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March  2021, 17(2): 549-573. doi: 10.3934/jimo.2019123

Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization

Sujit Kumar Samanta , and  Rakesh Nandi

Department of Mathematics, National Institute of Technology Raipur, Raipur-492010, India

* Corresponding author: Sujit Kumar Samanta

Received  September 2018 Revised  May 2019 Published  March 2021 Early access  October 2019

Fund Project: The first author acknowledges the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for partial support from the project grant 25(0271)/17/EMR-Ⅱ.

This paper analyzes an infinite-buffer single-server queueing system wherein customers arrive in batches of random size according to a discrete-time renewal process. The customers are served one at a time under discrete-time Markovian service process. Based on the censoring technique, the UL-type $ RG $-factorization for the Toeplitz type block-structured Markov chain is used to obtain the prearrival epoch probabilities. The random epoch probabilities are obtained with the help of classical principle based on Markov renewal theory. The system-length distributions at outside observer's, intermediate and post-departure epochs are obtained by making relations among various time epochs. The analysis of waiting-time distribution measured in slots of an arbitrary customer in an arrival batch has also been investigated. In order to unify the results of both discrete-time and its continuous-time counterpart, we give a brief demonstration to get the continuous-time results from those of the discrete-time ones. A variety of numerical results are provided to illustrate the effect of model parameters on the performance measures.

Keywords: Toeplitz type block-structured Markov chain, censored Markov chain, discrete-time Markovian service process (D-MSP), general independent batch arrival, queueing, UL-type $ RG $-factorization.
Mathematics Subject Classification: Primary: 60K25, 90B22, 68M20, 60K20.
Citation: Sujit Kumar Samanta, Rakesh Nandi. Analysis of $GI^{[X]}/D$-$MSP/1/\infty$ queue using $RG$-factorization. Journal of Industrial & Management Optimization, 2021, 17 (2) : 549-573. doi: 10.3934/jimo.2019123
References:
[1]

J. Abate, G. L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Comm. Statist. Stochastic Models, 10 (1994), 99-143.  doi: 10.1080/15326349408807290.  Google Scholar

[2]

A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer-Verlag, New York, 2016. doi: 10.1007/978-1-4939-3420-1.  Google Scholar

[3]

A. S. Alfa, J. Xue and Q. Ye, Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process, Queueing Syst., 36 (2000), 287-301.  doi: 10.1023/A:1011032718715.  Google Scholar

[4]

J. R. Artalejo, I. Atencia and P. Moreno, A discrete-time $Geo^{[X]}/G/1$ retrial queue with control of admission, Applied Mathematical Modelling, 29 (2005), 1100-1120.  doi: 10.1016/j.apm.2005.02.005.  Google Scholar

[5]

J. R. Artalejo and Q. L. Li, Performance analysis of a block-structured discrete-time retrial queue with state-dependent arrivals, Discrete Event Dyn. Syst., 20 (2010), 325-347.  doi: 10.1007/s10626-009-0075-6.  Google Scholar

[6]

F. Avram and D. F. Chedom, On symbolic $RG$-factorization of quasi-birth-and-death processes, TOP, 19 (2011), 317-335.  doi: 10.1007/s11750-011-0195-7.  Google Scholar

[7]

A. D. Banik and U. C. Gupta, Analyzing the finite buffer batch arrival queue under Markovian service process: $GI^{X}/MSP/1/N$, TOP, 15 (2007), 146-160.  doi: 10.1007/s11750-007-0007-2.  Google Scholar

[8]

P. P. Bocharov, C. D'Apice and S. Salerno, The stationary characteristics of the $G/MSP/1/r$ queueing system, Autom. Remote Control, 64 (2003), 288-301.  doi: 10.1023/A:1022219232282.  Google Scholar

[9]

H. Bruneel and B. G. Kim, Discrete-time Models for Communication Systems including ATM, The Springer International Series in Engineering and Computer Science, 205, Kluwer Academic Publishers, Boston, 1993. doi: 10.1007/978-1-4615-3130-2.  Google Scholar

[10]

M. L. Chaudhry, A. D. Banik and A. Pacheco, A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $GI^{[X]}/C$-$MSP/1/\infty $, Ann. Oper. Res., 252 (2017), 135-173.  doi: 10.1007/s10479-015-2026-y.  Google Scholar

