3.2. Main results

We now present a main result, which will be proved in Section 5. Recall that $L^{(r)}$ is a random variable subject to the stationary distribution of the rth two-level $GI/G/1$ queue. Define probability distribution $\nu^{(r)}$ under the assumptions (A1)–(A5) as

\begin{align*}\nu^{(r)}(B) = \mathbb{P}(rL^{(r)} \in B), \quad B \in {\mathcal{B}}(\mathbb{R}_{+}).\end{align*}

We refer to this $\nu^{(r)}$ as a scaled stationary distribution. Define finite measures $\nu^{(r)}_{i}$ and conditional probability distributions $\nu^{(r)}_{i|cd}$ on $(\mathbb{R}_{+}, {\mathcal B}(\mathbb{R}_{+}))$ for $i=1,2$ as

\begin{gather*}\nu^{(r)}_{1}(B) = \mathbb{P}(rL^{(r)} \in B;\ L^{(r)} \le \ell^{(r)}_{1}), \qquad \nu^{(r)}_{2}(B) = \mathbb{P}(rL^{(r)} \in B;\ \ell^{(r)}_{1} < L^{(r)}), \\\nu^{(r)}_{1|cd}(B) = \mathbb{P}(rL^{(r)} \in B| L^{(r)} \le \ell^{(r)}_{1}), \qquad \nu^{(r)}_{2|cd}(B) = \mathbb{P}(rL^{(r)} \in B| \ell^{(r)}_{1} < L^{(r)}),\end{gather*}

for $B \in {\mathcal B}(\mathbb{R}_{+})$ . Obviously, $\nu^{(r)}(B) =\nu^{(r)}_{1}(B) + \nu^{(r)}_{2}(B)$ .

Theorem 1. Assume the assumptions (A1)–(A6) hold. Let

(3.1) \begin{align}\sigma_{i}^{2} = \lambda_{i}^{2} \sigma_{e,i}^{2} + \mu^{2} \sigma_{s}^{2}, \qquad \beta_{i} = \frac {2 b_{i}} {c_{i} \sigma_{i}^{2}}, \quad i =1, 2.\end{align}

For each $r \in (0,1]$ , define $\widetilde{\nu}_{i|cd}$ as the probability measures on ${\mathcal B}(\mathbb{R}_{+})$ that have the density functions $h_{i|cd}$ for $i=1,2$ such that

(3.2) \begin{align}h_{1|cd}(x) = 1(0 \le x \le \ell_{1}) \times \left\{\begin{array}{l@{\qquad}l}\displaystyle \frac{1}{\ell_{1}}, & b_{1} = 0,\\[8pt]\displaystyle \frac {e^{\beta_{1} \ell_{1}} \beta_{1}}{e^{\beta_{1} \ell_{1}} -1} e^{-\beta_{1} x} , \quad & b_{1} \not= 0,\end{array}\right. \\[-24pt] \nonumber \end{align}

(3.3) \begin{align}h_{2|cd}(x) = 1 (x > \ell_{1}) \beta_{2} e^{-\beta_{2} (x-\ell_{1})}.\end{align}

1. (i) For each $i=1,2$ , if $\nu^{(r)}_{i}(\mathbb{R}_{+})$ is bounded away from 0 as $r \downarrow 0$ , that is,

   (3.4) \begin{align}\liminf_{r \downarrow 0} \varphi^{(r)}_{i}(0) > 0, \quad i=1,2,\end{align}
   then $\nu^{(r)}_{i|cd}$ weakly converges to $\widetilde{\nu}_{i|cd}$ , namely,
   (3.5) \begin{align}\lim_{r \downarrow 0} \nu_{i|cd}^{(r)}(B) = \widetilde{\nu}_{i|cd}(B), \quad B \in {\mathcal B}(\mathbb{R}_{+}).\end{align}

2. (ii) Define the probability measure $\widetilde{\nu}$ on $(\mathbb{R}_{+},{\mathcal B}(\mathbb{R}_{+}))$ and constant $\widetilde{\alpha}_{e}$ as

   (3.6) \begin{align}\widetilde{\nu}(B) = \widetilde{\nu}_{1|cd}(B) \widetilde{A}_{1} + \widetilde{\nu}_{2|cd}(B) \widetilde{A}_{2}, \qquad B \in {\mathcal B}(\mathbb{R}_{+}), \\[-24pt] \nonumber \end{align}
   (3.7) \begin{align}\widetilde{\alpha}_{e} = \lambda_{1} \widetilde{A}_{1} + \lambda_{2} \widetilde{A}_{2},\end{align}

   where $\widetilde{A}_{i}$ for $i=1,2$ are given by

   (3.8) \begin{align}& \widetilde{A}_{1} =\begin{cases}\displaystyle \frac {2b_{2} \ell_{1}} {c_{2} \sigma_{1}^{2} + 2b_{2} \ell_{1}}, & b_{1}=0,\\[10pt]\displaystyle \frac {c_{1}b_{2} (e^{\beta_{1} \ell_{1}} - 1)} {c_{2}b_{1} + c_{1}b_{2} (e^{\beta_{1} \ell_{1}} - 1)}, \; & b_{1} \not=0,\end{cases}\qquad \widetilde{A}_{2} = 1 - \widetilde{A}_{1}.\end{align}
   If either $\lambda^{(r)}_{1} - \lambda^{(r)}_{2} = O(r)$ , equivalently, $c^{(r)}_{1} - c^{(r)}_{2} = O(r)$ , or $T^{(r)}_{s}$ is exponentially distributed for all $r \in (0,1]$ , then
   (3.9) \begin{align}\lim_{r \downarrow 0} \nu^{(r)}(B) = \widetilde{\nu}(B), \qquad B \in {\mathcal B}(\mathbb{R}_{+}), \\[-24pt] \nonumber \end{align}
   (3.10) \begin{align}\lim_{r \downarrow 0} \alpha^{(r)}_{e} = \widetilde{\alpha}_{e}.\end{align}

3. (iii) If $\lambda_{1} > \lambda_{2}$ ( $\lambda_{1} < \lambda_{2}$ ) then $\nu^{(r)}_{1|cd}$ ( $\nu^{(r)}_{2|cd}$ ) weakly converges to $\widetilde{\nu}_{1|cd}$ ( $\widetilde{\nu}_{2|cd}$ , respectively), as $r\downarrow 0$ .

Remark 2. Note that, under one of the conditions of part (ii) of Theorem 1, we have

(3.11) \begin{align}\lim_{r \downarrow 0} \nu^{(r)}_{i}(\mathbb{R}_{+}) = \widetilde{A}_{i}, \quad i=1,2.\end{align}

Conversely, if (3.11) holds then we have the weak convergence (3.9) from (i) because $\widetilde{A}_{i}> 0$ . Thus, (3.11) is equivalent to (3.9). We conjecture that (3.9) holds without any condition in (ii). This is supported by the intuitive verification of (3.11), which is detailed in Section 6.2.

Let $S^{(r)}(n)$ be the sojourn time of the nth arriving customer in service, then the traffic intensity $\rho^{(r)}$ can be defined under the stationary setting as

(3.12) \begin{align}\rho^{(r)} = \alpha^{(r)}_{e} \lim_{n \to \infty} \frac 1n \sum_{k=1}^{n}S^{(r)}(k).\end{align}

Hence, by Little’s law, we have $\rho^{(r)} = \mathbb{P}(L^{(r)} > 0) = 1 -\mathbb{P}(L^{(r)}=0)$ . In diffusion applications of a single server queue, the mean queue length is estimated through $(1 - \rho^{(r)})\mathbb{E}[L^{(r)}]$ as $r \downarrow 0$ . This limit is computed in the following corollary, which is proved in Appendix A.2.

Corollary 1. Under the assumptions (A1)–(A6) and either one of the assumptions in part (ii) of Theorem 1,

(3.13) \begin{align}& \lim_{r \downarrow 0} (1 - \rho^{(r)}) \mathbb{E}[L^{(r)}] = \lim_{r \downarrow 0} \mathbb{P}(L^{(r)}=0) \mathbb{E}[L^{(r)}] \nonumber\\& \qquad =\begin{cases}\displaystyle \frac {\sigma_{1}^{2} (2b_{2} \ell_{1}(b_{2} \ell_{1} + c_{2} \sigma_{1}^{2}) + c_{2}^{2} \sigma_{1}^{2} \sigma_{2}^{2})} {2(2b_{2} \ell_{1} + c_{2} \sigma_{1}^{2})^{2}}, & b_{1}=0, \\ \displaystyle \frac {c_{1}^{2} b_{2}^{2} \sigma_{1}^{2} (e^{\beta_{1} \ell_{1}} - 1) + 2 b_{1} b_{2} \ell_{1} (c_{2} b_{1} - c_{1} b_{2}) + c_{2}^{2} b_{1}^{2} \sigma_{2}^{2}} {2 b_{1} b_{2} (c_{2}b_{1} + c_{1}b_{2} (e^{\beta_{1} \ell_{1}} - 1))}, \; & b_{1} \not= 0.\end{cases}\end{align}

Let us consider Theorem 1 when $T_{e,1}^{(r)},T_{e,2}^{(r)}$ , and $T_{s}^{(r)}$ are all exponentially distributed. We call this model the rth two-level $M/M/1$ queue. Note that (A6) is automatically satisfied in this exponential model.

Proposition 1. Under the assumptions (A1)–(A5), the scaled stationary distribution $\nu^{(r)}_{\exp}$ of the rth two-level $M/M/1$ queue weakly converges to $\widetilde{\nu}_{\exp}$ as $r \downarrow 0$ , where $\widetilde{\nu}_{\exp}$ is defined as $\widetilde{\nu}_{\exp}(B) =\widetilde{\nu}_{1|cd}(B) \widetilde{A}_{\exp,1} +\widetilde{\nu}_{2|cd}(B) \widetilde{A}_{\exp,2}$ for $B \in {\mathcal{B}}(\mathbb{R}_{+})$ in which

(3.14) \begin{align}& \widetilde{A}_{\exp,1} = \begin{cases}\displaystyle \frac {\beta_{2} \ell_{1}} {1 + \beta_{2} \ell_{1}}, & b_{1} =0,\\ \displaystyle \frac {\beta_{2} (e^{\beta_{1} \ell_{1}} - 1)} {\beta_{1} + \beta_{2} (e^{\beta_{1} \ell_{1}} - 1)}, \quad & b_{1} \not=0,\end{cases}\end{align}

and $\widetilde{A}_{\exp,2} = 1 - \widetilde{A}_{\exp,1}$ .

Of course, Proposition 1 is the special case of part (ii) of Theorem 1, so no proof is needed. However, for a double check, we give its direct proof in Appendix A.3, which is helpful to convince (ii) of Theorem 1. This proof is not obvious because the exponential model requires a Markov chain with a two-dimensional state space to correctly describe the case that an arrival finds the queue length less than $\ell_{1}$ when the previous arrival found the queue length greater than $\ell_{1}$ but multiple departures occurred (see the arrival at time $t^{(r)}_{e}(8)$ in Figure 1). This Markov chain will be detailed in Appendix A.3.

Figure 1.

Sample path of the queue length process of the rth model.

