text:
\section{Introduction}

Recently, there has been a great interest in queues with a large
number of servers, motivated by applications to telephone call
centers.  Since a customer can easily hang up after waiting for too
long, abandonment is a non-negligible aspect in the study of
many-server queues. In our study, a customer can leave the system
(without getting service) once has been waiting in queue for more than
his patience time. Both patience and service times are modeled using
random variables.  A recent statistical study by Brown et al.\
\cite{BGMSSZZ2005} suggests that the exponential assumption on service
time distribution, in many cases, is not valid.  In fact, the
distribution of service times at call centers may be log-normal in
some cases as shown in \cite{BGMSSZZ2005}.  This emphasizes the need
to look at the many-server model with generally distributed service
and patience times.

In this paper, we study many-server queues with general patience and
service times.  The queueing model is denoted by $G/GI/n$+$GI${}.  The $G$
represents a general stationary arrival process.  The first $GI$
indicates that service times come from a sequences of independent and
identically distributed (IID) random variables with a general
distribution.  The $n$ denotes the number of homogeneous servers.
There is an unlimited waiting space, called buffer, where customers
wait and can choose to abandon if their patience times expires before
their service starts.  Again, the patience times of each customer are
IID and with a general distribution (the $GI$ after the `+' sign).

Useful insights can be obtained by considering a many-server queue in
limit regimes where the number $n$ of servers increases along with the
arrival rate $\lambda^n$ such that the traffic intensity
\[\rho^n=\frac{\lambda^n}{n\mu}\to \rho\textrm{ as }n\to\infty,\]
where $\mu$ is the service rate of a single server (in other words,
the reciprocal of the mean service time), and $\rho\in[0,\infty)$.
Since the abandonment ensures stability, the limit $\rho$ in the above
need not to be less than 1.  In fact, according to $\rho$, the limit
regimes can be divided into three classes,
i.e. \emph{Efficiency-Driven} (ED) regime when $\rho>1$,
\emph{Quality-and-Efficiency-Driven} (QED) regime when $\rho=1$ and
\emph{Quality-Driven} (QD) regime when $\rho<1$. The QED regime is
also called \emph{Halfin-Whitt} regime due to the seminal work Halfin
and Whitt \cite{HalfinWhitt1981}.  With this motivation, we establish
the fluid (also called law of large number) limit for the $G/GI/n$+$GI${}
queue in all the ED, QED and QD limit regimes.

We show that the fluid model has an equilibrium, which yields
approximations for various performance quantities.  These fluid
approximations work pretty well in the ED and QD regime where $\rho$
is not that close to 1, as demonstrated in the numerical experiments
of Whitt \cite{Whitt2006}.  However, when $\rho$ is very close (say
within 5\%) to 1, the fluid approximations lose their accuracy and we
shall look at a more refined limit, the diffusion limit, in this case.
Diffusion limit is not within the scope of the current paper.

One of the challenges in studying many-server queue with general
service (as well as patience) time is that Markovian analysis can not
be used.  In a system where multiple customers are processed at the
same time, such as the many-server queue, how to describe the system
becomes an important issue. The number of customers in the system does
not give much information since they may all have large remaining
service times or all have small remaining service times, and this
information can affect future evolution of the system.  We choose
finite Borel measures on $(0,\infty)$ to describe the system.  At any
time $t\ge 0$, instead of recording the total number of customers in
service (i.e. the number of busy servers), we record all the remaining
patience times using measure $\ser(t)$. For any Borel set $C\in
(0,\infty)$, $\ser(t)(C)$ indicates the number of customers in server
with \emph{remaining service time} belongs to $C$ at that time.
Similar idea applies for the remaining patience times.  We first
introduce the \emph{virtual buffer}, which holds all the customers who
have arrived but not yet scheduled to receive service (assuming they
are infinitely patient).  We record all the remaining patience times
for those in the virtual buffer using finite Borel measure $\rbuf(t)$
on $\R=(-\infty,\infty)$.  At time $t\ge 0$, $\rbuf(t)(C)$ indicates
the number of customers in the virtual buffer with \emph{remaining
  patience time} belongs to the Borel set $C$.  The descriptor
$(\rbuf(\cdot),\ser(\cdot))$ contains very rich information, almost
all information about the system can be recovered from it.  Note that
a customer with negative remaining patience time has already
abandoned. So the actual number of customers in the buffer is
\[
Q(t)=\rbuf(t)((0,\infty))\textrm{ for all }t\ge0.
\]
More details will be discussed when we rigorously introduce the
mathematical model in Section~\ref{sec:stochastic-model}.  In
literature, another descriptor that keeps track of the ages of
customers in service and the ages of customers in waiting have been
used, e.g.\ \cite{KangRamanan2008,Whitt2006}; The age proceses have
the advantage of being observable, without requiring future
information, though their analysis is often more complicated. Both age
and residul descriptions of the system often results in the same
steady state insights. In this paper, we focus on residual processes
only.

The framework of using measure-valued process has been successfully
applied to study models where multiple customers are processed at the
same time.  Existing works include Gromoll and Kruk
\cite{GromollKurk2007}, Gromoll, Puha and Williams \cite{GPW2002} and
Gromoll, Robert and Zwart \cite{GRZ2008}, to name a few.  Most of
these works are on the processor sharing queue and related models
where there is no waiting buffer.  Recently, Zhang, Dai and Zwart
\cite{ZDZ2009,ZDZ2007b} apply the measure-valued process to study the
limited processor sharing queue, where only limited number of
customers can be served at any time with extra customers waiting in a
buffer.  Many techniques in this paper closely follows from those
developed in \cite{ZDZ2009}.
There has been a huge literature on many-server queue and related
models since the seminal work by Halfin and Whitt
\cite{HalfinWhitt1981}. But there are not many successes with the case
where the service time distribution is allowed to be
non-exponential. One exception is the work of Reed~\cite{Reed2007}, in
which fluid and diffusion limits of the customer-count process of many
server queues (without abandonment) are established where few
assumptions beyond a first moment are placed on the service time
distribution. Later, Puhalskii and Reed~\cite{PuhalskiiReed2009}
extend the aforementioned results to allow noncritical loading,
generally distributed service times, and general initial conditions.
Jelenkovi\'{c} et al.\ \cite{JMM2004} study the many-server queue with
deterministic service times; Garmarnik and Mom\v{c}ilovi\'{c}
\cite{GamarnikMomvcilovic2007} study the model with lattice-valued service
times; Puhalskii and Reiman \cite{PuhalskiiReiman2000} study the model
with phase-type service time distributions.  Mandelbaum and
Mom\v{c}ilovi\'{c} \cite{MandelbaumMomvcilovic2008} study the virtual
waiting time processes, and Kaspi and Ramanan \cite{KaspiRamanan}
study the fluid limit of measure-valued processes for many-server
queues with general service times.
For the many-server queue with abandonment, a version of the fluid
model have been established as a conjecture in Whitt~\cite{Whitt2004},
where a lot of insight was demonstrated, which help greatly in our
work. Recently, Kang and Ramanan also worked on the same topic and
summarized their result in the technical report
\cite{KangRamanan2008}.
Although we focus on the same topic, our work uses different
methodology from that in \cite{KangRamanan2008} and requires less
assumptions on the service time distribution.  In our work, the only
assumption on the service time distribution is continuity, while the
service time distribution in \cite{KangRamanan2008} is required to
have a density and the hazard rate function must be either bounded or
lower lower-semicontinuous.  From the modeling aspect, our approach
mainly based on tracking the ``residual'' processes, while
\cite{KangRamanan2008} tracks the ``age'' processes for studying the
queueing model.  Also, we propose a quite simple fluid model, which
facilitates the analysis.  The existence of solution to the fluid
model in \cite{KangRamanan2008} is proved by showing each fluid limit
satisfies the fluid model equations.  The current paper proves the
existence directly from the definition of the fluid model without
invoking fluid limits.
In addition, we verify in the end of this paper (c.f.\
Section~\ref{sec:special-case-with}) that our fluid model is
consistent with the special case where both service and patience times
are exponentially distributed, as established in
Whitt~\cite{Whitt2004} for the ED regime, Garnet et al.\
\cite{GarnetMandelbaumReiman2002} for QED regime and Pang and Whitt
\cite{PangWhitt2008a} and Puhalskii \cite{Puhalskii2008} for all three
regimes.  Additional works on many-server queues with abandonment
includes Dai, He and Tezcan \cite{DaiHeTezcan2009} for phase-type
service time distributions and exponential patience time distribution;
Zeltyn and Mandelbaum \cite{ZeltynMandelbaum2005} for exponential
service time distribution and general patience time distributions;
Mandelbaum and Mom\v{c}ilovi\'{c} \cite{MandelbaumMomvcilovic2009} for
both general service time distribution and general patience time
distribution. The difference between our work and
\cite{MandelbaumMomvcilovic2009} is that we study the fluid limit of
measure-valued processes in all three regimes, and
\cite{MandelbaumMomvcilovic2009} studies the diffusion limit of
customer count processes and virtual waiting processes in the QED
regime.  By assuming a convenient initial condition,
\cite{MandelbaumMomvcilovic2009} does not require a detailed fluid
model analysis.

The paper is organized as follows: We begin in
Section~\ref{sec:stochastic-model} by formulating the mathematical
model of the $G/GI/n$+$GI${} queue. The dynamics of the system are clearly
described by modeling with measure-valued processes; see
\eqref{eq:stoc-dyn-eqn-B} and \eqref{eq:stoc-dyn-eqn-S}. The main
results, including a characterization of the fluid model and the
convergence of the stochastic processes underlying the $G/GI/n$+$GI${} queue
to the fluid model solution are stated in
Section~\ref{sec:main-results}.  In
Section~\ref{sec:prop-fluid-model}, we explore the fluid model and
give proofs of all the results on the fluid model.
Section~\ref{sec:fluid-appr-stoch} is devoted to establishing the
convergence of stochastic processes, which includes the proof of
pre-compactness and the characterization of the limit as the fluid
model solution.

\subsection{Notation}
\label{subsec:notatioin}

The following notation will be used throughout.  Let $\N$, $\Z$ and
$\R$ denote the set of natural numbers, integers and real numbers
respectively.  Let $\R_+=[0,\infty)$.  For $a,b\in\R$, write $a^+$ for
the positive part of $a$, $\fl a$ for the integer part, $\cl a$ for
$\fl a+1$, $a\vee b$ for the maximum, and $a\wedge b$ for the minimum.
For any $A\subset\R$, denote $\mathscr B(A)$ the collection of all
Borel subsets which are subsets of $A$.

Let $\M$ denote the set of all non-negative finite Borel measures on
$\R$, and $\M_+$ denote the set of all non-negative finite Borel
measures on $(0,\infty)$.  To simplify the notation, let us take the
convention that for any Borel set $A\subset\R$, $\nu(A\cap
(-\infty,0])=0$ for any $\nu\in\M_+$. Also, by this convention, $\M_+$
is embedded as a subspace of $\M$.  For $\nu_1,\nu_2\in\M$, the
Prohorov metric is defined to be
\begin{equation}\nonumbe
  \begin{split}
    \pov[\nu_1,\nu_2]=\inf\Big\{\epsilon>0:
    \nu_1(A)\le\nu_2(A^\epsilon)+\epsilon&\text{ and }\\
    \nu_2(A)\le\nu_1(A^\epsilon)+\epsilon&\text{ for all closed Borel
      set }A\subset\R\Big\},
  \end{split}
\end{equation}
where $A^\epsilon=\{b\in\R:\inf_{a\in A}|a-b|<\epsilon\}$. This is the
metric that induces the topology of weak convergence of finite Borel
measures.  (See Section~6 in \cite{Billingsley1999}.)  For any Borel
measurable function $g:\R\to\R$, the integration of this function
with respect to the measure $\nu\in\M$ is denoted by $\inn{g}{\nu}$.

Let $\M_+\times\M$ denote the Cartesian product.  There are a number
of ways to define the metric on the product space.  For convenience
we define the metric to be the maximum of the Prohorov metric
between each component.  With a little abuse of notation, we still
use $\pov$ to denote this metric.

Let $(\mathbf E, \pi)$ be a general metric space.  We consider the
space $\D$ of all right-continuous $\mathbf E$-valued functions with
finite left limits defined either on a finite interval $[0,T]$ or the
infinite interval $[0,\infty)$.  We refer to the space as
$\D([0,T],\mathbf E)$ or $\D([0,\infty),\mathbf E)$ depending upon the
function domain.  The space $\D$ is also known as the space of \rcll{}
functions.  For $g(\cdot),g'(\cdot)\in\D([0,T],\mathbf E)$, the
uniform metric is defined as
\begin{equation}\label{eq:sup-norm}
  \upsilon_T[g,g']=\sup_{0\le t\le T} \pi[g(t),g'(t)].
\end{equation}
However, a more useful metric we will use is the following Skorohod
$J_1$ metric,
\begin{equation}\label{eq:skorohod-L-def}
  \sko_T[g,g']=\inf_{f\in\Lambda_T}(\|f\|^\circ_T\vee\upsilon_T[g,g'\circ f]),
\end{equation}
where $g\circ f(t)=g(f(t))$ for $t\ge 0$ and $\Lambda_T$ is the set of
strictly increasing and continuous mapping of $[0,T]$ onto itself and
\[
\|f\|^\circ_{T}=\sup_{0\le s<t\le T}\big|\log\frac{f(t)-f(s)}{t-s}\big|.
\]
If $g(\cdot)$ and $g'(\cdot)$ are in the space $\D([0,\infty),\mathbf E)$,
the Skorohod $J_1$ metric is defined as
\begin{equation}\label{eq:skorohod-def}
  \sko[g,g']=\int_0^\infty e^{-T}(\sko_T[g,g']\wedge 1)dT.
\end{equation}
By saying convergence in the space $\D$, we mean the convergence under
the Skorohod $J_1$ topology, which is the topology induced by the
Skorohod $J_1$ metric \cite{EthierKurtz1986}.

We use ``$\to$'' to denote the convergence in the metric space
$(\mathbf E,\pi)$, and use ``$\dto$'' to denote the convergence in
distribution of random variables taking value in the metric space
$(\mathbf E, \pi)$.

\section{Stochastic Model}
\label{sec:stochastic-model}

In this section, we first describe the $G/GI/n$+$GI${} queueing system and
then introduce a pair of measure-valued processes that capture the
dynamics of the system.

There are $n$ identical servers in the system.  Customers arrive
according to a general stationary arrival process (the initial G) with
arrival rate $\lambda$.  Let $a_i$ denote the arrival time of the
$i$th arriving customer, $i=1, 2,\cdots$.  An arriving customer enters
service immediately upon arrival if there is a server available.  If
all $n$ servers are busy, the arriving customer waits in a buffer,
which has infinite capacity.  Customers are served in the order of
their arrival by the first available server.  Waiting customers may
also elect to abandon.  We assume that each customer has a random
patience time.  A customer will abandon immediately when his waiting
time in the buffer exceeds his patience time. Once a customer starts
his service, the customer remains until the service is completed.
There are no retrials; abandoning customers leave without affecting
future arrivals.

