A law of iterated logarithm for multiclass queues with preemptive priority service discipline

Abstract

A law of iterated logarithm (LIL) is established for a multiclass queueing model, having a preemptive priority service discipline, one server and $K$ customer classes, with each class characterized by a renewal arrival process and i.i.d. service times. The LIL limits quantify the magnitude of asymptotic stochastic fluctuations of the stochastic processes compensated by their deterministic fluid limits. The LIL is established in three cases: underloaded, critically loaded, and overloaded, for five performance measures: queue length, workload, busy time, idle time, and number of departures. The proof of the LIL is based on a strong approximation approach, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.

Appendix: Overview

This appendix contains additional materials supplementing the main paper. In Sect. 1 we provide an alternative definition for the one-dimensional ORM in (14). In Sect. 1 we provide the analytic solutions to the fluid Eq. (13). In Sect. 1, more numerical examples are given to supplement Sect. 5. In Sect. 1 we further analyze Corollary 1 in two cases. Finally, in Sect. 1 we prove Lemma 2.

1.1 The one-dimensional oblique reflection mapping

We now provide an alternative definition of the ORM in (14).

Definition 1

For any function $x \in \mathbb {D}_0$, if there exists a unique pair of functions $z, y\in \mathbb {D}_0$ satisfying

• (i) $z(t)=x(t)+y(t)\ge 0;$

• (ii) $y$ is non-decreasing and $y(0)=0$;

• (iii) $\int _0^\infty z(t)\mathsf{d}y(t)=0$,

is called the one-dimensional oblique reflection mapping, denoted by $(z,y)=(\varPhi ,\varPsi )(x)$.

1.2 Fluid solution to (13)

We provide analytic solutions to (13) in three cases: (i) $\rho _1> 1$, (ii) there exists an integer $k_0: 1<k_0<K$ such that $\sum _{l=1}^{k_0}\rho _l\le 1$ and $\sum _{l=1}^{k_0+1}\rho _l>1$, and (iii) $\rho <1$.

Case $(i)$: If $\rho _1>1$, then the solution of (13) is

$$\begin{aligned} \bar{Q}_k(t)&= \mu _k \bar{Z}_k(t) = \left\{ \begin{array}{ll} (\lambda _k-\mu _k)t, &{}\quad k=1; \\ \lambda _kt, &{}\quad k=2,3,\dots ,K, \end{array} \right. \end{aligned}$$

(52)

$$\begin{aligned} \bar{B}_k(t)&= \left\{ \begin{array}{l@{\quad }l} t, &{} k=1; \\ 0, &{} k=2,3,\ldots ,K, \end{array} \right. \end{aligned}$$

(53)

$$\begin{aligned} \bar{I}_k(t)&= 0, \qquad k=1,2,\dots ,K, \end{aligned}$$

(54)

and $\bar{X}_k(t)=\bar{Q}_k(t)$ and $\bar{D}_1(t)=\mu _1t$ and $\bar{D}_k(t)=0$ for all $k=2,3,\dots ,K$.

Case $(ii)$: If there exists an integer $k_0: 1<k_0<K$ such that $\sum _{l=1}^{k_0}\rho _l\le 1$ and $\sum _{l=1}^{k_0+1}\rho _l>1$, then the solution of (13) is

$$\begin{aligned} \bar{Q}_k(t)&= \mu _k \bar{Z}_k(t) = \lambda _kt-\left[ \rho _k\wedge \left( 1-\sum _{l=1}^{k-1}\rho _l\right) ^+\right] \mu _k t\nonumber \\&= \left\{ \begin{array}{l@{\quad }l} 0, &{} k=1,2,\dots ,k_0; \\ \mu _{k_0+1}\left( \sum \limits _{l=1}^{k_0+1}\rho _l-1\right) t, &{} k=k_0+1; \\ \lambda _kt, &{} k=k_0+2,k_0+3,\dots ,K, \end{array} \right. \end{aligned}$$

(55)

$$\begin{aligned} \bar{B}_k(t)&= \left[ \rho _k\wedge \left( 1-\sum \limits _{l=1}^{k-1}\rho _l\right) ^+\right] t \!=\! \left\{ \begin{array}{l@{\quad }l} \rho _kt, &{} k\!=\!1,2,\dots ,k_0; \\ \left( 1\!-\!\sum \limits _{l=1}^{k-1}\rho _l\right) t, &{} k=k_0\!+\!1; \\ 0, &{} k\!=\!k_0+2,k_0\!+\!3,\dots ,K, \end{array} \right. \end{aligned}$$