[11]

M. L. Chaudhry, S. K. Samanta and A. Pacheco, Analytically explicit results for the $GI/C$-$MSP/1/\infty$ queueing system using roots, Probab. Engrg. Inform. Sci., 26 (2012), 221-244.  doi: 10.1017/S0269964811000349.  Google Scholar

[12]

E. Çinlar, Introduction to Stochastic Process, Prentice Hall, New Jersey, 1975.  Google Scholar

[13]

D. Freedman, Approximating Countable Markov Chains, 2$^{nd}$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8230-0.  Google Scholar

[14]

Y. Gao and W. Liu, Analysis of the $GI/Geo/c$ queue with working vacations, Applied Mechanics and Materials, 197 (2012), 534-541.  doi: 10.4028/www.scientific.net/AMM.197.534.  Google Scholar

[15]

V. Goswami, U. C. Gupta and S. K. Samanta, Analyzing discrete-time bulk-service $Geo/Geo^b/m$ queue, RAIRO Operations Research, 40 (2006), 267-284.  doi: 10.1051/ro:2006021.  Google Scholar

[16]

W. K. Grassmann and D. P. Heyman, Equilibrium distribution of block-structured Markov chains with repeating rows, J. Appl. Probab., 27 (1990), 557-576.  doi: 10.2307/3214541.  Google Scholar

[17]

U. C. Gupta and A. D. Banik, Complete analysis of finite and infinite buffer $GI/MSP/1$ queue — A computational approach, Oper. Res. Lett., 35 (2007), 273-280.  doi: 10.1016/j.orl.2006.02.003.  Google Scholar

[18]

A. Horváth, G. Horváth and M. Telek, A joint moments based analysis of networks of $MAP/MAP/1$ queues, Performance Evaluation, 67 (2010), 759-778.  doi: 10.1016/j.peva.2009.12.006.  Google Scholar

[19]

J. J. Hunter, Mathematical techniques of applied probability, in Discrete-Time Models: Techniques and Applications, Operations Research and Industrial Engineering, Academic Press, New York, 1983.  Google Scholar

[20]

T. Jiang and L. Liu, Analysis of a batch service multi-server polling system with dynamic service control, J. Ind. Manag. Optim., 14 (2018), 743-757.  doi: 10.3934/jimo.2017073.  Google Scholar

[21]

N. K. Kim, S. H. Chang and K. C. Chae, On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals, Oper. Res. Lett., 30 (2002), 25-32.  doi: 10.1016/S0167-6377(01)00110-9.  Google Scholar

[22]

Q. Li, Y. Ying and Y. Q. Zhao, A $BMAP/G/1$ retrial queue with a server subject to breakdowns and repairs, Ann. Oper. Res., 141 (2006), 233-270.  doi: 10.1007/s10479-006-5301-0.  Google Scholar

[23]

Q. L. Li, Constructive Computation in Stochastic Models with Applications: The RGfactorization, Springer, Berlin and Tsinghua University Press, Beijing, 2010. doi: 10.1007/978-3-642-11492-2.  Google Scholar

[24]

Q. L. Li and Y. Q. Zhao, Light-tailed asymptotics of stationary probability vectors of Markov chains of $GI/G/1$ type, Adv. in Appl. Probab., 37 (2005), 1075-1093.  doi: 10.1017/S0001867800000677.  Google Scholar

[25]

Q. L. Li and Y. Q. Zhao, A $MAP/G/1$ queue with negative customers, Queueing Syst., 47(1) (2004), 5–43. doi: 10.1023/B:QUES.0000032798.65858.19.  Google Scholar

[26]

D. M. Lucantoni and M. F. Neuts, Some steady-state distributions for the $MAP/SM/1$ queue, Comm. Statist. Stochastic Models, 10 (1994), 575-598.  doi: 10.1080/15326349408807311.  Google Scholar

[27]

C. D. Meyer, Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM Review, 31 (1989), 240-272.  doi: 10.1137/1031050.  Google Scholar

[28]

M. S. Mushtaq, S. Fowler and A. Mellouk, QoE in 5G cloud networks using multimedia services, in Proceeding of IEEE international Wireless Communication and Networking Conference (WCNC'16), Doha, Qatar, 2016. doi: 10.1109/WCNC.2016.7565173.  Google Scholar