The two GIs in the notation mean that the service times and patience
times come from two independent sequences of iid random variables;
these two sequences are assumed to be independent of the arrival
process.  Let $u_i$ and $v_i$ denote the patience and service time of
the $i$th arriving customer, $i=1, 2,\cdots$. In many applications
such as telephone call centers, customers cannot see the queue (the
case of invisible queues, c.f.\ \cite{MandelbaumShimkin2000}), thus do
not know the experience of other customers. In such a case, it is
natural to assume that patience times are iid.  Denote $F(\cdot)$ and
$G(\cdot)$ the distributions for the patience and service times,
respectively.

To describe the system using measure-valued process, we first
introduce the notion of \emph{virtual buffer}.  The virtual buffer
holds all customers in the real buffer and some of the abandoned
customers.  An abandoned customer continues to wait in the virtual
buffer when he first abandons until it were his turn for service had
he not abandoned. At this time, he leaves the virtual buffer.  At any
time $t\ge 0$, $\rbuf(t)$ denotes a measure in $\M$ such that
$\rbuf(t)(C)$ is the number of customers in the virtual buffer with
remaining patience time in $C\in\mathscr B(\R)$.  Please note that
this way of modeling requires $\rbuf(\cdot)$ to be a measure on $\R$,
not just $(0,\infty)$.  It is clear that
\begin{equation}
  \label{eq:QR}
  Q(t)=\rbuf(t)((0,\infty))  \text{  and }
  R(t)=\rbuf(t)(\R)
\end{equation}
represent the number of customers waiting in the real buffer and
number of customers in the virtual buffer, respectively.

We also use a measure to describe the server. At any time $t\ge 0$,
$\ser(t)$ denotes a measure in $\M_+$ such that $\ser(t)(C)$ is the
number of customers in service with remaining service time in
$C\in\mathscr B((0,\infty))$. Different from the virtual buffer, the servers
only hold customers with positive remaining service times, so we only
care about the subsets in $(0,\infty)$. The quantity
\begin{equation}
  \label{eq:Z}
  Z(t)=\ser(t)((0,\infty)),
\end{equation}
represents the number of customers in service at any time $t\ge0$.

The measure-valued (taking value in $\M\times\M_+$) stochastic process
$(\rbuf(\cdot),\ser(\cdot))$ serves as the descriptor for the $G/GI/n$+$GI${}
queueing model.  Before we use it to describe the dynamics of the
system, let us first talk about the initial condition, since the
system is allowed to be non-empty initially.  The initial state
specifies $R(0)$, the number of customers in the virtual buffer as
well as their remaining patience times $u_i$ and service times $v_i$,
$i=1-R(0), 2-R(0), \cdots, 0$.  The initial state also specifies
$Z(0)$, the number of customers in service as well as their remaining
service times $v_i$, $i=1-R(0)-Z(0), \cdots, -R(0)$.  Briefly, the
initial customers are given negative index, in order not to conflict
with the index of arriving customers. Those initial customers in the
buffer are also assumed to have i.i.d.\ service times with
distribution $G(\cdot)$.  For each $t\ge 0$, denote $E(t)$ the number
of customers that has arrived during time interval $(0, t]$. Arriving
customers are indexed by $1,2,\cdots$ according to the order of their
arrival.  By this way of indexing customers, it is clear that the
index of the first customer in the virtual buffer at time $t\ge0$ is
$B(t)+1$, where
\begin{equation}\label{eq:B(t)}
B(t)=E(t)-R(t).
\end{equation}
Denote $w_i$ the waiting time of the $i$th customers; then
$\tau_i=a_i+w_i$ is the time when the $i$th job starts \emph{service}
for all $i\ge 1-R(0)$. For $i<0$, $a_i$ may be a negative number
indicating how long the $i$th customer had been there by time 0. We
will impose some conditions on $a_i$'s with $i<0$ later on.  Let
$\delta_x$ and $\delta_{(x,y)}$ denote the Dirac point measure at
$x\in\R$ and $(x,y)\in\R^2$, respectively.  Denote $C+x=\{c+x:x\in
C\}$ for any subset $C\subset\R$ and $C_x=(x,\infty)$. For any subsets
$C, C'\subset\R$, let $C\times C'$ denote the Cartesian product. Using
the Dirac measure and the above introduced notations, the evolution of
the system can be captured by the following \emph{stochastic dynamic
  equations}:
\begin{align}
  \label{eq:stoc-dyn-eqn-B}
  \rbuf(t)(C)&=\sum_{i=1+B(t)}^{E(t)}\delta_{u_i}(C+t-a_i),
  \quad \textrm{for all }C\in\mathscr B(\R),\\
  \label{eq:stoc-dyn-eqn-S}
  \begin{split}
    \ser(t)(C)&=\sum_{i=1-R(0)-Z(0)}^{-R(0)}\delta_{v_i}(C+t)\\
    &\quad+\sum_{i=1-R(0)}^{B(t)}\delta_{(u_i,v_i)}
                   (C_0+\tau_i-a_i)\times(C+t-\tau_i),
  \end{split}
  \quad \textrm{for all }C\in\mathscr B((0,\infty)),
\end{align}
for all $t\ge 0$.  Denote the total number of customers in the
system by
\begin{equation*}
  \label{eq:X}
  X(t)=Q(t)+Z(t) \quad\text{for all }t\ge0.
\end{equation*}
The following \emph{policy constraints} must be satisfied at any time
$t\ge 0$,
\begin{align}
  Q(t)&=(X(t)-n)^+,\label{eq:constr-B}\\
  Z(t)&=(X(t)\wedge n),\label{eq:constr-S}
\end{align}
where $n$, as introduced above, denotes the number of servers in the
system.

\section{Main Results}
\label{sec:main-results}

The main results of this paper contains two parts. The first part is a
characterization of the fluid model, including the existence and
uniqueness of the fluid model solution, and the equilibrium of the fluid
model; these results are summarized in
Section~\ref{subsec:fluid-model}. The second part is the convergence
of the stochastic processes to the fluid model solution; this result
is stated in Section~\ref{sec:sequ-queu-models}.

\subsection{Fluid Model}
\label{subsec:fluid-model}

To study the stochastic model, we introduce a determinisitic fluid
model.  To simplify notations, let $F^c(\cdot)$ denote the complement
of the patience time distribution $F(\cdot)$, i.e. $F^c(x)=1-F(x)$ for
all $x\in\R$; the complement of the service time distribution, denoted
by $G^c(\cdot)$, is defined in the same way.  We introduce the
following \emph{fluid dynamic equations}:
\begin{align}
  \label{eq:fluid-dyn-eqn-B}
  \bar\rbuf(t)(C_x)&=\lambda\int_{t-\frac{\bar
      R(t)}{\lambda}}^{t}F^c(x+t-s)ds, \quad t\ge 0, \quad x\in \R,\\
  \label{eq:fluid-dyn-eqn-S}
  \bar\ser(t)(C_x)&=\bar\ser(0)(C_x+t)+\int_0^t F^c\left(\bar
    R(s)/\lambda\right)G^c(x+t-s)d\bar B(s),\quad t\ge 0, \quad x\in (0,\infty),
\end{align}
where $C_x=(x,\infty)$ and $\bar B(s)=\lambda s-\bar R(s)$.  Here, all
the time dependent quanities are assumed to be right continuous on
$[0, \infty)$ and to have left limits in $(0, \infty)$; furthermore,
$\bar{B}(\cdot)$ is a non-decreasing function, and the integral
$\int_0^t g(s)\, d\bar{B}(s)$ is interpreted as the Lebesgue-Stieltjes
integral on the interval $(0, t]$.  The quantities $\bar R(\cdot)$,
$\bar Q(\cdot)$, $\bar Z(\cdot)$ and $\bar X(\cdot)$ are defined in
the same way as their stochastic counterparts in (\ref{eq:QR}),
(\ref{eq:Z}) and (\ref{eq:X}).  The following policy constraints must
be satisfied for all $t\ge 0$,
\begin{align}
\bar Q(t)&=(\bar X(t)-1)^+,\label{eq:fluid-constr-B}\\
\bar Z(t)&=(\bar X(t)\wedge 1).\label{eq:fluid-constr-S}
\end{align}
The fluid dynamic equations \eqref{eq:fluid-dyn-eqn-B} and
\eqref{eq:fluid-dyn-eqn-S} and the policy constraints
\eqref{eq:fluid-constr-B} and \eqref{eq:fluid-constr-S} define a
\emph{fluid model}, which is denoted by $\fm$.

Denote $(\bar\rbuf_0, \bar\ser_0)=(\bar\rbuf(0), \bar\ser(0))$ the
initial condition of the fluid model. For the convenience of
notations, also denote $\bar Q_0=\bar Q(0)$, $\bar Z_0=\bar Z(0)$ and
$\bar X_0=\bar Q_0+\bar Z_0$.  We need to require that the initial
condition satisfies the dynamic equations and the policy constraints,
i.e.\
\begin{align}
  \label{eq:valid-init-buf}
  \bar\rbuf_0(C_x)&=\lambda\int_0^{\frac{\bar
      R_0}{\lambda}}F^c(x+s)ds, \quad x\in \R,\\
  \label{eq:valid-constr-B}
  \bar Q_0&=(\bar X_0-1)^+,\\
  \label{eq:valid-constr-S}
  \bar Z_0&=(\bar X_0 \wedge 1).
\end{align}
We also require that
\begin{equation}\label{eq:valid-0}
\bar\ser_0(\{0\})=0,
\end{equation}
which means that nobody with remaining service time 0 stays in the
server. We call any element $(\bar\rbuf_0, \bar\ser_0)\in\M\times\M_+$ a
\emph{valid} initial condition if it satisfies
\eqref{eq:valid-init-buf}--\eqref{eq:valid-0}.

We call
$(\bar\rbuf(\cdot),\bar\ser(\cdot))\in\D([0,\infty),\M\times\M_+)$ a
solution to the fluid model $\fm$ with a valid initial condition
$(\bar\rbuf_0,\bar\ser_0)$ if it satisfies the fluid dynamic equations
\eqref{eq:fluid-dyn-eqn-B} and \eqref{eq:fluid-dyn-eqn-S} and the
policy constraints \eqref{eq:fluid-constr-B} and
\eqref{eq:fluid-constr-S}.

Denote $\mu$ the reciprocal of first moment of the service time
distribution $G(\cdot)$.  Let
\begin{equation}\label{eq:support-F}
M_F=\inf\{x\ge 0: F(x)=1\}.
\end{equation}
By the right continuity, it is clear that $F(x)<1$ for all $x<M_F$ and
$F(x)=1$ for all $x\ge M_F$.  If the patience time distribution
$F(\cdot)$ has a density $f(\cdot)$, then define the hazard rate
$h_F(\cdot)$ of the distribution $F(\cdot)$ by
\begin{equation*}
  h_F(x)=\left\{
  \begin{array}{ll}
    \frac{f(x)}{1-F(x)} & x<M_F,\\
    0 & x\ge M_F.
  \end{array}\right.
\end{equation*}

\begin{thm}[Existence and Uniqueness]\label{thm:uniq-exit}
  Assume the service time distribution satisfies both that
  \begin{equation}\label{cond:G-no-atom}
    G(\cdot) \textrm{ is continuous,}
  \end{equation}
  and that
  \begin{equation}\label{cond:fl-1}
     0<\mu<\infty.
  \end{equation}
  Assume the patience time distribution satisfies either that
  \begin{equation}
    \label{cond:fl-3-opt}
    F(\cdot )\textrm{ is Lipschitz continuous},
  \end{equation}
  or that $F(\cdot)$ has a density $f(\cdot)$ such that the
  hazard rate is bounded, i.e.\
  \begin{equation}\label{cond:fl-3}
    \sup_{x\in[0,\infty)}h_F(x)<\infty.
  \end{equation}
  There exists a unique solution to the fluid model $\fm$ for any
  valid initial condition $(\bar\rbuf_0, \bar\ser_0)$.
\end{thm}

The above theorem provides the foundation to further study the fluid
model. A key property is that the fluid model  has an
equilibrium state. An equilibrium state is defined as the following:

\begin{defn}\label{def:eqm-state}
  An element $(\bar\rbuf_\infty,\bar\ser_\infty)\in\M\times\M_+$ is
  called an \emph{equilibrium state} for the fluid model $\fm$ if the
  solution to the fluid model with initial condition
  $(\bar\rbuf_\infty,\bar\ser_\infty)$ satisfies
  \[
    (\bar\rbuf(t),\bar\ser(t))=(\bar\rbuf_\infty,\bar\ser_\infty)\quad
    \textrm{for all }t\ge 0.
  \]
\end{defn}

This definition says that if a fluid model solution starts from an
equilibrium state, it will never change in the future. To present the
result about equilibrium state, we need to introduce some more
notation.  For the service time distribution function $G(\cdot)$ on
$\R_+$, the associated \emph{equilibrium} distribution is given by
\[G_e(x)=\mu\int_0^xG^c(y)dy,\quad\textrm{for all }x\ge 0.\]

\begin{thm}\label{thm:eqm}
  Assume the conditions in Theorem~\ref{thm:uniq-exit}. The state
  $(\bar\rbuf_\infty,\bar\ser_\infty)$ is an equilibrium state of the
  fluid model $\fm$ if and only if it satisfies
  \begin{align}
    \label{eq:eqm-B}
    \bar\rbuf_\infty(C_x)&=\lambda\int_0^{w}F^c(x+s)ds, \quad x\in \R,\\
    \label{eq:eqm-S}
    \bar\ser_\infty(C_x)&=\min\left(\rho,1\right)[1-G_e(x)], \quad
    x\in (0,\infty),
  \end{align}
 where $w$ is a solution to the equation
  \begin{equation}\label{eq:eqm-w}
    F(w)=\max\left(\frac{\rho-1}{\rho},0\right).
  \end{equation}
\end{thm}

\begin{rem}
  If equation \eqref{eq:eqm-w} has multiple solutions, then the
  equilibrium is not unique (any solution $w$ gives an
  equilibrium). If the equation has a unique solution (for example
  when $F(\cdot)$ is strictly increasing), then the equilibrium state
  is unique.
\end{rem}

The quantity $w$ is interpreted to be the \emph{offered} waiting time
for an arriving customer. If his patience time exceeds $w$, he will
not abandon. Thus, the probabilty of his abandonment is given by
$F(w)$, which is equal to $(\rho-1)/\rho$ when $\rho>1$; the latter
quantity is the fraction of traffic that has to be discarded due to
the overloading.  From (\ref{eq:eqm-B}),
$\bar\rbuf_\infty(C_x)=\lambda w$ for $x\le -w$. Thus, the average
number of customers in the virtual buffer is
 \begin{displaymath}
\bar R_\infty = \bar\rbuf_\infty(\R) = \lambda w,
 \end{displaymath}
 which is consistent with the Little's law. From (\ref{eq:eqm-S}), the
 average number of busy servers is
\begin{displaymath}
  \bar Z_\infty = \bar\ser_\infty((0,\infty)) = \min(\rho, 1),
\end{displaymath}
which is intuitively clear.  These observations and interpretations
were first made by Whitt~\cite{Whitt2006}, where approximation
formulas based on a conjectured fluid model were also given, and were
compared with extensive simulation results.  The approximation
formulas derived from our fluid model is consistent with those
formulas in Whitt~\cite{Whitt2006}.

