May  2020, 16(3): 1049-1076. doi: 10.3934/jimo.2018192

The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue

a. 

School of Science, Beijing University of Posts and Telecommunications, Beijing 100876, China

b. 

School of Engineering Science, University of the Chinese Academy of Sciences, Beijing, 100049, China

* Corresponding author

Received  September 2017 Revised  January 2018 Published  December 2018

Two types of the law of iterated logarithm (LIL) and one functional LIL (FLIL) are established for the sojourn time process for a multiclass queueing model, having a priority service discipline, one server and $K$ customer classes, with each class characterized by a batch renewal arrival process and independent and identically distributed (i.i.d.) service times. The LIL and FLIL limits quantify the magnitude of asymptotic stochastic fluctuations of the sojourn time process compensated by its deterministic fluid limits in two forms: the numerical and functional. The LIL and FLIL limits are established in three cases: underloaded, critically loaded and overloaded, defined by the traffic intensity. We prove the results by a approach based on strong approximation, which approximates discrete performance processes with reflected Brownian motions. We conduct numerical examples to provide insights on these LIL results.

Citation: Yongjiang Guo, Yuantao Song. The (functional) law of the iterated logarithm of the sojourn time for a multiclass queue. Journal of Industrial & Management Optimization, 2020, 16 (3) : 1049-1076. doi: 10.3934/jimo.2018192
References:
[1]

M. Bramson and J. G. Dai, Heavy traffic limits for some queueing networks, Annals of Applied Probability, 11 (2001), 49-90.  doi: 10.1214/aoap/998926987.  Google Scholar

[2]

L. Caramellino, Strassen's law of the iterated logarithm for diffusion processes for small time, Stochastic Processes and their Applications, 74 (1998), 1-19.  doi: 10.1016/S0304-4149(97)00100-2.  Google Scholar

[3]

H. Chen and A. Mandelbaum, Hierarchical modeling of stochastic network, part Ⅱ: Strong approximations, Stochastic Modeling and Analysis of Manufacturing Systems, Yao, D. D. (Eds), (1994), 107-131. Google Scholar

[4]

H. Chen and X. Shen, Strong approximations for multiclass feedforward queueing networks, Annals of Applied Probability, 10 (2000), 828-876.  doi: 10.1214/aoap/1019487511.  Google Scholar

[5]

H. Chen and D. D. Yao, Fundamentals of Queueing Networks, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-5301-1.  Google Scholar

[6]

H. Chen and H. Q. Zhang, A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines, Queueing Systems, 34 (2000), 237-268.  doi: 10.1023/A:1019113204634.  Google Scholar

[7]

M. Csörgő ang P. Révész, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981. Google Scholar

[8]

M. CsörgőP. Deheuvels and L. Horváth, An approximation of stopped sums with applications in queueing theory, Advances in Applied Probability, 19 (1987), 674-690.  doi: 10.2307/1427412.  Google Scholar

[9]

M. CsörgőZ. S. Hu and H. W. Mei, Strassen-type law of the iterated logarithm for self-normalized increments of sums, Journal of Mathematical Analysis and Applications, 393 (2012), 45-55.  doi: 10.1016/j.jmaa.2012.03.047.  Google Scholar

[10]

C. CunyF. Merlevéde and M. Peligrad, Law of the iterated logarithm for the periodogram, Stochastic Processes and their Applications, 123 (2013), 4065-4089.  doi: 10.1016/j.spa.2013.05.009.  Google Scholar

[11]

J. G. Dai, On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid limit models, Annals of Applied Probability, 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828.  Google Scholar

[12]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[13]

P. W. Glynn and W. Whitt, A new view of the heavy-traffic limit for infinite-server queues, Advances in Applied Probability, 23 (1991), 188-209.  doi: 10.2307/1427517.  Google Scholar

[14]

P. W. Glynn and W. Whitt, Departures from many queues in series, Annals of Applied Probability, 1 (1991), 546-572.  doi: 10.1214/aoap/1177005838.  Google Scholar

[15]

P. W. Glynn and W. Whitt, A central-limit-theorem version of L=λW, Queueing Systems, 1 (1986), 191-215.  doi: 10.1007/BF01536188.  Google Scholar

