ for all $i\in{\mathcal{I}}$. Recall the form of the drift in Eq. 2.29. Using Eq. 2.22, it is simple to show that the matrix $\widehat{\Phi}_{ij}[u]$ for this network is given by

[TABLE]

Using Eq. 2.32, the drift takes the following simple form:

[TABLE]

with $\eta_{ij}\coloneqq\frac{\mu_{ij}}{\mu_{i1}}$ for $j\in{\mathcal{J}}_{1}(i)$ and $i\in{\mathcal{I}}$. Note that $B_{1}=\operatorname{diag}(\mu_{11},\dotsc,\mu_{m1})$, and so $\ell=-\frac{\varrho}{m}B_{1}e$, where $\varrho$ is given by Eq. 2.35. We define

[TABLE]

2.4.2. Networks with class-dependent service rates

We consider in this part arbitrary tree networks where the service rates are dictated by the customer type; namely $\mu_{ij}\,=\,\mu_{i}$ for all $(i,j)\in{\mathcal{E}}$. Recall the definition in Eq. 2.28. Using Eqs. 2.22 and 2.29, the drift of this network takes the form

[TABLE]

Note then that $B_{1}=\operatorname{diag}(\mu_{1},\dots,\mu_{m})$ and $B_{2}=0$.

We remark that both classes of networks have one common feature: the matrix $B_{1}$ is diagonal.

2.5. A necessary and sufficient condition for uniform stability

We define the spare capacity (or the safety staffing) for the $n^{\mathrm{th}}$ system (prelimit) and the diffusion limit by

[TABLE]

respectively. Note, of course, that $\varrho_{n}\to\varrho$ as $n\to\infty$ by Eq. 2.2. Recall the expressions of $\ell^{n}$ and $\ell$ in Eq. 2.14. Recall that $\ell_{i}^{n}$ and $\ell_{i}$ can be regarded as the deficit or surplus in the number of servers (of order ${\mathscr{O}}(\sqrt{n}$) allocated to class $i$ in the diffusion scale. Hence $\varrho_{n}$ and $\varrho$ 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 summarize the main results of the paper.

Theorem 2.1**.**

Consider a network with a dominant server pool, or with class-dependent service rates. Then the conditions $\varrho>0$ and $\varrho_{n}>0$ are necessary and sufficient for the uniform stability of the limiting diffusion and the diffusion-scaled queueing processes, respectively. More precisely:

• (i)

if $\varrho<0$, the process ${\{X_{t}\}_{t\geq 0}}$ in Eq. 2.27 is transient under any stationary Markov control. In addition, if $\varrho=0$, then ${\{X_{t}\}_{t\geq 0}}$ cannot be positive recurrent.

• (ii)

if $\varrho_{n}<0$, the process ${\{X^{n}_{t}\}_{t\geq 0}}$ is transient under any stationary Markov scheduling policy. In addition, if $\varrho_{n}=0$, then ${\{X^{n}_{t}\}_{t\geq 0}}$ cannot be positive recurrent.

• (iii)

if $\varrho>0$, the processes ${\{X_{t}\}_{t\geq 0}}$ are uniformly exponentially ergodic over stationary Markov controls.

• (iv)

if $\varrho_{n}>0$, the processes ${\{X^{n}_{t}\}_{t\geq 0}}$ are uniformly exponentially ergodic over SWC scheduling policies, and the invariant distributions have exponential tails.

Parts (i) and (ii) of Theorem 2.1 follow from Propositions 3.1 and 3.2, respectively. Part (iii) follows from Theorem 4.2, and part (iv) from Theorem 5.1.

3. Two properties of the spare capacity and transience

In the first part of this section we prove the results in Theorem 2.1 (i) and (ii). It is important to note that for the models in Subsections 2.4.1 and 2.4.2 we have

[TABLE]

Note that for networks with a dominant server pool, we have $1=u_{1}^{s}+\sum_{i\in{\mathcal{I}}}\sum_{j\in{\mathcal{J}}_{1}(i)}u_{j}^{s}$. Hence

[TABLE]

Note also that for networks with a class dependent service rate we have established that $B_{2}=0$. Hence $1+\langle e,B_{1}^{-1}B_{2}u^{s}\rangle=1>0$.

Proposition 3.1**.**

Suppose that $\varrho=-\langle e,B_{1}^{-1}\ell\rangle<0$. Then the process ${\{X_{t}\}_{t\geq 0}}$ in Eq. 2.27 is transient under any stationary Markov control. In addition, if $\varrho=0$, then ${\{X_{t}\}_{t\geq 0}}$ cannot be positive recurrent.

Proof.

Recall that the first and second order derivatives of the hyperbolic tangent function are

[TABLE]

Let H(x)\coloneqq\tanh\bigl{(}\beta\langle e,B_{1}^{-1}x\rangle\bigr{)}, with $\beta>0$. Then

[TABLE]

We have

[TABLE]

Thus, for $0<\beta<\langle e,B_{1}^{-1}\ell\rangle\,\lvert\upsigma^{\mathsf{T}}B_{1}^{-1}e\rvert^{-2}$, we obtain ${\mathcal{L}}_{u}H(x)>0$ by Eq. 3.1. Therefore, \{H\bigl{(}X_{t}\bigr{)}\}_{t\geq 0} is a bounded submartingale, so it converges almost surely. Since $X$ is irreducible, it can be either recurrent or transient. If it is recurrent, then $H$ should be constant a.e. in $\mathbb{R}^{m}$, which is not the case. Thus $X$ is transient.

We now turn to the case where $\varrho=0$. Suppose that the process $\{X(t)\}_{t\geq 0}$ (under some stationary Markov control) has an invariant probability measure $\uppi(\mathrm{d}{x})$. It is well known that $\uppi$ must have a positive density. Let $h_{1}(x)$ and $h_{2}(x)$ denote respectively the first and the second terms on the right hand side of Eq. 3.2. Applying Itô’s formula to Eq. 3.2 with $X_{0}=x$ as in Eq. 2.27, we obtain

[TABLE]

where $\tau_{r}$ denotes the first exit time from ${\mathscr{B}}_{r}$, $r>0$. Note that $h_{1}(x)$ is bounded and $h_{2}(x)$ is non-negative. Thus using dominated and monotone convergence, we can take limits in Eq. 3.3 as $r\to\infty$ for the terms on the right side to obtain

[TABLE]

Since $H(x)$ is bounded, we can divide both sides by $t$ and $\beta$ and take the limit as $t\to\infty$ to get

[TABLE]

Since $\beta^{-1}h_{1}(x)$ tends to 0 uniformly in $x$ as $\beta\searrow 0$, the first term on the left hand side of Eq. 3.4 vanishes as $\beta\searrow 0$. However, since $\beta^{-1}h_{2}(x)$ is bounded away from [math] on the open set $\{x\in{\mathbb{R}^{m}}\colon\langle e,x\rangle^{-}>1\}$, this contradicts the fact that $\uppi(\mathrm{d}{x})$ has full support. ∎

Remark 3.1*.*

In the proof of Proposition 3.1, the function $H(x)$ is a bounded test function which was chosen so that ${\mathcal{L}}_{u}H(x)\geq 0$. Assume that $X_{t}$ is recurrent. Since $H(X_{t})$ converges a.s. (being a bounded submartingale), and since $X_{t}$ ‘visits every open neighborhood’ in ${\mathbb{R}^{m}}$ with probability $1$, it follows that $H$ must be a constant function which is a contradiction.