\subsection{Convergence of Stochastic Models}
\label{sec:sequ-queu-models}

We consider a sequence of queueing systems indexed by the number of
servers $n$, with $n\to\infty$.  Each model is defined in the same way
as in Section~\ref{sec:stochastic-model}.  The arrival rate of each
model is assumed be to proportional to $n$.  To distinguish models
with different indices, quantities of the $n$th model are accompanied
with superscript $n$.  Each model may be defined on a different
probability space $(\Omega^n,\mathcal F^n,\mathbb P^n)$.  Our results
concern the asymptotic behavior of the descriptors under the
\emph{fluid} scaling, which is defined by
\begin{equation}
  \fs\rbuf(t)=\frac{1}{n} \rbuf^n(t),\quad
  \fs\ser(t)=\frac{1}{n} \ser^n(t),
\end{equation}
for all $t\ge 0$.  The fluid scaling for the arrival process
$E^n(\cdot)$ is defined in the same way, i.e.
\[\fs E(t)=\frac{1}{n}E^n(t),\]
for all $t\ge 0$. We assume that
\begin{equation}\label{eq:cond-A}
  \fs E(\cdot)\dto\lambda \cdot\quad\text{as }n\to\infty.
\end{equation}
Since the limit is deterministic,  the convergence in distribution in
(\ref{eq:cond-A}) is equivalent to
convergence in probability; namely, for each $T>0$ and each
$\epsilon>0$,
\begin{equation}
  \nonumbe
  \lim_{n\to\infty}  \mathbb{P}^n\Bigl( \sup_{0\le t \le T}
  |\bar{E}^n(t)-\lambda t|
  > \epsilon\Bigr) =0.
\end{equation}
Denote $\nu^n_F$ and $\nu^n_G$ the probability measures corresponding to
the patience time distribution $F^n$ and the service time distribution
$G^n$, respectively. Assume that as $n\to\infty$,
\begin{equation}\label{eq:cond-measures}
  \nu^n_F\to \nu_F, \quad \nu^n_G\to \nu_G,
\end{equation}
where $\nu_F$ and $\nu_G$ are some probability measures with
associated distribution functions $F$ and $G$.  Also, the following
initial condition will be assumed:
\begin{align}
  \label{eq:cond-initial}
  (\fs\rbuf(0),\fs\ser(0))&\dto(\bar\rbuf_0,\bar\ser_0)
  \quad\textrm{as }n\to\infty,
\end{align}
where, almost surely, $(\bar\rbuf_0,\bar\ser_0)$ is a valid initial
condition and
\begin{align}
  \label{eq:cond-initial-noatom}
  \bar\rbuf_0 \textrm{ and }\bar\ser_0 \textrm{ has no atoms}.
\end{align}

\begin{thm}\label{thm:fluid-limit}
  In addition to the assumptions \eqref{cond:G-no-atom}--\eqref{cond:fl-3}
  in Theorem~\ref{thm:uniq-exit}, if the sequence of many-server
  queues satisfies
  \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}, then
  \[(\fs\rbuf(\cdot),\fs\ser(\cdot))\dto(\bar\rbuf(\cdot),\bar\ser(\cdot))
    \quad\textrm{ as }n\to\infty,\]
  where, almost surely,
  $(\bar\rbuf(\cdot),\bar\ser(\cdot))$ is the
  unique solution to the fluid model $\fm$ with initial condition
  $(\bar\rbuf_0,\bar\ser_0)$.
\end{thm}

\section{Properties of the Fluid Model}
\label{sec:prop-fluid-model}

In this section, we analyze the proposed fluid model and establish
some basic properties of the fluid model solution.  The proof of
Theorem~\ref{thm:uniq-exit} for existence and uniqueness and the proof
of Theorem~\ref{thm:eqm} for characterization of the equilibrium will
be presented in Section~\ref{subsec:exist-uniq} and
Section~\ref{sec:equil-state-fluid}, respectively.

\subsection{Existence  and Uniqueness of Fluid Model Solutions}
\label{subsec:exist-uniq}

We first present some calculus on the fluid dynamic equations
\eqref{eq:fluid-dyn-eqn-B} and \eqref{eq:fluid-dyn-eqn-S}, which
define the fluid model.  It follows from \eqref{eq:fluid-dyn-eqn-B}
that
\begin{align*}
  \bar Q(t)=\bar \rbuf(t)(C_0)
  =\lambda\int_{t-\frac{\bar R(t)}{\lambda}}^{t}F^c(t-s)ds
  =\lambda\int_{0}^{\frac{\bar R(t)}{\lambda}}F^c(s)ds.
\end{align*}
Let
\begin{equation*}
  F_d(x)=\int_0^x[1-F(y)]dy\quad\textrm{for all }x\ge 0.
\end{equation*}
Please note that the density of $F_d(\cdot)$ is not scaled by the mean
of $F(\cdot)$. Thus, this is not exactly the equilibrium distribution
associated with $F(\cdot)$.  In fact, we do not need the mean
\begin{equation}
  \label{eq:def-N-F}
  N_F=\int_0^\infty[1-F(y)]dy
\end{equation}
to be finite.  Now we have
\begin{equation}\label{eq:fluid-buffer}
  \frac{\bar Q(t)}{\lambda}=F_d(\frac{\bar R(t)}{\lambda}).
\end{equation}
It follows from \eqref{eq:fluid-dyn-eqn-S} that
\begin{align*}
  \bar Z(t)=\bar\ser(t)(C_0)
  &=\bar\ser_0(C_0+t)+\lambda\int_0^tF^c(\frac{\bar R(s)}{\lambda})G^c(t-s)ds\\
  &\quad-\int_0^tF^c(\frac{\bar R(s)}{\lambda})G^c(t-s)d\bar R(s).
\end{align*}
Note that by \eqref{eq:fluid-buffer}, $d\bar Q(s)=F^c(\frac{\bar
  R(s)}{\lambda})d\bar R(s)$.  So
\begin{align*}
  \bar Z(t)&=\bar\ser_0(C_0+t)+\frac{\lambda}{\mu}
    \int_0^tF^c(\frac{\bar R(s)}{\lambda})dG_e(t-s)
   -\int_0^tG^c(t-s)d\bar Q(s).
\end{align*}
Performing change of variable and integration by parts, we have
\begin{equation}\label{eq:fluid-server}
  \begin{split}
    \bar Z(t)&=\bar \ser_0(C_t)+\frac{\lambda}{\mu}
      \int_0^tF^c(\frac{\bar R(t-s)}{\lambda})dG_e(s)\\
    &\quad-\bar Q(t)G^c(0)+\bar Q(0)G^c(t)+\int_0^t\bar Q(t-s)d G(s).
  \end{split}
\end{equation}
We wish to represent the term $F^c(\frac{\bar R(\cdot)}{\lambda})$
using $\bar Q(\cdot)$.  Recall $M_F$ and $N_F$, which are defined in
\eqref{eq:support-F} and \eqref{eq:def-N-F}, respectively. It is clear
that $F_d(x)$ is strictly monotone for $x\in [0, M_F)$. Thus,
$F_d^{-1}(y)$ is well defined for each $y\in [0, N_F)$.  We define
$F_d^{-1}(y)=M_F$ for all $y\ge N_F$.  Thus, (\ref{eq:fluid-buffer})
implies that
\begin{equation}\label{eq:fluid-buffer-inv}
F^c(\frac{\bar R(t)}{\lambda})
=F^c\left(F_d^{-1}(\frac{\bar Q(t)}{\lambda})\right).
\end{equation}
Note that $G^c(0)=1$ by assumption~\eqref{cond:G-no-atom}.  Combining
\eqref{eq:fluid-constr-B}, \eqref{eq:fluid-constr-S},
\eqref{eq:fluid-server}, and \eqref{eq:fluid-buffer-inv}, we obtain
\begin{equation}\nonumber
  \begin{split}
    \bar X(t)&=\bar\ser_0(C_t)+\bar Q_0G^c(t)\\
    &\quad+\frac{\lambda}{\mu}\int_0^tF^c
    \Big(F_d^{-1}(\frac{(\bar X(t-s)-1)^+}{\lambda})\Big)dG_e(s)\\
    &\quad+\int_0^t(\bar X(t-s)-1)^+ d G(s).
  \end{split}
\end{equation}
Now, introduce
\begin{equation*}
H(x)=
\begin{cases}
F^c(F_d^{-1}(\frac{x}{\lambda})) & \text{ if } 0\le x<\lambda, \\
0 & \text{ if } x \ge \lambda,
\end{cases}
\end{equation*}
and
$\zeta_0(\cdot)=\bar\ser_0(C_0+\cdot)+\bar Q_0G^c(\cdot)$.  It then
follows that
\begin{equation}\label{eq:fluid-key}
  \begin{split}
    \bar X(t)&=\zeta_0(t)+\rho\int_0^t H\big((\bar
    X(t-s)-1)^+\big)dG_e(s)+\int_0^t(\bar X(t-s)-1)^+ d G(s).
  \end{split}
\end{equation}
Please note that $\zeta_0(\cdot)$ depends only on the initial
condition and $H(\cdot)$ is a function defined by the arrival rate
$\lambda$ and the patience time distribution $F(\cdot)$.  The equation
\eqref{eq:fluid-key} serves as a key to the analysis of the fluid
model.

\begin{proof}[Proof of Theorem~\ref{thm:uniq-exit}]
  We first prove the existence.  Given a valid initial condition
  $(\bar\rbuf_0,\bar\ser_0)$ (i.e. an element in $\M\times\M_+$ that
  satisfies \eqref{eq:valid-init-buf}--~\eqref{eq:valid-0}), we now
  construct a solution $(\bar\rbuf(\cdot),\bar\ser(\cdot))$ to the
  fluid model $\fm$ with this initial condition.  If the patience time
  distribution $F(\cdot)$ is Lipschitz continuous, then it is clear
  that $H(\cdot)$ is also Lipschitz continuous; if $F(\cdot)$ has a
  density, then the function $H(\cdot)$ is differentiable and has
  derivative
  \begin{equation*}
  H'(x)=-f(y)\frac{1}{F_d'(y)}=\frac{-f(y)}{1-F(y)}=-h_F(y),
  \end{equation*}
  on the interval $(0,\lambda N_F)$ if $y=F_d^{-1}(x)$ and $H(x)=0$
  for all $x\ge \lambda N_F$.  By condition \eqref{cond:fl-3},
  \begin{displaymath}
    \sup_{0< x <\lambda N_F} |H'(x)| = \sup_{y\in [0, M_F)} h_F(y),
  \end{displaymath}
  which implies that $H(\cdot)$ is Lipschitz continuous.  It follows
  from Lemma~\ref{reg-map} that the equation \eqref{eq:fluid-key} has
  a unique solution $\bar X(\cdot)$.  Denote $\bar Q(t)=(\bar X(t)-1)^+$.

  We first claim that $\bar Q(t)/\lambda\le N_F$ for all $t\ge 0$. The
  claim is automatically true if $N_F=\infty$. Now, let us consider the
  case where $N_F<\infty$. Since $(\bar\rbuf_0,\bar\ser_0)$ is a valid
  initial condition, $\bar Q(0)/\lambda\le N_F$. Suppose there exists
  $t_1> 0$ such that $\bar Q(t_1)/\lambda>N_F$. Let $t_0=\sup\{s:\bar
  Q(s)/\lambda\le N_F,s\le t_1\}$. So we have that $\lim_{t\to t_0}\bar
  Q(t)/\lambda\le N_F$, since $\bar Q(\cdot)$ has left limit. Let
  $\delta=(Q(t_1)/\lambda-N_F)/4$ and pick $t_\delta\in[t_0-\delta,t_0]$
  such that $\bar Q(t_\delta)/\lambda\le N_F+\delta$.  By
  Lemma~\ref{lem:reg-map--incre},
  \begin{equation}\label{ineq:non-desc-aux}
    \frac{\bar Q(t')}{\lambda}-\frac{\bar Q(t)}{\lambda}\le \int_t^{t'}
    F^c(F_d^{-1}(\frac{\bar Q(s)}{\lambda}))ds
  \end{equation}
  for any $t<t'$.  This gives that
  \begin{align*}
    \frac{\bar Q(t_1)}{\lambda}
    &\le \frac{\bar Q(t_\delta)}{\lambda}
    +\int_{t_\delta}^{t_1}
    [1-F(F_d^{-1}(\frac{\bar Q(s)}{\lambda}))]ds\\
    &\le N_F+\delta+\int_{t_\delta}^{t_0}1 ds+\int_{t_0}^{t_1}0ds\\
    &\le N_F+2\delta<\frac{\bar Q(t_1)}{\lambda},
  \end{align*}
  which is a contradiction. This proves the claim.
  Let
  \begin{align*}
    \bar Z(t)&=\min(\bar X(t),1),\\
    \bar R(t)&=\lambda F_d^{-1}(\frac{\bar Q(t)}{\lambda}),\\
    \bar B(t)&=\lambda t-\bar R(t),
  \end{align*}
  for all $t\ge 0$.

  Next, we claim that the process $\bar B(\cdot)$ is
  non-decreasing.  To prove this claim, it is enough show that
  \begin{equation}\label{eq:B-non-decreasing-aux-1}
    F_d^{-1}(\frac{\bar Q(t')}{\lambda})-
    F_d^{-1}(\frac{\bar Q(t)}{\lambda})
    \le t'-t
  \end{equation}
  for any $t\le t'$.

  Since $F_d^{-1}$ is a non-decreasing function, the inequality holds
  trivially when $\bar Q(t')\le \bar Q(t)$. We now focus on the case
  where $\bar Q(t')>\bar Q(t)$.  Note that the function
  $F_d^{-1}(\cdot)$ is convex, since the derivative is
  non-decreasing. This together with
  \eqref{ineq:non-desc-aux} implies that
  \begin{equation*}
    F_d^{-1}(\frac{\bar Q(t')}{\lambda})
    \le F_d^{-1}(\frac{\bar Q(t)}{\lambda})+
    {F_d^{-1}}'(\frac{\bar Q(t)}{\lambda})
    \int_t^{t'}F^c(F_d^{-1}(\frac{\bar Q(s)}{\lambda}))ds.
  \end{equation*}
  If $\bar Q(t)\le \bar Q(s)$ for all $s\in[t,t']$, then due to the
  fact that $F^c(F_d^{-1}(\cdot))$ is non-increasing, we have
  \begin{align*}
    F_d^{-1}(\frac{\bar Q(t')}{\lambda})&\le F_d^{-1}(\frac{\bar Q(t)}{\lambda})+
    \frac{1}{F^c(F_d^{-1}(\frac{\bar Q(t)}{\lambda}))}
    F^c(F_d^{-1}(\frac{\bar Q(t)}{\lambda}))(t'-t)\\
    &\le F_d^{-1}(\frac{\bar Q(t)}{\lambda})+ t'-t,
  \end{align*}
  which gives \eqref{eq:B-non-decreasing-aux-1};
  otherwise, let $t^*\in(t,t')$ be the point where $\bar Q(\cdot)$
  achieves minimum. Since $\bar Q(t)>\bar Q(t^*)$, we have
  \begin{equation*}
    F_d^{-1}(\frac{\bar Q(t^*)}{\lambda})-
    F_d^{-1}(\frac{\bar Q(t)}{\lambda})
    \le t^*-t.
  \end{equation*}
  Since $\bar Q(t^*)\le \bar Q(s)$ for all $s\in[t^*,t']$, by the same
  reasoning in the above, we also have
  \begin{equation*}
    F_d^{-1}(\frac{\bar Q(t')}{\lambda}
    \le F_d^{-1}(\frac{\bar Q(t^*)}{\lambda})+t'-t^*.
  \end{equation*}
  The above two inequalities also leads to
  \eqref{eq:B-non-decreasing-aux-1}. So the claim is proved.