4.1. Palm distribution and BAR

Recall that we are working on stochastic basis $(\Omega,{\mathcal{F}},\mathbb{F},\mathbb{P})$ with time-shift operator semi-group $\Theta_{\bullet}$ . Furthermore, by the stability assumption (A5), we can assume that this framework is stationary, that is,

\begin{align*}\mathbb{P}(\Theta_{t}^{-1} A) = \mathbb{P}(A), \qquad A \in {\mathcal F}.\end{align*}

Then, for each $r \in (0,1]$ , $X^{(r)}({\cdot})$ and the increments of $N^{(r)}_{e}$ and $N^{(r)}_{s}$ are jointly stationary. Since $\alpha^{(r)}_{e}$ and $\alpha^{(r)}_{s}$ are finite and positive by Lemma 1, we can define probability distributions $\mathbb{P}^{(r)}_{e}$ and $\mathbb{P}^{(r)}_{s}$ on $(\Omega,{\mathcal F})$ by

(4.15) \begin{align}\mathbb{P}^{(r)}_{v}(A) = (\alpha^{(r)}_{v})^{-1} \mathbb{E}\left[ \int_{(0,1]} 1_{\Theta_{u}^{-1} A} N^{(r)}_{v}({\rm d} u) \right], \qquad A \in {\mathcal F}, v=e,s,\end{align}

where $1_{A}$ is the indicator function of the event $A \in {\mathcal F}$ . For $v=e,s$ , $\mathbb{P}^{(r)}_{v}$ is called a Palm distribution concerning the counting process $N^{(r)}_{v}$ .

The Palm distributions play a major role for the derivation of the stationary equations in our analysis (see, e.g., (4.17)). Intuitively, $\mathbb{P}^{(r)}_{v}$ is considered as the conditional probability distribution given the event that $N^{(r)}_{v}$ counts arrivals for $v=e$ (service completion for $v=s$ ) at time 0. This event occurs with probability 0 under the stationary framework, so such an event is often handled as a boundary condition in the stationary equation (see, e.g., [Reference Davis5]). On the other hand, Palm distributions are defined through time averages under the stationary framework. This enables us to derive the stationary equation without any boundary condition, where the boundary conditions are handled by Palm distributions in the stationary equation. This is quite convenient because the boundary conditions may be only asymptotically satisfied. This is exactly our case.

We now derive a BAR for $X^{(r)}({\cdot})$ using the framework proposed in [Reference Miyazawa12]. For this, we introduce notations for describing the dynamics of $X^{(r)}({\cdot})$ . Let S be its state space, that is, $S =\mathbb{Z}_{+} \times \mathbb{R}_{+}^{2}$ , and let $C_{b}^{p}(S)$ be the set of all bounded and continuous functions from S to $\mathbb{R}_{+}$ that are partially differentiable from the right concerning the second and third variables and in which $ ({\partial^{+}}/{\partial x_{i}})f$ is the ith partial derivative from the right. Define operator ${\mathcal H}$ on $C_{b}^{p}(S)$ that maps $f \in C_{b}^{p}(S)$ to ${\mathcal H} f \in C_{b}^{p}(S)$ by

\begin{align*}& {\mathcal H} f({\boldsymbol x}) = - \frac {\partial^{+}}{\partial x_{2}} f({\boldsymbol x}) - (c_{1} 1(1 \le x_{1} \le \ell_{1}) + c_{2} 1(x_{1} > \ell_{1})) \frac {\partial^{+}}{\partial x_{3}} f({\boldsymbol x}), \qquad {\boldsymbol x} \in S,\end{align*}

where ${\boldsymbol x} = (x_{1},x_{2},x_{3}) \in \mathbb{R}_{+}^{2}$ .

In what follows, we often use a difference operator $\Delta$ , which is defined as $\Delta g(t) = g(t) - g(t-)$ for function g from $\mathbb{R}_{+}$ to $\mathbb{R}$ , which is right continuous with the left limit. For $t \ge 0$ , define $\mathbb{F}$ -adapted process $X^{(r)}_{1}({\cdot})$ and $L^{(r)}_{1}({\cdot})$ as

\begin{align*}& X^{(r)}_{1}(t) =\begin{cases}X^{(r)}(t-), & \Delta N^{(r)}_{e}(t) = 0,\\(L^{(r)}(t-) + 1, T^{(r)}_{e,1}(n_{e,1}(t)), R^{(r)}_{s}(t-)), \quad & \Delta N^{(r)}_{e}(t) = 1, L^{(r)}(t-) < \ell^{(r)}_{1},\\(L^{(r)}(t-) + 1, T^{(r)}_{e,2}(n_{e,2}(t)), R^{(r)}_{s}(t-)), \quad & \Delta N^{(r)}_{e}(t) = 1, L^{(r)}(t-) \ge \ell^{(r)}_{1},\end{cases}\\& L^{(r)}_{1}(t) = L^{(r)}(t-) + 1(\Delta N^{(r)}_{e}(t) = 1), \qquad R^{(r)}_{1}(t) = (R^{(r)}_{e}(t), R^{(r)}_{s}(t-)),\end{align*}

where $n_{e,1}(t) = \sum_{n=1}^{N^{(r)}_{e}(t)} 1(L^{(r)}(t^{(r)}_{e,n}-) <\ell^{(r)}_{1})$ and $n_{e,2}(t) = \sum_{n=1}^{N^{(r)}_{e}(t)}1(L^{(r)}(t^{(r)}_{e,n}-) \ge \ell^{(r)}_{1})$ . We call $X^{(r)}_{1}(t)$ an intermediate state. Note that

\begin{align*}X^{(r)}(t) = (L^{(r)}_{1}(t) -1, R^{(r)}_{e}(t), T^{(r)}_{s}(N^{(r)}_{s}(t)))\end{align*}

when $\Delta N^{(r)}_{s}(t) = 1$ , and $X^{(r)}(t) = X^{(r)}_{1}(t)$ when $\Delta N^{(r)}_{s}(t) = 0$ .

In these definitions, $X^{(r)}(0-)$ is required, but formally not defined. However, this does not cause any problem; see part (i) of Remark 4 for its details. For convenience, we define the difference operators $\Delta_{e}$ and $\Delta_{s}$ as

\begin{align*}& \Delta_{e} f (X^{(r)})(t) = f(X^{(r)}_{1}(t)) - f(X^{(r)}(t-)),\\& \Delta_{s} f (X^{(r)})(t) = f(X^{(r)}(t)) - f(X^{(r)}_{1}(t)).\end{align*}

Furthermore, let $f(X)(t) = f(X(t))$ . Then, from the time evolution of $X^{(r)}(t)$ , we have

(4.16) \begin{align}&f(X^{(r)}(t)) \nonumber\\&\qquad= f(X^{(r)}(0)) & + \int_{0}^{t} {\mathcal H} f(X^{(r)}(u)) \, {\rm d} u + \sum_{u \in (0,t]} [\Delta_{e} f(X^{(r)})(u) + \Delta_{s} f(X^{(r)})(u)],\end{align}

where the summation is well defined because $\Delta_{e} f(X^{(r)})(u)$ and $\Delta_{s} f(X^{(r)})(u)$ vanish except for finitely many $u \in (0,t]$ . Take the expectation of (4.16) for $t=1$ by $\mathbb{P}$ , then the stationarity of $X({\cdot})$ and the definition of the Palm distribution yield

\begin{gather*}\mathbb{E}\left[\int_{0}^{1} {\mathcal H} f(X^{(r)}(u)) \,{\rm d} u\right] = \int_{0}^{1} \mathbb{E}\left[{\mathcal H} f(X^{(r)}(u))\right] {\rm d} u = \mathbb{E}\left[{\mathcal H} f(X^{(r)}(0))\right], \\\mathbb{E}\left[\sum_{u \in (0,1]} \Delta_{v} f(X^{(r)})(u)]\right] = \alpha^{(r)}_{v} \mathbb{E}^{(r)}_{v}\left[\Delta_{v} f(X^{(r)})(0) \right], \quad v= e,s.\end{gather*}

Hence, we have a stationary equation:

(4.17) \begin{align}\mathbb{E}\left[ {\mathcal H}f(X^{(r)}(0)) \right] + \alpha^{(r)}_{e}\mathbb{E}^{(r)}_{e}\left[\Delta_{e} f(X^{(r)})(0) \right] +\alpha^{(r)}_{s} \mathbb{E}^{(r)}_{s}\left[\Delta_{s} f(X^{(r)})(0) \right]= 0, \qquad f \in C_{b}^{p}.\end{align}

Equation (4.17) is a special case of the rate conservation law (see, e.g., [Reference Miyazawa9]) and referred to as a BAR, we then call the analysis based on this BAR a BAR approach. A major difficulty in this BAR approach is to evaluate the expectations under the Palm distributions (the last two terms in (4.17)). Their evaluations will be done for the two-level $GI/G/1$ queue in Section 4.3.

1. (i) $X^{(r)}(0-)$ is not defined under $\mathbb{P}$ , but its distribution is well defined under the Palm distributions $\mathbb{P}^{(r)}_{e}$ and $\mathbb{P}^{(r)}_{e}$ because the definition (4.15) only uses $X^{(r)}(t)$ for $t > 0$ under $\mathbb{P}$ . So, $X^{(r)}(0-)$ is a random variable that can be defined under $\mathbb{P}^{(r)}_{e}$ and $\mathbb{P}^{(r)}_{s}$ . Hence, $\Delta_{e} f(X^{(r)})(0)$ and $\Delta_{s}f(X^{(r)})(0)$ in (4.17) are well defined under the Palm distributions.

2. (ii) Under the Palm distributions, $X^{(r)}_{1}(0)$ is an intermediate state between just after $N^{(r)}_{e}$ is counted at time 0 and just before $N^{(r)}_{s}$ is counted at time 0. This means that we take a so-called arrival first framework. As for the intermediate state, see [Reference Miyazawa12] for its details. We consider $L^{(r)}(0-)$ under $\mathbb{P}^{(r)}_{e}$ , while $L^{(r)}(0)$ under $\mathbb{P}^{(r)}_{s}$ . This prevents us directly referring to the intermediate state $X^{(r)}_{1}(0)$ in our arguments. Of course, the intermediate state is not needed if $N^{(r)}_{e}$ and $N^{(r)}_{s}$ do not have a simultaneous count, but such a restriction is avoided by introducing $X^{(r)}_{1}(0)$ .

We note basic facts on $\alpha^{(r)}_{e}$ and $L^{(r)}({\cdot})$ under the Palm distributions.

Lemma 2. Under conditions (A2) and (A5),

(4.18) \begin{align} & \mathbb{P}^{(r)}_{e}[(L^{(r)}(0-) = k] =\mathbb{P}^{(r)}_{s}[L^{(r)}(0) = k], \quad k \ge 0, \\[-6pt] \nonumber \end{align}

(4.19) \begin{align} & \alpha^{(r)}_{e} = \frac {\lambda^{(r)}_{1}\lambda^{(r)}_{2}} {\lambda^{(r)}_{2} \mathbb{P}^{(r)}_{e}(L^{(r)}(0-) <\ell^{(r)}_{1}) + \lambda^{(r)}_{1} \mathbb{P}^{(r)}_{e}(L^{(r)}(0-) \ge\ell^{(r)}_{1})}, \\[-6pt] \nonumber\end{align}

(4.20) \begin{align} & \alpha^{(r)}_{e} = c^{(r)}_{1} \mu^{(r)}\mathbb{P}(1 \le L^{(r)} \le \ell^{(r)}_{1}) + c^{(r)}_{2} \mu^{(r)}\mathbb{P}(L^{(r)} > \ell^{(r)}_{1}). \\[6pt] \nonumber\end{align}

Proof. For each integer $k \ge 0$ , let $f(z,{\boldsymbol x}) = z \wedge (k+1)$ for $(z,{\boldsymbol x}) \in S$ , then obviously $f \in C_{b}^{p}(S)$ . Since $\Delta_{e} f(X^{(r)})(u) = 1(L^{(r)}(u-) \le k, N^{(r)}_{e}(u) -N^{(r)}_{e}(u-) = 1)$ and $\Delta_{s} f(X^{(r)})(u) = - 1(L^{(r)}(u) \le k,N^{(r)}_{s}(u) - N^{(r)}_{s}(u-) = 1)$ , we have

\begin{align*}& \alpha^{(r)}_{e} \mathbb{E}^{(r)}_{e}\left[\Delta_{e} f(X^{(r)})(0) \right] + \alpha^{(r)}_{s} \mathbb{E}^{(r)}_{s}\left[\Delta_{s} f(X^{(r)})(0) \right]\\& \qquad = \alpha^{(r)}_{e} \mathbb{P}^{(r)}_{e}\left[L^{(r)}(0-) \le k\right] - \alpha^{(r)}_{s} \mathbb{P}^{(r)}_{s}\left[L^{(r)}(0) \le k\right].\end{align*}