The formalism behind the above argument is as follows: Let $\uptau$ be the first hitting time to the unit ball $B$ centered at $x=0$. If $\operatorname{\mathbb{E}}_{x}$ denotes the expectation operator on the canonical space of the Markov process $\{X_{t}\}_{t\geq 0}$ with initial condition $X_{0}=x$, then by Dynkin’s formula we obtain $\operatorname{\mathbb{E}}_{x}[H(X_{\uptau})]\geq H(x)$. If the process is recurrent, then of course $\operatorname{\mathbb{P}}_{x}(\uptau<\infty)=1$, which gives $\operatorname{\mathbb{E}}_{x}[H(X_{\uptau})]\leq\sup_{y\in B}\,H(y)$. Thus $\sup_{y\in B}\,H(y)\geq H(x)$ for all $x\in{\mathbb{R}^{m}}$ which is not true. Moving the center of the the ball $B$ to an arbitrary point $z$, and denoting it as $B(z)$, we similarly have

[TABLE]

This implies that $H(x)=$ constant which is a contradiction and hence $X$ is transient.

Proposition 3.2**.**

Suppose that $\varrho_{n}<0$. Then the state process ${\{X^{n}_{t}\}_{t\geq 0}}$ of the $n^{\mathrm{th}}$ system is transient under any stationary Markov scheduling policy. In addition, if $\varrho^{n}=0$, then ${\{X^{n}_{t}\}_{t\geq 0}}$ cannot be positive recurrent.

Proof.

The proof mimics that of Proposition 3.1. We apply the function $H$ in that proof to the operator $\widehat{\mathscr{L}}_{\hat{z}}^{n}$ in Eq. 2.15, and use the identity

[TABLE]

to express the first and second order incremental quotients, together with Eq. 2.25 which implies that

[TABLE]

The rest follows exactly as in the proof of Proposition 3.1. ∎

3.1. Spare capacity and average idleness

It is shown in [AHP18, APS19] that if the diffusion limit of the ‘V’ network with no abandonment has a invariant distribution $\uppi$ under some stationary Markov control, then $\varrho$ represents the ‘average idleness’ of the system, that is, $\varrho=\int_{{\mathbb{R}^{m}}}\langle e,x\rangle^{-}\uppi(\mathrm{d}{x})$. In calculating this average for multi-pool networks, idle servers are not weighted equally across different pools and the term \bigl{\langle}e,B_{1}^{-1}B_{2}u^{s}(x)\bigr{\rangle} appears in the expression, see Eq. 3.6. It is important to note that only the control on the idleness allocations among server pools $u^{s}$ appears in the identity, and the control component $u^{c}$ does not.

Theorem 3.1**.**

Consider a network with a dominant server pool, or with class-dependent service rates, and suppose that $\varrho>0$. Let $\uppi_{u}$ denote the invariant invariant probability measure corresponding to a stationary Markov control $u\in\mathfrak{U}_{\mathsf{sm}}$, whose existence follows from Theorem 2.1 (iii). Then

[TABLE]

Proof.

The proof is similar to that of [APS19, Corollary 5.1], but more involved for the multiclass multi-pool networks. We first recall some definitions and notations. Let $\chi_{r}(t)$, $\breve{\chi}_{r}(t)$, $r>1$ be smooth, concave and convex functions, respectively, defined by

[TABLE]

Let g_{r}(x)=\breve{\chi}_{r}\bigl{(}\langle e,B_{1}^{-1}x\rangle\bigr{)}, and f_{r}(x)=\chi_{r}\bigl{(}g_{r}(x)\bigr{)}. A straightforward calculation shows that

[TABLE]

where

[TABLE]

We note that $g_{r}(x)$ is positive and bounded below away from [math], and $f_{r}(x)$ is smooth, bounded, and has bounded derivatives. Also note that $h_{i}$, $i=1,2,3$, are bounded, and $h_{2}$ is nonnegative. Therefore, if $\{X(t)\}_{t\geq 0}$ is positive recurrent with an invariant probability measure $\uppi_{u}(\mathrm{d}{x})$, a straightforward application of Itô’s formula shows that \uppi_{u}\bigl{(}{\mathcal{L}}_{u}f_{r}\bigr{)}=0. Therefore, we obtain

[TABLE]

By the definition of $\chi_{r}$ and $\breve{\chi}_{r}$, it is straightforward to verify that

[TABLE]

In addition, using dominated convergence theorem

[TABLE]

Combining Eqs. 3.7, 3.8, and 3.9, we obtain Eq. 3.6. ∎

Remark 3.2*.*

Using Theorem 3.1 and the drift in Subsections 2.4.1 and 2.4.2, it is easy to verify that in the case of a network with a dominant server pool we have

[TABLE]

whereas in the case of a network with class-dependent service rates, Eq. 3.6 takes the form

[TABLE]

4. Uniform exponential ergodicity of the diffusion limit

In this section we show that if $\varrho>0$ then the diffusion limit of a network with a dominant server pool, or with class-dependent service rates, is uniformly exponentially ergodic and the invariant distributions have exponential tails.

We start by reviewing the notion of uniform exponential ergodicity for a controlled diffusion. We do this under fairly general hypotheses. Consider a controlled diffusion process $X=\{X_{t},\,t\geq 0\}$ which takes values in the $m$-dimensional Euclidean space $\mathbb{R}^{m}$, and is governed by the Itô equation

[TABLE]

All random processes in Eq. 4.1 live in a complete probability space $(\Omega,{\mathcal{F}},\operatorname{\mathbb{P}})$. The process $W$ is a $m$-dimensional standard Wiener process independent of the initial condition $X_{0}$. The function $v$ maps $\mathbb{R}^{m}$ to a compact, metrizable set $\mathbb{U}$ and is Borel measurable. The collection of such functions comprising of the set of stationary Markov controls is denoted by $\mathfrak{U}_{\mathsf{sm}}$.

The parameters of the equation (4.1) satisfy the following:

• (1)

Local Lipschitz continuity: The functions $b\colon\mathbb{R}^{m}\times\mathbb{U}\to\mathbb{R}^{m}$ and $\upsigma\colon\mathbb{R}^{m}\to\mathbb{R}^{m\times m}$ are continuous, and satisfy

[TABLE]

for some constant $C_{R}>0$ depending on $R>0$.

• (2)

Affine growth condition: For some $C_{0}>0$, we have

[TABLE]

• (3)

Nondegeneracy: For each $R>0$, it holds that

[TABLE]

and for all $\xi=(\xi_{1},\dotsc,\xi_{m})^{\mathsf{T}}\in\mathbb{R}^{m}$, where $a=\frac{1}{2}\upsigma\upsigma^{\mathsf{T}}$.

It is well known that, under hypotheses (1)–(2), Eq. 4.1 has a unique strong solution which is also a strong Markov process for any $v\in\mathfrak{U}_{\mathsf{sm}}$ [Gyongy-96]. We let $\operatorname{\mathbb{E}}^{v}_{x}$ denote the expectation operator on the canonical space of the process controlled by $v$, with initial condition $X_{0}=x$. Let $\uptau(A)$ denote the first exit time from the set $A\in{\mathbb{R}^{m}}$.

We say that the process $\{X_{t}\}_{t\geq 0}$ is uniformly exponentially ergodic if for some ball ${\mathscr{B}}_{\circ}$ there exist $\delta_{\circ}>0$ and $x_{\circ}\in\bar{{\mathscr{B}}}_{\circ}^{\mathsf{c}}$ such that $\sup_{v\in\mathfrak{U}_{\mathsf{sm}}}\,\operatorname{\mathbb{E}}_{x_{\circ}}^{v}[\mathrm{e}^{\delta_{\circ}\,\uptau({\mathscr{B}}^{\mathsf{c}}_{\circ})}]<\infty$.

We let $\widehat{\mathcal{A}}$ denote the operator

[TABLE]

for $\phi\in\mathcal{C}^{2}({\mathbb{R}^{m}})$. For a locally bounded, Borel measurable function $f\colon{\mathbb{R}^{m}}\to\mathbb{R}$, which is bounded from below in ${\mathbb{R}^{m}}$, i.e., $\inf_{\mathbb{R}^{m}}f>-\infty$, we define the generalized principal eigenvalue of $\widehat{\mathcal{A}}+f$ by

[TABLE]

where $\mathscr{W}_{\mathrm{loc}}^{2,m}$ is a local Sobolev space. We have the following equivalent characterizations of uniform exponential ergodicity. This is a straightforward extension of [APS19, Theorem 3.1] for controlled diffusions, and is stated without proof. Recall that a map $f\colon{\mathbb{R}^{m}}\to\mathbb{R}$ is called coercive, or inf-compact, if $\inf_{{\mathscr{B}}_{r}^{\mathsf{c}}}\,f\to\infty$ as $r\to\infty$.