[29]

T. Ozawa, Analysis of queues with Markovian service processes, Stochastic Models, 20 (2004), 391-413.  doi: 10.1081/STM-200033073.  Google Scholar

[30]

A. Pacheco, S. K. Samanta and M. L. Chaudhry, A short note on the $GI/Geo/1$ queueing system, Statist. Probab. Lett., 82 (2012), 268-273.  doi: 10.1016/j.spl.2011.09.022.  Google Scholar

[31]

S. K. Samanta, Sojourn-time distribution of the $GI/MSP/1$ queueing system, OPSEARCH, 52 (2015), 756-770.  doi: 10.1007/s12597-015-0202-0.  Google Scholar

[32]

S. K. Samanta, M. L. Chaudhry and A. Pacheco, Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.  doi: 10.1007/s11009-014-9429-0.  Google Scholar

[33]

S. K. Samanta, M. L. Chaudhry, A. Pacheco and U. C. Gupta, Analytic and computational analysis of the discrete-time $GI/D$-$MSP/1$ queue using roots, Comput. Oper. Res., 56 (2015), 33-40.  doi: 10.1016/j.cor.2014.10.017.  Google Scholar

[34]

S. K. Samanta, U. C. Gupta and M. L. Chaudhry, Analysis of stationary discrete-time $GI/D$-$MSP/1$ queue with finite and infinite buffers, 4OR, 7 (2009), 337-361.  doi: 10.1007/s10288-008-0088-2.  Google Scholar

[35]

S. K. Samanta and Z. G. Zhang, Stationary analysis of a discrete-time $GI/D$-$MSP/1$ queue with multiple vacations, Appl. Math. Model., 36 (2012), 5964-5975.  doi: 10.1016/j.apm.2012.01.049.  Google Scholar

[36]

K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and and S. Wittevrongel, Efficient performance analysis of newly proposed sleep-mode mechanisms for IEEE 802.16m in case of correlated downlink traffic, Wireless Networks, 19 (2013), 831-842.  doi: 10.1007/s11276-012-0504-6.  Google Scholar

[37]

Y. C. Wang, J. H. Chou and S. Y. Wang, Loss pattern of $DBMAP/DMSP/1/K$ queue and its application in wireless local communications, Appl. Math. Model., 35 (2011), 1782-1797.  doi: 10.1016/j.apm.2010.10.009.  Google Scholar

[38]

Y. Wang, C. Linb and Q. L. Li, Performance analysis of email systems under three types of attacks, Performance Evaluation, 67 (2010), 485-499.  doi: 10.1016/j.peva.2010.01.003.  Google Scholar

[39]

M. Yu and A. S. Alfa, Algorithm for computing the queue length distribution at various time epochs in $DMAP/G^{(1, a, b)}/1/N$ queue with batch-size-dependent service time, European J. Oper. Res., 244 (2015), 227-239.  doi: 10.1016/j.ejor.2015.01.056.  Google Scholar

[40]

M. Zhang and Z. Hou, Performance analysis of $MAP/G/1$ queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560.  doi: 10.1016/j.apm.2010.09.031.  Google Scholar

[41]

J. A. Zhao, B. Li, C. W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146.  doi: 10.1023/B:WINE.0000013078.74259.13.  Google Scholar

[42]

Y. Q. Zhao, Censoring technique in studying block-structured Markov chains, in Advance in Algorithmic Methods for Stochastic Models, Notable Publications, 2000, 417–433. Google Scholar

[43]

Y. Q. Zhao, W. Li and W. J. Braun, Infinite block-structured transition matrices and their properties, Adv. in Appl. Probab., 30 (1998), 365-384.  doi: 10.1239/aap/1035228074.  Google Scholar

[44]

Y. Q. Zhao and D. Liu, The censored Markov chain and the best augmentation, Journal of Applied Probability, 33 (1996), 623-629.  doi: 10.1017/S0021900200100063.  Google Scholar

show all references

References:
[1]

J. Abate, G. L. Choudhury and W. Whitt, Asymptotics for steady-state tail probabilities in structured Markov queueing models, Comm. Statist. Stochastic Models, 10 (1994), 99-143.  doi: 10.1080/15326349408807290.  Google Scholar