(56)

$$\begin{aligned} \bar{I}_k(t)&= \left( 1-\sum \limits _{l=1}^k\rho _l\right) ^+t= \left\{ \begin{array}{l@{\quad }l} \left( 1-\sum \limits _{l=1}^k\rho _l\right) t, &{} k\!=\!1,2,\dots ,k_0; \\ 0, &{} k\!=\!k_0+1,k_0+2,\dots ,K. \end{array} \right. \end{aligned}$$

(57)

Finally, we also note that

$$\begin{aligned} \bar{X}_k(t)\!&= \!\mu _k\left[ \rho _kt\!+\!\sum \limits _{l=1}^{k-1}\bar{B}_l(t)\!-\!t\right] \!=\!\left\{ \begin{array}{l@{\quad }l} \mu _k\left( \sum \limits _{l=1}^k\rho _l\!-\!1\right) t, &{} k\!=\!1,2,\dots ,k_0\!+\!1; \\ \lambda _kt, &{} k\!=\!k_0+2,k_0\!+\!3,\dots ,K, \end{array} \right. \nonumber \\\end{aligned}$$

(58)

$$\begin{aligned} \bar{D}_k(t)\!&= \! \mu _k\bar{B}_k(t)\!=\! \left\{ \begin{array}{l@{\quad }l} \lambda _kt, &{} k=1,2,\dots ,k_0; \\ \mu _k\left( 1-\sum \limits _{l=1}^{k-1}\rho _l\right) t, &{} k=k_0+1; \\ 0, &{} k=k_0+2,k_0+3,\dots ,K. \end{array} \right. \end{aligned}$$

(59)

Case $(iii)$: If $\rho <1$, all fluid functions satisfy the fluid functions in case (ii) with $k_0$ replaced by $K$.

1.3 More numerical examples

1.3.1 A type-1 OL example

Example 3

(Discontinuities of the LIL limits in the class index) Consider a 6-queue example $\lambda _{k}=1$, $\mu _{k}=3$, $c_{a,k}=c_{s,k}=1$ for all $1\le k\le K=6$. This example belongs to the type-1 OL case because $k_0=3$, $\sum _{k=1}^{3} \rho _k =1$ and $\sum _{k=1}^{4} \rho _k= 4/3>1$.

According to Theorem 4, we compute the LIL limits in Table 2 and plot these limits as functions of $k$ in Fig. 4. The vertical line in 4 serving as a “benchmark” for $\rho =1$ (because $\sum _{k=1}^{3} \rho _k =1$ and $\sum _{k=1}^{4}\rho _k > 1$). We see that all LIL limits jump at $k_0$ and $k_0+1$. The LIL limits $Q^{*}_{k}$, $Z^{*}_{k}$, $B^{*}_{k}$, and $D^{*}_{k}$ all peak at $k_0=3$ and $k_0+1=4$, where stochastic processes $Q_k$, $Z_k$, $B_k$, and $D_k$ experiencing the largest asymptotical stochastic variability. The LIL $I^{*}_k$ increases in $k$ and peak at $k=k_0=3$, it then drops to 0. This is so because the variability of $I_k$ is cumulative (thus increasing) for $1\le k\le k_0$ and then $I_k$ becomes asymptotically negligible for all $k_0< k\le K$. See Remark 5 for more discussions.

Fig. 4

LIL limits of Example 2 as functions of $k$, $1\le k\le 6$, with $k_0=3$, $\sum _{k=1}^{3}\rho _{k}=1$ and $\sum _{k=1}^{4}\rho _{k}>1$

We plot the LIL limits as functions of $k$ in Fig. 4.

1.3.2 LIL formulas for Example 2

We next provide the explicit LIL limits for Example 2. These limits are piecewise functions. The LIL limits for $Q$ are

$$\begin{aligned} Q_{1}^{*}(\rho _{2})&= 0,\quad \text {for all}\quad 0\le \rho _{2}\le 1,\\ Q_{2}^{*}(\rho _{2})&= \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0.5\}} + \sqrt{1+\sqrt{0.5}+\rho _{2}}\cdot \mathbf{1}_{\{0.5<\rho _{2}\le 1\}},\\ Q_{3}^{*}(\rho _{2})&= \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0\}} \!+\! \sqrt{1.5\!+\!2\rho _{2}\!+\!\sqrt{0.5-\rho _{2}}}\cdot \mathbf{1}_{\{0<\rho _{2}\le 0.5\}} \!+\! \sqrt{0.5}\cdot \mathbf{1}_{\{0.5<\rho _{2}\le 1\}},\\ Q_{4}^{*}(\rho _{2})&= \sqrt{2.5}\cdot \mathbf{1}_{\{\rho _{2}=0\}} + \sqrt{0.5}\cdot \mathbf{1}_{\{0<\rho _{2}\le 1\}}. \end{aligned}$$