Theorem 4.1**.**

The following are equivalent.

1. (a)

For some ball ${\mathscr{B}}_{\circ}$ there exists $\delta_{\circ}>0$ and $x_{\circ}\in\bar{{\mathscr{B}}}_{\circ}^{\mathsf{c}}$ such that $\sup_{v\in\mathfrak{U}_{\mathsf{sm}}}\,\operatorname{\mathbb{E}}_{x_{\circ}}[\mathrm{e}^{\delta_{\circ}\,\uptau({\mathscr{B}}^{\mathsf{c}}_{\circ})}]<\infty$. 2. (b)

For every ball ${\mathscr{B}}$ there exists $\delta>0$ such that $\sup_{v\in\mathfrak{U}_{\mathsf{sm}}}\,\operatorname{\mathbb{E}}_{x}^{v}[\mathrm{e}^{\delta\,\uptau({\mathscr{B}}^{\mathsf{c}})}]<\infty$ for all $x\in{\mathscr{B}}^{\mathsf{c}}$. 3. (c)

For every ball ${\mathscr{B}}$, there exists a coercive function ${\mathscr{V}}\in\mathscr{W}_{\mathrm{loc}}^{2,p}({\mathbb{R}^{m}})$, $p>d$, with $\inf_{\mathbb{R}^{m}}\,{\mathscr{V}}\geq 1$, and positive constants $\kappa_{0}$ and $\delta$ such that

[TABLE] 4. (d)

Equation 4.1* is recurrent, and $\Lambda(\mathds{1}_{{\mathscr{B}}^{\mathsf{c}}})<1$ for every ball ${\mathscr{B}}$.*

Remark 4.1*.*

Recall Eq. 1.2. Let $P^{v}_{t}(x,\mathrm{d}{y})$ denote the transition probability of ${\{X_{t}\}_{t\geq 0}}$ in Eq. 4.1 under a control $v\in\mathfrak{U}_{\mathsf{sm}}$. It is well known that Eq. 4.2 implies that there exist constants $\gamma$ and $C_{\gamma}$ which do not depend on the control $v$ chosen, such that

[TABLE]

where $\uppi_{v}$ denotes the invariant probability measure of ${\{X_{t}\}_{t\geq 0}}$ under the control $v$. In addition, for any control $v\in\mathfrak{U}_{\mathsf{sm}}$ we have

[TABLE]

In particular for ${\mathscr{V}}(x)$ being an exponential function, the moment generating function of ${\{X_{t}\}_{t\geq 0}}$ is finite.

4.1. A class of intrinsic Lyapunov functions for the queueing network model

As seen in Theorem 4.1, uniform exponential ergodicity is equivalent to the Foster–Lyapunov inequality in Eq. 4.2. In establishing this property for the diffusion limit of stochastic networks, a proper choice of a Lyapunov function is of tantamount importance. We first describe an intrinsic class of such functions.

We fix a convex function $\psi\in\mathcal{C}^{2}(\mathbb{R})$ with the property that $\psi(t)$ is constant for $t\leq-1$, and $\psi(t)=t$ for $t\geq 0$. This is defined by

[TABLE]

For ${\varepsilon}>0$ we define

[TABLE]

Thus $\psi_{\varepsilon}(t)={\varepsilon}t$ for $t>0$. A simple calculation also shows that $\psi^{\prime\prime}_{\varepsilon}(t)\leq\frac{3}{2}{\varepsilon}^{2}$.

Suppose that $B_{1}=\operatorname{diag}(\tilde{\mu}_{1},\dotsc,\tilde{\mu}_{m})$. Using the function $\psi_{\varepsilon}$ introduced above, we let

[TABLE]

with

[TABLE]

We also define

[TABLE]

with $\theta$ a positive constant.

As a result of fixing the value of ${\varepsilon}$ in Eq. 4.6, $\Psi$ depends only on the parameter $\theta$. This simplifies the statements of the results in the rest of the paper.

We review some useful properties of the function $\psi_{\varepsilon}$. Note that for ${\varepsilon}>0$ we have

[TABLE]

The minimum of $\psi^{\prime}_{\varepsilon}(t)t$ when ${\varepsilon}t\in[-1,0]$ is $-\frac{3(39+55\sqrt{33})}{4096}\geq-\frac{1}{2}$ for ${\varepsilon}t=-\frac{1+\sqrt{33}}{16}$. Therefore

[TABLE]

and similarly $-\sum_{i\in{\mathcal{I}}}\psi^{\prime}_{\varepsilon}(-x_{i})x_{i}\,\geq\,{\varepsilon}\lVert x^{-}\rVert_{1}-\frac{m}{2}$, where ${\mathcal{I}}\coloneqq\{1,2,\dotsc,m\}$. Note also that

[TABLE]

The function $V$ in Eq. 4.7, scaled by the parameter $\theta$ which are suppressed in the notation, is our choice of a Lyapunov function when $B_{1}$ is a diagonal matrix. It is constructed in an intricate manner in order to capture both the total workload (using $\Psi(x)$) on the positive half-space and the idleness (using $\Psi(-x)$) on the negative half-space. As one cannot simply use $x^{+}$ or $x^{-}$, we must construct mollified smooth functions for them. In addition, one must recognize that the effects of the workload and idleness on the system are not the same (not symmetric), so we also introduce a parameter $\theta$ in the definition of $V_{1}(\cdot)$. The reader will notice the similarities in Eq. 4.7 and [AHP18, Definition 2.2]. However the function $V$ used in this paper is the sum of the two exponential functions $V_{1}$ and $V_{2}$, whereas their product is used in [AHP18, Lemma 2.1]. As will be seen later, in the case of multiclass multi-pool networks, the analysis is considerably more complex. In Subsection 4.3.1 for example, the proofs required $V_{1}\geq V_{2}^{2}$ on a subset of the state space while requiring the opposite inequality on its complement in order to establish the Foster-Lyapunov inequality in Eq. 4.15. See the discussions following Eqs. 4.16 and 4.22. This is mainly the reason behind using the sum instead of the product when defining the function $V$.

In the following subsections, we establish the uniform exponential ergodicity of the networks under consideration. To help with the exposition, we study the ‘N’ network in detail and then proceed to the more general networks with a dominant server pool, and networks with class-dependent service rates.

In establishing the desired drift inequalities, we often partition the space appropriately, and focus on the subsets of the partition. The following cones appear quite often in the analysis.

For $\delta\in[0,1]$, we define the cones

[TABLE]

It is clear that ${\mathcal{K}}_{0}^{+}$ (${\mathcal{K}}_{0}^{-}$) corresponds to the nonnegative (nonpositive) canonical half-space, and ${\mathcal{K}}_{1}^{+}$ (${\mathcal{K}}_{1}^{-}$) is the nonnegative (nonpositive) closed orthant.

The following identities are very useful.

[TABLE]

In addition, it is straightforward to show that

[TABLE]

Also, the following inequality is true in general for any ${\mathcal{I}}^{\prime}\subset{\mathcal{I}}$.

[TABLE]

Remark 4.2*.*

There is an important scaling of the drift which we employ. Note that if we let $\zeta=\frac{\varrho}{m}e+B_{1}^{-1}\ell$, with $\varrho$ as in Eq. 2.35, then a mere translation of the origin of the form $\tilde{X}_{t}=X_{t}+\zeta$ results in a diffusion of with the same drift as Eq. 2.29, except that the vector $\ell$ gets replaced by $\ell=-\frac{\varrho}{m}B_{1}e$. Therefore, we may assume without any loss of generality that the drift in Eq. 2.30 takes the form

[TABLE]

4.2. The Foster–Lyapunov inequality

Recall the definition of the operator ${\mathcal{L}}_{u}$ in Eq. 2.31. We start with the following simple assertion.