[2]

A. S. Alfa, Applied Discrete-Time Queues, 2$^{nd}$ edition, Springer-Verlag, New York, 2016. doi: 10.1007/978-1-4939-3420-1.  Google Scholar

[3]

A. S. Alfa, J. Xue and Q. Ye, Perturbation theory for the asymptotic decay rates in the queues with Markovian arrival process and/or Markovian service process, Queueing Syst., 36 (2000), 287-301.  doi: 10.1023/A:1011032718715.  Google Scholar

[4]

J. R. Artalejo, I. Atencia and P. Moreno, A discrete-time $Geo^{[X]}/G/1$ retrial queue with control of admission, Applied Mathematical Modelling, 29 (2005), 1100-1120.  doi: 10.1016/j.apm.2005.02.005.  Google Scholar

[5]

J. R. Artalejo and Q. L. Li, Performance analysis of a block-structured discrete-time retrial queue with state-dependent arrivals, Discrete Event Dyn. Syst., 20 (2010), 325-347.  doi: 10.1007/s10626-009-0075-6.  Google Scholar

[6]

F. Avram and D. F. Chedom, On symbolic $RG$-factorization of quasi-birth-and-death processes, TOP, 19 (2011), 317-335.  doi: 10.1007/s11750-011-0195-7.  Google Scholar

[7]

A. D. Banik and U. C. Gupta, Analyzing the finite buffer batch arrival queue under Markovian service process: $GI^{X}/MSP/1/N$, TOP, 15 (2007), 146-160.  doi: 10.1007/s11750-007-0007-2.  Google Scholar

[8]

P. P. Bocharov, C. D'Apice and S. Salerno, The stationary characteristics of the $G/MSP/1/r$ queueing system, Autom. Remote Control, 64 (2003), 288-301.  doi: 10.1023/A:1022219232282.  Google Scholar

[9]

H. Bruneel and B. G. Kim, Discrete-time Models for Communication Systems including ATM, The Springer International Series in Engineering and Computer Science, 205, Kluwer Academic Publishers, Boston, 1993. doi: 10.1007/978-1-4615-3130-2.  Google Scholar

[10]

M. L. Chaudhry, A. D. Banik and A. Pacheco, A simple analysis of the batch arrival queue with infinite-buffer and Markovian service process using roots method: $GI^{[X]}/C$-$MSP/1/\infty $, Ann. Oper. Res., 252 (2017), 135-173.  doi: 10.1007/s10479-015-2026-y.  Google Scholar

[11]

M. L. Chaudhry, S. K. Samanta and A. Pacheco, Analytically explicit results for the $GI/C$-$MSP/1/\infty$ queueing system using roots, Probab. Engrg. Inform. Sci., 26 (2012), 221-244.  doi: 10.1017/S0269964811000349.  Google Scholar

[12]

E. Çinlar, Introduction to Stochastic Process, Prentice Hall, New Jersey, 1975.  Google Scholar

[13]

D. Freedman, Approximating Countable Markov Chains, 2$^{nd}$ edition, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4613-8230-0.  Google Scholar

[14]

Y. Gao and W. Liu, Analysis of the $GI/Geo/c$ queue with working vacations, Applied Mechanics and Materials, 197 (2012), 534-541.  doi: 10.4028/www.scientific.net/AMM.197.534.  Google Scholar

[15]

V. Goswami, U. C. Gupta and S. K. Samanta, Analyzing discrete-time bulk-service $Geo/Geo^b/m$ queue, RAIRO Operations Research, 40 (2006), 267-284.  doi: 10.1051/ro:2006021.  Google Scholar

[16]

W. K. Grassmann and D. P. Heyman, Equilibrium distribution of block-structured Markov chains with repeating rows, J. Appl. Probab., 27 (1990), 557-576.  doi: 10.2307/3214541.  Google Scholar

[17]

U. C. Gupta and A. D. Banik, Complete analysis of finite and infinite buffer $GI/MSP/1$ queue — A computational approach, Oper. Res. Lett., 35 (2007), 273-280.  doi: 10.1016/j.orl.2006.02.003.  Google Scholar

[18]