Applying this to (4.17) yields

\begin{align*}\alpha^{(r)}_{e} \mathbb{P}^{(r)}_{e}\left[L^{(r)}(0-) \le k\right] - \alpha^{(r)}_{s} \mathbb{P}^{(r)}_{s}\left[L^{(r)}(0) \le k\right] = 0, \quad k \ge 0.\end{align*}

Since $\alpha^{(r)}_{e} = \alpha^{(r)}_{s}$ by Lemma 1, the above equation yields (4.18). We next let $f(z,{\boldsymbol x}) = x_{1}$ and apply (4.17), then

\begin{align*}- 1 + \alpha^{(r)}_{e} \mathbb{E}^{(r)}_{e}\left[ T^{(r)}_{e,1}1(L^{(r)}(0-) <\ell^{(r)}_{1}) + T^{(r)}_{e,2} 1(L^{(r)}(0-) \ge\ell^{(r)}_{1}) \right] = 0\end{align*}

because $(R^{(r)}_{e})'(t) = -1$ . This proves (4.19). Similarly, let $f(z,{\boldsymbol x}) = x_{2}$ , then

(4.21) \begin{align}(R^{(r)}_{s})'(t) = - c_{1}^{(r)} 1(1 \le L^{(r)}(0) \le \ell^{(r)}_{1}) -c_{2}^{(r)} 1(L^{(r)}(0) > \ell^{(r)}_{1}),\end{align}

and $\mathbb{E}^{(r)}_{s}[\Delta_{s} R^{(r)}_{s}(0)] =\mathbb{E}[T^{(r)}_{s}]$ . Hence, (4.17) yields (4.20).

Since it follows from (4.19) and (A2) that

(4.22) \begin{align} 0 < \lambda^{(r)}_{1} \wedge \lambda^{(r)}_{2}\le \alpha^{(r)}_{e} \le \lambda^{(r)}_{1} \vee \lambda^{(r)}_{2} < \infty,\quad r \in (0,1],\end{align}

we have the following corollary.

To prove Theorem 1, we fully use the BAR (4.17) replacing $\alpha^{(r)}_{s}$ by $\alpha^{(r)}_{e}$ because $\alpha^{(r)}_{s} = \alpha^{(r)}_{e}$ by Lemma 1. We first apply an exponential type of a test function to this BAR, and derive various properties on it under the stationary distribution $\mathbb{P}$ and the Palm distributions $\mathbb{P}^{(r)}_{e}$ and $\mathbb{P}^{(r)}_{s}$ in Section 4.2, then Theorem 1 is proved in Section 5.

4.2. Asymptotic BAR for the prelimit process $X^{(r)}({\cdot})$

We prove Theorem 1 using the moment generating functions $\varphi^{(r)}_{i}(\theta_{i})$ and $\varphi^{(r)}_{i|cd}(\theta_{i})$ of $\nu^{(r)}_{i}$ and $\nu^{(r)}_{i|cd}$ , respectively, for $i=1,2$ . Namely, for $\theta_{1} \in \mathbb{R}$ and $\theta_{2} \le 0$ ,

\begin{align*} & \varphi^{(r)}_{1}(\theta_{1}) = \mathbb{E}[e^{\theta_{1} rL^{(r)}}1(L^{(r)} \le \ell^{(r)}_{1})], \qquad \varphi^{(r)}_{2}(\theta_{2}) = \mathbb{E}[e^{\theta_{2} rL^{(r)}}1(\ell^{(r)}_{1} < L^{(r)})],\\ & \varphi^{(r)}_{1|cd}(\theta_{1}) = \mathbb{E}[e^{\theta_{1} rL^{(r)}}|L^{(r)} \le \ell^{(r)}_{1}], \qquad \varphi^{(r)}_{2|cd}(\theta_{2}) = \mathbb{E}[e^{\theta_{2} rL^{(r)}}|\ell^{(r)}_{1} < L^{(r)}].\end{align*}

To this end, we introduce test functions for the BAR (4.17). We first define functions $w^{(r)}_{i}$ and $g^{(r)}_{i,\theta}$ for $i=1,2$ and $\theta \in \mathbb{R}$ as

\begin{gather*}w^{(r)}_{1}(z) = 1(z \le \ell^{(r)}_{1}), \qquad w^{(r)}_{2}(z) = 1(z > \ell^{(r)}_{1}), \qquad z \in \mathbb{Z}_{+}, \\g^{(r)}_{i,\theta}(x_{1},x_{2}) = \exp\biggl( - \eta^{(r)}_{i}(\theta) \biggl(x_{1} \wedge \frac{1}{r}\biggr) - \zeta^{(r)}(\theta) \biggl(x_{2} \wedge \frac{1}{r}\biggr)\biggr), \qquad (x_{1},x_{2}) \in \mathbb{R}_{+}^{2},\end{gather*}

where $\eta^{(r)}_{i}(\theta), \zeta^{(r)}(\theta)$ are the unique solutions of the following equations for each $\theta \in \mathbb{R}$ :

(4.23) \begin{align}e^{\theta} \mathbb{E}\left[e^{-\eta^{(r)}_{i}(\theta) (T^{(r)}_{e,i} \wedge 1/r)}\right] = 1, \quad i=1,2, \qquad e^{-\theta} \mathbb{E}\left[e^{-\zeta^{(r)}(\theta) (T^{(r)}_{s} \wedge 1/r)}\right] = 1,\end{align}

because $\eta^{(r)}_{i}(\theta)$ is the inverse function of the Laplace transform of the finite positive random variable $T^{(r)}_{e,i} \wedge1/r$ , taking the value $e^{-\theta}$ and $\zeta^{(r)}(\theta)$ is similarly determined. Then, for each ${\boldsymbol \theta} \equiv(\theta_{1},\theta_{2}) \in \mathbb{R}^{2}$ , define the test function $f^{(r)}_{{\boldsymbol \theta}}$ for the BAR as

(4.24) \begin{align} & f^{(r)}_{{\boldsymbol \theta}}(z,x_{1},x_{2}) = \sum_{i=1,2} e^{\theta_{i} z} g^{(r)}_{i,\theta_{i}}(x_{1},x_{2}) w^{(r)}_{i}(z), \qquad (z,x_{1},x_{2}) \in S.\end{align}

Functions $\eta^{(r)}_{i}(\theta)$ and $\zeta^{(r)}(\theta)$ play key roles in our analysis. Consider their properties. First, it is easy to see that they are infinitely many differentiable functions of $\theta$ . Hence, by twice differentiating the equations in (4.23), the following lemma can be obtained by their Taylor expansions around the origin. It is formally proved in [Reference Braverman, Dai and Miyazawa4, Lemma 5.8].

Lemma 3. Under the assumptions (A1)–(A4), we have the following expansions as $r\downarrow 0$ for each fixed $\theta \in \mathbb{R}$ :

(4.25) \begin{align} & \eta_{i}^{(r)}(r\theta) = \lambda^{(r)}_{i} r \theta + \tfrac 12 (\lambda^{(r)}_{i})^{3} (\sigma^{(r)}_{e,i})^{2} r^{2} \theta^{2} + o((\theta r)^{2}), \qquad i = 1,2, \\[-6pt] \nonumber\end{align}

(4.26) \begin{align} & \zeta^{(r)}(r\theta) = - \mu^{(r)} r \theta + \tfrac 12 (\mu^{(r)})^{3} (\sigma^{(r)}_{s})^{2} r^{2} \theta^{2} + o((\theta r)^{2}). \\[6pt] \nonumber\end{align}

Furthermore, for $i=1,2$ , there are constant $d_{e,i}, d_{s}, a > 0$ such that, for $r \in (0,1]$ and $\theta \in \mathbb{R}$ satisfying $r|\theta| < a$ ,

(4.27) \begin{align} & \left|\eta^{(r)}_{i}(r\theta) (u_{1} \wedge 1/r) \right| \le |r \theta| d_{e,i} (u_{1} \wedge 1/r), \quad i=1,2, \\[-6pt] \nonumber\end{align}

(4.28) \begin{align} & \left|\zeta^{(r)}(r\theta) (u_{2} \wedge 1/r)\right| \le |r \theta| d_{s} (u_{2} \wedge 1/r). \\[6pt] \nonumber\end{align}

Recall that (A1)–(A5) are always assumed in Section 4 if particular assumptions among them are specified. By Lemma 3, $g^{(r)}_{i,r\theta_{i}}(R^{(r)})$ is bounded by $e^{|\theta_{i}|(d_{e,i} + d_{s})}$ for sufficiently small r for each $\theta_{1} \in \mathbb{R}$ and $\theta_{2} \le 0$ , so we have the following facts.

Corollary 3. Let $R^{(r)} = (R^{(r)}_{e},R^{(r)}_{s})$ , then $g^{(r)}_{i,r\theta_{i}}(R^{(r)})$ converges to 1 as $r \downarrow 0$ , and

(4.29) \begin{align}\lim_{r \downarrow 0} \mathbb{E}\left[g^{(r)}_{i,r\theta_{i}}(R^{(r)})\right] = 1, \qquad \lim_{r \downarrow 0} \mathbb{E}_{u}\left[g^{(r)}_{i,r\theta_{i}}(R^{(r)})\right] = 1, \quad u = e, s.\end{align}

We next introduce auxiliary functions $\psi^{(r)}_{i}(\theta_{i})$ for $i=1,2$ as

(4.30) \begin{align} & \psi^{(r)}_{i}(\theta_{i}) =\mathbb{E}\left[e^{r\theta_{i} L^{(r)}}g^{(r)}_{i,r \theta_{i}}(R^{(r)})w^{(r)}_{i}(L^{(r)}) \right], \quad \theta_{1} \in \mathbb{R}, \theta_{2}\le 0.\end{align}

Then, by Lemma 3 and Corollary 3, we have

(4.31) \begin{align}\psi^{(r)}_{i}(\theta_{i}) < \infty, \qquad \mathbb{E}_{u}\left[f^{(r)}_{r{\boldsymbol \theta}}(L^{(r)},R^{(r)})\right] < \infty, \quad u=e,s, \quad \mbox{uniformly in $r \in (0,1]$},\end{align}

because $r \theta_{1} L^{(r)}1(L^{(r)} \le \ell^{(r)}_{1}) \le \theta_{1}(\ell + o(r))$ for $\theta_{1} > 0$ . These finiteness guarantees our computations for $\theta_{1}\in \mathbb{R}$ and $\theta_{2} \le 0$ . We also note the following fact.

Corollary 4. For $\theta_{1} \in \mathbb{R}$ and $\theta_{2} \le 0$ ,

(4.32) \begin{align}\lim_{r \downarrow 0} \left|\psi^{(r)}_{i}(\theta_{i}) - \varphi^{(r)}_{i}(\theta_{i})\right| = 0, \quad i=1,2.\end{align}

Proof. By Corollary 3, we have

\begin{align*}|\psi^{(r)}_{i}(\theta_{i}) - \varphi^{(r)}_{i}(\theta_{i})| & = |\mathbb{E}[e^{r\theta_{i} L^{(r)}}(g_{1,\theta_{i}}(R^{(r)}_{e}) - 1)1( L^{(r)} \le \ell^{(r)}_{1})]| \nonumber\\& \le e^{|\theta_{1}| r \ell^{(r)}_{1}}|\mathbb{E}[g_{i,\theta_{i}}(R^{(r)}_{e}) - 1]| \nonumber\\&\to 0, \quad r \downarrow0.\end{align*}

This proves (4.32).