Lemma 4.1**.**

Let $V$ be the function in Eq. 4.7 with ${\varepsilon}$ as in Eq. 4.6, and $\theta\geq\theta_{0}$ where $\theta_{0}$ is a constant. Suppose that for any $\delta\in(0,1)$ there exist positive constants $c_{0}$ and $c_{1}$ such that the drift $b$ in Eq. 4.13 satisfies

[TABLE]

Then, there exists a constant $C_{0}$ such that

[TABLE]

Proof.

A straightforward calculation, using the fact that $\psi^{\prime\prime}_{\varepsilon}(t)\leq\frac{3}{2}{\varepsilon}^{2}$, shows that

[TABLE]

Therefore, the choice of ${\varepsilon}$ in Eq. 4.6 implies that \frac{1}{2}\operatorname{trace}\bigl{(}a\nabla^{2}V_{2}(x)\bigr{)}\leq\frac{\varrho{\varepsilon}}{4m}V_{2} for all $x\in\mathbb{R}^{2}$, and thus

[TABLE]

by Eq. 4.14b. Since $\lvert x^{+}\rvert\geq\frac{1+\delta}{1-\delta}\lvert x^{-}\rvert$ for all $x\in{\mathcal{K}}_{\delta}^{+}$, we may select $\delta$ sufficiently close to $1$ such that $V_{2}\geq V_{1}^{2}$ on ${\mathcal{K}}_{\delta}^{+}\cap{\mathcal{K}}_{r}^{\mathsf{c}}$ for some $r>0$. Since $V_{2}$ has exponential growth in $\lVert x\rVert_{1}$ on ${\mathcal{K}}_{\delta}^{+}$ and ${\mathcal{L}}_{u}V_{1}(x)\,\leq\,C(1+\lvert x\rvert_{1})V_{1}(x)$ on ${\mathcal{K}}_{\delta}^{+}\times\varDelta$, it then follows that Eq. 4.15 holds on ${\mathcal{K}}_{\delta}^{+}\times\varDelta$ by Eq. 4.16. It is also clear that Eq. 4.15 also holds on \bigl{(}{\mathcal{K}}_{\delta}^{+}\bigr{)}^{\mathsf{c}}\times\varDelta by Eq. 4.14a. This completes the proof. ∎

We now have the following result.

Theorem 4.2**.**

Assume that $\varrho>0$. Let $V$ be the function in Eq. 4.7 with ${\varepsilon}$ as in Eq. 4.6, and $\theta\geq\theta_{0}$ where $\theta_{0}$ is a constant. Then the diffusion limit of any network with a dominant server pool or with class-dependent service rates is uniformly exponentially ergodic and the invariant distributions have exponential tails. In particular, there exists $C_{0}$ such that

[TABLE]

In particular,

[TABLE]

where $P^{v}_{t}(x,\mathrm{d}{y})$ denote the transition probability of ${\{X_{t}\}_{t\geq 0}}$ under $v=(u^{c},u^{s})$ and $\uppi_{v}$ denotes the invariant probability measure of ${\{X_{t}\}_{t\geq 0}}$ under the control $v$.

Proof.

In Lemmas 4.3 and 4.4 in the section which follows, we establish Eqs. 4.14a and 4.14b for these networks. Thus the proof of Eq. 4.17 follows directly from Lemma 4.1. ∎

4.3. Three technical lemmas

In this section, we prove Eqs. 4.14a and 4.14b for the networks under consideration which implies Theorem 4.2. Even though the ‘N’ network is a special case of the networks with a dominant server pool, we first establish the result for this network in Lemma 4.2 as understanding the results in $\mathbb{R}^{2}$ will definitely help the reader in understanding the equations in Lemmas 4.3 and 4.4.

4.3.1. The case of the ‘N’ network

Here, $m=2$, and the matrices $B_{i}$, $i=1,2$, in Eq. 2.30 are given by

[TABLE]

Thus, using Eq. 4.13, the drift $b\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ for the ‘N’ network is given by

[TABLE]

Note that for the ‘N’ network, we have $\Psi(x)=\frac{\psi_{\varepsilon}(x_{1})}{\mu_{11}}+\frac{\psi_{\varepsilon}(x_{2})}{\mu_{21}}$ by Eq. 4.5. Recall the definition of the cube $K_{r}$ in Eq. 1.1. We have the following lemma that verifies the drift inequalities Eqs. 4.14a and 4.14b for the ‘N’ network.

Lemma 4.2**.**

Consider an ‘N’ network satisfying $\varrho>0$. Let $\delta\in(0,1)$, $\theta\geq\theta_{0}\coloneqq 2(\eta\vee\eta^{-1})$, with $\eta\coloneqq\frac{\mu_{12}}{\mu_{11}}$, and $V(x)$ be as in Eq. 4.7. Then, Eqs. 4.14a and 4.14b hold with $m=2$.

Proof.

To simplify the notation we define

[TABLE]

We use Eq. 4.19, and apply Eq. 4.8 and the inequalities $\frac{\varrho}{2}\sum_{i\in{\mathcal{I}}}\psi^{\prime}_{\varepsilon}\leq\varrho{\varepsilon}$, and

[TABLE]

to obtain

[TABLE]

Similarly, we have

[TABLE]

Note that, due to Eq. 4.11 and the choice of $\theta$, we have $V_{1}\geq V_{2}^{2}$ on ${\mathcal{K}}_{0}^{-}$. Thus, since $V_{1}$ has exponential growth in $\lVert x\rVert_{1}$ on ${\mathcal{K}}_{0}^{-}$, combining Eqs. 4.21 and 4.22 and choosing an appropriate cube $K_{r}$, we obtain

[TABLE]

We continue with estimates on ${\mathcal{K}}_{0}^{+}$. A straightforward calculation shows that

[TABLE]

Again using Eq. 4.8, we have

[TABLE]

We break the estimate of $F_{2}$ in two parts. First, for any $\delta\in(0,1)$, using Eq. 4.8, we obtain

[TABLE]

Combining Eqs. 4.24 and 4.25, we get

[TABLE]

and $\theta\geq 1$, where we use the fact that $\lVert x^{-}\rVert_{1}\geq\frac{1-\delta}{2}\lVert x\rVert_{1}$ on ${\mathcal{K}}_{0}^{+}\setminus{\mathcal{K}}_{\delta}^{+}$ by Eq. 4.10. Thus Eq. 4.14a follows by Eqs. 4.23 and 4.26.

Next, using Eq. 4.8, we have

[TABLE]

and this completes the proof. ∎

4.3.2. The case of networks with a dominant server pool

Consider the class of networks described in Subsection 2.4.1. We have the following lemma.

Lemma 4.3**.**

Consider a network with a dominant server pool, such that $\varrho>0$. Let $\delta\in(0,1)$, $\theta\geq\theta_{0}\coloneqq 2\,\frac{\max_{i}{\mu_{i1}}}{\min_{i}{\mu_{i1}}}$, and $V(x)$ be as in Eq. 4.7. Then, Eqs. 4.14a and 4.14b hold.

Proof.

The method we follow is analogous to the proof of Lemma 4.2. Recall the definitions in Eq. 4.20. A straightforward calculation using Eq. 2.33 shows that

[TABLE]

Let $\underline{\eta}\coloneqq\min_{ij}\eta_{ij}$, and $\bar{\eta}\coloneqq\max_{ij}\eta_{ij}$. Noting that

[TABLE]

it is easy to verify using Eqs. 4.9 and 4.10 that

[TABLE]

Note that the drift equations on ${\mathcal{K}}_{0}^{+}$ are similar to those of the ‘N’ model, with the only exception that $\frac{\varrho}{2}$ is replaced by $\frac{\varrho}{m}$, and the sum ranges from $i=1,\dotsc,m$ instead of $i=1,2$. Hence, we obtain

[TABLE]

and

[TABLE]

for any $\delta\in(0,1)$.