  We now construct a fluid model solution by letting
  \begin{align*}
    \bar\rbuf(t)(C_x)&=\lambda\int_{t-\frac{\bar R(t)}{\lambda}}^{t}F^c(x+t-s)ds,\\
    \bar\ser(t)(C_x)&=\bar\ser_0(C_x+t)+\int_0^tF^c(\frac{\bar
      R(s)}{\lambda})G^c(x+t-s)d\bar B(s),
  \end{align*}
  for all $t\ge 0$.  It is clear that the above defined
  $(\bar\rbuf(\cdot),\bar\ser(\cdot))$ satisfies the fluid dynamic
  equations \eqref{eq:fluid-dyn-eqn-B} and \eqref{eq:fluid-dyn-eqn-S}
  and constraints \eqref{eq:fluid-constr-B} and
  \eqref{eq:fluid-constr-S}.  So we conclude that
  $(\bar\rbuf(\cdot),\bar\ser(\cdot))$ is a fluid model solution.

  It now remains to show the uniqueness.  Suppose there is another
  solution to the fluid model $\fm$ with initial condition
  $(\bar\rbuf_0,\bar\ser_0)$, denoted by
  $(\bar\rbuf^\dagger(\cdot),\bar\ser^\dagger(\cdot))$. Similarly,
  denote
  \begin{align*}
    \bar R^\dagger(t)&=\bar\rbuf^\dagger(\R),\\
    \bar Z^\dagger(t)&=\bar\ser^\dagger((0,\infty)),
  \end{align*}
  for all $t\ge 0$.  It must satisfy the fluid dynamic equations
  \eqref{eq:fluid-dyn-eqn-B} and \eqref{eq:fluid-dyn-eqn-S} and
  constraints \eqref{eq:fluid-constr-B} and \eqref{eq:fluid-constr-S}.
  For all $t\ge 0$, let
  \[
    \bar Q^\dagger(t)=\lambda\bar F_d(\frac{\bar R^\dagger(t)}{\lambda}).
  \]
  According to the algebra at the beginning of
  Section~\ref{subsec:exist-uniq}, $\bar X^\dagger(\cdot)$ must also
  satisfy equation \eqref{eq:fluid-key}.  By the uniqueness of the
  solution to the equation \eqref{eq:fluid-key} in
  Lemma~\ref{reg-map},
  \[\bar X^\dagger(t)=\bar X(t)\quad\text{for all }t\ge 0.\]
  This implies that $\bar R^\dagger(t)=\bar R(t)$.  By the dynamic
  equations \eqref{eq:fluid-dyn-eqn-B} and \eqref{eq:fluid-dyn-eqn-S},
  we must have that
  \[
  (\bar\rbuf^\dagger(t),\bar\ser^\dagger(t))
  =(\bar\rbuf(t),\bar\ser(t))\quad \text{for all }t\ge 0.
  \]
  This completes the proof.
\end{proof}

\subsection{Equilibrium State of the Fluid Model Solution}
\label{sec:equil-state-fluid}

In this section, we first intuitively explain what an equilibrium
should be. Then we rigorously prove it in Theorem~\ref{thm:eqm}.
To provide some intuition, note that  in the equilibrium, by equation
\eqref{eq:fluid-dyn-eqn-B}, one should have
\[
  \bar \rbuf_\infty(C_x)={\lambda}\int_0^{\bar R_\infty/\lambda}F^c(x+s)ds,
\]
for the buffer. This immediately implies that
\[
\bar\rbuf_\infty(C_x)=\lambda[F_d(x+\frac{\bar R_\infty}\lambda)-F_d(x)].
\]
So the rate at which customers leave the buffer due to abandonment is:
\begin{align*}
  \lim_{x\to 0}\frac{\bar \rbuf_\infty(C_0)-\bar
    \rbuf_\infty(C_x)}{x} =\lambda F(\frac{\bar R_\infty}\lambda).
\end{align*}
In the equilibrium, intuitively, the number of customers in service
should not change and the distribution for the remaining service time
should be the equilibrium distribution $G_e(\cdot)$, i.e.
\[\bar \ser_\infty(C_x)=\bar Z_\infty [1-G_e(x)].\]
The rate at which customers depart from the server is:
\begin{align*}
  \lim_{x\to 0}\frac{\bar \ser_\infty(C_0)-\bar
    \ser_\infty(C_x)}{x} =\bar Z_\infty\mu.
\end{align*}
The arrival rate must be equal to the summation of the departure rate
from server (due to service completion) and the one from buffer (due
to abandonment), i.e.
\begin{equation}\label{eq:tech-4-1}
\lambda=\lambda F(\frac{\bar R_\infty}\lambda)+\bar Z_\infty\mu.
\end{equation}
It follows directly from \eqref{eq:fluid-buffer} that
\begin{equation}\label{eq:tech-4-2}
\bar Q_\infty=\lambda F_d(\frac{\bar R_\infty}{\lambda}).
\end{equation}
If $\bar R_\infty>0$, then according to \eqref{eq:tech-4-2} we have
$\bar Q_\infty>0$. Thus $\bar Z_\infty=1$ according to policy
constraints. By \eqref{eq:tech-4-1}, $\rho>1$ and $\frac{\bar
  R_\infty}{\lambda}$ is a solution to the equation
$F(w)=\frac{\rho-1}{\rho}$.  If $\bar R_\infty=0$, then according to
\eqref{eq:tech-4-1} we have $\rho=\bar Z_\infty\le 1$. In summary, we
have that
\begin{align*}
  \bar Q_\infty&=\lambda F_d(w),\\
  \bar Z_\infty&=\min(\rho,1),
\end{align*}
where $w$ is a solution to the equation
$F(w)=\max(\frac{\rho-1}{\rho},0)$.  This is consistent with the one
in \cite{Whitt2006}, which is derived from a conjecture of a fluid
model. Now, we rigorously prove this result.

\begin{proof}[Proof of Theorem~\ref{thm:eqm}]
  If $(\bar\rbuf_\infty,\bar\ser_\infty)$ is an equilibrium state,
  then according to the definition, it must satisfies
  \begin{align}
    \label{eq:tech-3-eqm-B}\bar\rbuf_\infty(C_x)
    &=\lambda\int_{t-\frac{\bar R_\infty}{\lambda}}^{t}F^c(x+t-s)ds,
    \quad t\ge 0,\\
    \label{eq:tech-3-eqm-S}\bar\ser_\infty(C_x)
    &=\bar\ser_\infty(C_x+t) +\int_0^tF^c(\frac{\bar
      R_\infty}{\lambda})G^c(x+t-s)d\lambda s,
    \quad t\ge 0.
  \end{align}
  It follows from \eqref{eq:tech-3-eqm-S} that
  \begin{align*}
    \bar\ser_\infty(C_x)-\bar\ser_\infty(C_x+t) &=\rho F^c(\frac{\bar
      R_\infty}{\lambda})\mu\int_0^tG^c(x+t-s)ds\\
    &=\rho F^c(\frac{\bar R_\infty}{\lambda})[G_e(x+t)-G_e(x)], \quad
    t\ge 0.
  \end{align*}
  Taking $t\to\infty$, one has
  \begin{equation}\label{eq:tech-4-ser-infty}
    \bar\ser_\infty(C_x)=\rho F^c(\frac{\bar R_\infty}{\lambda})G_e^c(x).
  \end{equation}
  Thus $\bar Z_\infty=\rho F^c(\frac{\bar R_\infty}{\lambda})$.
  According to \eqref{eq:fluid-buffer}, we have that
  \[
  \bar Q_\infty=\lambda F_d(\frac{\bar R_\infty}{\lambda}).
  \]
  First assume that $\bar R_\infty>0$. Then $\bar Q_\infty>0$, and
  thus $\bar Z_\infty=1$ by the policy constraints
  \eqref{eq:fluid-constr-B} and \eqref{eq:fluid-constr-S}. Therefore,
  $\rho F^c(\frac{\bar R_\infty}{\lambda})=1$, which implies that
  $F(\frac{\bar R_\infty}{\lambda})=\frac{\rho-1}{\rho}$ and $\rho>1$.
  Now assume that $\bar R_\infty=0$. Then $\bar Z_\infty=\rho$, which
  must be less than or equal to 1 by the policy
  constraints. Summarizing the cases where $\rho>1$ and $\rho\le 1$,
  we have that the equilibrium state must satisfy
  \eqref{eq:eqm-B}--\eqref{eq:eqm-w}.

  If a state $(\bar\rbuf_\infty,\bar\ser_\infty)$ satisfies
  \eqref{eq:eqm-B}--\eqref{eq:eqm-w}, then let
  \begin{align*}
    (\bar\rbuf(t),\bar\ser(t))=(\bar\rbuf_\infty, \bar\ser_\infty),
  \end{align*}
  for all $t\ge 0$.  If $\rho\le 1$, then $\bar\rbuf(\cdot)\equiv\0$
  and $\bar Z(\cdot)\equiv\rho$;
  if $\rho> 1$, then $\bar R(\cdot)\equiv\lambda w$ and $\bar
  Z(\cdot)\equiv 1$, where $w$ is a solution to equation
  \eqref{eq:eqm-w}.  It is easy to check that
  $(\bar\rbuf(\cdot),\bar\ser(\cdot))$ is a fluid model solution in
  both cases.  So by definition, the state
  $(\bar\rbuf_\infty,\bar\ser_\infty)$ is a equilibrium state.
\end{proof}

\section{Fluid Approximation of the Stochastic Models}
\label{sec:fluid-appr-stoch}

Similar to \eqref{eq:B(t)}, let
\begin{equation}\label{eq:def-B}
B^n(t)=E^n(t)-R^n(t).
\end{equation}
It follows from \eqref{eq:stoc-dyn-eqn-B} and
\eqref{eq:stoc-dyn-eqn-S} that the dynamics for the fluid scaled
processes can be written as
\begin{align}
  \label{eq:stoc-dyn-eqn-B-fs}
  \fs\rbuf(t)(C)&=\frac{1}{n}\sum_{i=B^n(t)+1}^{E^n(t)}\delta_{u^n_i}(C+t-a^n_i),
  \quad\textrm{for all }C\in\mathscr B(\R),\\
  \label{eq:stoc-dyn-eqn-S-fs}
  \begin{split}
    \fs\ser(t)(C)&=\fs\ser(s)(C+t-s)\\
      &\quad+\frac{1}{n}\sum_{i=B^n(s)+1}^{B^n(t)}
       \delta_{(u^n_i,v^n_i)}(C_0+\tau^n_i-a^n_i)\times(C+t-\tau^n_i),
  \end{split}
  \quad\textrm{for all }C\in\mathscr B((0,\infty)),
\end{align}
for all $0\le s\le t$.

\subsection{Precompactness}

We first establish the following precompactness for the sequence of
fluid scaled stochastic processes
$\{(\fs\rbuf(\cdot),\fs\ser(\cdot))\}$.

\begin{thm}\label{thm:procompactness}
  Assume \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}.  The
  sequence of the fluid scaled stochastic processes
  $\{(\fs\rbuf(\cdot),\fs\ser(\cdot))\}_{N\in\N}$ is precompact as
  $n\to\infty$; namely, for each subsequence
  $\{(\bar\rbuf^{n_k}(\cdot),\bar\ser^{n_k}(\cdot))\}_{n_k}$ with
  $n_k\to\infty$, there exists a further subsequence
  $\{(\bar\rbuf^{n_{k_j}}(\cdot),\bar\ser^{n_{k_j}}(\cdot))\}_{n_{k_j}}$
  such that
  \[
  (\bar\rbuf^{n_{k_j}}(\cdot),\bar\ser^{n_{k_j}}(\cdot))
  \dto(\tilde\rbuf(\cdot),\tilde\ser(\cdot))\quad\text{as }j\to\infty,
  \]
  for some
  $(\tilde\rbuf(\cdot),\tilde\ser(\cdot))\in\D([0,\infty),\M\times\M_+)$.
\end{thm}

The remaining of this section is devoted to proving the above theorem.
By Theorem~$3.7.2$ in \cite{EthierKurtz1986}, it suffices to verify
($a$) the compact containment property, Lemma~\ref{lem:comp-cont} and
($b$) the oscillation bound, Lemma~\ref{lem:osc-bound} below.

\subsubsection{Compact Containment}

A set $\mathbf K\subset\M$ is relatively compact if
$\sup_{\xi\in\mathbf K}\xi(\R)<\infty$, and there exists a sequence of
nested compact sets $A_j\subset \R$ such that $\cup A_j=\R$ and
\[\lim_{j\to\infty}\sup_{\xi\in\mathbf K}\xi(A_j^c)=0,\]
where $A_j^c$ denotes the complement of $A_j$; see
\cite{Kallenberg1986}, Theorem~A$7.5$.  The first major step to prove
Theorem~\ref{thm:procompactness} is to establish the following
\emph{compact containment} property.

\begin{lem}\label{lem:comp-cont}
  Assume \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}.
  Fix $T>0$.  For each $\eta>0$ there exists a compact set $\mathbf
  K\subset\M$ such that
  \begin{equation}\nonumber
    \liminf_{n\to\infty}\prr{(\fs\rbuf(t),\fs\ser(t))
      \in \mathbf K\times\mathbf K\text{ for all }t\in[0,T]}\ge 1-\eta.
  \end{equation}
\end{lem}

To prove this result, we first need to establish some bound
estimations. For the convenience of notation, denote $\fs E(s,t)=\fs
E(t)-\fs E(s)$ for any $0\le s\le t$. Fix $T>0$.  It follows
immediately from condition \eqref{eq:cond-A} that for each
$\epsilon>0$ there exists an $n_0$ such that when $n>n_0$,
\begin{equation}\label{ineq:arrival-bound}
  \prr{\sup_{0\le s<t\le T}|\fs E(s,t)-\lambda(t-s)|<\epsilon}\ge 1-\epsilon.
\end{equation}
To facilitate some arguments later on, we derive the following result
from the above inequality.

\begin{lem}\label{lem:epsilon-function}
  Fix $T>0$. There exists a function $\epsilon_E(\cdot)$, with
  $\lim_{n\to\infty}\epsilon_E(n)=0$ such that
  \begin{equation}\nonumber
    \prr{\sup_{0\le s<t\le T}|\fs E(s,t)-\lambda(t-s)|<\epsilon_E(n)}
    \ge 1-\epsilon_E(n),
  \end{equation}
  for each $n\ge 0$.
\end{lem}

The derivation of the above lemma from \eqref{ineq:arrival-bound}
follows the same as the proof of Lemma~$5.1$ in \cite{ZDZ2009}. We
omit the proof for brevity.  Based on the above lemma, we construct
the following event,
\begin{equation}\label{eq:omegaE-def}
  \Omega^n_E=\{\sup_{t\in[0,T]}|\fs E(s,t)-\lambda (t-s)|<\epsilon_E(n)\}.
\end{equation}
We have that on this event, the arrival process is regular,
i.e. $\fs E(s,t)$ is ``close'' to $\lambda(t-s)$.
And this event has ``large'' probability, i.e.
\begin{equation}\label{eq:omegaE}
  \lim_{n\to\infty}\prr{\Omega^n_E}=1.
\end{equation}

\begin{proof}[Proof of Lemma~\ref{lem:comp-cont}]
  By the convergence of the initial condition \eqref{eq:cond-initial},
  for any $\epsilon>0$, there exists a relatively compact set $\mathbf
  K_0\subset \M$ such that
  \begin{equation}\label{eq:tech-5-init-compact-prob}
      \liminf_{n\to\infty}\prr{\fs\rbuf(0)\in\mathbf K_0
        \textrm{ and }\fs\ser(0)\in\mathbf K_0}>1-\epsilon.
  \end{equation}
  Denote the event in the above probability by $\Omega^n_0$. On this
  event, by the definition of relatively compact set in the space
  $\M$, there exists a function $\kappa_0(\cdot)$ with
  $\lim_{x\to\infty}\kappa_0(x)=0$ such that
  \begin{equation}\label{eq:tech-5-init-0}
    \fs\rbuf(0)(C_x)\le \kappa_0(x),\quad\fs\ser(0)(C_x)\le \kappa_0(x),
  \end{equation}
  and
  \begin{equation}\label{eq:tech-5-init-1}
    \fs\rbuf(0)(C^-_{x})\le \kappa_0(x),
  \end{equation}
  for all $x\ge 0$, where $C^-_x=(-\infty, -x)$ for any
  $y\in\R$. (Remember that $\fs\ser(0)$ is a measure on $(0,\infty)$,
  so we do not need to consider its measure of $C^-_x$.)