[16]

P. W. Glynn and W. Whitt, Sufficient conditions for functional limit theorem versions of L=λW, Queueing Systems, 1 (1987), 279-287.  doi: 10.1007/BF01149539.  Google Scholar

[17]

P. W. Glynn and W. Whitt, An LIL version of L=λW, Mathematics of Operations Research, 13 (1988), 693-710.  doi: 10.1287/moor.13.4.693.  Google Scholar

[18]

Y. Guo and Z. Li, Asymptotic variability analysis for a two-stage tandem queue, part Ⅰ: The functional law of the iterated logarithm, J. Math. Anal. Appl., 450 (2017), 1479-1509.  doi: 10.1016/j.jmaa.2017.01.062.  Google Scholar

[19]

Y. Guo and Z. Li, Asymptotic variability analysis for a two-stage tandem queue, part Ⅱ: The law of the iterated logarithm, J. Math. Anal. Appl., 450 (2017), 1510-1534.  doi: 10.1016/j.jmaa.2016.10.054.  Google Scholar

[20]

Y. Guo and Y. Liu, A law of iterated logarithm for multiclass queues with preemptive priority service discipline, Queueing Systems, 79 (2015), 251-291.  doi: 10.1007/s11134-014-9419-5.  Google Scholar

[21]

Y. GuoY. Liu and R. Pei, Functional law of iterated logarithm for multi-server queues with batch arrivals and customer feedback, Annals of Operations Research, 264 (2018), 157-191.  doi: 10.1007/s10479-017-2529-9.  Google Scholar

[22]

J. M. Harrison, Brownian Motion and Stochastic Flow System, Wiley, New York, 1985.  Google Scholar

[23]

L. Horváth, Strong approximation of renewal processes, Stochastic Process. Appl., 18 (1984), 127-138.  doi: 10.1016/0304-4149(84)90166-2.  Google Scholar

[24]

L. Horváth, Strong approximation of extended renewal processes, The Annals of Probability, 12 (1984), 1149-1166.  doi: 10.1214/aop/1176993145.  Google Scholar

[25]

L. Horváth, Strong approximations of open queueing networks, Mathematics of Operations Research, 17 (1992), 487-508.  doi: 10.1287/moor.17.2.487.  Google Scholar

[26]

G. L. Iglehart, Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm, Z.Wahrscheinlichkeitstheorie verw. Geb., 17 (1971), 168-180.  doi: 10.1007/BF00538869.  Google Scholar

[27]

P. Lévy, Théorie de L'addition des Variables Aléatories, Gauthier-Villars, Paris, 1937. Google Scholar

[28]

P. Lévy, Procesus Stochastique et Mouvement Brownien, Gauthier-Villars, Paris, 1948. Google Scholar

[29]

E. Löcherbach and D. Loukianova, The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes, Stochastic Processes and their Applications, 119 (2009), 2312-2335.  doi: 10.1016/j.spa.2008.11.006.  Google Scholar

[30]

A. Mandelbaum and W. A. Massey, Strong approximations for time-dependent queues, Mathematics of Operations Research, 20 (1995), 33-64.  doi: 10.1287/moor.20.1.33.  Google Scholar

[31]

A. MandelbaumW. A. Massey and M. Reiman, Strong approximations for Markovian service networks, Queueing Systems, 30 (1998), 149-201.  doi: 10.1023/A:1019112920622.  Google Scholar

[32]

S. Minkevi$\check{c}$ius and S. Stei$\check{s}\bar{u}$nas, A law of the iterated logarithm for global values of waiting time in multiphase queues, Statistics and Probability Letters, 61 (2003), 359-371.  doi: 10.1016/S0167-7152(02)00393-0.  Google Scholar

[33]

S. Minkevi$\check{c}$ius, On the law of the iterated logarithm in multiserver open queueing networks, Stochastics, 86 (2014), 46-59.  doi: 10.1080/17442508.2012.755625.  Google Scholar

[34]

S. Minkevi$\check{c}$iusV. Dolgopolovas and L. L. Sakalauskas, A law of the iterated logarithm for the sojourn time process in queues in series, Methodology and Computing in Applied Probability, 18 (2016), 37-57.  doi: 10.1007/s11009-014-9402-y.  Google Scholar