The choice of $\theta$ implies that $V_{1}\geq V_{2}^{2}$ on ${\mathcal{K}}_{0}^{-}$. Thus Eq. 4.14a holds by Eqs. 4.27 and 4.28, while Eq. 4.29 is equivalent to Eq. 4.14b. ∎

4.3.3. The case of networks with class-dependent service rates

Consider the class of networks described in Subsection 2.4.2. Such networks have a limiting diffusion with the same drift structure studied in [AHP18], and that paper shows that when $\varrho>0$, then the diffusion (and the prelimit) is uniformly exponentially ergodic in the presence or absence of abandonment. However, the proof of uniform exponential ergodicity of the prelimit for models with class-dependent service rates does not seem to carry through with the Lyapunov function used in [AHP18]. Thus for the sake of proving the result for the $n^{\mathrm{th}}$ system in Section 5, we adopt here the Lyapunov function in Eq. 4.7.

Lemma 4.4**.**

Consider a network satisfying $\mu_{ij}=\mu_{i}$ for all $i\in{\mathcal{I}}$, and $\varrho>0$. Let $\delta\in(0,1)$, and $\theta\geq\theta_{0}\coloneqq 2\,\frac{\mu_{\mathsf{max}}}{\mu_{\mathsf{min}}}$. Then, Eqs. 4.14a and 4.14b hold.

Proof.

A simple calculation using Eq. 2.34 shows that

[TABLE]

Using Eq. 4.30, we obtain

[TABLE]

Therefore, Eq. 4.14a holds on ${\mathcal{K}}_{0}^{-}\times\varDelta$ by this inequality and the choice of $\theta$.

On ${\mathcal{K}}_{0}^{+}\times\varDelta$, the equations in Eq. 4.30 are identical to the corresponding ones for a network with a dominant server pool, for which the result has already been established in Lemma 4.3. This completes the proof. ∎

5. Uniform exponential ergodicity of the $n^{\mathrm{th}}$

system

In this section we show that if $\varrho_{n}>0$ then the prelimit of a network with a dominant server pool, or with class-dependent service rates, is uniformly exponentially ergodic and the invariant distributions have exponential tails.

Recall that $\{\tilde{\mu}_{i}\,,\;i\in{\mathcal{I}}\}$ are the elements of the diagonal matrix $B_{1}^{n}$ in Eq. 2.25. Throughout this section $V$ denotes the function in Eq. 4.7, with ${\varepsilon}$ given by

[TABLE]

Recall the definition of the operator $\widehat{\mathscr{L}}_{\hat{z}}^{n}$ in Eq. 2.15, and the definitions of ${\mathscr{S}}^{n}$ and ${\widetilde{\mathcal{Z}}^{n}}(\hat{x})$ in Eqs. 2.12 and 2.1. We start with the following simple assertion.

Lemma 5.1**.**

Let $V$ be the function in Eq. 4.7 with ${\varepsilon}$ as in Eq. 5.1, and $\theta$ fixed at some value. Suppose that for any $\delta\in(0,1)$ there exist positive constants $c_{0}$ and $c_{1}$ such that the drift $b^{n}$ in Eq. 2.25 satisfies

[TABLE]

Then, there exists a constant $\widehat{C}_{0}$ such that

[TABLE]

Proof.

A simple calculation shows that

[TABLE]

Thus, using Eq. 3.5 to express the first and second order incremental quotients in Eq. 2.15, we obtain

[TABLE]

The rest follows as in the proof of Lemma 4.1 by selecting $\delta$ sufficiently close to $1$. ∎

Remark 5.1*.*

Recall Eq. 2.25. In direct analogy to Remark 4.2, if we let $\zeta^{n}=\frac{\varrho_{n}}{m}e+(B_{1}^{n})^{-1}\ell^{n}$, with $\varrho_{n}$ as in Eq. 2.35, then a mere translation of the origin of the form $\tilde{X}^{n}=\hat{X}^{n}+\zeta^{n}$ results in a diffusion of with the same drift as Eq. 2.25, except that the vector $\ell^{n}$ gets replaced by $\ell^{n}=-\frac{\varrho_{n}}{m}(B_{1})^{n}e$. Therefore, we may assume without any loss of generality that the drift in Eq. 2.25 takes the form

[TABLE]

Note that this centering has the effect of translating the ‘equilibrium’ allocations $\overline{z}^{n}_{ij}$ given in Eq. 2.8. Since this translation is of ${\mathscr{O}}(n^{-\nicefrac{{1}}{{2}}})$, it has no effect on the results for large $n$. However, in the interest of providing precise estimates we calculate the new values of $\overline{z}^{n}_{ij}$. Note that $\langle e,\zeta^{n}\rangle=0$, and recall the map $\Phi$ in Eq. 2.22. Let $\check{z}^{n}_{ij}=\Phi(\zeta^{n},0)$. Then, the centering of $\hat{x}$ that results in Eq. 5.4 is given by (compare with Eq. 2.8)

[TABLE]

Throughout this section the family $\{\overline{z}^{n}_{ij}\;(i,j)\in{\mathcal{E}}\}$ is as given in Eq. 5.5.

Remark 5.2*.*

Recall Definitions 2.1 and 2.26. Let $\hat{z}\in{\widetilde{\mathcal{Z}}^{n}}(\hat{x})$. Then, $\hat{\vartheta}^{n}(\hat{x},\hat{z})=\hat{\vartheta}^{n}_{\ast}(\hat{x})=0$ for all $\hat{x}\in\breve{{\mathscr{X}}}^{n}$, and in view of Eq. 5.4, for any $\hat{x}\in\breve{{\mathscr{X}}}^{n}$, there exists $u=u(\hat{x},\hat{z})\in\varDelta$ such that

[TABLE]

In view of Lemma 5.1, and using Lemmas 4.2, 4.3, and 4.4, it is clear that Foster–Lyapunov equations for ${\mathcal{L}}_{u}$ carry over to analogous equations for $\widehat{\mathscr{L}}_{\hat{z}}^{n}$ on $\breve{{\mathscr{X}}}^{n}$ uniformly over SWC policies. However, even though $\breve{{\mathscr{X}}}^{n}$ fills the whole space as $n\to\infty$, $b$ and $b^{n}$ differ in functional form when $\hat{\vartheta}^{n}(\hat{x},\hat{z})\neq 0$, and this makes the stability analysis of multiclass multi-pool networks much harder than the ‘V’ network studied in [AHP18].

Notation 5.1**.**

Let ${\varepsilon}_{n}$ and $\bar{z}^{n}_{ij}$ as in Eqs. 5.1 and 5.5, respectively. For a network with a dominant server pool as in Subsection 4.3.2 define

[TABLE]

while for network with class-dependent service rates, we let

[TABLE]

Since $\{{\varepsilon}_{n}\}$ and $\{\bar{z}^{n}_{i1}\}$ are bounded away from [math] by the convergence of the parameters in Eq. 2.2, the number $n_{0}$ is finite.

The next theorem is the main result for the uniform exponential ergodicity of the prelimit processes. Recall Eq. 2.35. Notice the similarities between the results in Theorem 5.1 for the diffusion-scaled processes and the results in Theorems 4.2 and 4.1 for the diffusion limit of the networks under consideration.

Theorem 5.1**.**

Assume that $\varrho_{n}>0$, and let $n_{0}$ be as in Notation 5.1. Then the prelimit dynamics of any network with a dominant server pool or with class-dependent service rates are uniformly exponentially ergodic and the invariant distributions have exponential tails for all $n>n_{0}$. In particular, due to the convergence of the parameters, there exists $\widehat{C}_{0}$ independent of $n$ such that

[TABLE]

where $V$ and ${\varepsilon}$ are as in Eqs. 4.7 and 5.1.