The LIL limits for $Z$ are

$$\begin{aligned} Z_{1}^{*}(\rho _{2})&= 0,\quad \text {for all}\quad 0\le \rho _{2}\le 1,\\ Z_{2}^{*}(\rho _{2})&= \sqrt{1+2\rho _{2}}\cdot \mathbf{1}_{\{0.5\le \rho _{2}\le 1\}},\\ Z_{3}^{*}(\rho _{2})&= \sqrt{2+2\rho _{2}}\cdot \mathbf{1}_{\{0\le \rho _{2}\le 0.5\}} + \mathbf{1}_{\{0.5<\rho _{2}\le 1\}},\\ Z_{4}^{*}(\rho _{2})&= \sqrt{3}\cdot \mathbf{1}_{\{\rho _{2}=0\}} + \mathbf{1}_{\{0<\rho _{2}\le 1\}}. \end{aligned}$$

The LIL limits for $B$ are

$$\begin{aligned} B_{1}^{*}(\rho _{2})&= 1,\quad \text {for all}\quad 0\le \rho _{2}\le 1,\\ B_{2}^{*}(\rho _{2})&= \sqrt{2\rho _{2}}\cdot \mathbf{1}_{\{0\le \rho _{2}< 0.5\}}+ \mathbf{1}_{\{0.5\le \rho _{2} \le 1\}},\\ B_{3}^{*}(\rho _{2})&= \sqrt{1+2\rho _{2}}\cdot \mathbf{1}_{\{0\le \rho _{2}\le 0.5\}},\\ B_{4}^{*}(\rho _{2})&= \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0\}}. \end{aligned}$$

The LIL limits for $I$ are

$$\begin{aligned} I_{1}^{*}(\rho _{2})&= 1,\quad \text {for all}\quad 0\le \rho _{2}\le 1,\\ I_{2}^{*}(\rho _{2})&= \sqrt{1+2\rho _{2}}\cdot \mathbf{1}_{\{0\le \rho _{2}\le 0.5\}},\\ I_{3}^{*}(\rho _{2})&= \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0\}},\\ I_{4}^{*}(\rho _{2})&= 0,\quad \text {for all}\quad 0\le \rho _{2}\le 1. \end{aligned}$$

The LIL limits for $D$ are

$$\begin{aligned} D_{1}^{*}(\rho _{2})&= \sqrt{0.5},\quad \text {for all}\quad 0\le \rho _{2}\le 1,\\ D_{2}^{*}(\rho _{2})&= \sqrt{\rho _{2}}\cdot \mathbf{1}_{\{0\le \rho _{2}<0.5\}} + \sqrt{1.5}\cdot \mathbf{1}_{\{\rho _{2}=0.5\}}+\sqrt{1+\sqrt{0.5}}\cdot \mathbf{1}_{\{0.5< \rho _{2}\le 1\}},\\ D_{3}^{*}(\rho _{2})&= \sqrt{1.5}\cdot \mathbf{1}_{\{\rho _{2} = 0\}} + \sqrt{1+2\rho _{2}+\sqrt{0.5-\rho _{2}}}\cdot \mathbf{1}_{\{0<\rho _{2}<0.5\}} + \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0.5\}},\\ D_{4}^{*}(\rho _{2})&= \sqrt{2}\cdot \mathbf{1}_{\{\rho _{2}=0\}}. \end{aligned}$$

1.4 More discussion on Corollary 1

Using the analytic solutions to (13), we transform the results of Corollary 1 to more detailed formulas. We consider two cases: (1) $\rho <1$ and (2) there exists $k_0: 1<k_0<K$ such that $\sum _{l=1}^{k_0}\rho _l\le 1$ and $\sum _{l=1}^{k_0+1}\rho _l> 1$.