[35]

K. Miyabea and A. Takemura, The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges, Stochastic Processes and their Applications, 123 (2013), 3132-3152.  doi: 10.1016/j.spa.2013.03.018.  Google Scholar

[36]

W. P. Peterson, A heavy traffic limit theorem for networks of queues with multiple customer types, Mathematics of Operations Research, 16 (1991), 90-118.  doi: 10.1287/moor.16.1.90.  Google Scholar

[37]

L. L. Sakalauskas and S. Minkevi$\check{c}$ius, On the law of the iterated logarithm in open queueing networks, European Journal of Operational Research, 120 (2000), 632-640.  doi: 10.1016/S0377-2217(99)00003-X.  Google Scholar

[38]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie Verw. Geb., 3 (1964), 211-226.  doi: 10.1007/BF00534910.  Google Scholar

[39]

T. H. Tsai, Empirical law of the iterated logarithm for Markov chains with a countable state space, Stochastic Processes and their Applications, 89 (2000), 175-191.  doi: 10.1016/S0304-4149(00)00019-3.  Google Scholar

[40]

Y. Wang, The law of the iterated logarithm for p-random sequences. In: Proc. 11th IEEE Conference on Computational Complexity (CCC), (1996), 180-189. Google Scholar

[41]

W. Whitt, Weak convergence theorems for priority queues: Preemptive-Resume discipline, Journal of Applied Probability, 8 (1971), 74-94.  doi: 10.2307/3211839.  Google Scholar

[42]

H. Q. Zhang and G. X. Hsu, Strong approximations for priority queues: Head-of-the-line-first discipline, Queueing Systems, 10 (1992), 213-233.  doi: 10.1007/BF01159207.  Google Scholar

[43]

H. Q. ZhangG. X. Hsu and R. X. Wang, Strong approximations for multiple channels in heavy traffic, Journal of Applied Probability, 27 (1990), 658-670.  doi: 10.2307/3214549.  Google Scholar

[44]

H. Q. Zhang, Strong approximations of irreducible closed queueing networks, Advances in Applied Probability, 29 (1997), 498-522.  doi: 10.2307/1428014.  Google Scholar

show all references

References:
[1]

M. Bramson and J. G. Dai, Heavy traffic limits for some queueing networks, Annals of Applied Probability, 11 (2001), 49-90.  doi: 10.1214/aoap/998926987.  Google Scholar

[2]

L. Caramellino, Strassen's law of the iterated logarithm for diffusion processes for small time, Stochastic Processes and their Applications, 74 (1998), 1-19.  doi: 10.1016/S0304-4149(97)00100-2.  Google Scholar

[3]

H. Chen and A. Mandelbaum, Hierarchical modeling of stochastic network, part Ⅱ: Strong approximations, Stochastic Modeling and Analysis of Manufacturing Systems, Yao, D. D. (Eds), (1994), 107-131. Google Scholar

[4]

H. Chen and X. Shen, Strong approximations for multiclass feedforward queueing networks, Annals of Applied Probability, 10 (2000), 828-876.  doi: 10.1214/aoap/1019487511.  Google Scholar

[5]

H. Chen and D. D. Yao, Fundamentals of Queueing Networks, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4757-5301-1.  Google Scholar

[6]

H. Chen and H. Q. Zhang, A sufficient condition and a necessary condition for the diffusion approximations of multiclass queueing networks under priority service disciplines, Queueing Systems, 34 (2000), 237-268.  doi: 10.1023/A:1019113204634.  Google Scholar

[7]

M. Csörgő ang P. Révész, Strong Approximations in Probability and Statistics, Academic Press, New York, 1981. Google Scholar

[8]

M. CsörgőP. Deheuvels and L. Horváth, An approximation of stopped sums with applications in queueing theory, Advances in Applied Probability, 19 (1987), 674-690.  doi: 10.2307/1427412.  Google Scholar

[9]

M. CsörgőZ. S. Hu and H. W. Mei, Strassen-type law of the iterated logarithm for self-normalized increments of sums, Journal of Mathematical Analysis and Applications, 393 (2012), 45-55.  doi: 10.1016/j.jmaa.2012.03.047.  Google Scholar