In addition, with $P_{t}^{n,\hat{z}}$ and $\uppi^{n}_{\hat{z}}$ denoting respectively the transition probability and the stationary distribution of $\hat{X}^{n}(t)$ under a policy $\hat{z}\in{\widetilde{\mathfrak{Z}}^{n}}$, there exist positive constants $\gamma$ and $C_{\gamma}$ not depending on $n\geq 0$ or $\hat{z}$, such that

[TABLE]

Proof.

In Lemmas 5.4 and 5.5 in the section which follows, we establish Eq. 5.2 for these networks. Thus the proof of Eq. 5.9 follows directly from Lemma 5.1.

Since the process $\hat{X}^{n}$ is irreducible and aperiodic under any stationary Markov scheduling $\hat{z}\in{\widetilde{\mathfrak{Z}}^{n}}$ (see Definition 2.1), a convergence property completely analogous to Eq. 4.3 follows from Eq. 5.9. The proof of this fact is identical to [AHP18, Theorem 2.1(b)]. ∎

Remark 5.3*.*

Using Eq. 4.18 and Eq. 5.10, it is clear that under any scheduling policy $\hat{z}\in{\widetilde{\mathfrak{Z}}^{n}}$ with a corresponding control $v$, the stationary distribution of the diffusion-scaled process $\hat{X}^{n}(t)$ converges to that of the limiting diffusion ${\{X_{t}\}_{t\geq 0}}$ for the two classes of networks, that is,

[TABLE]

That is, the interchange of limits property holds.

5.1. Four technical lemmas

In this section, we establish the technical results used in the proof of Theorem 5.1.

Let

[TABLE]

with $\hat{\vartheta}_{*}^{n}$ as in Definition 2.1. As seen in Subsection 2.3, the set $\breve{{\mathscr{X}}}^{n}$ in Eq. 2.26 is contained in ${\mathscr{S}}^{n}\setminus\widetilde{{\mathscr{X}}}^{n}$. In establishing Eq. 5.2 on ${\mathscr{S}}^{n}\setminus\widetilde{{\mathscr{X}}}^{n}$, the results in Section 4 pave the way, since the drift of of the controlled generator $\widehat{\mathscr{L}}_{\hat{z}}^{n}$ over the class of SWC stationary Markov policies ${\widetilde{\mathfrak{Z}}^{n}}$ (see Eq. 5.6) has the same functional form as the drift of the diffusion in Eq. 2.31. So it remains to establish Eq. 5.2 in $\widetilde{{\mathscr{X}}}^{n}$. We start by establishing a bound for $\hat{\vartheta}^{n}$ in Eq. 2.18 over all SWC policies.

As done earlier in the interest of notational economy, we suppress the dependence on $n$ in the diffusion scaled variables $\hat{x}^{n}$ and $\hat{z}^{n}$ in Eq. 2.11.

Lemma 5.2**.**

There exists a number $\varkappa^{n}_{\circ}<1$ depending only on the parameters of the network such that

[TABLE]

In addition, due to the convergence of the parameters in Eq. 2.2, such a constant $\varkappa_{\circ}<1$ may be selected which does not depend on $n$.

Before proceeding to the proof of Lemma 5.2, we provide an interpretation of Eq. 5.13. Recall from Eq. 2.10 that $\hat{x}_{i}=\frac{1}{\sqrt{n}}(x_{i}-\sum_{j\in{\mathcal{J}}}\xi^{*}_{ij}N_{j}^{n})$, and observe that $\hat{x}_{i}$ is positive if the total number of class $i$ customers exceeds the total number of servers assigned to class $i$ in the fluid equilibrium \bigl{(}\sum_{j\in{\mathcal{J}}}\xi^{*}_{ij}N_{j}^{n}\bigr{)} and is negative otherwise. This means that the quantity $\lVert\hat{x}^{+}\rVert_{1}\wedge\lVert\hat{x}^{-}\rVert_{1}$ on the right hand side of Eq. 5.13 represents the minimum of the total number of customers in the queues and the total number of idle servers if the servers are assigned to customer classes according to the fluid equilibrium. Recall also that $\hat{\vartheta}^{n}_{*}$ represents the minimum of the total number of customers in the queues and the total number of idle servers under a SWC policy. The result in Lemma 5.2 is now clear, that is, the minimum of the total number of customers in the queues and idle servers is smaller under a SWC policy compared to a policy that assigns servers according to the fluid equilibrium.

Proof.

Let $\hat{x}\in\widetilde{{\mathscr{X}}}^{n}$, $\hat{z}\in{\widetilde{\mathcal{Z}}^{n}}(\hat{x})$, and define

[TABLE]

and \widetilde{\mathcal{E}}\coloneqq\bigl{\{}(i,j)\in{\mathcal{E}}\,\colon(i,j)\in\widetilde{\mathcal{I}}\times\widetilde{\mathcal{J}}\bigr{\}}. Work conservation implies that $x_{i}^{n}=\sum_{j\in{\mathcal{J}}(i)}z_{ij}^{n}$ for all $i\in\widetilde{\mathcal{I}}$. Let $\hat{\imath}\in{\mathcal{I}}$ be such that $\hat{q}_{\hat{\imath}}>0$, and consider the unique path (since the graph of the network is a tree) connecting $\hat{\imath}$ and $\widetilde{\mathcal{I}}$, that is, a path $\hat{\imath}\rightarrow j_{1}\rightarrow i_{1}\rightarrow j_{2}\dotsc\rightarrow j_{k}\rightarrow\tilde{\imath}$, with $j_{\ell}\in{\mathcal{J}}\setminus\widetilde{\mathcal{J}}$ for $\ell=1,\dotsc,k$, $i_{\ell}\in{\mathcal{I}}\setminus\widetilde{\mathcal{I}}$ for $\ell=1,\dotsc,k-1$, and $\tilde{\imath}\in\widetilde{\mathcal{I}}$. We claim that $z_{\tilde{\imath},j_{k}}^{n}=0$, or equivalently, that $\hat{z}_{\tilde{\imath},j_{k}}=-\nicefrac{{\bar{z}_{ij}^{n}}}{{\sqrt{n}}}$, with $\bar{z}_{ij}^{n}$ as defined in Eq. 5.5. If not, then we can move a job of class $\tilde{\imath}$ from pool $j_{k}$ to some pool in $\widetilde{\mathcal{J}}$, and proceeding along the path to place one additional job from class $\hat{\imath}$ into service, thus contradicting the hypothesis that $\hat{z}\in{\widetilde{\mathcal{Z}}^{n}}(\hat{x})$. Removing all such paths, we are left with a strict subnetwork (possibly disconnected) {\mathcal{G}}_{\circ}=\bigl{(}{\mathcal{I}}_{\circ}\cup{\mathcal{J}}_{\circ},{\mathcal{E}}_{\circ}), with ${\mathcal{I}}_{\circ}\supset\widetilde{\mathcal{I}}$, ${\mathcal{J}}_{\circ}\supset\widetilde{\mathcal{J}}$, and {\mathcal{E}}_{\circ}\coloneqq\bigl{\{}(i,j)\in{\mathcal{E}}\,\colon(i,j)\in{\mathcal{I}}_{\circ}\times{\mathcal{J}}_{\circ}\bigr{\}}, such that

[TABLE]

Let {\mathcal{E}}_{\circ}^{\prime}\coloneqq\bigl{(}{\mathcal{I}}_{\circ}\times({\mathcal{J}}\setminus{\mathcal{J}}_{\circ})\bigr{)}\cap{\mathcal{E}}. By Eq. 5.14 we have

[TABLE]

Thus we have

[TABLE]

by the construction above. By Eq. 5.15, we obtain

[TABLE]

Similarly,

[TABLE]