Case 1. If $\rho \le 1$, then $\bar{Q}_k(t)=0$ and $\bar{B}_k(t)=\rho _kt$ for $k=1,2,\dots ,K$. Hence (29) and (30) are the following

$$\begin{aligned} \widetilde{X}_k(t)&\mathop {=}\limits ^\mathrm{d}\mu _k\left( \sum _{l=1}^k\rho _l-1\right) t -\sum _{l=1}^{k-1} \frac{\mu _k}{\mu _l}\widetilde{Q}_l(t)\nonumber \\&\quad +\sum _{l=1}^k\frac{\mu _k}{\mu _l} \left[ \lambda _l^{1/2}c_{a,l}W_{a,l}(t)- \lambda _l^{1/2}c_{s,l}W_{s,l}(t)\right] ,\end{aligned}$$

(60)

$$\begin{aligned} \bar{B}_k(t)-\widetilde{B}_k(t)&\mathop {=}\limits ^\mathrm{d}\rho _kt-\widetilde{B}_k(t)=\frac{1}{\mu _k}\widetilde{Q}_k(t)\nonumber \\&\quad -\frac{1}{\mu _k}\left[ \lambda _k^{1/2}c_{a,k}W_{a,k}(t)- \lambda _k^{1/2}c_{s,k}W_{s,k}(t)\right] . \end{aligned}$$

(61)

Case 2. If there exists $1<k_0<K$ such that $\sum _{l=1}^{k_0}\rho _l\le 1$ and $\sum _{l=1}^{k_0+1}\rho _l> 1$, then $\bar{Q}_k(t)=0$ and $\bar{B}_k(t)=\rho _kt$ for $k=1,2,\dots ,k_0$, and (60) and (61) hold for $1,2,\dots ,k_0$. For $k=k_0+1$, we note that $\bar{Q}_{k_0+1}(t)=\bar{X}_{k_0+1}(t)=\mu _{k_0+1} (\sum _{l=1}^{k_0+1}\rho _l-1)t$ and $\bar{B}_{k_0+1}(t)=(1-\sum _{l=1}^{k_0}\rho _l)t$. Hence,

$$\begin{aligned} \begin{aligned} \widetilde{X}_{k_0+1}(t)&\mathop {=}\limits ^\mathrm{d}\mu _{k_0+1}\left( \sum _{l=1}^{k_0+1}\rho _l-1\right) t\\&\quad -\sum _{l=1}^{k_0} \frac{\mu _{k_0+1}}{\mu _l} \left[ \widetilde{Q}_l(t) +\lambda _l^{1/2}c_{a,l}W_{a,l}(t)-\lambda _l^{1/2}c_{s,l}W_{s,l}(t)\right] \\&\quad +\left[ \lambda _{k_0+1}^{1/2}c_{a,k_0+1}W_{a,k_0+1}(t) -\mu _{k_0+1}^{1/2}c_{s,k_0+1}\sqrt{1\!-\!\sum _{l=1}^{k_0}\rho _i}W_{s,k_0+1}(t)\right] \!, \end{aligned} \end{aligned}$$

(62)

and

$$\begin{aligned}&\bar{B}_{k_0+1}(t)-\widetilde{B}_{k_0+1}(t)\nonumber \\&\quad =\left( 1-\sum _{l=1}^{k_0}\rho _l\right) t-\widetilde{B}_{k_0+1}(t)\nonumber \\&\quad \mathop {=}\limits ^\mathrm{d}\frac{1}{\mu _{k_0+1}}\left[ \widetilde{Q}_{k_0+1}(t) -\mu _{k_0+1}\left( \sum \limits _{l=1}^{k_0+1}\rho _l-1\right) t\right] \nonumber \\&\qquad -\frac{1}{\mu _{k_0+1}}\left[ \lambda _{k_0+1}^{1/2}c_{a,k_0+1} W_{a,k_0+1}(t)- \mu _{k_0+1}^{1/2}c_{s,k_0+1} \sqrt{1-\sum _{l=1}^{k_0}\rho _i}W_{s,k_0+1}(t)\right] .\nonumber \\ \end{aligned}$$

(63)