[10]

C. CunyF. Merlevéde and M. Peligrad, Law of the iterated logarithm for the periodogram, Stochastic Processes and their Applications, 123 (2013), 4065-4089.  doi: 10.1016/j.spa.2013.05.009.  Google Scholar

[11]

J. G. Dai, On the positive Harris recurrence for multiclass queueing networks: a unified approach via fluid limit models, Annals of Applied Probability, 5 (1995), 49-77.  doi: 10.1214/aoap/1177004828.  Google Scholar

[12]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence, Wiley, New York, 1986. doi: 10.1002/9780470316658.  Google Scholar

[13]

P. W. Glynn and W. Whitt, A new view of the heavy-traffic limit for infinite-server queues, Advances in Applied Probability, 23 (1991), 188-209.  doi: 10.2307/1427517.  Google Scholar

[14]

P. W. Glynn and W. Whitt, Departures from many queues in series, Annals of Applied Probability, 1 (1991), 546-572.  doi: 10.1214/aoap/1177005838.  Google Scholar

[15]

P. W. Glynn and W. Whitt, A central-limit-theorem version of L=λW, Queueing Systems, 1 (1986), 191-215.  doi: 10.1007/BF01536188.  Google Scholar

[16]

P. W. Glynn and W. Whitt, Sufficient conditions for functional limit theorem versions of L=λW, Queueing Systems, 1 (1987), 279-287.  doi: 10.1007/BF01149539.  Google Scholar

[17]

P. W. Glynn and W. Whitt, An LIL version of L=λW, Mathematics of Operations Research, 13 (1988), 693-710.  doi: 10.1287/moor.13.4.693.  Google Scholar

[18]

Y. Guo and Z. Li, Asymptotic variability analysis for a two-stage tandem queue, part Ⅰ: The functional law of the iterated logarithm, J. Math. Anal. Appl., 450 (2017), 1479-1509.  doi: 10.1016/j.jmaa.2017.01.062.  Google Scholar

[19]

Y. Guo and Z. Li, Asymptotic variability analysis for a two-stage tandem queue, part Ⅱ: The law of the iterated logarithm, J. Math. Anal. Appl., 450 (2017), 1510-1534.  doi: 10.1016/j.jmaa.2016.10.054.  Google Scholar

[20]

Y. Guo and Y. Liu, A law of iterated logarithm for multiclass queues with preemptive priority service discipline, Queueing Systems, 79 (2015), 251-291.  doi: 10.1007/s11134-014-9419-5.  Google Scholar

[21]

Y. GuoY. Liu and R. Pei, Functional law of iterated logarithm for multi-server queues with batch arrivals and customer feedback, Annals of Operations Research, 264 (2018), 157-191.  doi: 10.1007/s10479-017-2529-9.  Google Scholar

[22]

J. M. Harrison, Brownian Motion and Stochastic Flow System, Wiley, New York, 1985.  Google Scholar

[23]

L. Horváth, Strong approximation of renewal processes, Stochastic Process. Appl., 18 (1984), 127-138.  doi: 10.1016/0304-4149(84)90166-2.  Google Scholar

[24]

L. Horváth, Strong approximation of extended renewal processes, The Annals of Probability, 12 (1984), 1149-1166.  doi: 10.1214/aop/1176993145.  Google Scholar

[25]

L. Horváth, Strong approximations of open queueing networks, Mathematics of Operations Research, 17 (1992), 487-508.  doi: 10.1287/moor.17.2.487.  Google Scholar

[26]

G. L. Iglehart, Multiple channel queues in heavy traffic: IV. Law of the iterated logarithm, Z.Wahrscheinlichkeitstheorie verw. Geb., 17 (1971), 168-180.  doi: 10.1007/BF00538869.  Google Scholar

[27]

P. Lévy, Théorie de L'addition des Variables Aléatories, Gauthier-Villars, Paris, 1937. Google Scholar

[28]

P. Lévy, Procesus Stochastique et Mouvement Brownien, Gauthier-Villars, Paris, 1948. Google Scholar

[29]