  It is clear that on the event $\Omega^n_E\cap\Omega^n_0$, for any
  $t\le T$ and all large $n$,
  \begin{align*}
    \fs\rbuf(t)(\R)&\le\sup_n\fs\rbuf(0)(\R)+2\lambda T,\\
    \fs\ser(t)((0,\infty))&\le 1,
  \end{align*}
  where the last inequality is due to the fact that $Z^n(\cdot)\le n$.
  Again, by the definition of relative compact set in $\M$, we have
  that $\sup_n\fs\rbuf(0)(\R)=M_0<\infty$.

  It follows from the dynamic equation \eqref{eq:stoc-dyn-eqn-B-fs}
  and \eqref{eq:stoc-dyn-eqn-S-fs} that for all $x>0$,
  \begin{align*}
    \fs\rbuf(t)(C_x)&\le\fs\rbuf(0)(C_x)
        +\frac{1}{n}\sum_{i=1}^{E^n(t)}\delta_{u^n_i}(C_x),\\
    \fs\ser(t)(C_x)&\le\fs\ser(0)(C_x)
        +\frac{1}{n}\sum_{i=1}^{E^n(t)}\delta_{v^n_i}(C_x).
  \end{align*}
  Denote $\fs{\mathcal L}_1(t)=\frac{1}{n}\sum_{i=1}^{E^n(t)}
  \delta_{u^n_i}$ and $\fs{\mathcal L}_2(t)=\frac{1}{n}\sum_{i=1}^{E^n(t)}
  \delta_{v^n_i}$. Let us first study these two terms.  Recall the
  definition of the event $\Omega^n_{\text{GC}}(M,L)$ and the envelope
  function $\bar f$ (which increases to infinity) in
  \eqref{def:omega-GC}.  For the application here, it is enough to set
  $M=1$ and $L=2\lambda T$.  On the event
  $\Omega^n_E\cap\Omega^n_{\text{GC}}(M,L)$, we have
  \begin{align*}
    \inn{\bar f}{\fs{\mathcal L}_1(t)}
    \le\inn{\bar f}{\frac{1}{n}\sum_{i=1}^{2\lambda Tn}\delta_{u^n_i}}
    \le 2\lambda T\inn{\bar f}{\nu_F}+1,
  \end{align*}
  for all large enough $n$.  Similarly, on the same event we have that
  \begin{align*}
    \inn{\bar f}{\fs{\mathcal L}_2(t)}
    \le\inn{\bar f}{\frac{1}{n}\sum_{i=1}^{2\lambda Tn}\delta_{v^n_i}}
    \le 2\lambda T\inn{\bar f}{\nu_G}+1,
  \end{align*}
  for all large enough $n$. Denote $M_B=2\lambda T\max(\inn{\bar
    f}{\nu_F}, \inn{\bar f}{\nu_G})+1$.  By Markov's inequality, for
  all $x> 0$ (again, on the same event and for all large $n$)
  \[
  \fs{\mathcal L}_1(t)(C_x)<M_b/\bar f(x), \quad
  \fs{\mathcal L}_2(t)(C_x)<M_b/\bar f(x).
  \]

  Unlike the measure $\ser(t)\in\M_+$, the measure $\rbuf(t)\in\M$.
  So we need to consider all the test set $C^-_x=(-\infty, -x)$ for
  $x\ge 0$. The following inequality again follows from
  \eqref{eq:stoc-dyn-eqn-B-fs},
  \begin{equation}\nonumber
    \fs\rbuf(t)(C^-_x)\le\fs\rbuf(0)(C^-_x+t)+
    \frac{1}{n}\sum_{i=1}^{E^n(t)}\delta_{u^n_i}(C^-_x+t).
  \end{equation}
  Note that if we take $x>T$, then $\delta_{u^n_i}(C^-_x+t)=0$. So we
  have that
  \begin{equation}
    \fs\rbuf(t)(C^-_x)\le \fs\rbuf(0)(C^-_x+T)=\fs\rbuf(0)(C^-_{x-T}),
    \quad\textrm{for all }t\le T.
  \end{equation}

  Now, define the set $\mathbf K\subset \M$ by
  \begin{equation*}
    \begin{split}
      \mathbf K=\Big\{\xi\in\M: &\xi(\R)<1+M_0+2\lambda T,\\
          &\xi(C_x)<\kappa_0(x)+M_b/\bar f(x) \textrm{ for all }x>0,\\
          &\xi(C^-_x)\le\kappa_0(x-T) \textrm{ for all }x\ge T\Big\}.
    \end{split}
  \end{equation*}
  It is clear that $\mathbf K$ is relatively compact and on the event
  $\Omega^n_E\cap\Omega^n_{\text{GC}}(M,L)\cap\Omega^n_0$,
  \[
    (\fs\rbuf(t),\fs\ser(t)) \in \mathbf K\times\mathbf K
    \text{ for all }t\in[0,T].
  \]
  The result of this lemma then follows immediately from
  \eqref{eq:omegaE}, \eqref{eq:tech-5-init-compact-prob} and
  \eqref{eq:omega-GC}.
\end{proof}

\subsubsection{Oscillation Bound}

The second major step to prove precompactness is to obtain the
oscillation bound in Lemma~\ref{lem:osc-bound} below.  The oscillation
of a \rcll{} function $\zeta(\cdot)$ (taking values in a metric space
$(\mathbf E,\pi)$) on a fixed interval $[0,T]$ is defined as
\begin{equation}\nonumber
  \osc{\zeta}{\delta}{T}=\sup_{s,t\in[0,T],|s-t|<\delta}\pi[\zeta(s),\zeta(t)].
\end{equation}
If the metric space is $\R$, we just use the Euclidean metric; if the
space is $\M$ or $\M_+$, we use the Prohorov metric $\pov$ defined in
Section~\ref{subsec:notatioin}.  For the measure-valued processes in our
model, oscillations mainly result from sudden departures of a large
number of customers.  To control the departure process, we show that
$\fs\ser(\cdot)$ and $\fs\rbuf(\cdot)$ assign arbitrarily small mass
to small intervals.

\begin{lem}\label{lem:asymp-reg}
  Assume \eqref{cond:G-no-atom},
  \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}.  Fix $T>0$.  For
  each $\epsilon,\eta>0$ there exists a $\kappa>0$ (depending on
  $\epsilon$ and $\eta$) such that
  \begin{align}

    &\liminf_{n\to\infty}\prr{\sup_{t\in[0,T]}\sup_{x\in\R_+}
      \fs\ser(t)([x,x+\kappa])\le\epsilon}\ge1-\eta.
  \end{align}
\end{lem}
\begin{proof}
First, We have that for any $\epsilon, \eta>0$, there exists a
$\kappa$ such that
\begin{equation}\label{ineq:initial-regular}
  \liminf_{n\to\infty}\prr{\sup_{x\in\R_+}\fs\ser(0)([x,x+\kappa])\le\epsilon/2}
  \ge 1-\eta.
\end{equation}
This inequality is derived from the initial condition.  The derivation
is exactly the same as in the proof of (5.14) in \cite{ZDZ2009}, so we
omit it here for brevity.

Now we need to extend this result to the interval $[0,T]$.  Denote the
event in \eqref{ineq:initial-regular} by $\Omega^n_0$, and the event
in Lemma~\ref{lem:comp-cont} by $\Omega^n_C(\mathbf K)$.  Fix $M=1$
and $L=2\lambda T$, Let
\begin{equation}\label{eq:omega-reg}
  \Omega^n_1(M,L)=\Omega^n_0\cap\Omega^n_C(\mathbf K)
  \cap\Omega^n_E\cap\Omega^n_{\text{GC}}(M,L).
\end{equation}
By \eqref{ineq:initial-regular}, Lemma~\ref{lem:comp-cont},
\eqref{eq:omegaE} and \eqref{eq:omega-GC}, for any fixed $M,L>0$,
\[\liminf_{n\to\infty}\prr{\Omega^n_1(M,L)}\ge1-\eta.\]
In the remainder of the proof, all random objects are evaluated at a
fixed sample path in $\Omega^n_1(M,L)$.

It follows from the fluid scaled stochastic dynamic equation
\eqref{eq:stoc-dyn-eqn-S-fs} that
\begin{align*}
  \fs\ser(t)([x,x+\kappa])&\le\fs\ser(0)([x,x+\kappa]+t)\\
  &\quad+\frac{1}{n}\sum_{i=B(0)+1}^{B(t)}
  \delta_{v^n_i}([x,x+\kappa]+t-\tau^n_i),
\end{align*}
for each $x,\kappa\in\R_+$. By \eqref{ineq:initial-regular}, the first
term on the right hand side of the above equation is always upper
bounded by $\epsilon/2$.  Let $S$ denote the second term on the right
hand side of the preceding equation.  Now it only remains to show
that $S<\epsilon/2$.

Let $0=t_0< t_1< \cdots< t_J=t$ be a partition of the interval $[0,t]$
such that $|t_{j+1}-t_j|<\delta$ for all $j=0, \cdots, J-1$, where
$\delta$ and $N$ are to be chosen below.  Write $S$ as the summation
\[
S=\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=B(t_j)+1}^{B(t_{j+1})}
\delta_{v^n_i}([x,x+\kappa]+t-\tau^n_i).
\]
Recall that $\tau^n_i$ is the time that the $i$th job starts service,
so on each sub-interval $[t_j,t_{j+1}]$ those $i$'s to be summed must
satisfy $t_j\le \tau^n_i\le t_{j+1}$.  This implies that
\[t-t_{j+1}\le t-\tau_i\le t-t_j.\]
Then
\[
S\le\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=B(t_j)+1}^{B(t_{j+1})}
\delta_{v^n_i}([x+t-t_{j+1},x+t-t_{j}+\kappa]).
\]
By \eqref{eq:def-B},
we have for all $j=0,\cdots, J-1$
\begin{align*}
  -\fs R(0)\le \fs B(t_j)&\le \fs E(T),\\
  0\le \fs B(t_{j+1})-\fs B(t_{j})&\le \fs E(T)+\fs R(0).
\end{align*}
By Lemmas~\ref{lem:comp-cont} and \ref{lem:epsilon-function}, $\fs
R(0)<M_0$ and $\fs E(T)\le 2\lambda T$ on $\Omega^n_C(\mathbf
K)\cap\Omega^n_E$ for some constant $M_0$. Take $M=\max(M_0,2\lambda
T)$ and $L=M_0+2\lambda T$, it follows from the Glevenko-Cantelli
estimate \eqref{def:omega-GC} that
\begin{align*}
  &\quad\frac{1}{n}\sum_{i=B^n(t_j)+1}^{B^n(t_{j+1})}
        \delta_{v^n_i}([x+t-t_{j+1},x+t-t_{j}+\kappa])\\
  &\le\Big(\fs B(t_{j+1})-\fs
  B(t_j)\Big)\nu^n([x+t-t_{j+1},x+t-t_{j}+\kappa])+\frac{\epsilon}{4J},
\end{align*}
for each $j<J$.
By condition \eqref{eq:cond-measures}, for any $\epsilon_2>0$,
\[\pov[\nu^n_G,\nu_G]<\epsilon_2,\]
for all large $n$. By the definition of Prohorov metric, we have
\[\nu^n_G([x+t-t_{j+1},x+t-t_{j}+\kappa])\le
\nu_G([x+t-t_{j+1}-\epsilon_2,x+t-t_{j}+\kappa+\epsilon_2]),\] for
all large $n$.  Since
$[x+t-t_{j+1}-\epsilon_2,x+t-t_{j}+\kappa+\epsilon_2]$ is a close
interval with length less than $\kappa+\delta+2\epsilon_2$, by
condition \eqref{cond:G-no-atom}, we can choose
$\kappa,\delta,\epsilon_2$ small enough such that
\[
\nu([x+t-t_{j+1}-\epsilon_2,x+t-t_{j}+\kappa+\epsilon_2])
\le\frac{\epsilon}{4M}.
\]
Thus, we conclude that
\begin{align*}
  S\le\frac{\epsilon}{4J}[\fs B(T)-\fs B(0)]+\frac{\epsilon}{4}
  \le\epsilon/2.
\end{align*}
This completes the proof.
\end{proof}