For $k=k_0+i,i=2,3,\dots ,K-k_0$, $\bar{Q}_{k_0+i}(t)=\lambda _{k_0+i}t$, $\bar{B}_{k_0+i}(t)=0$, we have

$$\begin{aligned}&\widetilde{X}_{k_0+i}(t) \mathop {=}\limits ^\mathrm{d}\lambda _{k_0+i}t -\sum _{l=1}^{k_0}\frac{\mu _{k_0+i}}{\mu _l}\widetilde{Q}_l(t) \!-\!\frac{\mu _{k_0+i}}{\mu _{k_0+1}}\left[ \widetilde{Q}_{k_0+1}(t) -\mu _{k_0+1}\left( \sum \limits _{l=1}^{k_0+1}\rho _l-1\right) t\right] \nonumber \\&\quad -\sum _{l=k_0+2}^{k_0+i-1} \frac{\mu _{k_0+i}}{\mu _l}\left[ \widetilde{Q}_l(t)-\lambda _{l}t\right] +\sum _{l=1}^{k_0}\frac{\mu _{k_0+i}}{\mu _l} \left[ \lambda _l^{1/2}c_{a,l}W_{a,l}(t)- \lambda _l^{1/2}c_{s,l}W_{s,l}(t)\right] \nonumber \\&\quad +\frac{\mu _{k_0+i}}{\mu _{k_0+1}}\left[ \lambda _{k_0+1}^{1/2}c_{a,k_0+1} W_{a,k_0+1}(t)- \mu _{k_0+1}^{1/2}c_{s,k_0+1} \sqrt{1-\sum _{l=1}^{k_0}\rho _i}W_{s,k_0+1}(t)\right] \nonumber \\&\quad +\sum _{l=k_0+2}^{k_0+i}\frac{\mu _{k_0+i}}{\mu _l}\lambda _l^{1/2}c_{a,l} W_{a,l}(t), \end{aligned}$$

(64)

and

$$\begin{aligned} \bar{B}_{k_0+i}(t)-\widetilde{B}_{k_0+i}(t) \mathop {=}\limits ^\mathrm{d}\frac{\widetilde{Q}_{k_0+i}(t)-\lambda _{k_0+i}t}{\mu _{k_0+i}} -\frac{\lambda _{k_0+i}^{1/2}c_{a,k_0+i}W_{a,k_0+i}(t)}{\mu _{k_0+i}}. \end{aligned}$$

(65)

1.5 Proof of Lemma 2

According to (13), $\bar{B}_k(t)=\rho _kt$ and $\bar{X}_k(t)=-\theta _k t<0$ for $k=1,2,\dots ,K$. Next, (27) implies that $\widetilde{Q}_1(t)=\varPhi (\widetilde{X}_1)(t)$ is a reflected BM, where $\widetilde{X}_1(t)=\bar{X}_1(t)+W_1(t)$ is a BM with negative drift $-\theta _1$ and variance parameter $\mu _1\sigma _1$. Theorem 6.2 in [18] implies that

$$\begin{aligned} \mathsf{P}\left\{ \sup _{0\le t\le T}\widetilde{Q}_1(t)\ge z\right\} \le \exp \left\{ -\frac{2\theta _1}{\mu _1^2\sigma _1^2}z\right\} \le \exp \left\{ -2\gamma z\right\} , \quad z\ge 0. \end{aligned}$$

(66)

We next consider $k= 2, 3,\dots ,K$. For $z\ge 0$,

$$\begin{aligned}&\mathsf{P}\left\{ \sup _{0\le t\le T}\widetilde{Q}_k(t)\ge z\right\} \nonumber \\&\quad = \mathsf{P}\left\{ \sup _{0\le t\le T}\left\{ \widetilde{X}_k(t)+\sup _{0\le s\le t}\left[ -\widetilde{X}_k(s)\right] \right\} \ge z\right\} \nonumber \\&\quad =\mathsf{P}\left\{ \sup _{0\le t\le T}\sup _{0\le s\le t}\left[ \widetilde{X}_k(t)-\widetilde{X}_k(s)\right] \ge z\right\} \nonumber \\&\quad \le \mathsf{P}\left\{ \sup _{0\le t\le T}\sup _{0\le s\le t}\left[ (\bar{X}_k(t)+W_k(t))-(\bar{X}_k(s)+W_k(s))\right] \ge \frac{z}{2}\right\} \nonumber \\&\qquad +\,\mathsf{P}\left\{ \sup _{0\le t\le T}\sup _{0\le s\le t}\left[ \sum _{l=1}^{k-1} \frac{\mu _k}{\mu _l}\widetilde{Q}_l(s)-\sum _{l=1}^{k-1} \frac{\mu _k}{\mu _l}\widetilde{Q}_l(t)\right] \ge \frac{z}{2}\right\} \nonumber \\&\quad \le \mathsf{P}\left\{ \sup _{0\le s\le T}[\bar{X}_k(s)+W_k(s)]\ge \frac{z}{2}\right\} +\mathsf{P}\left\{ \sup _{0\le s\le T}\sum _{l=1}^{k-1} \frac{\mu _k}{\mu _l}\widetilde{Q}_l(s)\ge \frac{z}{2}\right\} ,\qquad \end{aligned}$$