Based on these facts, we compute $\mathbb{E}\left[ {\mathcal{H}}f(X^{(r)}(0))\right]$ in (4.17) for $f^{(r)}_{r{\boldsymbol\theta}}$ . Then,

(4.33) \begin{align}\mathbb{E}\left[ {\mathcal H}f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}(0))\right] = &\sum_{i=1,2} \eta_{i}^{(r)}(r\theta_{i}) \psi^{(r)}_{i}(\theta_{i}) + \sum_{i=1,2} c^{(r)}_{i} \zeta^{(r)} (r\theta_{i}) \psi^{(r)}_{i}(\theta_{i})\nonumber\\& - c^{(r)}_{1} \zeta^{(r)}(r\theta_{1}) \mathbb{E}\left[1(L^{(r)}=0) g^{(r)}_{1,r \theta_{1}}(R^{(r)}) )\right].\end{align}

Since $\lambda^{(r)}_{i} - c^{(r)}_{i} \mu^{(r)} = - b_{i} \mu^{(r)} r +o(r)$ , it follows from Lemma 3, $\beta_{i} =2b_{i}/(c_{i} \sigma_{i}^{2})$ , and $\lambda_{i} = c_{i} \mu$ that, for each fixed $\theta_{i} \in \mathbb{R}$ ,

\begin{align*}\eta^{(r)}_{i}(r\theta_{i}) + c^{(r)}_{i} \zeta^{(r)}(r\theta_{i}) & = - b_{i} r^{2} \mu \theta_{i} + \tfrac 12 c_{i} (\lambda_{i}^{2} \sigma_{e,i}^{2} + \mu^{2} \sigma_{s}^{2}) r^{2} \mu \theta_{i}^{2} + o((\theta_{i} r)^{2})\\& = - \tfrac 12 c_{i} \sigma_{i}^{2} (\beta_{i} - \theta_{i}) \mu \theta_{i} r^{2} + o((\theta_{i} r)^{2}), \quad r \downarrow 0.\end{align*}

Hence, from (4.33), we have

(4.34) \begin{align}& \mathbb{E}\left[ {\mathcal H}f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}(0))\right] \nonumber\\&\qquad = \frac 12 \sum_{i=1,2} c_{i} \sigma_{i}^{2} \left(- \beta_{i} + \theta_{i} \right) \mu \theta_{i} r^{2} \psi^{(r)}_{i}(\theta_{i}) \nonumber\\&\qquad\hskip4mm + c^{(r)}_{1} \left(1 - \frac 12 \mu^{2} \sigma_{s}^{2} r\theta_{1}\right) \mu r \theta_{1} \mathbb{E}\left[1(L^{(r)}=0)g^{(r)}_{1,r \theta_{1}}(R^{(r)})\right] + o((\theta_{1} r)^{2}) +o((\theta_{2} r)^{2}).\end{align}

We next compute the Palm expectation terms in (4.17). In what follows, we use the following abbreviated notations:

(4.35) \begin{align}\widehat{T}^{(r)}_{e,i} \equiv T^{(r)}_{e,i} \wedge 1/r, \quad i=1,2, \qquad \widehat{T}^{(r)}_{s} \equiv T^{(r)}_{s} \wedge 1/r, \\[-24pt] \nonumber \end{align}

(4.36) \begin{align} \widehat{R}^{(r)}_{e} \equiv R^{(r)}_{e}(0) \wedge 1/r, \qquad \widehat{R}^{(r)}_{s} \equiv R^{(r)}_{s}(0) \wedge 1/r, \qquad \widehat{R}^{(r)}_{s-} \equiv R^{(r)}_{s}(0-) \wedge 1/r.\end{align}

Since $g^{(r)}_{i,\theta_{i}}(R^{(r)}(t))$ changes at $t=0$ only when $L^{(r)}(0-) = \ell^{(r)}_{1}$ under $\mathbb{P}^{(r)}_{e}$ or $L^{(r)}(0) = \ell^{(r)}_{1}$ under $\mathbb{P}^{(r)}_{s}$ , (4.23) implies that

\begin{align*}& \mathbb{E}^{(r)}_{e}\left[ \Delta_{e} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0) 1(L^{(r)}(0-) \not= \ell^{(r)}_{1})\right] = 0,\\& \mathbb{E}^{(r)}_{s}\left[\Delta_{s} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0) 1(L^{(r)}(0) \not= \ell^{(r)}_{1})\right] = 0.\end{align*}

Hence, from the definitions of $X^{(r)}_{1}(0)$ and $\Delta_{e} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0)$ , we have

(4.37) \begin{align}\mathbb{E}^{(r)}_{e}[\Delta_{e} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0)] & = \mathbb{E}^{(r)}_{e}[(f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}_{1}(0)) - f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}(0-))) 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& = \mathbb{E}^{(r)}_{e} [e^{r\theta_{2} (\ell^{(r)}_{1} + 1)} g^{(r)}_{2,r\theta_{2}}(R^{(r)}_{1}(0))1(L^{(r)}_{1}(0) = \ell^{(r)}_{1}+1) \nonumber\\& \quad - e^{r\theta_{1} \ell^{(r)}_{1}} g^{(r)}_{1,r\theta_{1}}(R^{(r)}(0-))) 1(L^{(r)}(0-) = \ell^{(r)}_{1}) ] \nonumber\\& = e^{r\theta_{2} \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-}} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \quad - e^{r\theta_{1} \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta_{1}) \widehat{R}^{(r)}_{s-}} 1(L^{(r)}(0-) = \ell^{(r)}_{1}))], \nonumber\\& = (e^{r\theta_{2} \ell^{(r)}_{1}} - e^{r\theta_{1}\ell^{(r)}_{1}}) \mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-}} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \hskip4mm + e^{r\theta_{1} \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[(e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-}} - e^{-\zeta^{(r)}(r\theta_{1}) \widehat{R}^{(r)}_{s-}}) 1(L^{(r)}(0-) = \ell^{(r)}_{1}))],\end{align}

where the third equality is obtained by applying (4.23). Similarly, we have

(4.38) \begin{align}\mathbb{E}^{(r)}_{s}[\Delta_{s} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0)] & = \mathbb{E}^{(r)}_{s}[(f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}(0)) - f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)}_{1}(0))) 1(L^{(r)}(0) = \ell^{(r)}_{1})]\nonumber\\& = \mathbb{E}^{(r)}_{s} [e^{r\theta_{1} \ell^{(r)}_{1}} g^{(r)}_{1,r\theta_{1}}(R^{(r)}(0))1(L^{(r)}(0) = \ell^{(r)}_{1}) \nonumber\\& \quad - e^{r\theta_{2} (\ell^{(r)}_{1}+1)} g^{(r)}_{2,r\theta_{2}}(R^{(r)}_{1}(0))) 1(L^{(r)}_{1}(0-) = \ell^{(r)}_{1}+1)] \nonumber\\& = e^{r\theta_{1} (\ell^{(r)}_{1}+1)} \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})] \nonumber\\& \hskip4mm - e^{r\theta_{2} (\ell^{(r)}_{1}+1)} \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{2}(r\theta_{2}) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})] \nonumber\\& = (e^{r\theta_{1} (\ell^{(r)}_{1}+1)} - e^{r\theta_{2} (\ell^{(r)}_{1}+1)}) \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})] \nonumber\\&\hskip4mm + e^{r\theta_{2} (\ell^{(r)}_{1}+1)} \mathbb{E}^{(r)}_{s}[(e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} - e^{-\eta^{(r)}_{2}(r\theta_{2}) \widehat{R}^{(r)}_{e}}) 1(L^{(r)}(0) = \ell^{(r)}_{1})].\end{align}

Define $D^{(r)}(\theta_{1},\theta_{2}) = \alpha^{(r)}_{e} ( \mathbb{E}^{(r)}_{e}[\Delta_{e} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0) ] + \mathbb{E}^{(r)}_{s}[\Delta_{s} f^{(r)}_{r{\boldsymbol \theta}}(X^{(r)})(0) ] )$ . Then, by (4.37) and (4.38),

(4.39) \begin{align}D^{(r)}(\theta_{1},\theta_{2}) = \ & \alpha^{(r)}_{e} (e^{r\theta_{2} \ell^{(r)}_{1}} - e^{r\theta_{1} \ell^{(r)}_{1}}) \mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-})} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& + \alpha^{(r)}_{e} e^{r\theta_{1} \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[(e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-}} - e^{-\zeta^{(r)}(r\theta_{1}) \widehat{R}^{(r)}_{s-}}) 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& + \alpha^{(r)}_{e} (e^{r\theta_{1} (\ell^{(r)}_{1}+1)} - e^{r\theta_{2} (\ell^{(r)}_{1} + 1)}) \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})] \nonumber\\& + \alpha^{(r)}_{e} e^{r\theta_{2} (\ell^{(r)}_{1} + 1)} \mathbb{E}^{(r)}_{s}[(e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} - e^{-\eta^{(r)}_{2}(r\theta_{2}) \widehat{R}^{(r)}_{e}}) 1(L^{(r)}(0) = \ell^{(r)}_{1})].\end{align}

Thus, applying (4.34), (4.37), and (4.38) to (4.17), we have a prelimit BAR for the two-level $GI/G/1$ queue.

Lemma 4. For each fixed $\theta_{1} \in \mathbb{R}$ and $\theta_{2} \le 0$ , we have, as $r \downarrow 0$ ,

(4.40) \begin{align}& \frac 12 \sum_{i=1,2} c_{i} \sigma_{i}^{2} \big((\beta_{i} - \theta_{i}) \mu \theta_{i} r^{2} + o((\theta_{i} r)^{2})\big) \psi^{(r)}_{i}(\theta_{i}) \nonumber\\& - c_{1} \biggl(\mu r\theta_{1} - \frac{\mu^{3} \sigma_{s}^{2} r^{2}\theta_{1}^{2}}{2} + o((\theta_{1} r)^{2})\biggr) \mathbb{E}[ 1(L^{(r)} = 0)g^{(r)}_{1, r \theta_{1}}(R^{(r)})] \nonumber\\&\qquad = D^{(r)}(\theta_{1},\theta_{2}).\end{align}

This BAR is called an asymptotic BAR, and a starting point of our proof of Theorem 1.

4.3. Expansion of the asymptotic BAR

We aim to derive the limits of $\psi^{(r)}_{1}(\theta_{1})$ and $\psi^{(r)}_{2}(\theta_{2})$ from the BAR (4.40). For this, we divide it for either $\theta_{2} = 0$ or $\theta_{1} = 0$ by $r^{2}$ , and take its limit as $r \downarrow0$ , expanding the exponential functions in (4.40). We perform this computation in the following three steps. Recall that assumptions (A1)–(A5) are assumed but (A6) is not assumed throughout Section 4.

Step 1: In this step we show that the moments of $\widehat{T}^{(r)}_{v}$ for $v=(e,1), (e,2), s$ and $\widehat{R}^{(r)}_{v'}$ for $v'=e,s$ are well behaved as $r \downarrow 0$ . This is also examined for $\widehat{R}^{(r)}_{s-} 1(L^{(r)} = \ell^{(r)}_{1})$ and $\widehat{R}^{(r)}_{e} 1(L^{(r)} = \ell^{(r)}_{0-})$ under $\mathbb{P}^{(r)}_{e}$ and $\mathbb{P}^{(r)}_{s}$ , respectively. These are basis for the expansion of the BAR. We then introduce the set of key quantities, $\Delta^{(r)}_{i}$ for $i=1,2$ .