A. Horváth, G. Horváth and M. Telek, A joint moments based analysis of networks of $MAP/MAP/1$ queues, Performance Evaluation, 67 (2010), 759-778.  doi: 10.1016/j.peva.2009.12.006.  Google Scholar

[19]

J. J. Hunter, Mathematical techniques of applied probability, in Discrete-Time Models: Techniques and Applications, Operations Research and Industrial Engineering, Academic Press, New York, 1983.  Google Scholar

[20]

T. Jiang and L. Liu, Analysis of a batch service multi-server polling system with dynamic service control, J. Ind. Manag. Optim., 14 (2018), 743-757.  doi: 10.3934/jimo.2017073.  Google Scholar

[21]

N. K. Kim, S. H. Chang and K. C. Chae, On the relationships among queue lengths at arrival, departure, and random epochs in the discrete-time queue with D-BMAP arrivals, Oper. Res. Lett., 30 (2002), 25-32.  doi: 10.1016/S0167-6377(01)00110-9.  Google Scholar

[22]

Q. Li, Y. Ying and Y. Q. Zhao, A $BMAP/G/1$ retrial queue with a server subject to breakdowns and repairs, Ann. Oper. Res., 141 (2006), 233-270.  doi: 10.1007/s10479-006-5301-0.  Google Scholar

[23]

Q. L. Li, Constructive Computation in Stochastic Models with Applications: The RGfactorization, Springer, Berlin and Tsinghua University Press, Beijing, 2010. doi: 10.1007/978-3-642-11492-2.  Google Scholar

[24]

Q. L. Li and Y. Q. Zhao, Light-tailed asymptotics of stationary probability vectors of Markov chains of $GI/G/1$ type, Adv. in Appl. Probab., 37 (2005), 1075-1093.  doi: 10.1017/S0001867800000677.  Google Scholar

[25]

Q. L. Li and Y. Q. Zhao, A $MAP/G/1$ queue with negative customers, Queueing Syst., 47(1) (2004), 5–43. doi: 10.1023/B:QUES.0000032798.65858.19.  Google Scholar

[26]

D. M. Lucantoni and M. F. Neuts, Some steady-state distributions for the $MAP/SM/1$ queue, Comm. Statist. Stochastic Models, 10 (1994), 575-598.  doi: 10.1080/15326349408807311.  Google Scholar

[27]

C. D. Meyer, Stochastic complementation, uncoupling Markov chains, and the theory of nearly reducible systems, SIAM Review, 31 (1989), 240-272.  doi: 10.1137/1031050.  Google Scholar

[28]

M. S. Mushtaq, S. Fowler and A. Mellouk, QoE in 5G cloud networks using multimedia services, in Proceeding of IEEE international Wireless Communication and Networking Conference (WCNC'16), Doha, Qatar, 2016. doi: 10.1109/WCNC.2016.7565173.  Google Scholar

[29]

T. Ozawa, Analysis of queues with Markovian service processes, Stochastic Models, 20 (2004), 391-413.  doi: 10.1081/STM-200033073.  Google Scholar

[30]

A. Pacheco, S. K. Samanta and M. L. Chaudhry, A short note on the $GI/Geo/1$ queueing system, Statist. Probab. Lett., 82 (2012), 268-273.  doi: 10.1016/j.spl.2011.09.022.  Google Scholar

[31]

S. K. Samanta, Sojourn-time distribution of the $GI/MSP/1$ queueing system, OPSEARCH, 52 (2015), 756-770.  doi: 10.1007/s12597-015-0202-0.  Google Scholar

[32]

S. K. Samanta, M. L. Chaudhry and A. Pacheco, Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.  doi: 10.1007/s11009-014-9429-0.  Google Scholar

[33]

S. K. Samanta, M. L. Chaudhry, A. Pacheco and U. C. Gupta, Analytic and computational analysis of the discrete-time $GI/D$-$MSP/1$ queue using roots, Comput. Oper. Res., 56 (2015), 33-40.  doi: 10.1016/j.cor.2014.10.017.  Google Scholar

[34]

S. K. Samanta, U. C. Gupta and M. L. Chaudhry, Analysis of stationary discrete-time $GI/D$-$MSP/1$ queue with finite and infinite buffers, 4OR, 7 (2009), 337-361.  doi: 10.1007/s10288-008-0088-2.  Google Scholar