Using the bound $\hat{\vartheta}^{n}_{*}(\hat{x})\leq\sqrt{n}\sum_{(i,j)\in{\mathcal{E}}_{\circ}}\bar{z}^{n}_{ij}$ we obtain from Eq. 5.17 that

[TABLE]

It should be now clear how to construct $\varkappa_{\circ}^{n}$. For any given subset ${\mathcal{J}}^{\prime}\subsetneq{\mathcal{J}}$, let

[TABLE]

and {\mathcal{E}}^{\prime}_{{\mathcal{J}}^{\prime}}\coloneqq\bigl{(}{\mathcal{I}}_{{\mathcal{J}}^{\prime}}\times({\mathcal{J}}\setminus{\mathcal{J}}^{\prime})\bigr{)}\cap{\mathcal{E}}, and define

[TABLE]

Then the result clearly follows from Eqs. 5.16 and 5.18 since

[TABLE]

This completes the proof. ∎

Remark 5.4*.*

We provide an example of a SWC policy for the ‘N’ network with two classes of customers and two server pools where class 1 can be served by both server pools while class 2 can only served by pool 2 to explain the analysis in the proof of Lemma 5.2. The SWC policy is given by

[TABLE]

This is a priority policy in which customer class $1$ prefers pool $1$ over pool $2$. This means that customer class $1$ can use servers in pool $2$ only if there are no idle servers in pool $1$.

Recall from Eq. 2.9 and Eq. 2.10 that $\hat{q}_{i}\geq 0$ for all $i\in{\mathcal{I}}$. Recall also that we are only considering work-conserving policies and that $\hat{\vartheta}^{n}$ represents the minimum of the total number of customers in the queues and the total number of idle servers. For the ‘N’ network with two classes of customers and two server pools, we have the following cases:

• Case 1

: $\hat{q}_{1}>0$ and $\hat{q}_{2}\geq 0$, in which case we have no idle servers and hence $\hat{\vartheta}^{n}_{*}(\hat{x})=0$.

• Case 2

: $\hat{q}_{1}=0$ and $\hat{q}_{2}=0$. Again, this is a trivial case and means that we have no customers in the queues which implies that $\hat{\vartheta}^{n}_{*}(\hat{x})=0$.

• Case 3

: $\hat{q}_{1}=0$ and $\hat{q}_{2}>0$. This is actually the case that requires some analysis. Here again we have the following cases:

• a

: $\hat{y}_{1}=0$ which means that there are no idle servers in pool 1 and hence $\hat{\vartheta}^{n}_{*}(\hat{x})=0$. This is because $\hat{q}_{2}>0$ which implies that $\hat{y}_{2}=0$ under work conservation.

• b

: $\hat{y}_{1}>0$ which means that there are some idle servers in pool $1$. This is the case analyzed in the proof of Lemma 5.2. (Observe the following notation in the proof: $\widetilde{\mathcal{J}}=\{1\}$, $\widetilde{\mathcal{I}}=\{1\}$, $\hat{\imath}=2$, the path is $\text{class 2}\rightarrow\text{pool 2}\rightarrow\text{class 1}$, and ${\mathcal{G}}_{\circ}=\{(1,1)\}$.) Note that since we are using a SWC policy, this means that no class $1$ customers are being served in pool $2$ because otherwise, one can move a class $1$ customer from pool $2$ to pool $1$ and get a smaller $\hat{\vartheta}^{n}$. Hence,

[TABLE]

where $(N_{1}^{n}-x_{1})$ is the total number of idle server and $(x_{2}-N_{2}^{n})$ is the total number of customers in the queues. This is because the only pool having idle servers is pool $1$ and the only class having customers in the queue is class $2$ ($\hat{q}_{1}=0$).

Recall that $\hat{x}_{i}=\frac{1}{\sqrt{n}}(x_{i}-\sum_{j\in{\mathcal{J}}}\xi^{*}_{ij}N_{j}^{n})$. This means in this case that

[TABLE]

Therefore, we have the following equation

[TABLE]

Note also that Eq. 2.4 ($\xi^{*}_{12}+\xi^{*}_{22}=1$) implies that

[TABLE]

It is now clear that there exists a constant $\varkappa_{\circ}^{n}<1$ such that \hat{\vartheta}^{n}_{*}<\varkappa_{\circ}^{n}\bigl{(}\lVert\hat{x}^{+}\rVert_{1}\wedge\lVert\hat{x}^{-}\rVert_{1}\bigr{)} where

[TABLE]

This completes the analysis of all the cases.

Remark 5.5*.*

It is easy to see that the estimates of the bounds on $\hat{\vartheta}^{n}$ can be improved. It is clear from Eqs. 5.15 and 5.16, that

[TABLE]

Also, since there can be at most $\sum_{j\in{\mathcal{J}}}N_{j}^{n}$ idle servers, it follows that $\widetilde{\varkappa}_{\circ}^{n}\in(0,1)$, such that

[TABLE]

where the constant $\widetilde{\varkappa}_{\circ}^{n}\in(0,1)$ can be selected as

[TABLE]

Due to the convergence of the parameters in Eq. 2.2, $\widetilde{\varkappa}_{\circ}^{n}$ is bounded away from [math] uniformly in $n\in\mathbb{N}$. Note that for the ‘N’ network this translates to

[TABLE]

Even though the ‘N’ network is a special case of networks with a dominant server pool we first establish the result for this network in Lemma 5.3 in order to exhibit with simpler calculations how Lemma 5.2 is applied.

Throughout the proofs of Lemmas 5.3, 5.4, and 5.5 we use the functions (compare with Eq. 4.20)

[TABLE]

and let $n_{0}$ be as in Notation 5.1. Moreover, we suppress the dependence on $n$ in the variables $\hat{q}^{n}$, $\hat{y}^{n}$, and $\hat{\vartheta}^{n}$ in Eqs. 2.17 and 2.18, and from ${\varepsilon}_{n}$ in Eq. 5.1.

5.1.1. The diffusion-scale of the ‘N’ network

We recall here [Stolyar-15b]. Recall also that we label the non-leaf server node as $j=1$ without loss of generality and hence we present Stolyar’s work in [Stolyar-15b] using our notation. In this work, Stolyar considers the ‘N’ network with ${\mathscr{O}}(\sqrt{n})$ safety staffing in pool $1$, under the priority discipline that class $2$ has priority in pool $1$ and class $1$ prefers pool $2$, and shows tightness of the invariant distributions. First note that for any stationary Markov scheduling policy $z$, such that class $2$ has priority in pool $1$ we have $z_{21}^{n}(x)=x_{2}^{n}\wedge N_{1}^{n}$, and it is clear that such a policy is SWC. The same applies to Markov policies under which class $1$ prefers pool $2$ (here $z_{12}^{n}(x)=x_{1}^{n}\wedge N_{2}^{n}$). As a result, SWC policies are more general than the particular policy considered in [Stolyar-15b]. Recall that the matrices $B_{1}^{n}$ and $B_{2}^{n}$ in the drift Eq. 5.4 are given by

[TABLE]

It is also worth noting here, that the spare capacity $\varrho_{n}$ of the $n^{\mathrm{th}}$ system is given by

[TABLE]

with

[TABLE]

This is clear by Eqs. 2.14, 5.19, and 2.35, together with [AP18, Equation (2)]. We let $\eta^{n}\coloneqq\frac{\mu_{12}^{n}}{\mu_{11}^{n}}$.

Lemma 5.3**.**

Consider the ‘N’ network, and assume that $\varrho_{n}>0$. Then for any $\theta\geq\theta_{0}^{n}\coloneqq\frac{\mu_{11}^{n}\,\vee\,\mu_{21}^{n}}{\mu_{11}^{n}\,\wedge\,\mu_{21}^{n}}$, and $\delta\in(0,1)$, there exist positive constants $c_{0}$ and $c_{1}$ such that Eq. 5.2 holds for all $n\geq n_{0}$.

Proof.