Lemma 5. For $k =0,1,2$ and $r \in (0,1]$ ,

(4.41) \begin{align}(k+1) \mathbb{E}[(R^{(r)}_{e})^{k}] \le \alpha^{(r)}_{e} \max_{i=1,2} \mathbb{E}[(T^{(r)}_{e,i})^{k+1}], \\[-24pt] \nonumber \end{align}

(4.42) \begin{align}(k+1) (\min_{i=1,2} c^{(r)}_{i}) \mathbb{E}[(R^{(r)}_{s})^{k}] \le \alpha^{(r)}_{e} \mathbb{E}[(T^{(r)}_{s})^{k+1}] + (k+1) (\min_{i=1,2} c^{(r)}_{i}) \mathbb{E}[T_{s}^{k}],\end{align}

and, for $i=1,2$ , as $r \downarrow 0$ ,

(4.43) \begin{align}& |\mathbb{E}[(\widehat{T}^{(r)}_{u})^{k}] - \mathbb{E}[(T^{(r)}_{u})^{k}]| = \mathbb{E}[(T^{(r)}_{u})^{k} 1(T^{(r)}_{u} > 1/r)] = o(r^{3-k}), \quad u = (e,i), s, \\[-6pt] \nonumber\end{align}

(4.44) \begin{align} & |\mathbb{E}[(\widehat{R}^{(r)}_{v})^{k}] - \mathbb{E}[(R^{(r)}_{v})^{k}]| = \mathbb{E}[(R^{(r)}_{v})^{k} 1(R^{(r)}_{v} > 1/r)] = o(r^{2-k}), \quad v = e, s. \\[6pt] \nonumber\end{align}

In particular, $\{(R^{(r)}_{e})^{k};\, r \in (0,1]\}$ and $\{(R^{(r)}_{s})^{k} 1(L^{(r)}(0) \ge 1);\, r \in (0,1]\}$ are uniformly bounded for $k=1,2$ by (A1).

This lemma is easily proved using the BAR (4.17), but its proof is lengthy, so we defer it to Appendix A.4. We prepare one more lemma and its corollary, which are helpful for the expansion of the BAR.

Lemma 6. For $k=1,2,3$ and $r \in (0,1]$ ,

(4.45) \begin{align}& c^{(r)}_{1} k \mathbb{E}[(R^{(r)}_{s})^{k-1} (1 - 1(R^{(r)}_{s} \ge 1/r)) 1(1\le L^{(r)} \le \ell^{(r)}_{1})] \nonumber\\& \qquad = \alpha^{(r)}_{e} ( - \mathbb{E}^{(r)}_{e}[(\widehat{R}^{(r)}_{s-})^{k} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] + \mathbb{E}[(\widehat{T}^{(r)}_{s})^{k}] \mathbb{P}^{(r)}_{s}[L^{(r)}(0) \le \ell^{(r)}_{1}]),\end{align}

(4.46) \begin{align} & c^{(r)}_{2} k \mathbb{E}[(R^{(r)}_{s})^{k-1} (1 - 1(R^{(r)}_{s} \ge 1/r)) 1( L^{(r)} > \ell^{(r)}_{1})]\nonumber\\& \qquad = \alpha^{(r)}_{e} ( \mathbb{E}^{(r)}_{e}[(\widehat{R}^{(r)}_{s-})^{k} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] + \mathbb{E}[(\widehat{T}^{(r)}_{s})^{k}] \mathbb{P}^{(r)}_{s}[L^{(r)}(0) > \ell^{(r)}_{1}]), \\[-6pt] \nonumber\end{align}

(4.47) \begin{align}& k \mathbb{E}[(R^{(r)}_{e})^{k-1} (1 - 1(R^{(r)}_{e} \ge 1/r)) 1(L^{(r)} \le \ell^{(r)}_{1})] \nonumber\\& \qquad = \alpha^{(r)}_{e}( \mathbb{E}_{s}[(\widehat{R}^{(r)}_{e})^{k} 1(L^{(r)}(0) = \ell^{(r)}_{1})] + \mathbb{E}[(\widehat{T}^{(r)}_{e,1})^{k}] \mathbb{P}^{(r)}_{e}[L^{(r)}(0-)< \ell^{(r)}_{1}]), \\[-6pt] \nonumber\end{align}

(4.48) \begin{align} & k \mathbb{E}[(R^{(r)}_{e})^{k-1} (1 - 1(R^{(r)}_{e} \ge 1/r)) 1(L^{(r)} > \ell^{(r)}_{1})] \nonumber\\& \qquad = \alpha^{(r)}_{e}(- \mathbb{E}_{s}[(\widehat{R}^{(r)}_{e})^{k} 1(L^{(r)}(0) = \ell^{(r)}_{1})] + \mathbb{E}[(\widehat{T}^{(r)}_{e,2})^{k}] \mathbb{P}^{(r)}_{e}[L^{(r)}(0-) \ge \ell^{(r)}_{1}] ). \\[6pt] \nonumber\end{align}

Proof. For $k \ge 1$ , we first prove (4.45). We apply $f(z,{\boldsymbol x}) = (x_{2} \wedge 1/r)^{k} 1(z \le \ell^{(r)}_{1})$ to (4.17), then we have (4.45) because

\begin{align*}\frac {\rm d}{{\rm d} t} f(X^{(r)}(t)) = k (R^{(r)}_{s}(t))^{k-1} 1(R^{(r)}_{s} < 1/r, 1 \le L^{(r)}(t) \le \ell^{(r)}_{1}).\end{align*}

Similarly, (4.46), (4.47), and (4.48) are obtained from (4.17) by applying $f(z,{\boldsymbol x}) =(x_{2} \wedge 1/r)^{k} 1(z > \ell^{(r)}_{1})$ , $f(z,{\boldsymbol x}) =(x_{1} \wedge 1/r)^{k} 1(z \le \ell^{(r)}_{1})$ , and $f(z,{\boldsymbol x})= (x_{1} \wedge 1/r)^{k} 1(z > \ell^{(r)}_{1})$ , respectively.

The following corollary is immediate from (4.45), (4.48), (A1), and Lemma 5.

Note that the limits of $\psi^{(r)}_{1}(\theta_{1})$ and $\psi^{(r)}_{2}(\theta_{2})$ should be obtained from (4.40). However, we see that it requires the relation between the limits of $\varphi^{(r)}_{1}(0)$ and $\varphi^{(r)}_{2}(0)$ , which is not easy to get from (4.40). To consider this problem, we introduce the following quantities. Define, for $i=1,2$ ,

(4.49) \begin{align}\Delta^{(r)}_{i} = \mu^{(r)} \mathbb{E}^{(r)}_{e}[\widehat{R}^{(r)}_{s-} 1(L^{(r)}(0-) = \ell^{(r)}_{1})]+ \lambda^{(r)}_{i} \mathbb{E}^{(r)}_{s}[\widehat{R}^{(r)}_{e} 1(L^{(r)}(0) = \ell^{(r)}_{1})] - \mathbb{P}^{(r)}_{e}[L^{(r)}(0-) = \ell^{(r)}_{1}].\end{align}

Then, $\Delta^{(r)}_{i}$ is well defined and finite by Corollary 5, and we have the following lemma.

Lemma 7. We have

(4.50) \begin{align}\alpha^{(r)}_{e} \Delta^{(r)}_{1} = - rb_{1} \mu \mathbb{P}[L^{(r)} \le \ell^{(r)}_{1}] + c^{(r)}_{1}\mu^{(r)} \mathbb{P}[L^{(r)} = 0] + o(r), \\[-24pt]\nonumber \end{align}

(4.51) \begin{align}\alpha^{(r)}_{e} \Delta^{(r)}_{2} = r\mu b_{2} \mathbb{P}[L^{(r)} > \ell^{(r)}_{1}] + o(r).\end{align}

Proof. For $k=1$ , multiply both sides of (4.45) and (4.47) by $\mu^{(r)}$ and $\lambda^{(r)}_{1}$ , respectively, and take their difference, then, noting that $c^{(r)}_{1} \mu^{(r)} - \lambda^{(r)}_{1} =r \mu^{(r)} b_{1} + o(r)$ , we have

\begin{align*}& (r \mu^{(r)} b_{1} + o(r)) \mathbb{P}[L^{(r)} \le \ell^{(r)}_{1}] - c^{(r)}_{1}\mu^{(r)} \mathbb{P}[L^{(r)} = 0] \nonumber\\& \qquad = \alpha^{(r)}_{e} (\mathbb{P}^{(r)}_{e}[L^{(r)}(0-) = \ell^{(r)}_{1}] - \mu^{(r)} \mathbb{E}^{(r)}_{e}[\widehat{R}^{(r)}_{s-} 1(L^{(r)}(0-) = \ell^{(r)}_{1})]\\&\qquad\hskip10mm - \lambda^{(r)}_{1}\mathbb{E}_{s}^{(r)}[\widehat{R}^{(r)}_{e}1(L^{(r)}(0) = \ell^{(r)}_{1})])+ o(r),\end{align*}

because $\mathbb{P}^{(r)}_{s}[L^{(r)}(0) \le \ell^{(r)}_{1}] -\mathbb{P}^{(r)}_{e}[L^{(r)}(0-) < \ell^{(r)}_{1}] =\mathbb{P}^{(r)}_{e}[L^{(r)}(0-) = \ell^{(r)}_{1}]$ by Lemma 2. This proves (4.50). Similarly, (4.51) is obtained from (4.46) and (4.48),

We use Lemma 7 to determine the limits of $\mathbb{P}[L^{(r)} \le \ell^{(r)}_{1}]$ and $\mathbb{P}[L^{(r)} >\ell^{(r)}_{1}]$ , equivalently, $\varphi^{(r)}_{1}(0)$ and $\varphi^{(r)}_{2}(0)$ , as $r \downarrow 0$ under some conditions.

Step 2: In this step we expand $D^{(r)}(\theta,0)$ and $D^{(r)}(0,\theta)$ concerning r as $r \downarrow 0$ . We will see that the $\Delta^{(r)}_{i}$ s describe the expansion terms of $D^{(r)}(\theta,0)$ of order r. We introduce here the ${\mathcal{E}}^{(r)}_{i}$ s for describing their expansion terms of order $r^{2}$ .

We start with the following elementary expansions. Recall Remark 5 for our convention of the order terms.

Lemma 8. For each fixed $\theta_{1}$ and $\theta_{2}$ , as $r \downarrow 0$ ,

(4.52) \begin{align}e^{r \theta_{i} \ell^{(r)}_{1}} = e^{\theta_{i} \ell_{1}} (1 + o(\theta_{i} r)), \quad e^{r\theta_{i} (\ell^{(r)}_{1}+1)} = e^{\theta_{i} \ell_{1}} (1 + \theta_{i}r + o(\theta_{i} r)), \quad i=1,2, \\[-24pt] \nonumber \end{align}

(4.53) \begin{align}e^{r\theta_{1}\ell^{(r)}_{1}} - e^{r\theta_{2} \ell^{(r)}_{1}} = e^{\theta_{1} \ell_{1}} - e^{\theta_{2} \ell_{1}} + e^{(\theta_{1} \vee \theta_{2}) \ell_{1}} ( o(\theta_{1} r) + o(\theta_{2} r)), \\[-24pt] \nonumber \end{align}

(4.54) \begin{align}e^{r\theta_{1}(\ell^{(r)}_{1}+1)} - e^{r\theta_{2} (\ell^{(r)}_{1}+1)} = e^{\theta_{1}\ell_{1}} - e^{\theta_{2} \ell_{1}} + r (e^{\theta_{1} \ell_{1}} \theta_{1} - e^{\theta_{2} \ell_{1}} \theta_{2}) + e^{(\theta_{1} \vee \theta_{2}) \ell_{1}} ( o(\theta_{1} r) + o(\theta_{2} r)).\end{align}

Lemma 9. Let $S_{n}(x) = \sum_{k=1}^{n} x^{k}$ for $x \in \mathbb{R}$ and integer $n\ge 1$ , then, for each fixed $\theta_{1}$ and $\theta_{2}$ , as $r\downarrow 0$ , for $i=1,2$ ,