\begin{lem}\label{lem:osc-bound}
  Assume \eqref{cond:G-no-atom},
  \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}.
  Fix $T>0$.  For each $\epsilon,\eta>0$ there exists a $\delta>0$
  (depending on $\epsilon$ and $\eta$) such that
\begin{equation}
  \liminf_{n\to\infty}\prr{\osc{(\fs\rbuf,\fs\ser)}{\delta}{T}\le
    3\epsilon}\ge 1-\eta.
\end{equation}
\end{lem}
\begin{proof}
Define
\begin{equation}\nonumber
  \Omega^n_{\text{Reg}}(\epsilon,\kappa)
  =\Big\{\sup_{t\in[0,T]}\sup_{x\in\R_+}\fs\ser(t)([x,x+\kappa])\le\epsilon\Big\}.
\end{equation}
By \eqref{eq:omegaE} and Lemma~\ref{lem:asymp-reg}, for each
$\epsilon,\eta>0$ there exists a $\kappa>0$ such that
\begin{equation}\label{ineq:omega-reg-E}
  \liminf_{n\to\infty}\prr{\Omega^n_E\cap\Omega^n_{\text{Reg}}(\epsilon,\kappa)}
  >1-\eta.
\end{equation}
On the event $\Omega^n_E\cap\Omega^n_{\text{Reg}}(\epsilon,\kappa)$,
we have some control over the dynamics of the system.
First, note that the number of customers (in the virtual buffer,
including those who have abandoned but ought to get service if they
did not) that enter the server during time interval $(s,t]$ can be
upper bounded by
\begin{equation}\nonumber
  \fs B(s,t)\le\fs E(s,t)+\fs\ser(s)([0,t-s]).
\end{equation}
When $t-s\le \min(\frac{\epsilon}{2\lambda},\kappa)$, by the
definition of $\Omega^n_E$ and
$\Omega^n_{\text{Reg}}(\epsilon,\kappa)$, we have
\begin{align}
  \label{eq:tech-osc-E-bd}
  \fs E(s,t)&\le \epsilon\\
  \label{eq:tech-osc-B-bd}
  \fs B(s,t)&\le2\epsilon.
\end{align}
Second, by the dynamic equation \eqref{eq:stoc-dyn-eqn-B-fs}, for any
$s<t$ and any set $C\in\mathscr B(\R)$,
\begin{align*}
  \fs\rbuf(t)(C)-\fs\rbuf(s)(C^{3\epsilon}))&\le\fs B(s,t)+\fs E(s,t)\\
  &\quad+\frac{1}{n}\sum_{1+B^n(t)}^{E^n(s)}
   [\delta_{u^n_i}(C+t-a^n_i)-\delta_{u^n_i}(C^{3\epsilon}+s-a^n_i)],
\end{align*}
where $C^a$ is the $a$-enlargement of the set $C$ as defined in
Section~\ref{subsec:notatioin}.  Note that when $t-s\le3\epsilon$,
$C+t-a^n_i\subseteq C^{3\epsilon}+s-a^n_i$ for all $i\in\Z$, which
implies that the second term in the above inequality is less than
zero.  By \eqref{eq:tech-osc-E-bd} and \eqref{eq:tech-osc-B-bd},
\[\fs\rbuf(t)(C)-\fs\rbuf(s)(C^{3\epsilon}))\le 3\epsilon.\]
By Property~(ii) on page 72 in \cite{Billingsley1999},
we have
\begin{equation}\label{eq:osc-bound-buf}
  \pov[\fs\rbuf(t),\fs\rbuf(s)]\le3\epsilon.
\end{equation}
Finally, by the dynamic equation \eqref{eq:stoc-dyn-eqn-S-fs},
\[\fs\ser(t)(C)\le\fs\ser(s)(C+t-s))+\fs B(s,t).\]
Note that when $t-s\le2\epsilon$, $C+t-s\subseteq C^{2\epsilon}$,
where $C^a$ is the $a$-enlargement of the set $C$ as defined in
Section~\ref{subsec:notatioin}.  By \eqref{eq:tech-osc-B-bd}, we have
\[\fs\ser(t)(C)\le\fs\ser(s)(C^{2\epsilon})+2\epsilon.\]
By Property~(ii) on page 72 in \cite{Billingsley1999}, we have
\begin{equation}\label{eq:osc-bound-ser}
\pov[\fs\ser(s),\fs\ser(t)]\le2\epsilon.
\end{equation}
The result of this lemma follows immediately from \eqref{ineq:omega-reg-E},
\eqref{eq:osc-bound-buf} and \eqref{eq:osc-bound-ser}.
\end{proof}

\subsection{Convergence to the Fluid Model Solution}
\label{sec:conv-fluid-model}

We have established the precompactness in
Theorem~\ref{thm:procompactness}.  So every subsequence of the fluid
scaled processes has a further subsequence which converges to some
limit.
For simplicity of notations, we index the convergent subsequence again
by $n$.  So we have that
\begin{equation}\label{eq:tech-5-sub-conv}
(\bar\rbuf^{n}(\cdot),\bar\ser^{n}(\cdot))\dto
(\tilde\rbuf(\cdot),\tilde\ser(\cdot))\quad\text{as }n\to\infty.
\end{equation}
By the oscillation bound in Lemma~\ref{lem:osc-bound}, the limit
$(\tilde\rbuf(\cdot),\tilde\ser(\cdot))$ is almost surely
continuous. We have the following result that further characterizes
the above limit.

\begin{lem}\label{lem:fluid-limit}
  Assume \eqref{cond:G-no-atom}--\eqref{cond:fl-3} and
  \eqref{eq:cond-A}--\eqref{eq:cond-initial-noatom}.  The limit
  $(\tilde\rbuf(\cdot),\tilde\ser(\cdot))$ in
  \eqref{eq:tech-5-sub-conv} is almost surely the solution to the
  fluid model $\fm$ with initial condition $(\bar\rbuf_0,\bar\ser_0)$.
\end{lem}

The rest of this section is devoted to characterizing the limits.  To
better structure the proof, we first provide some preliminary
estimates based on the dynamic equations \eqref{eq:stoc-dyn-eqn-B-fs}
and \eqref{eq:stoc-dyn-eqn-S-fs}.

\begin{lem}\label{lem:bound-sko-prep}
  Let $\{t_j\}_{j=0}^J$ be a partition of the interval $[s,t]$ such
  that $s=t_0<t_1<\ldots<t_J=t$.  We have for any $x\in \R$,
  \begin{align}
    \label{eq:buffer-sko-prep-le}\fs\rbuf(t)(C_x)
    &\le\sum_{i=0}^{J-1}\frac{1}{n}\sum_{i=1+E^n(t_j)}^{E^n(t_{j+1})}
    \delta_{u^n_i}(C_x+t-t_j)+|\fs E(s)-\fs B(t)|,\\
    \label{eq:buffer-sko-prep-ge}\fs\rbuf(t)(C_x)
    &\ge\sum_{i=0}^{J-1}\frac{1}{n}\sum_{i=1+E^n(t_j)}^{E^n(t_{j+1})}
    \delta_{u^n_i}(C_x+t-t_{j+1})-|\fs E(s)-\fs B(t)|.
  \end{align}
  If in addition that $\sup_{\tau\in[s,t]}|\fs
  E(\tau)-\lambda\tau|<\epsilon$, then for any $x>0$,
  \begin{align}
    \begin{split}\label{eq:server-sko-prep-le}
      \fs\ser(t)(C_x)&\le\fs\ser(s)(C_x+t-s)\\
      &\quad+\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=1+B^n(t_j)}^{B^n(t_{j+1})}
      \delta_{u^n_i}(C_0+\frac{\fs R_{L,j}-2\epsilon}{\lambda})
      \delta_{v^n_i}(C_x+t-t_j),
    \end{split}\\
    \begin{split}\label{eq:server-sko-prep-ge}
      \fs\ser(t)(C_x)&\ge\fs\ser(s)(C_x+t-s)\\
      &\quad+\sum_{j=0}^{J-1}\frac{1}{n}\sum_{i=1+B^n(t_j)}^{B^n(t_{j+1})}
      \delta_{u^n_i}(C_0+\frac{\fs R_{U,j}+2\epsilon}{\lambda})
      \delta_{v^n_i}(C_x+t-t_{j+1}),
    \end{split}
  \end{align}
  where $\fs R_{L,j}=\inf_{t\in[t_j,t_{j+1}]}\fs R(t)$ and $\fs
  R_{U,j}=\sup_{t\in[t_j,t_{j+1}]}\fs R(t)$.
\end{lem}
\begin{proof}
  Note that $0\le\delta_{u^n_i}(C)\le1$ for any Borel set $C$ and any
  random variable $u^n_i$. So by the dynamic equation
  \eqref{eq:stoc-dyn-eqn-B-fs}, we have
  \[
  \Big|\fs\rbuf(t)(C)-\frac{1}{n}\sum_{i=E^n(s)+1}^{E^n(t)}
  \delta_{u^n_i}(C+t-a^n_i)\Big|\le|\fs E(s)-\fs B(t)|.
  \]
  For those $i$'s such that $E^n(t_j)<i\le E^n(t_{j+1})$, we have that
  \begin{equation}\label{eq:arrival-times}
    t_j<a^n_i\le t_{j+1}.
  \end{equation}
  This implies that $C_x+t-a_i\subseteq C_x+t-t_j$. So we have
  \begin{align*}
    \sum_{i=1+E^n(t_j)}^{E^n(t_{j+1})}
    \delta_{u^n_i}(C_x+t-a_i)
    &\le
    \sum_{i=1+E^n(t_j)}^{E^n(t_{j+1})}
    \delta_{u^n_i}(C_x+t-t_j).
  \end{align*}
  This establishes
  \eqref{eq:buffer-sko-prep-le}. Also, \eqref{eq:arrival-times} implies
  $C_x+t-t_{j+1}\subseteq C_x+t-a_i$. So \eqref{eq:buffer-sko-prep-ge}
  follows in the same way.

  For those $i$'s such that $B^n(t_j)<i\le B^n(t_{j+1})$, we have that
  \begin{equation*
    t_j<\tau^n_j\le t_{j+1}.
  \end{equation*}
  Note that $\fs R(\tau^n_i)=\fs E(\tau^n_i)-\fs E(a^n_i)$ for each
  $i$.  So, by the closeness between $\fs E(\cdot)$ and
  $\lambda\cdot$, we have
  \begin{align*}
    &\quad|\fs R(\tau^n_i)-\lambda(\tau^n_i-a^n_i)|\\
    &\le|\fs R(\tau^n_i)-\fs E(\tau^n_i)+\fs E(a^n_i)|
    +|\fs E(\tau^n_i)-\fs E(a^n_i)-\lambda(\tau^n_i-a^n_i)|\\
    &\le 2\epsilon.
  \end{align*}
  So
  \begin{equation}\nonumber
    \fs R_{L,j}-2\epsilon\le \lambda(\tau^n_i-a^n_i)\le \fs R_{U,j}+2\epsilon,
  \end{equation}
  for all $i$'s such that $B^n(t_j)<i\le B^n(t_{j+1})$.  Thus,
  \[
  \sum_{i=1+B^n(t_j)}^{B^n(t_{j+1})}
    \delta_{u^n_i}(C_0+\tau^n_i-a^n_i)
    \delta_{v^n_i}(C_x+t-\tau^n_j)
  \le\sum_{i=1+B^n(t_j)}^{B^n(t_{j+1})}
    \delta_{u^n_i}(C_0+\frac{\fs R_{L,j}-2\epsilon}{\lambda})
    \delta_{v^n_i}(C_x+t-t_j).
  \]
  This implies \eqref{eq:server-sko-prep-le}. And
  \eqref{eq:server-sko-prep-ge} can be proved in the same way.
\end{proof}

Recall the notations $\fs\gc(m,l),\fs\gc_p(m,l)$ and $\fs\gc_S(m,l)$
are defined in \eqref{eq:def-gc-p}--\eqref{eq:def-gc} in the appendix.
Using these notations, Lemma~\ref{lem:bound-sko-prep} can be written
as the following:
\begin{lem}\label{lem:bound-sko-prep-GC}
  Let $\{t_j\}_{j=0}^J$ be a partition of the interval $[s,t]$ such
  that $s=t_0<t_1<\ldots<t_J=t$.  We have for any $x\in \R$,
  \begin{align}
    \label{eq:buffer-sko-prep-le-GC}\fs\rbuf(t)(C_x)
    &\le\sum_{i=0}^{J-1}\inn{1_{(C_x+t-t_j)}}
    {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}+|\fs E(s)-\fs B(t)|,\\
    \label{eq:buffer-sko-prep-ge-GC}\fs\rbuf(t)(C_x)
    &\ge\sum_{i=0}^{J-1}\inn{1_{(C_x+t-t_{j+1})}}
    {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}-|\fs E(s)-\fs B(t)|.
  \end{align}
  If in addition that $\sup_{\tau\in[s,t]}|\fs
  E(\tau)-\lambda\tau|<\epsilon$, then for any $x>0$,
  \begin{align}
    \begin{split}\label{eq:server-sko-prep-le-GC}
      \fs\ser(t)(C_x)&\le\fs\ser(s)(C_x+t-s)\\
      &\quad+\sum_{j=0}^{J-1}
      \inn{1_{(C_0+\frac{\fs R_{L,j}-2\epsilon}{\lambda})\times(C_x+t-t_j)}}
      {\fs\gc(B^n(t_j),\fs B(t_j,t_{j+1}))},
    \end{split}\\
    \begin{split}\label{eq:server-sko-prep-ge-GC}
      \fs\ser(t)(C_x)&\ge\fs\ser(s)(C_x+t-s)\\
      &\quad+\sum_{j=0}^{J-1}
      \inn{1_{(C_0+\frac{\fs R_{U,j}+2\epsilon}{\lambda})\times(C_x+t-t_{j+1})}}
      {\fs\gc(B^n(t_j),\fs B(t_j,t_{j+1}))}.
    \end{split}
  \end{align}
\end{lem}

Fix a constant $T>0$ and let $M=1$ and $L=2\lambda T$.  Denote the
random variable
\begin{equation}\label{eq:empirical-V-ML}
  \fs V_{M,L}=\max_{-nM< m< nM}\sup_{l\in[0,L]}\sup_{x,y\in\R}\left\{
    \begin{array}{l}
      \big|\fs\gc(m,l)(C_x\times C_y)-l\nu_F^n(C_x)\nu_G^n(C_y)\big|\\
      +\big|\fs\gc_F(m,l)(C_x)-l\nu_F^n(C_x)\big|\\
      +\big|\fs\gc_G(m,l)(C_x)-l\nu_G^n(C_x)\big|
    \end{array}
  \right\}.
\end{equation}
By Lemma~\ref{lem:glivenko-cantelli}, for any fixed constants $M,L>0$,
\[\fs V_{M,L}\dto 0\quad \textrm{as }n\to\infty.\]
By the assumption \eqref{eq:cond-A}, we have
\[\fs E(\cdot)\dto\lambda \cdot\quad\text{as }n\to\infty.\]
Since both the above two limits are deterministic, those convergences
are joint with the convergence of
$(\bar\rbuf^{n}(\cdot),\bar\ser^{n}(\cdot))$.  Now, for each $n\ge 1$,
we can view $(\fs E(\cdot),
\bar\rbuf^{n}(\cdot),\bar\ser^{n}(\cdot),V_{M,L})$ as a random
variable in the space $\mathbf E_1$, which is the product space of
three $\D([0,\infty),\R)$ spaces and the space $\R$. And
$(\fs\gc(m,\cdot), \fs\gc_F(m,\cdot), \fs\gc_G(m,\cdot): m\in\Z)$ in
the product space $\mathbf E_2$ of countable many $\D([0,\infty),\M)$
spaces. It is clear that both $\mathbf E_1$ and $\mathbf E_2$ are
complete and separable metric spaces.  Using the extension of Skorohod
representation Theorem, Lemma~\ref{lem:ext-sko}, we assume without
loss of generality that $\fs E(\cdot),
\bar\rbuf^{n}(\cdot),\bar\ser^{n}(\cdot), \fs V_{M,L},
\fs\gc(m,\cdot), \fs\gc_F(m,\cdot), \fs\gc_G(m,\cdot), m\in\Z$, and
$(\tilde\rbuf(\cdot),\tilde\ser(\cdot))$ are defined on a common
probability space $(\tilde\Omega, \tilde{\mathcal F}, \tilde{\mathbb
  P})$ such that, almost surely,
\begin{equation}\label{eq:sko-rep}
  \Big((\bar\rbuf^{n}(\cdot),\bar\ser^{n}(\cdot)),\fs V_{M,L},\fs E(\cdot)\Big)
  \to
  \Big((\tilde\rbuf(\cdot),\tilde\ser(\cdot)),0,\lambda \cdot\Big)
  \textrm{\quad as }n\to\infty,
\end{equation}
and inequalities
\eqref{eq:buffer-sko-prep-le-GC}--\eqref{eq:server-sko-prep-ge-GC} and
equation \eqref{eq:empirical-V-ML} also hold almost surely.  Note that
the convergence of each function component in the above is in the
Skorohod $J_1$ topology.  Since the limit is continuous, the
convergence is equivalent to the convergence in the uniform norm on
compact intervals.  Thus as $n\to\infty$,
\begin{align}
  \label{eq:buf-limit}
  \sup_{t\in[0,T]}\pov[\fs\rbuf(t),\tilde\rbuf(t)]\to0,\\
  \label{eq:ser-limit}
  \sup_{t\in[0,T]}\pov[\fs\ser(t),\tilde\ser(t)]\to0,\\
  \label{eq:arrive-limit}
  \sup_{t\in[0,T]}\big|\fs E(t)-\lambda t\big|\to0,
\end{align}
where $\pov$ is the Skorohod metric defined in
Section~\ref{subsec:notatioin}.  Same as on the original probability
space, let
\begin{align*}
  \bar R^n(\cdot)=\inn{1}{\bar \rbuf^n(\cdot)},
  &\quad\bar Q^n(\cdot)=\inn{1_{(0,\infty)}}{\bar \rbuf^n(\cdot)},\\
  \bar Z^n(\cdot)=\inn{1}{\bar \ser^n(\cdot)}, &\quad \bar
  X^n(\cdot)=\bar Q^n(\cdot)+\bar Z^n(\cdot),
\end{align*}
and
\[\fs B(\cdot)=\fs E(\cdot)-\fs R(\cdot).\]
According to \eqref{eq:buf-limit} and \eqref{eq:arrive-limit}, we have
\begin{equation}\label{eq:b-arr-limit}
  \sup_{t\in[0,T]}\big|\fs B(t)-\tilde B(t)\big|\to0.
\end{equation}