Recall here that $j=1$ is the non-leaf server pool. As discussed earlier, it suffices to establish Eq. 5.2 in $\widetilde{{\mathscr{X}}}^{n}$. It is clear that $\hat{x}_{1}^{-}=\hat{y}_{2}+\sqrt{n}\bar{z}_{11}$, and $\hat{x}_{2}=\hat{q}_{2}+\sqrt{n}\bar{z}_{11}$ for all $\hat{z}\in{\hat{\mathcal{Z}}^{n}}(\hat{x})$ and $\hat{x}\in\widetilde{{\mathscr{X}}}^{n}$, with $\widetilde{{\mathscr{X}}}^{n}$ as defined in Eq. 5.12. Hence $u_{1}^{c}=0$, $u_{2}^{c}=1$, $u_{2}^{s}=1$, and $u_{1}^{s}=0$. Note here that SWC policies are interpreted through $u^{c}$ and $u^{s}$ as follows: idle servers are only allowed in pool 1 and customer queues are only allowed in class 2 which means that class $1$ must use all the servers in pool $2$ before using servers in pool $1$. Also, by the definitions of $\psi_{\varepsilon}$ and $n_{0}$ we have

[TABLE]

By Eqs. 5.4 and 5.19, we have

[TABLE]

[TABLE]

Using the fact that $\hat{x}_{1}^{-}-\hat{x}_{2}=\hat{y}_{2}-\hat{q}_{2}$, and $\hat{\vartheta}^{n}=\hat{q}_{2}$ when $\langle e,\hat{x}\rangle\leq 0$, we obtain from Eqs. 5.20 and 5.21 that

[TABLE]

Similarly, from Eq. 5.22, we obtain

[TABLE]

where we also use Eqs. 4.8 and 5.2.

We continue with the estimate on ${\mathcal{K}}_{0}^{+}$. We have

[TABLE]

where in the last inequality we also use Lemma 5.2.

We break the estimate of $F_{2}^{n}$ in two parts. First, using (4.8), (4.10), (5.20) and Lemma 5.2, we obtain

[TABLE]

Thus, Eq. 5.2 follows by Eqs. 5.23, 5.24, 5.25, 5.26, and 4.10. This completes the proof. ∎

5.1.2. The diffusion scale of networks with a dominant pool

We describe these networks exactly as in Subsection 4.3.2 where the dominant server pool is $j=1$. We first note that the spare capacity $\varrho_{n}$ of the $n^{\mathrm{th}}$ system is given by

[TABLE]

where $\xi^{*}_{ij}$ satisfies

[TABLE]

This is again due to Eqs. 2.14, 2.4, 2.35, and 2.34.

Recall from Eq. 5.4 that the drift reduces to the following form:

[TABLE]

with $\eta_{ij}^{n}\coloneqq\frac{\mu_{ij}^{n}}{\mu_{i1}^{n}}$ for $j\in{\mathcal{J}}_{1}(i)\coloneqq{\mathcal{J}}(i)\setminus\{1\}$ and $i\in{\mathcal{I}}$. In analogy to Subsection 4.3.2, we define We define

[TABLE]

Lemma 5.4**.**

Consider a network with a dominant server pool, and assume $\varrho_{n}>0$. Then for any $\theta\geq\theta_{0}^{n}\coloneqq 2\,\frac{\max_{i}{\mu_{i1}^{n}}}{\min_{i}{\mu_{i1}^{n}}}$, and $\delta\in(0,1)$, there exist positive constants $c_{0}$ and $c_{1}$ such that Eq. 5.2 holds for all $n\geq n_{0}$.

Proof.

Suppose $\hat{x}\in\widetilde{{\mathscr{X}}}^{n}$. A simple calculation using Eq. 5.27 shows that

[TABLE]

and

[TABLE]

By Eq. 5.28 we obtain

[TABLE]

where we used Eq. 4.8 in the first inequality, Lemma 5.2 in the second, and Eq. 4.10 in the fourth.

Next, we estimate a bound for $F_{2}^{n}(\hat{x},\hat{z})$. Recall the definitions of ${\mathcal{I}}_{\circ}$, ${\mathcal{J}}_{\circ}$, ${\mathcal{E}}_{\circ}$, and ${\mathcal{E}}_{\circ}^{\prime}$ in the proof of Lemma 5.2. Since $x\in\widetilde{\mathscr{X}}^{n}$, we have $u_{i}^{c}=0$ for all $i\in{\mathcal{I}}_{\circ}$, and $u_{j}^{s}=0$ for all $j\in{\mathcal{J}}_{\circ}^{\mathsf{c}}$. SWC policies are interpreted here as follows: customer classes must use all the servers in the leaf pools available to them before using servers in pool $j=1$. Additionally, $\hat{x}_{i}\leq-\sum_{(i,j)\in{\mathcal{E}}_{\circ}^{\prime}}\bar{z}_{ij}$ for $i\in{\mathcal{I}}_{\circ}$, which implies that $\psi_{\varepsilon}^{\prime}(x_{i})=0$ for all $i\in{\mathcal{I}}_{\circ}$ and $n>n_{0}$, by Notation 5.1. Hence, since $\sum_{i\in{\mathcal{I}}_{\circ}^{\mathsf{c}}}\hat{x}_{i}>0$, where ${\mathcal{I}}_{\circ}^{\mathsf{c}}\equiv{\mathcal{I}}\setminus{\mathcal{I}}_{\circ}$, we have

[TABLE]

by Eq. 4.12. Using Eq. 5.29 together with Remarks 5.5 and 5.31, we obtain

[TABLE]

for all $n\geq n_{0}$. Thus, the result follows by Eqs. 5.30 and 5.32, noting also that the choice of $\theta$ implies that $V_{1}\geq V_{2}^{2}$ on ${\mathcal{K}}_{0}^{-}$. ∎

5.1.3. The diffusion-scale of networks with class-dependent service rates

Recall from Subsection 4.3.2 that the drift in Eq. 5.4 reduces to

[TABLE]

where $B_{1}^{n}=\operatorname{diag}(\mu_{1}^{n},\dotsc,\mu_{m}^{n})$. Thus, the spare capacity $\varrho_{n}$ is given by

[TABLE]

Lemma 5.5**.**

Suppose that $\mu_{ij}^{n}=\mu_{i}^{n}$, for all $i\in{\mathcal{I}}$, and $\varrho_{n}>0$. Then, for any $\theta\geq\theta_{0}^{n}\coloneqq 2\,\frac{\mu^{n}_{\mathsf{max}}}{\mu^{n}_{\mathsf{min}}}$, and $\delta\in(0,1)$, the conclusions of Lemma 5.4 follow.

Proof.

Suppose $\hat{x}\in\widetilde{{\mathscr{X}}}^{n}$. A simple calculation using Eq. 5.33 shows that

[TABLE]

By Eq. 5.34, we obtain

[TABLE]

In computing the analogous bound to Eq. 5.32, there is a difference here. It is not the case here that $\psi_{\varepsilon}^{\prime}(x_{i})=0$ for all $i\in{\mathcal{I}}_{\circ}$ and $n>n_{0}$.

So instead, recalling that $u_{i}^{c}=0$ for all $i\in{\mathcal{I}}_{\circ}$, and since $\hat{x}\in{\mathcal{K}}_{0}^{+}$, we write

[TABLE]

The third term on the right-hand side is nonpositive by Eq. 4.12. We also have

[TABLE]

and

[TABLE]

Therefore, by Eq. 5.35, Eqs. 5.36, 5.37, 5.38, and 5.5, we obtain

[TABLE]

The rest follows as in Lemma 5.4. ∎

Acknowledgment

This work is supported in part by the Army Research Office through grant W911NF-17-1-0019, and in part by NSF grants DMS-1715210, CMMI-1635410, and DMS/CMMI-1715875, and in part by the Office of Naval Research through grant N00014-16-1-2956 and was approved for public release under DCN #43-5454-19.