(4.55) \begin{align} & e^{-\eta^{(r)}_{i}(r\theta_{i}) \widehat{R}^{(r)}_{e}} = 1 - r\theta_{i} (\lambda^{(r)}_{i} \widehat{R}^{(r)}_{e}) - 2^{-1} (r\theta_{i})^{2} [ (\lambda^{(r)}_{i} \sigma^{(r)}_{e,i})^{2} (\lambda^{(r)}_{i} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{i} \widehat{R}^{(r)}_{e})^{2} ] \nonumber\\& \hskip21.5mm + S_{3}(\widehat{R}^{(r)}_{e}) o((r\theta)^{2}), \\[-6pt] \nonumber\end{align}

(4.56) \begin{align} & e^{-\zeta^{(r)}(r\theta_{i}) \widehat{R}^{(r)}_{s-}} = 1 + r\theta_{i} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - 2^{-1} (r\theta_{i})^{2} [ (\mu^{(r)} \sigma^{(r)}_{s})^{2} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - (\mu^{(r)} \widehat{R}^{(r)}_{s-})^{2} ] \nonumber\\& \hskip21.5mm + S_{3}(\widehat{R}^{(r)}_{s-}) o((r\theta)^{2}), \\[-6pt] \nonumber\end{align}

(4.57) \begin{align} & e^{-\eta^{(r)}_{1}(r\theta_{1}) \widehat{R}^{(r)}_{e}} - e^{-\eta^{(r)}_{2}(r\theta_{2}) \widehat{R}^{(r)}_{e}} \nonumber\\& \qquad = r (\theta_{2} \lambda^{(r)}_{2} - \theta_{1} \lambda^{(r)}_{1}) \widehat{R}^{(r)}_{e} + 2^{-1} r^{2} [\theta_{2}^{2} (\lambda^{(r)}_{2} \sigma^{(r)}_{e,2})^{2} \lambda^{(r)}_{2} - \theta_{1}^{2} (\lambda^{(r)}_{1} \sigma^{(r)}_{e,1})^{2} \lambda^{(r)}_{1}] \widehat{R}^{(r)}_{e} \nonumber\\& \qquad\hskip3.5mm - 2^{-1} r^{2} [\theta_{2}^{2} (\lambda^{(r)}_{2})^{2} - \theta_{1}^{2} (\lambda^{(r)}_{1})^{2}] (\widehat{R}^{(r)}_{e})^{2} + S_{3}(\widehat{R}^{(r)}_{e}) o((r\theta)^{2}), \\[-6pt] \nonumber\end{align}

(4.58) \begin{align} & e^{-\zeta^{(r)}(r\theta_{2}) \widehat{R}^{(r)}_{s-}} - e^{-\zeta^{(r)}(r\theta_{1}) \widehat{R}^{(r)}_{s-}} \nonumber\\& \qquad = r (\theta_{2} - \theta_{1}) (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - 2^{-1} r^{2} (\theta_{2}^{2} - \theta_{1}^{2}) [(\mu^{(r)} \sigma^{(r)}_{s})^{2} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - (\mu^{(r)} \widehat{R}^{(r)}_{s-})^{2}] \nonumber\\& \qquad\hskip3.5mm + S_{3}(\widehat{R}^{(r)}_{s-}) o((r\theta)^{2}). \\[6pt] \nonumber\end{align}

These lemmas are proved in Appendix A.5. We first compute $D^{(r)}(\theta,0)$ :

(4.59) \begin{align}D^{(r)}(\theta,0) & = \alpha^{(r)}_{e} (1 - e^{r\theta \ell^{(r)}_{1}}) \mathbb{P}^{(r)}_{e}[L^{(r)}(0-) = \ell^{(r)}_{1}] \nonumber\\ &\qquad + \alpha^{(r)}_{e} e^{r\theta \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[(1 - e^{-\zeta^{(r)}(r\theta) \widehat{R}^{(r)}_{s-}}) 1(L^{(r)}(0-) = \ell^{(r)}_{1})]\nonumber\\&\qquad + \alpha^{(r)}_{e} (re^{r\theta (\ell^{(r)}_{1}+1)} - 1) \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{1}(r\theta) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})]\\ & \qquad + \alpha^{(r)}_{e} \mathbb{E}^{(r)}_{s}[(e^{-\eta^{(r)}_{1}(r\theta) \widehat{R}^{(r)}_{e}} - 1) 1(L^{(r)}(0) = \ell^{(r)}_{1})]\nonumber\\& = - \alpha^{(r)}_{e} e^{r\theta \ell^{(r)}_{1}} \mathbb{E}^{(r)}_{e}[ e^{-\zeta^{(r)}(r\theta) \widehat{R}^{(r)}_{s-}} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \qquad + \alpha^{(r)}_{e} e^{r\theta \ell^{(r)}_{1}} e^{r \theta} \mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{1}(r\theta) \widehat{R}^{(r)}_{e}} 1(L^{(r)}(0) = \ell^{(r)}_{1})].\end{align}

Define ${\mathcal{E}}^{(r)}_{1}$ as

(4.60) \begin{align}{\mathcal{E}}^{(r)}_{1} & = 2^{-1}\mathbb{E}^{(r)}_{e}[ \{(\mu^{(r)} \sigma^{(r)}_{s})^{2} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - (\mu^{(r)} \widehat{R}^{(r)}_{s-})^{2} \}1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \hskip3.5mm - 2^{-1} \mathbb{E}^{(r)}_{s}[ \{ ((\lambda^{(r)}_{1} \sigma^{(r)}_{e,1})^{2} + 2) (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e})^{2} \} 1(L^{(r)}(0) = \ell^{(r)}_{1})].\end{align}

Applying (4.56) and

\begin{align*}e^{r \theta} e^{-\eta^{(r)}_{1}(r\theta) \widehat{R}^{(r)}_{e}} & = (1 + r \theta + o(r\theta)) \{1 - r\theta (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e}) \nonumber\\&\hskip4mm - 2^{-1} (r\theta)^{2} [ (\lambda^{(r)}_{1} \sigma^{(r)}_{e,1})^{2} (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e})^{2} ] + o((r\theta)^{2})\} \nonumber\\& = 1 - r\theta (-1 + \lambda^{(r)}_{1} \widehat{R}^{(r)}_{e}) - 2^{-1} (r\theta)^{2} [ ((\lambda^{(r)}_{1} \sigma^{(r)}_{e,1})^{2} + 2) (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{1} \widehat{R}^{(r)}_{e})^{2} ] \nonumber\\& \hskip4mm + S_{3}(\widehat{R}^{(r)}_{e}) o((r\theta)^{2})\end{align*}

to (4.59) and recalling the definition of $\Delta^{(r)}_{1}$ , we have

(4.61) \begin{align}& D^{(r)}(\theta,0) = \alpha^{(r)}_{e} e^{\theta \ell_{1}} (- r\theta \Delta^{(r)}_{1} + (r \theta)^{2} {\mathcal{E}}^{(r)}_{1}) + o((r\theta)^{2}),\end{align}

because $\mathbb{E}^{(r)}_{s}[S_{3}(\widehat{R}^{(r)}_{e}) 1(L^{(r)}(0) = \ell^{(r)}_{1})]$ is uniformly bounded for $r \in (0,1]$ by Corollary 5.

Similarly, define ${\mathcal{E}}^{(r)}_{2}$ as

(4.62) \begin{align}{\mathcal{E}}^{(r)}_{2} & = - 2^{-1} \mathbb{E}^{(r)}_{e} [ (\mu^{(r)} \sigma^{(r)}_{s})^{2} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - (\mu^{(r)} \widehat{R}^{(r)}_{s-})^{2} ] 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \hskip4mm + 2^{-1} \mathbb{E}^{(r)}_{s}[\{ (\lambda^{(r)}_{2} (\sigma^{(r)}_{e,2})^{2} + 2) (\lambda^{(r)}_{2} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{2} \widehat{R}^{(r)}_{e})^{2} \} 1(L^{(r)}(0) = \ell^{(r)}_{1})].\end{align}

Then, $D^{(r)}(0,\theta)$ is computed from (4.39) as

(4.63) \begin{align}&D^{(r)}(0,\theta) \nonumber\\&\quad = \alpha^{(r)}_{e} (e^{r\theta \ell^{(r)}_{1}} - 1) \mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta) \widehat{R}^{(r)}_{s})} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\&\quad\hskip4mm + \alpha^{(r)}_{e} \mathbb{E}^{(r)}_{e}[(e^{-\zeta^{(r)}(r\theta) \widehat{R}^{(r)}_{s}} - 1) 1(L^{(r)}(0-) = \ell^{(r)}_{1})] + \alpha^{(r)}_{e} (1 - e^{r\theta (\ell^{(r)}_{1} + 1)}) \mathbb{P}^{(r)}_{s}[L^{(r)}(0) = \ell^{(r)}_{1}] \nonumber\\&\quad\hskip4mm + \alpha^{(r)}_{e} e^{r\theta (\ell^{(r)}_{1} + 1)} \mathbb{E}^{(r)}_{s}[(1 - e^{-\eta^{(r)}_{2}(r\theta) \widehat{R}^{(r)}_{e}}) 1(L^{(r)}(0) = \ell^{(r)}_{1})] \nonumber\\&\quad = \alpha^{(r)}_{e} e^{r\theta \ell^{(r)}_{1}} \big\{\mathbb{E}^{(r)}_{e}[e^{-\zeta^{(r)}(r\theta) \widehat{R}^{(r)}_{s}} 1(L^{(r)}(0-) = \ell^{(r)}_{1})] - e^{r\theta}\mathbb{E}^{(r)}_{s}[e^{-\eta^{(r)}_{2}(r\theta) \widehat{R}^{(r)}_{e}}1(L^{(r)}(0) = \ell^{(r)}_{1})] \} \nonumber\\&\quad = \alpha^{(r)}_{e} e^{\theta \ell_{1}} \{\mathbb{E}^{(r)}_{e}[1 + r\theta \mu^{(r)} \widehat{R}^{(r)}_{s-} \nonumber\\& \quad\hskip4mm - 2^{-1} (r \theta)^{2} [ (\mu^{(r)} \sigma^{(r)}_{s})^{2} (\mu^{(r)} \widehat{R}^{(r)}_{s-}) - (\mu^{(r)} \widehat{R}^{(r)}_{s-})^{2} ] 1(L^{(r)}(0-) = \ell^{(r)}_{1})] \nonumber\\& \quad\hskip4mm - (1 + r \theta + o(r\theta)) \mathbb{E}^{(r)}_{s}[1 - r\theta (\lambda^{(r)}_{2} \widehat{R}^{(r)}_{e}) \nonumber\\& \quad\hskip4mm - 2^{-1} (r\theta)^{2} [ (\lambda^{(r)}_{2} \sigma^{(r)}_{e,2})^{2} (\lambda^{(r)}_{2} \widehat{R}^{(r)}_{e}) - (\lambda^{(r)}_{2} \widehat{R}^{(r)}_{e})^{2} ] 1(L^{(r)}(0) = \ell^{(r)}_{1})]\big\} + o((r\theta)^{2}) \nonumber\\&\quad = \alpha^{(r)}_{e} e^{\theta \ell_{1}} (r \theta \Delta^{(r)}_{2} + (r \theta)^{2} {\mathcal{E}}^{(r)}_{2}) + o((r\theta)^{2}).\end{align}

Step 3: In this step we derive asymptotic expansions for $\psi^{(r)}_{i}(\theta_{i})$ for $i=1,2$ . First, note that (4.50) and (4.51) can be written as

(4.64) \begin{align} \alpha^{(r)}_{e} \Delta^{(r)}_{1} = & - rb_{1} \mu \varphi^{(r)}_{1}(0) + c^{(r)}_{1}\mu^{(r)} \mathbb{P}[L^{(r)} = 0] + o(r), \\[-6pt] \nonumber\end{align}