For each $n$, let $\tilde\Omega_{n,2}$ be an event of probability one
on which the stochastic dynamic equations \eqref{eq:stoc-dyn-eqn-B-fs}
and \eqref{eq:stoc-dyn-eqn-S-fs} and the policy constraints
\eqref{eq:constr-B} and \eqref{eq:constr-S} hold.  Define
$\tilde\Omega_0=\tilde\Omega_1\cap(\cap_{n=0}^\infty\tilde\Omega^n_{n,2})$,
where $\tilde\Omega_1$ is the event of probability one on which
\eqref{eq:sko-rep} holds.  Then $\tilde\Omega_0$ also has probability
one.  Based on Lemma~\ref{lem:bound-sko-prep} and the above argument
using Skorohod Representation theorem, we can now prove
Lemma~\ref{lem:fluid-limit}.

\begin{proof}[Proof of Lemma~\ref{lem:fluid-limit}]
  For any $t\ge 0$, fix a constant $T> t$. Let us now study
  $(\tilde\rbuf(\cdot),\tilde\ser(\cdot))$ on the time interval
  $[0,T]$.  It is enough to show that on the event $\tilde\Omega_0$,
  $(\tilde\rbuf(t),\tilde\ser(t))$ satisfies the fluid model equation
  \eqref{eq:fluid-dyn-eqn-B}--\eqref{eq:fluid-dyn-eqn-S} and the
  constraints \eqref{eq:fluid-constr-B}--\eqref{eq:fluid-constr-S}.
  Assume for the remainder of this proof that all random objects are
  evaluated at a sample path in the event $\tilde\Omega_0$.

  We first verify \eqref{eq:fluid-dyn-eqn-B}.  For any $\epsilon>0$,
  consider the difference
  \begin{align*}
    &\quad\tilde\rbuf(t)(C_x)-
    \int_{t-\frac{\tilde R(t)}{\lambda}}^tF^c(x+t-s)d\lambda s\\
    &=\tilde\rbuf(t)(C_x)-\fs\rbuf(t)(C^\epsilon_x) +
    \fs\rbuf(t)(C^\epsilon_x) -\int_{t-\frac{\tilde
        R(t)}{\lambda}}^tF^c(x+t-s)d\lambda s,
  \end{align*}
  where $C^\epsilon_x$ is the $\epsilon$-enlargement of the set $C_x$
  as defined in Section~\ref{subsec:notatioin}, which is essentially
  $C_{x-\epsilon}$.  Let $t_0=t-\tilde R(t)/\lambda$.  According to
  \eqref{eq:buffer-sko-prep-le-GC}, we have that
  \begin{equation}\label{eq:diff-buf-ub}
    \begin{split}
      &\quad\tilde\rbuf(t)(C_x)- \int_{t-\frac{\tilde
          R(t)}{\lambda}}^t
      F^c(x+t-s)d\lambda s\\
      &\le\tilde\rbuf(t)(C_x)-\fs\rbuf(t)(C^\epsilon_x)+|\fs E(t_0)-\fs B(t)|\\
      &\quad\sum_{i=0}^{J-1}\inn{1_{(C^\epsilon_x+t-t_j)}}
      {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}
        -\int_{t_0}^tF^c(x+t-s)d\lambda s,
    \end{split}
  \end{equation}
  where $\{t_j\}_{j=0}^J$ is a partition of the interval $[t_0,t]$
  such that $t_0<t_1<\ldots<t_J=t$ and $\max_j(t_{j+1}-t_j)<\delta$
  for some $\delta>0$.  By the definition of Prohorov metric and the
  convergence in \eqref{eq:buf-limit}, the first term on the right
  hand side of \eqref{eq:diff-buf-ub} is bounded by $\epsilon$ for
  all large $n$.  By \eqref{eq:buf-limit} and \eqref{eq:arrive-limit}
  \begin{align*}
    |\fs B(t)-\fs E(t_0)|&=|\fs E(t)-\fs R(t)-\fs E(t_0)|\\
    &\le |\fs E(t)-\lambda t|+|\fs R(t)-\tilde R(t)|+|\fs
    E(t_0)-\lambda t_0|<3\epsilon,
  \end{align*}
  for all large $n$.  So
  \begin{equation}\label{eq:diff-buf-u}
    \begin{split}
      &\quad\tilde\rbuf(t)(C_x)- \int_{t-\frac{\tilde
          R(t)}{\lambda}}^t
      F^c(x+t-s)d\lambda s\\
      &\le4\epsilon+\sum_{i=0}^{J-1}\inn{1_{(C^\epsilon_x+t-t_j)}}
      {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}
        -\int_{t_0}^tF^c(x+t-s)d\lambda s,
    \end{split}
  \end{equation}
  for all large $n$. Similarly, according to
  \eqref{eq:buffer-sko-prep-ge-GC}, we have
  \begin{equation}\label{eq:diff-buf-l}
    \begin{split}
      &\quad\tilde\rbuf(t)(C_x)- \int_{t-\frac{\tilde
          R(t)}{\lambda}}^t
      F^c(x+t-s)d\lambda s\\
      &\ge-4\epsilon+\sum_{i=0}^{J-1}\inn{1_{(C^\epsilon_x+t-t_{j+1})}}
      {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}
        -\int_{t_0}^tF^c(x+t-s)d\lambda s,
    \end{split}
  \end{equation}
  for all large $n$.
  Note that for each $j$, we have
  \begin{align*}
    &\quad\inn{1_{(C_x+t-t_j)}}
    {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}\\
    &\le
    \inn{1_{(C_x+t-t_j)}}
    {\fs\gc_p(E^n(t_j),\lambda(t_{j+1}-t_j)+2\epsilon}\\
    &\le
    [\lambda(t_{j+1}-t_j)+2\epsilon]\nu^n_F(C^\epsilon_x+t-t_j)+\epsilon\\
    &\le
    [\lambda(t_{j+1}-t_j)+2\epsilon][\nu_F(C_x+t-t_j)+\epsilon]+\epsilon\\
    &\le
    \lambda(t_{j+1}-t_j)\nu_F(C_x+t-t_j)+(3+\lambda\delta)\epsilon
  \end{align*}
  for all large $n$, where the first inequality is due to
  \eqref{eq:arrive-limit}, the second one is due to \eqref{eq:sko-rep}
  (the component of $\fs V_{M,L}$), the third one is due to
  \eqref{eq:cond-measures}, and the last one is due to algebra.
  Similarly, we can show that
  \begin{align*}
    &\quad\inn{1_{(C_x+t-t_{j+1})}}
    {\fs\gc_p(E^n(t_j),\fs E(t_j,t_{j+1})}\\
    &\ge
    \lambda(t_{j+1}-t_j)\nu_F(C_x+t-t_{j+1})-(3+\lambda\delta)\epsilon
  \end{align*}
  for all large $n$.  Note that
  $\sum_{j=0}^{J-1}\lambda(t_{j+1}-t_j)F^c(x+t-t_{j})$ and
  $\sum_{j=0}^{J-1}\lambda(t_{j+1}-t_j)F^c(x+t-t_{j+1})$ serve as the
  upper and lower Reimann sum of the integral
  $\int_{t_0}^tF^c(x+t-s)d\lambda s$, which converge to the
  integration as $n\to\infty$.  So by \eqref{eq:diff-buf-u} and
  \eqref{eq:diff-buf-l}, we have that for all large $n$,
  \begin{equation*}
    \big|\tilde\rbuf(t)(C_x)-\int_{t-\frac{\tilde R(t)}{\lambda}}^t
      F^c(x+t-s)d\lambda s\big|\le(3+\lambda\delta)J\epsilon+5\epsilon.
  \end{equation*}
  We conclude that $\tilde\rbuf(t)(C_x)-\int_{t-\frac{\tilde
      R(t)}{\lambda}}^t F^c(x+t-s)d\lambda s=0$ since $\epsilon$ in
  the above can be arbitrary.  This verifies
  \eqref{eq:fluid-dyn-eqn-B}.

  Next, we verify \eqref{eq:fluid-dyn-eqn-S}.  For any $\epsilon>0$,
  consider the difference
  \begin{equation}\label{eq:diff-ser}
    \begin{split}
      &\quad\Big|\tilde\ser(t)(C_x)-\bar\ser_0(C_x+t)
      -\int_0^tF^c(\frac{\tilde R(s)}{\lambda})
      G^c(x+t-s)d[\lambda s-\tilde R(s)]\Big|\\
      &\le|\tilde\ser(t)(C_x)-\fs\ser(t)(C_x^\epsilon)|
      +|\tilde\ser_0(C_x+t)-\fs\ser(0)(C_x^\epsilon+t)|\\
      &\quad+\Big|\fs\ser(t)(C_x^\epsilon)-\fs\ser(0)(C_x^\epsilon+t)
      -\int_0^tF^c(\frac{\tilde R(s)}
      {\lambda})G^c(x+t-s)d[\lambda s-\tilde R(s)]\Big|,
    \end{split}
  \end{equation}
  where the above inequality is due to the fluid scaled stochastic
  dynamic equation \eqref{eq:stoc-dyn-eqn-S-fs}.  Again, by the
  definition of Prohorov metric and the convergence in
  \eqref{eq:ser-limit}, each of the first two terms on the right hand
  side in the above inequality is less than $\epsilon$ for all large
  $n$.  Let $\{t_j\}_{j=0}^J$ be a partition of the interval $[0,t]$
  such that $0=t_0<t_1<\ldots<t_J=t$ and $\max_j(t_{j+1}-t_j)<\delta$
  for some $\delta>0$.  Let
  \[
  \tilde R_{U,j}=\sup_{t\in[t_j,t_{j+1}]}\tilde R(t),\quad
  \tilde R_{L,j}=\inf_{t\in[t_j,t_{j+1}]}\tilde R(t).
  \]
  By \eqref{eq:buf-limit}, we have that
  \begin{equation*}
    |\fs R_{U,j}-\tilde R_{U,j}|\le \epsilon,\quad |\fs R_{L,j}-\tilde R_{L,j}|\le \epsilon,
  \end{equation*}
  for all large $n$.  So for each $j$, we have
  \begin{align*}
    &\quad\inn{1_{(C_0+\frac{\fs R_{L,j}-2\epsilon}{\lambda})\times(C^\epsilon_x+t-t_j)}}
    {\fs\gc(B^n(t_j),\fs B(t_j,t_{j+1}))}\\
    &\le\inn{1_{(C_0+\frac{\tilde R_{L,j}-3\epsilon}{\lambda})\times(C^\epsilon_x+t-t_j)}}
    {\fs\gc(B^n(t_j),\tilde B(t_{j+1})-\tilde B(t_j)+2\epsilon)}\\
    &\le[\tilde B(t_{j+1})-\tilde B(t_j)+2\epsilon]
    \nu^n_F(C_0+\frac{\tilde R_{L,j}-3\epsilon}{\lambda})
    \nu^n_G(C^\epsilon_x+t-t_j)+\epsilon\\
    &\le[\tilde B(t_{j+1})-\tilde B(t_j)+2\epsilon]
    [\nu_F(C_0+\frac{\tilde R_{L,j}}{\lambda})+\frac{3\epsilon}{\lambda}]
    [\nu_G(C_x+t-t_j)+\epsilon]+\epsilon
  \end{align*}
  for all large $n$, where the first inequality is due to
  \eqref{eq:b-arr-limit}, the second one is due to \eqref{eq:sko-rep}
  (the component of $\fs V_{M,L}$), the third one is due to
  \eqref{eq:cond-measures}.  Let $M_B$ be a finite upper bound of
  $\tilde B(t_{J})-\tilde B(t_0)$, the above inequality can be further
  bounded by
  \begin{align*}
    [\tilde B(t_{j+1})-\tilde B(t_j)]
    \nu_F(C_0+\frac{\tilde R_{L,j}}{\lambda})
    \nu_G(C_x+t-t_j)+(\frac{3}{\lambda}+2)M_B\epsilon+3\epsilon.
  \end{align*}
  Similarly, we can show that
  \begin{align*}
    &\quad\inn{1_{(C_0+\frac{\fs R_{U,j}+2\epsilon}{\lambda})\times(C_x+t-t_{j+1})}}
    {\fs\gc(B^n(t_j),\fs B(t_j,t_{j+1}))}\\
    &\ge
    [\tilde B(t_{j+1})-\tilde B(t_j)]
    \nu_F(C_0+\frac{\tilde R_{L,j}}{\lambda})
    \nu_G(C_x+t-t_j)-(\frac{3}{\lambda}+2)M_B\epsilon-3\epsilon.
  \end{align*}
  Note that $\sum_{j=0}^{J-1}[\tilde B(t_{j+1})-\tilde B(t_j)]
  F^c(\frac{\tilde R_{U,j}}{\lambda})G^c(x+t-t_{j})$ and
  $\sum_{j=0}^{J-1}[\tilde B(t_{j+1})-\tilde B(t_j)]
  F^c(\frac{\tilde R_{L,j}}{\lambda})G^c(x+t-t_{j+1})$ serve as the upper and
  lower Reimann sum of the integral $\int_{t_0}^tF^c(\frac{\tilde
    R(s)}{\lambda})G^c(x+t-s)d\tilde B(s)$, which converge to the
  integration as $n\to\infty$.  So, by
  \eqref{eq:server-sko-prep-le-GC} and
  \eqref{eq:server-sko-prep-ge-GC}, we have that for all large $n$,
  \[
  \Big|\fs\ser(t)(C_x^\epsilon)-\fs\ser(0)(C_x^\epsilon+t)
  -\int_{t_0}^tF^c(\frac{\tilde R(s)}{\lambda})G^c(x+t-s)d\tilde B(s)\Big|
  \le (\frac{3}{\lambda}+2)M_B\epsilon+3\epsilon+\epsilon.
  \]
  In summary, the right hand side of \eqref{eq:diff-ser} can be
  bounded by a finite multiple of $\epsilon$.  We conclude that the
  left hand side of \eqref{eq:diff-ser} must be 0 since it does not
  depend on $\epsilon$, which can be arbitrary.  This verifies
  \eqref{eq:fluid-dyn-eqn-S}.