E. Löcherbach and D. Loukianova, The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes, Stochastic Processes and their Applications, 119 (2009), 2312-2335.  doi: 10.1016/j.spa.2008.11.006.  Google Scholar

[30]

A. Mandelbaum and W. A. Massey, Strong approximations for time-dependent queues, Mathematics of Operations Research, 20 (1995), 33-64.  doi: 10.1287/moor.20.1.33.  Google Scholar

[31]

A. MandelbaumW. A. Massey and M. Reiman, Strong approximations for Markovian service networks, Queueing Systems, 30 (1998), 149-201.  doi: 10.1023/A:1019112920622.  Google Scholar

[32]

S. Minkevi$\check{c}$ius and S. Stei$\check{s}\bar{u}$nas, A law of the iterated logarithm for global values of waiting time in multiphase queues, Statistics and Probability Letters, 61 (2003), 359-371.  doi: 10.1016/S0167-7152(02)00393-0.  Google Scholar

[33]

S. Minkevi$\check{c}$ius, On the law of the iterated logarithm in multiserver open queueing networks, Stochastics, 86 (2014), 46-59.  doi: 10.1080/17442508.2012.755625.  Google Scholar

[34]

S. Minkevi$\check{c}$iusV. Dolgopolovas and L. L. Sakalauskas, A law of the iterated logarithm for the sojourn time process in queues in series, Methodology and Computing in Applied Probability, 18 (2016), 37-57.  doi: 10.1007/s11009-014-9402-y.  Google Scholar

[35]

K. Miyabea and A. Takemura, The law of the iterated logarithm in game-theoretic probability with quadratic and stronger hedges, Stochastic Processes and their Applications, 123 (2013), 3132-3152.  doi: 10.1016/j.spa.2013.03.018.  Google Scholar

[36]

W. P. Peterson, A heavy traffic limit theorem for networks of queues with multiple customer types, Mathematics of Operations Research, 16 (1991), 90-118.  doi: 10.1287/moor.16.1.90.  Google Scholar

[37]

L. L. Sakalauskas and S. Minkevi$\check{c}$ius, On the law of the iterated logarithm in open queueing networks, European Journal of Operational Research, 120 (2000), 632-640.  doi: 10.1016/S0377-2217(99)00003-X.  Google Scholar

[38]

V. Strassen, An invariance principle for the law of the iterated logarithm, Z. Wahrscheinlichkeitstheorie Verw. Geb., 3 (1964), 211-226.  doi: 10.1007/BF00534910.  Google Scholar

[39]

T. H. Tsai, Empirical law of the iterated logarithm for Markov chains with a countable state space, Stochastic Processes and their Applications, 89 (2000), 175-191.  doi: 10.1016/S0304-4149(00)00019-3.  Google Scholar

[40]

Y. Wang, The law of the iterated logarithm for p-random sequences. In: Proc. 11th IEEE Conference on Computational Complexity (CCC), (1996), 180-189. Google Scholar

[41]

W. Whitt, Weak convergence theorems for priority queues: Preemptive-Resume discipline, Journal of Applied Probability, 8 (1971), 74-94.  doi: 10.2307/3211839.  Google Scholar

[42]

H. Q. Zhang and G. X. Hsu, Strong approximations for priority queues: Head-of-the-line-first discipline, Queueing Systems, 10 (1992), 213-233.  doi: 10.1007/BF01159207.  Google Scholar

[43]

H. Q. ZhangG. X. Hsu and R. X. Wang, Strong approximations for multiple channels in heavy traffic, Journal of Applied Probability, 27 (1990), 658-670.  doi: 10.2307/3214549.  Google Scholar

[44]

H. Q. Zhang, Strong approximations of irreducible closed queueing networks, Advances in Applied Probability, 29 (1997), 498-522.  doi: 10.2307/1428014.  Google Scholar