(67)

where the first equality holds because $\widetilde{X}_k(0)=0$ and the first inequality holds by (29). To bound the second term in (67), we have

$$\begin{aligned} \mathsf{P}\left\{ \sup _{0\le s\le T}\sum _{l=1}^{k-1} \frac{\mu _k}{\mu _l}\widetilde{Q}_l(s)\ge \frac{z}{2}\right\}&\le \sum _{l=1}^{k-1} \mathsf{P}\left\{ \sup _{0\le s\le T} \widetilde{Q}_l(s)\ge \frac{\min \{\mu _1,\dots ,\mu _{k-1}\}}{(k-1)\mu _k}\cdot \frac{z}{2}\right\} \nonumber \\&\le \sum _{l=1}^{k-1} \mathsf{P}\left\{ \sup _{0\le s\le T} \widetilde{Q}_l(s)\ge \delta _kz\right\} \end{aligned}$$

(68)

with $\delta _k$ given in Lemma 2.

We are now ready to prove (33) for $2\le k\le K$. We use induction. First, when $k=2$, using (67), (68) and the fact that $\delta _k\le \frac{1}{2}$, we have

$$\begin{aligned}&\mathsf{P}\left\{ \sup _{0\le t\le T}\widetilde{Q}_2(t)\ge z\right\} \le \mathsf{P}\left\{ \sup _{0\le s\le T}[\bar{X}_2(s)+W_2(s)]\ge \delta _2z\right\} \\&\quad +\, \mathsf{P}\left\{ \sup _{0\le s\le T} \widetilde{Q}_1(s)\ge \delta _2z\right\} \\&\le \exp \left\{ -\frac{2\theta _2}{\mu _2^2\sigma _2^2}\delta _2z\right\} + \exp \left\{ -\frac{2\theta _1}{\mu _1^2\sigma _1^2}\delta _2z\right\} \le N_2\exp \left\{ -2\gamma \delta _2z\right\} , \end{aligned}$$

where the second inequality holds by (66) and Lemma 5.5 in [18] (with $\bar{X}_2(t)+W_2(t)$ being a BM with negative drift $-\theta _2$ and variance parameter $\mu _2\sigma _2$).

Next, assume (33) holds for classes $2,\ldots ,k$. For class $k+1$, we have

$$\begin{aligned}&\mathsf{P}\left\{ \sup _{0\le t\le T}\widetilde{Q}_{k+1}(t)\ge z\right\} \\&\quad \le \mathsf{P}\left\{ \sup _{0\le s\le T}[\bar{X}_{k+1}(t)+W_{k+1}(s)]\ge \delta _{k+1}z\right\} +\sum _{l=1}^k \mathsf{P}\left\{ \sup _{0\le s\le T} \widetilde{Q}_l(s)\ge \delta _{k+1}z\right\} \\&\quad \le \exp \left\{ -\frac{2\theta _{k+1}\delta _{k+1}z}{\mu _{k+1}^2\sigma _{k+1}^2}\right\} \!+\!\exp \left\{ -\frac{2\theta _1\delta _{k+1}z}{\mu _1^2\sigma _1^2}\right\} \!+\!\sum _{l=2}^k N_l\exp \left\{ -2\gamma \delta _2\delta _3\cdots \delta _l\delta _{k+1}z\right\} \\&\quad \le N_{k+1}\exp \left\{ -2\gamma \delta _2\delta _3\cdots \delta _{k+1}z\right\} =N_{k+1}\,\exp \left\{ -2\gamma \prod _{j=1}^{k+1}\delta _{j}\,z\right\} , \end{aligned}$$

where the first inequality holds by (67), (50), and the fact that $\delta _k\le 1/2$, and the second inequality holds by the induction hypothesis and Lemma 5.5 in [18] (with $\bar{X}_{k+1}(t)+W_{k+1}(t)$ being a BM with negative drift $-\theta _{k+1}$ and variance parameter $\mu _{k+1}\sigma _{k+1}$). $\square$