(4.65) \begin{align} & \alpha^{(r)}_{e} \Delta^{(r)}_{2} = r\mu b_{2} \varphi^{(r)}_{2}(0) + o(r). \\[6pt] \nonumber\end{align}

Then, substituting (4.61) into the asymptotic BAR (4.40) with $\theta_{2} = 0$ and applying (4.64), we have

(4.66) \begin{align}& \tfrac 12 r c_{1} \mu \sigma_{1}^{2} \big(\beta_{1} - \theta_{1}\big) \psi^{(r)}_{1}(\theta_{1}) \nonumber\\&\qquad = c_{1} \mu \mathbb{P}[L^{(r)} = 0] - e^{\theta_{1} \ell_{1}} \alpha^{(r)}_{e} (\Delta^{(r)}_{1} - r \theta_{1} {\mathcal{E}}^{(r)}_{1}) + \theta_{1} o(r), \nonumber\\& \qquad = c_{1} \mu (1 - e^{\theta_{1} \ell_{1}}) \mathbb{P}[L^{(r)} = 0]+ r e^{\theta_{1}\ell_{1}} \big( b_{1} \mu \varphi^{(r)}_{1}(0) +\theta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}\big) + \theta_{1} o(r),\end{align}

Similarly, substituting (4.63) into (4.40) with $\theta_{1} = 0$ and applying (4.65), we have

(4.67) \begin{align}\tfrac 12 r c_{2} \mu \sigma_{2}^{2} \big(\beta_{2} - \theta_{2}\big) \psi^{(r)}_{2}(\theta_{2}) & = \alpha^{(r)}_{e} e^{\theta_{2}\ell_{1}} \big(\Delta^{(r)}_{2} + r \theta_{2} {\mathcal{E}}^{(r)}_{2} \big) + \theta_{2} o(r) \nonumber\\& = r e^{\theta_{2}\ell_{1}} \big( \mu b_{2} \varphi^{(r)}_{2}(0) + \theta_{2} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{2} \big) + \theta_{2} o(r).\end{align}

We next compute $\mathbb{P}[L^{(r)} = 0]$ for (4.66).

Lemma 10. We have

(4.68) \begin{align}\mathbb{P}[L^{(r)} = 0] =\begin{cases}\displaystyle\frac {r} {2 c_{1} \mu \ell_{1}} ( c_{1} \mu \sigma_{1}^{2} \varphi^{(r)}_{1}(0) + 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}) + o(r), & b= 0,\\[8pt]\displaystyle\frac {r e^{\beta_{1} \ell_{1}}}{c_{1} \mu (e^{\beta_{1}\ell_{1}} - 1)} \big(b_{1} \mu \varphi^{(r)}_{1}(0) + \beta_{1}\alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} \big) + o(r), & b \not= 0.\end{cases}\end{align}

Proof. From (4.66), we have

(4.69) \begin{align}\mathbb{P}[L^{(r)} = 0] & = r \frac {c_{1} \mu \sigma_{1}^{2} \big(\theta_{1} - \beta_{1}\big) \psi^{(r)}_{1}(\theta_{1}) + 2 e^{\theta_{1} \ell_{1}} (b_{1} \mu \varphi^{(r)}_{1}(0) + \theta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1})} {2 c_{1} \mu (e^{\theta_{1} \ell_{1}} - 1)} + o(r).\end{align}

If $b_{1}=0$ then $\beta_{1} = b_{1} = 0$ , and therefore, (4.69) yields

\begin{align*}\mathbb{P}[L^{(r)} = 0] & = r \lim_{\theta_{1} \to 0} \frac {\theta_{1} (c_{1} \mu \sigma_{1}^{2} \psi^{(r)}_{1}(\theta_{1}) + 2 e^{\theta_{1} \ell_{1}}\alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1})} {2 c_{1} \mu (e^{\theta_{1} \ell_{1}} - 1)} + o(r) \nonumber\\& = \frac {r} {2 c_{1} \mu \ell_{1}} ( c_{1} \mu \sigma_{1}^{2} \varphi^{(r)}_{1}(0) + 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}) + o(r),\end{align*}

which proves (4.68) for $b_{1} = 0$ . If $b_{1} \not= 0$ then substituting $\theta_{1} = \beta_{1}$ into (4.69) yields

\begin{align*}\mathbb{P}[L^{(r)} = 0] = \frac {r e^{\beta_{1} \ell_{1}}}{c_{1} \mu (e^{\beta_{1} \ell_{1}} - 1)} (b_{1} \mu \varphi^{(r)}_{1}(0) + \beta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} ) + o(r),\end{align*}

which is (4.68) for $b_{1} \not= 0$ .

We also compute $\alpha^{(r)}_{e} \Delta^{(r)}_{1}$ to be used later.

Lemma 11. We have

(4.70) \begin{align}& \alpha^{(r)}_{e} \Delta^{(r)}_{1} =\begin{cases}\displaystyle\frac {r} {2 \ell_{1}} ( c_{1} \mu \sigma_{1}^{2} \varphi^{(r)}_{1}(0) + 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}) + o(r), & b_{1} = 0,\\\displaystyle\frac {r } {e^{\beta_{1} \ell_{1}} - 1} \Big(b_{1} \mu \varphi^{(r)}_{1}(0) + \frac 1{c_{1} \mu} \beta_{1} e^{\beta_{1} \ell_{1}} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} \Big) + o(r), & b \not= 0.\end{cases}\end{align}

This lemma is immediate from substituting $\mathbb{P}^{(r)}_{e}[L^{(r)} = 0]$ of Lemma 10 into (4.64). We are now ready to compute $\psi^{(r)}_{i}(\theta_{i})$ for $i=1,2$ , which will be a key to prove Theorem 1.

Lemma 12. We have

(4.71) \begin{align}\psi^{(r)}_{1}(\theta_{1}) & =\begin{cases}\displaystyle\frac {e^{\theta_{1} \ell_{1}} - 1} {\ell_{1} \theta_{1}} \Big(\varphi^{(r)}_{1}(0) + \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}}{c_{1} \mu \sigma_{1}^{2}} \Big) - \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}}{c_{1} \mu \sigma_{1}^{2}} e^{\theta_{1} \ell_{1}} + o(1), & b_{1} = 0,\\[9pt]\displaystyle\frac {\beta_{1} }{ e^{\beta_{1} \ell_{1}} - 1} \frac {e^{\beta_{1} \ell_{1}} - e^{\theta_{1} \ell_{1}}} {\beta_{1} - \theta_{1}} \Big( \varphi^{(r)}_{1}(0) + \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}} {c_{1} \mu \sigma_{1}^{2}} \Big) - \frac {2\alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} } { c_{1} \mu \sigma_{1}^{2}} e^{\theta_{1} \ell_{1}} + o(1), & b_{1} \not= 0,\end{cases}\end{align}

(4.72) \begin{align} \psi^{(r)}_{2}(\theta_{2}) & = \frac {\beta_{2} e^{\theta_{2} \ell_{1}}} {\beta_{2} - \theta_{2}} \Big( \varphi^{(r)}_{2}(0) + \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{2}} {c_{2} \mu \sigma_{2}^{2} } \Big) - \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{2}} {c_{2} \mu \sigma_{2}^{2} } e^{\theta_{2} \ell_{1}} + o(1).\end{align}

Proof. We first compute $\psi^{(r)}_{1}(\theta_{1})$ . Assume that $b_{1} = 0$ . Substituting $\mathbb{P}^{(r)}_{e}[L^{(r)} = 0]$ of Lemma 10 into (4.66) for $\beta_{1} =b_{1} = 0$ , we have

\begin{align*}& - r c_{1} \mu \sigma_{1}^{2} \theta_{1} \psi^{(r)}(\theta_{1}) \nonumber= (1 - e^{\theta_{1} \ell_{1}}) \frac {r} {\ell_{1}} ( c_{1} \mu \sigma_{1}^{2} \varphi^{(r)}_{1}(0) + 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}) + r 2 e^{\theta_{1} \ell_{1}} \theta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} + o(r),\end{align*}

which proves (4.71) for $b_{1} = 0$ . Next assume that $b_{1} \not= 0$ . Similarly to the case of $b_{1} = 0$ , it follows from Lemma 10 and (4.66) for $b_{1} \not= 0$ that

\begin{align*}& \frac {r}{2} c_{1} \mu \sigma_{1}^{2} (\beta_{1} - \theta_{1}) \psi^{(r)}(\theta_{1}) \nonumber\\& \qquad = \frac {r e^{\beta_{1} \ell_{1}}(1 - e^{\theta_{1} \ell_{1}}) }{ (e^{\beta_{1} \ell_{1}} - 1)} \big(b_{1} \mu \varphi^{(r)}_{1}(0) + \beta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} \big) + r e^{\theta_{1} \ell_{1}} \big( b_{1} \mu \varphi^{(r)}(0) + \theta_{1} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} \big) + o(r) \nonumber\\& \qquad = \frac {r b_{1} \mu \varphi^{(r)}(0) }{ e^{\beta_{1} \ell_{1}} - 1} \big(e^{\beta_{1} \ell_{1}} - e^{\theta_{1} \ell_{1}}) + r \frac {\beta_{1} e^{\beta_{1} \ell_{1}}(1 - e^{\theta_{1} \ell_{1}}) + \theta_{1} e^{\theta_{1} \ell_{1}} (e^{\beta_{1} \ell_{1}} - 1)} { (e^{\beta_{1} \ell_{1}} - 1)} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} + o(r).\end{align*}

Hence, we have

\begin{align*}\psi^{(r)}_{1}(\theta_{1}) & = \frac {\beta_{1} \varphi^{(r)}(0) }{ e^{\beta_{1} \ell_{1}} - 1} \frac {e^{\beta_{1} \ell_{1}} - e^{\theta_{1} \ell_{1}}} {\beta_{1} - \theta_{1}} \nonumber\\& \hskip4mm + \frac { \beta_{1} e^{\beta_{1} \ell_{1}} - \theta_{1} e^{\theta_{1} \ell_{1}} + (\theta_{1} - \beta_{1}) e^{\theta_{1} \ell_{1}} e^{\beta_{1} \ell_{1}} } { c_{1} \mu \sigma_{1}^{2} (e^{\beta_{1} \ell_{1}} - 1) (\beta_{1} - \theta_{1}) } 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} + o(r) \nonumber\\& = \frac {\beta_{1} \varphi^{(r)}(0) }{ e^{\beta_{1} \ell_{1}} - 1} \frac {e^{\beta_{1} \ell_{1}} - e^{\theta_{1} \ell_{1}}} {\beta_{1} - \theta_{1}} \nonumber\\& \hskip4mm + \frac { \beta_{1} (e^{\beta_{1} \ell_{1}} - e^{\theta_{1} \ell_{1}} ) + (\beta_{1} - \theta_{1}) e^{\theta_{1} \ell_{1}} - (\beta_{1} -\theta_{1}) e^{\theta_{1} \ell_{1}} e^{\beta_{1} \ell_{1}} } { c_{1} \mu \sigma_{1}^{2} (e^{\beta_{1} \ell_{1}} - 1) (\beta_{1} - \theta_{1}) } 2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} + o(r) \nonumber\\& = \frac {\beta_{1} }{ e^{\beta_{1} \ell_{1}} - 1} \frac {e^{\beta_{1}\ell_{1}} - e^{\theta_{1} \ell_{1}}} {\beta_{1} - \theta_{1}} \Big(\varphi^{(r)}(0) + \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1}} {c_{1}\mu \sigma_{1}^{2}} \Big) - \frac {2 e^{\theta_{1} \ell_{1}} } { c_{1} \mu\sigma_{1}^{2}} \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{1} + o(r).\end{align*}

Thus, we have (4.71) for $b_{1} \not= 0$ . We next compute $\psi^{(r)}_{2}(\theta_{2})$ . From (4.67), we have

\begin{align*}& \psi^{(r)}_{2}(\theta_{2}) = \frac {e^{\theta_{2}\ell_{1}} }{\beta_{2} - \theta_{2}}\Big(\beta_{2} \varphi^{(r)}_{2}(0) + (\beta_{2} - (\beta_{2} - \theta_{2} )) \frac {2 \alpha^{(r)}_{e} {\mathcal{E}}^{(r)}_{2}} {c_{2} \mu \sigma_{2}^{2}} \Big) + \theta_{2} o(r),\end{align*}

which proves (4.72).