Figure 1.  The LIL limits in Example 3
Table 1.  The LIL and FLIL limits for (Ⅰ) in Example 1
$k$ 1 2 3 4 5 6
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3}$ $\sqrt{3.9}$ $\sqrt{4.8}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $0$ $-\sqrt{3.9}$ $-\sqrt{4.8}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (\sqrt{3}))$ $\mathcal{G} (\sqrt{3.9})$ $\mathcal{G} (\sqrt{4.8})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $10\sqrt{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $0$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (10\sqrt{3}))$
$k$ 1 2 3 4 5 6
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3}$ $\sqrt{3.9}$ $\sqrt{4.8}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $0$ $-\sqrt{3.9}$ $-\sqrt{4.8}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (\sqrt{3}))$ $\mathcal{G} (\sqrt{3.9})$ $\mathcal{G} (\sqrt{4.8})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $10\sqrt{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $0$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (10\sqrt{3}))$
Table 2.  The LIL and FLIL limits for (Ⅱ) in Example 1
$k$ 1 2 3 4 5 6
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3.6}$ $\sqrt{4.5}$ $\sqrt{5.4}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $-\sqrt{3.6}$ $-\sqrt{4.5}$ $-\sqrt{5.4}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (3.6)$ $\mathcal{G} (4.5)$ $\mathcal{G} (5.4)$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $20\sqrt{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $-20\sqrt{3}$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (20\sqrt{3})$
$k$ 1 2 3 4 5 6
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $0$ $\sqrt{3.6}$ $\sqrt{4.5}$ $\sqrt{5.4}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $-\sqrt{3.6}$ $-\sqrt{4.5}$ $-\sqrt{5.4}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (3.6)$ $\mathcal{G} (4.5)$ $\mathcal{G} (5.4)$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $0$ $20\sqrt{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$ $-20\sqrt{3}$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\{0\}$ $\mathcal{G} (20\sqrt{3})$
Table 3.  The LIL and FLIL limits for (Ⅱ) in Example 2
$k$ 1 2 3 4 5
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $C_{3}$ $C_{4}$ $C_{5}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $-C_{4}$ $-C_{5}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (C_{3}))$ $\mathcal{G} (C_{4})$ $\mathcal{G} (C_{5})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $5C_{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (5C_{3}))$
$k$ 1 2 3 4 5
$Z^*_k=Z^*_{sup, k}$ $0$ $0$ $C_{3}$ $C_{4}$ $C_{5}$
$Z^*_{inf, k}$ $0$ $0$ $0$ $-C_{4}$ $-C_{5}$
$\mathcal{K}_{Z_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (C_{3}))$ $\mathcal{G} (C_{4})$ $\mathcal{G} (C_{5})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ $0$ $0$ $5C_{3}$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $0$
$\mathcal{K}_{\mathcal{S}_{k}}$ $\{0\}$ $\{0\}$ $\Phi(\mathcal{G} (5C_{3}))$
Table 4.  The LIL and FLIL limits for (Ⅲ) in Example 2
$k$ 1 2 3 4 5
$Z^*_{k}=Z^*_{sup, k}$ $0$ $0$ $D_{3}$ $D_{4}$ $D_{5}$
$Z^*_{inf, k}$ $0$ $0$ $-D_{3}$ $-D_{4}$ $-D_{5}$
$\mathcal{K}_{Z_{k}}$ $0$ $0$ $\mathcal{G}(D_{3})$ $\mathcal{G}(D_{4})$ $\mathcal{G}(D_{5})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ 0 0 $D$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $-D$
$\mathcal{K}_{\mathcal{S}_{k}}$ $0$ $0$ $\mathcal{G}(D)$
$k$ 1 2 3 4 5
$Z^*_{k}=Z^*_{sup, k}$ $0$ $0$ $D_{3}$ $D_{4}$ $D_{5}$
$Z^*_{inf, k}$ $0$ $0$ $-D_{3}$ $-D_{4}$ $-D_{5}$
$\mathcal{K}_{Z_{k}}$ $0$ $0$ $\mathcal{G}(D_{3})$ $\mathcal{G}(D_{4})$ $\mathcal{G}(D_{5})$
$\mathcal{S}^*_{k}=\mathcal{S}^*_{sup, k}$ 0 0 $D$
$\mathcal{S}^*_{inf, k}$ $0$ $0$ $-D$
$\mathcal{K}_{\mathcal{S}_{k}}$ $0$ $0$ $\mathcal{G}(D)$
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