  The verification of fluid constrains \eqref{eq:fluid-constr-B} and
  \eqref{eq:fluid-constr-S} is quite straightforward. Basically, it is
  just passing the fluid scaled stochastic constraints
  \begin{align*}
    \fs Q(t)&=(\fs X(t)-1)^+,\\
    \fs Z(t)&=(\fs X(t)\wedge1),
  \end{align*}
  to $n\to\infty$. We omit it for brevity.
\end{proof}

\section{The Special Case with Exponential Distribution}
\label{sec:special-case-with}

In this section, we verify that the fluid model developed in this
paper for the general patience and service time distributions is
consistent with the one in \cite{Whitt2004}, that was obtained in the
special case where both distributions are assumed to be exponential.

Our fluid model equations implies the key relationship
\eqref{eq:fluid-key}. Now, we specialize in the case with exponential
distribution, i.e.
\[
F(t)=F_e(t)=1-e^{-\alpha t},\quad G(t)=G_e(t)=1-e^{-\mu t}, \quad
\textrm{ for all } t\ge 0.
\]
Now \eqref{eq:fluid-key} becomes
\begin{equation*
  \begin{split}
    \bar X(t)&=\zeta_0(t)+\rho\int_0^t \big[1-\frac{\alpha}{\lambda}\big((\bar
    X(t-s)-1)^+\big)\big]\mu e^{-\mu s}ds+\int_0^t(\bar X(t-s)-1)^+ \mu
    e^{-\mu s}ds.
  \end{split}
\end{equation*}
In the case of exponential service time distribution, the remaining
service time of those initially in service and the service times of
those initially waiting in queue are also assumed to be exponentially
distributed. So we have
\[\zeta_0(t)=\bar\ser_0(C_0+t)+\bar Q_0e^{-\mu t}=\bar X_0e^{-\mu t},\]
where $\bar X_0=\bar Z_0+\bar Q_0$ is the initial number of customers
in the system. By some algebra, the above two equations can be
simplified as the following,
\begin{equation}\label{eq:fluid-key-expl}
\bar X(t)=\bar X_0e^{-\mu t}+\rho[1-e^{-\mu t}]+(\mu-\alpha)
\int_0^t(\bar X(t-s)-1)^+e^{-\mu s}ds.
\end{equation}
By the change of variable $t-s\to s$, the above integration can be
written as
\[
\int_0^t(\bar X(t-s)-1)^+e^{-\mu s}ds
=e^{-\mu t}\int_0^t(\bar X(s)-1)^+e^{\mu s}ds.
\]
Taking the derivative on both sides of \eqref{eq:fluid-key-expl}
yields
\begin{align*}
\bar X'(t)&=-\mu X_0e^{-\mu t}+\mu\rho e^{\mu t}\\
&\quad +(\mu-\alpha)[-\mu e^{-\mu t}\int_0^t(\bar X(s)-1)^+e^{\mu s}ds
+e^{-\mu t}(\bar X(t)-1)^+e^{\mu t}]\\
&=-\mu X_0e^{-\mu t}-\mu\rho[1-e^{\mu t}]+\mu\rho\\
&\quad -\mu(\mu-\alpha)
e^{-\mu t}\int_0^t(\bar X(s)-1)^+e^{\mu s}ds+(\mu-\alpha)(\bar X(t)-1)^+\\
&=-\mu \bar X(t)+\mu\rho+(\mu-\alpha)(\bar X(t)-1)^+.
\end{align*}
Using the notation in \cite{Whitt2004}, $a^{-}=-\min(0,a)$ for any
$a\in\R$. Note that $a=\min(a,1)+(a-1)^+=1-(a-1)^-+(a-1)^+$. So the
above equation further implies
\[\bar X'(t)=\mu(\rho-1)-\alpha(\bar X(t)-1)^++\mu(\bar X(t)-1)^-,
\quad\textrm{for all }t\ge 0.\] This equation is consistent with
Theorem~2.2 in \cite{Whitt2004} ($\mu$ is assumed to be 1 in that
paper).

\section*{Acknowledgements}

The author would like to express the gratitude to his Ph.D
supervisors, Professor Jim Dai and Professor Bert Zwart, for many
useful discussions.  The author is grateful to Professor Christian
Gromoll from the department of mathematics at University of Virginia
for suggesting a nice method on using Skorohod representation theorem
to make the presentation in Section~\ref{sec:conv-fluid-model}
rigorous.  This research is supported in part by National Science
Foundation grants CMMI-0727400 and CNS-0718701.

\bibliographystyle{../bib/acmtrans-ims}

redpajama_set_name: RedPajamaArXiv


text:
Bloodstone: Information, Meaning, Value & How to Care for
Sabrina Yuquan Chen (陈玉泉)
Rough Blood Stone
What Is Bloodstone & Where Can It Be Found
Bloodstone is dickite or kaolinite found in cinnabar bands, it is a type of stone featuring the scarlet color that is redder than vermilion. Because the redness is exactly the same as chicken blood, people name it heliotrope (chicken blood stone by the Chinese). Cinnabar is found in all localities that yield mercury, notably Puerto Princesa in Philippines; Almadén in Spain; New Almaden and New Idria in California; Idrija in Slovenia; Giza, Egypt; Huancavelica in Peru; Murfreesboro in Arkansas; Terlingua inTexas, and few other places. In China, bloodstone was first found in Yuyan mountain in Changhua, Zhejiang. Later Bahrain blood stone was found in Balinyouqi, Chifeng city of Inner Mongolia. In the 20th century, bloodstone was also discovered in Shaanxi, Gansu, Sichuan, Hunan, Yunnan and other places in China. Now that the Changhua cinnabar (mercury) mining is near its end, the blood stone output is very limited, which lead to ever increasing market prices.
What Factors Affect Colors of Bloodstone
When cinnabar (mercuric sulfide) penetrates into kaolinite and dickite, the result is a naturally blended stone from the two which is very rare around the world. The blood part is composed of mainly mercuric sulfide, while the base stone is either kaolinite or dickite. Its hardness is 2-3.The stone is often mixed with a small amount of spotted mercury and grains of quartz on the surface. In addition to mercuric sulfide, the blood also contains other coloration elements like iron and titanium, they are the darker dying ingredients and the main factors bloodstones showing different reds. Additionally, the bloodstone also contains different photosensitive elements such as selenium and tellurium, they are the main reason why the color fades or changes under the light and heat.
Bloodstone Jewelry
A Beautiful High Grade Bloodstone
High Grade Chicken Blood Stone Sculpture
Bloostone: Used for Seal, Crafting & Jewelry
Bloodstone has various colors, it can be red, pink, purple, dark red, etc., the most valuable is the bright red with the "blood" appears to be actively flowing. In China, it is mainly used as a seal or crafting material. High grade bloodstone is mostly appropriate as a seal, so to speak, any bloodstone has been carved as a sculpture is not of high quality, since the purpose of carving it is to cover the natural defects.
The stone has eight basic colors, which blend or combine in different layers: red, black, white, yellow, green, blue, gray, and purple. The ingredients vary from 2 to 7 in hardness, making it a very difficult stone to work with.
The best jewelry-grade bloodstone is solid green with deeply saturated color and highly visible red spots, yet most pieces have only tiny red spots, some have no spots at all. The more blood the higher grade in general. Most bloodstone is treated to a highly polished finish to enhance its value. Bloodstone is often cut into round or convex (cabochon) beads to make necklaces and pendants. It is also made into rings and bracelets.
Polished Rough Bloodstone Display
Bloodstone Seal
Chicken Blood Stone: Meaning & Value
Bloodstone is called chicken blood stone in China, it gets its name for its bright red color, resembling fresh chicken blood. Chicken blood stone found at Yuyan mountain of Changhua, China began its development back to the Ming Dynasty, flourished in the Qianlong period of Qing Dynasty, it's been 600 years of history. Bloodstone is beautiful and rare, and has always been crowned as "queen of stone", "princess of seal stone", and "queen of seal stone".
During the establishment of diplomatic relations between China and Japan in September, 1972, Premier Zhou Enlai of the Chinese Government presented a pair of valuable seal, originally stored in National Palace Museum, to the Japanese Prime Minister Tanaka. This symbolic action caused great repercussions in Japan, Southeast Asia, Hong Kong and Macao region of China, and set off a chicken blood stone boom. Later, large number of Japanese tourists must purchase this chicken blood stone as the preferred gift and take them back home. The Chinese had also built up the greatest enthusiasm in collecting and invest in chicken blood stone ever since.
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As with other ornamental objects, the value will be based primarily on artistry and workmanship. You also need to take any antique value into consideration.
Chicken Blood Stone Seal
Bloodstone Display with Crane
A Legend from Ancient China
Once upon a time, there was a bird called "Bird Lion" that was naturally aggressive. One day, the bird saw a female phoenix hatching when he was flying over Yuyan mountain, and immediately came up with some evil thoughts and began to attack her. The leg of the unprepared female phoenix was snapped apart. Male phoenix arrived after hearing the news, and began to fight against the evil bird. The bird lion was defeated in the end. Although the phoenix claimed victory, the female phoenix was bleeding, so much that the blood stained the entire Yuyan mountain, and the mountain gradually turned into chicken blood stone.
Chicken Blood Stone - Traditional Gift in China
In China, chicken blood stone is a wonderful piece of art in ornamental stones and one of the most prized ornamental materials. Since the color red is the symbol of good luck, the Chinese tend to present everything in red color including giving red seals or chicken blood stones to bless people for events marriages, birthdays, promotions, and success. Chicken blood stone is always one of the greatest gift ideas in such cases and it never goes out of style. Appreciation and evaluation of chicken blood stone in the modern world are mainly constrained by human's visual, psychological factor and quality of a stone. In the past, the value was pretty much seal-based, and nowadays it is much more for visual appreciation based on a stone's appearance and grade. There are many different angles and evaluation methods, it's all a matter of opinion.
Bloodstone Benefits
Bloodstone is thought to cure blood disorders and seal wounds. The stone can supposedly renew relationships and bring courage, calm, rational thoughts, abundance and good fortune to its wearer. Its use in powder form in Indian folk remedies is why gem-quality bloodstone is so hard to find.
Care for Bloodstone
Never expose the stone to the sun - Since the color will turn brown when exposed to the sunlight, you should always be careful not to put the stone under the sun, it is the bright red gives bloodstone its distinct appearance and any darkening will lower its value.
Care need to be taken when cutting this material. Cinnabar contains mercury sulfide (HgS), which is highly toxic.
Munkhuu on January 14, 2018:
bolorchuluun on November 30, 2012:
thak u .good luck
Phil C. Ouano on June 17, 2011:
Thank you very much for the useful information on Chicken Bloodstones. Sincerely, Phil C. Ouano
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redpajama_set_name: RedPajamaCommonCrawl


SUMMER CAMPS at LSC are Set and Ready to Play… ARE YOU?
Campers should bring a water bottle as well as any allergy or emergency medicine if needed. We strongly encourage campers to write their name on everything they bring to camp – including soccer balls, water bottles, lunch boxes, etc.
Campers should wear comfortable clothes to run and play in, as well as socks and athletic shoes. NO outdoor cleats are allowed at Longevity Sports Center.
How about snacks and lunch?
We provide campers with 2 snacks/day. Full-day campers need to bring a lunch.
Well qualified and trained coaches are with the kids at all times, they love what they do and are great with kids of all ages! Coach to camper ratio is 10:1.
We group kids into the proper age ranges, taking into consideration social and cognitive levels since they will be matched up to appropriate and interactive activities accordingly. If siblings are close in age but they are divided, they do have the option to join each other. However, we recommend that the children remain in their proper age groups so that they benefit the most from their time at summer camp.
Camps are non-refundable, however you may transfer your child to a different week of camp. No transfers or refunds will be issued the day of camp.
After Care starts promptly at 4:00pm and will end at 5:00pm,this is an additional $10 charge. Children will be given coloring activities, card games, etc. and will be sat at a table with Coach Supervision.
Good morning I have a question for a summer all day camp what time does it start in the Morning and what time does it end?


Twitter has prohibited misgendering and deadnaming on its platform in an effort to curtail anti-trans abuse.
The social media company has changed its rules to ban the practices and has warned that any user who deliberately targets a trans person in these ways may face permanent suspension.
These offensive techniques—which involve using the wrong gender to refer to a trans person or a trans person's old name—are often used on Twitter to insult and erase trans people's identities and right to exist.
In Twitter's updated terms of service, the company states: "We prohibit targeting individuals with repeated slurs, tropes or other content that intends to dehumanise, degrade or reinforce negative or harmful stereotypes about a protected category.


text:
Klinik Aarau
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Hirslanden Group
Centre for Cosmetic and Reconstructive Surgery
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Doctors & Care Find a doctor Nursing care
About us Portrait Facts & Figures Quality Image Gallery
The Centre for Aesthetic and Reconstructive Surgery at the Hirslanden Klinik Aarau specialises in the treatment of external abnormalities that are either congenital or caused by an accident or advanced age. The procedures include breast reconstructions, breast enlargements and reductions, liposuction and cosmetic surgery of scars caused by injuries. Honest and open consultations, the latest surgical techniques and the experience of our medical specialists ensure the best possible results.
A person's inner and outer self are linked by complex interactions: satisfaction with your appearance increases self-confidence and this, in the end, is the secret behind that radiance which is perceived as beauty in the eye of the beholder.
Many elements, such as congenital malformations, the consequences of accidents, a missing breast after an operation or changes resulting from age, cause people to lose their sense of wellbeing. However, even characteristics such as not having the right nose, having a certain chin or an unattractive breast can damage the Psyche.
Welcome to the Centre for Cosmetic and Reconstructive Surgery.
Hirslanden Klinik Aarau
Hirslanden Klinik Aarau offers first-class medical care with highly experienced specialists in a welcoming environment.
© Hirslanden Private Hospital Group 2021
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What was the top pro sports team in Denver in 2017, revenue wise?
The Rockies were No. 1 in the city based on total revenue for 2017!
When it came to the top revenue-producing professional sports team in Denver, the Colorado Rockies came out on top in 2017.
That's according to a report by seat-broker Vivid Seats, which ranked the Rockies No. 1 based on total revenue for 2017.
Baseball teams came out on top in most cities with multiple sports teams, but there were exceptions, according to Vivid Seats. In Oakland, for example, the NBA's Warriors came out on top. In Pittsburgh, the NHL's Penguins were No. 1. And in cities like Minneapolis, Detroit, Philadelphia, Atlanta and Tampa Bay, the NFL franchise there bested the baseball team.
When it came to the most priciest concert in Denver this year, Chris Rock's show was determined by Vivid Seats to be the top show in Denver this year, with a median ticket price of $312.
Read about other money-makers in town at the Denver Business Journal.


Chamberlain is excited beyond reason about his Switch preorder. Chance and Alex do there best to bring him back down to reality and get his to act his age. Watch_Dogs 2 is suprisingly good, Dragon Quest 8 is a little skeevy and somebody played something that wasn't Overwatch. And then he played more Overwatch.