6.1. Process limit

Up to now, we have considered the limit of the sequence of the scaled stationary distributions in heavy traffic. From the assumptions (A1)–(A6), one can guess this limiting distribution must be the stationary distribution of the reflected diffusion on $[0,\infty)$ whose Brownian component changes its drift and variance at the level $\ell_{1}$ .

Recall that, if one of the conditions in part (ii) of Theorem 1 is satisfied then, for $b_{1} \not= 0$ , the stationary distribution $\widetilde{\nu}$ is characterized by the stationary equation, that is, BAR, i.e.

(6.92) \begin{align}& \sum_{i=1}^{2} \bigg(- b_{i} \theta_{i} + \frac 12 c_{i} \sigma_{i}^{2} \theta_{i}^{2}\bigg) \widetilde{\varphi}_{i}(\theta_{i}) + \frac {\beta_{1} e^{\beta_{1} \ell_{1}}} {e^{\beta_{1} \ell_{1}} - 1} \theta_{1} \frac {c_{1} \sigma^{2}_{1}}{2} \widetilde{\varphi}_{1}(0) - \frac {\beta_{1}e^{\theta_{1} \ell_{1}}} {e^{\beta_{1} \ell_{1}} - 1} \theta_{1} \frac {c_{1} \sigma^{2}_{1}}{2} \widetilde{\varphi}_{1}(0) \nonumber\\& + \beta_{2} e^{\theta_{2}\ell_{1}} \theta_{2} \frac {c_{2}\sigma^{2}_{2}}{2} \widetilde{\varphi}_{2}(0) = 0, \qquad \theta \equiv(\theta_{1}, \theta_{2}) \in \mathbb{R} \times \mathbb{R}_{-},\end{align}

which can be obtained from (4.40) and (4.39) similarly to $D^{(r)}(\theta,\theta)$ from (A.135) and (A.136), recall that $\beta_{i} = 2 b_{i}/(c_{i}\sigma_{i}^{2})$ for $i=1,2$ . The BAR for $b_{1} = 0$ is obtained from (6.92) by letting $b_{1} \downarrow 0$ . So, we only consider the case that $b_{1} \not= 0$ .

We like to identify this stationary distribution $\widetilde{\nu}$ by that of a reflected diffusion process. Assume that the diffusion scaled process $L^{(r)}_\textrm{dif}({\cdot}) \equiv \{r L^{(r)}(r^{-2} t);\, t \ge 0\}$ weakly converges to a nonnegative valued stochastic process $Z({\cdot})\equiv \{Z(t);\, t \ge 0\}$ as $r \downarrow 0$ . We aim to characterize this $Z({\cdot})$ by a stochastic integral equation. To this end, define real-valued functions $\sigma(x)$ and b(x) for $x \in \mathbb{R}_{+}$ as

\begin{gather*}b(x) = b_{1} \mu 1(x \le \ell_{1}) + b_{2} \mu 1(x > \ell_{1}),\\\sigma(x) = \sqrt{c_{1}} \mu \sigma_{1} 1(x \le \ell_{1}) + \sqrt{c_{2}} \mu \sigma_{2} 1(x > \ell_{1}),\end{gather*}

then one may expect that $Z({\cdot})$ satisfies the following stochastic integral equation:

(6.93) \begin{align} Z(t) = Z(0) & + \int_{0}^{t} \sigma(Z(u)) \,{\rm d} W(u)- \int_{0}^{t} b(Z(u)) \,{\rm d} u + Y(t), \quad t \ge 0.\end{align}

Here $W({\cdot})$ is the standard Brownian motion, $Y({\cdot}) \equiv \{Y(t);\, t\ge 0\}$ is a nondecreasing process satisfying $\int_{0}^{t} 1(Z(u)> 0) Y({\rm d} u) = 0$ for $t \ge 0$ . Recently, Miyazawa [Reference Miyazawa11] (see also [Reference Atar, Castiel and Reiman1, Reference Atar and Wolansky2, Reference Miyazawa13]) showed that (6.93) has a unique weak solution and Z(t) has a stationary distribution if only if $b_{2} > 0$ , which corresponds to $\rho^{(r)}_{2} < 1$ .

Assume that $Z({\cdot})$ is stationary, then, applying the Ito formula to (6.93) for test function $f(x) = e^{\theta x}$ , we have

(6.94) \begin{align} & \sum_{i=1}^{2} \Big(\! - b_{i} \theta + \frac 12c_{i} \sigma_{i}^{2} \theta^{2}\Big) \widehat{\varphi}_{i}(\theta) + \frac{\beta_{1} e^{\beta_{1} \ell_{1}}} {e^{\beta_{1} \ell_{1}} - 1} \theta\frac {c_{1} \sigma^{2}_{1}}{2} \widehat{\varphi}_{1}(0) = 0, \quad \theta\le 0,\end{align}

where $\widehat{\varphi}_{1}(\theta) = \mathbb{E}[e^{\theta Z} 1(Z \le\ell_{1})]$ and $\widehat{\varphi}_{2}(\theta) = \mathbb{E}[e^{\theta Z}1(Z > \ell_{1})]$ for random variable Z subject to the stationary distribution of the $Z({\cdot})$ . Equation (6.94) is the stationary equation for $Z({\cdot})$ . Then, we can see that (6.92) for $\theta_{1} = \theta_{2} = \theta$ agrees with (6.94) if and only if $c_{1} = c_{2}$ , equivalently, $\lambda_{1} = \lambda_{2}$ . Hence, if $L^{(r)}_\textrm{dif}({\cdot})$ weakly converges to $Z({\cdot})$ as $r \downarrow 0$ then we must have $\lambda_{1} = \lambda_{2}$ . Of course, this does not exclude the possibility that $L^{(r)}_\textrm{dif}({\cdot})$ weakly converges to some process other than $Z({\cdot})$ . We conjecture that this is indeed possible because the stationary distribution of $L^{(r)}_\textrm{dif}({\cdot})$ weakly converges to the $\widetilde{\nu}$ of (3.7) for $\lambda_{1} \not= \lambda_{2}$ at least when $T_{s}$ is exponentially distributed by part (ii) of Theorem 1. Thus, we have two challenging problems.

1. (a) Prove that $L^{(r)}_\textrm{dif}({\cdot})$ weakly converges to $Z({\cdot})$ when $\lambda_{1} = \lambda_{2}$ .

2. (b) Find a process limit of $L^{(r)}_\textrm{dif}({\cdot})$ when $\lambda_{1}\not= \lambda_{2}$ .

Both problems require theoretical investigations based on process limits, which is beyond the present framework. So, we leave them for future study.

6.2. Intuitive verification of (3.11)

We intuitively show (3.11). First assume that $b_{1} \not= 0$ . Under the stationary setting, let $\tau^{(r)}_{1}$ be the first time for $L^{(r)}(t)$ to get into $\ell^{(r)}_{1}$ from below or above. Suppose that $L^{(r)}(t)$ first moves to $\ell^{(r)}_{1} + 1$ after time $\tau^{(r)}_{1}$ , and returns to $\ell^{(r)}_{1}$ at time $\tau^{(r)}_{2}$ after time $B^{(r)}_{2}$ . Since the relative frequency of moving up at level $\ell^{(r)}_{1}$ is guessed to be $f^{(r)}_\textrm{up} \equiv\lambda^{(r)}_{2}/(\lambda^{(r)}_{2} + c^{(r)}_{1} \mu^{(r)})$ and $B^{(r)}_{2}$ is stochastically almost identical with the busy period of the $GI/G/1$ queue with interarrival time $T^{(r)}_{e,2}$ and service time $T^{(r)}_{s}/c^{(r)}_{2}$ , the mean sojourn time for $L^{(r)}(t)$ to be above $\ell^{(r)}_{1}$ in the cycle $(\tau^{(r)}_{1}, \tau^{(r)}_{2}]$ is

(6.95) \begin{align}f^{(r)}_\textrm{up} \mathbb{E}[B^{(r)}_{2}] \sim \frac {\lambda^{(r)}_{2}} {\lambda^{(r)}_{2} + c^{(r)}_{1} \mu^{(r)}} \times \frac 1{c^{(r)}_{2} \mu^{(r)} (1 - \rho^{(r)}_{2})} = \frac {c_{2}}{c_{1} + c_{2}} \times \frac 1{r b_{2} \mu} + o(1),\end{align}

where ‘ $\sim$ ’ stands for the same order as $r \downarrow 0$ , that is, for nonzero valued functions $f_{1}$ and $f_{2}$ , $f_{1}(r) \sim f_{2}(r)$ means that $\lim_{r \downarrow 0} f_{1}(r)/f_{2}(r) = 1$ .

On the other hand, when $L^{(r)}(t)$ first moves to $\ell^{(r)}_{1} - 1$ after time $\tau^{(r)}_{1}$ , let $B^{(r)}_{1}$ be the time to return to $\ell^{(r)}_{1}$ from below. Let $f^{(r)}_\textrm{down} = 1 -f^{(r)}_\textrm{up}$ , then, similarly to $B^{(r)}_{2}$ but, taking $b_{1}\not = 0$ and the reflecting boundary at level 0 into account, it follows from (4.25) of [Reference Miyazawa12] that the mean sojourn time for $L^{(r)}(t)$ to be below $\ell^{(r)}_{1}$ in the cycle $(\tau^{(r)}_{1},\tau^{(r)}_{2}]$ is

(6.96) \begin{align} f^{(r)}_\textrm{down} \mathbb{E}[B^{(r)}_{1}] & \sim\frac {c^{(r)}_{1} \mu^{(r)}} {\lambda^{(r)}_{2} + c^{(r)}_{1} \mu^{(r)}}\times \frac 1{\lambda^{(r)}_{1}\mathbb{P}^{(r)}_{s}[L^{(r)}(0)=\ell^{(r)}_{1}]} \nonumber\\& = \frac {c_{1}}{c_{1} + c_{2}} \times \frac {e^{\beta_{1} \ell_{1}} - 1}{r b_{1} \mu} + o(1).\end{align}

Since $f^{(r)}_\textrm{up} B^{(r)}_{2} + f^{(r)}_\textrm{down} B^{(r)}_{1}$ is the mean cycle time to return to $\ell^{(r)}_{1}$ , it follows from the cycle formula of a regenerative process that

\begin{align*}\lim_{r \downarrow 0} \nu^{(r)}_{1}(\mathbb{R}_{+}) = \lim_{r \downarrow 0} \frac {f^{(r)}_\textrm{down} \mathbb{E}[B^{(r)}_{1}]} {f^{(r)}_\textrm{down} \mathbb{E}[B^{(r)}_{1}] + f^{(r)}_\textrm{up} \mathbb{E}[B^{(r)}_{2}]} = \frac {c_{1} b_{2} (e^{\beta_{1} \ell_{1}} - 1)}{c_{2} b_{1} + c_{1} b_{2} (e^{\beta_{1} \ell_{1}} - 1)} = \widetilde{A}_{1}.\end{align*}

Thus, for $b_{1} \not= 0$ , (3.11) is intuitively obtained for $i=1$ , and it is similarly obtained for $i=2$ . The case for $b_{1} = 0$ is similarly derived. Thus, we have support of (3.11).