[35]

S. K. Samanta and Z. G. Zhang, Stationary analysis of a discrete-time $GI/D$-$MSP/1$ queue with multiple vacations, Appl. Math. Model., 36 (2012), 5964-5975.  doi: 10.1016/j.apm.2012.01.049.  Google Scholar

[36]

K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel and and S. Wittevrongel, Efficient performance analysis of newly proposed sleep-mode mechanisms for IEEE 802.16m in case of correlated downlink traffic, Wireless Networks, 19 (2013), 831-842.  doi: 10.1007/s11276-012-0504-6.  Google Scholar

[37]

Y. C. Wang, J. H. Chou and S. Y. Wang, Loss pattern of $DBMAP/DMSP/1/K$ queue and its application in wireless local communications, Appl. Math. Model., 35 (2011), 1782-1797.  doi: 10.1016/j.apm.2010.10.009.  Google Scholar

[38]

Y. Wang, C. Linb and Q. L. Li, Performance analysis of email systems under three types of attacks, Performance Evaluation, 67 (2010), 485-499.  doi: 10.1016/j.peva.2010.01.003.  Google Scholar

[39]

M. Yu and A. S. Alfa, Algorithm for computing the queue length distribution at various time epochs in $DMAP/G^{(1, a, b)}/1/N$ queue with batch-size-dependent service time, European J. Oper. Res., 244 (2015), 227-239.  doi: 10.1016/j.ejor.2015.01.056.  Google Scholar

[40]

M. Zhang and Z. Hou, Performance analysis of $MAP/G/1$ queue with working vacations and vacation interruption, Applied Mathematical Modelling, 35 (2011), 1551-1560.  doi: 10.1016/j.apm.2010.09.031.  Google Scholar

[41]

J. A. Zhao, B. Li, C. W. Kok and I. Ahmad, MPEG-4 video transmission over wireless networks: A link level performance study, Wireless Networks, 10 (2004), 133-146.  doi: 10.1023/B:WINE.0000013078.74259.13.  Google Scholar

[42]

Y. Q. Zhao, Censoring technique in studying block-structured Markov chains, in Advance in Algorithmic Methods for Stochastic Models, Notable Publications, 2000, 417–433. Google Scholar

[43]

Y. Q. Zhao, W. Li and W. J. Braun, Infinite block-structured transition matrices and their properties, Adv. in Appl. Probab., 30 (1998), 365-384.  doi: 10.1239/aap/1035228074.  Google Scholar

[44]

Y. Q. Zhao and D. Liu, The censored Markov chain and the best augmentation, Journal of Applied Probability, 33 (1996), 623-629.  doi: 10.1017/S0021900200100063.  Google Scholar

Figure 1.  Various time epochs in LAS-DA
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Figure 2.  Various time epochs in EAS
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Table 1.  System-length distribution at prearrival epoch (LAS-DA)
$ n $ $ \pi^{-}_1(n) $ $ \pi^{-}_2(n) $ $ \pi^{-}_3(n) $ $ \pi^{-}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{-}(n){\bf e} $
0 0.147931 0.087562 0.141983 0.215337 0.592813
1 0.017322 0.008928 0.017209 0.019057 0.062516
2 0.008854 0.004021 0.009012 0.007208 0.029094
3 0.007634 0.003498 0.007749 0.006361 0.025243
4 0.007169 0.003300 0.007241 0.006043 0.023753
5 0.012534 0.006493 0.012352 0.013954 0.045334
10 0.005820 0.002639 0.005923 0.004720 0.019101
30 0.000272 0.000123 0.000275 0.000218 0.000887
60 0.000002 0.000001 0.000002 0.000002 0.000007
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.268062 0.144522 0.263029 0.324388 1.000000
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$ n $ $ \pi^{-}_1(n) $ $ \pi^{-}_2(n) $ $ \pi^{-}_3(n) $ $ \pi^{-}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{-}(n){\bf e} $
0 0.147931 0.087562 0.141983 0.215337 0.592813
1 0.017322 0.008928 0.017209 0.019057 0.062516
2 0.008854 0.004021 0.009012 0.007208 0.029094
3 0.007634 0.003498 0.007749 0.006361 0.025243
4 0.007169 0.003300 0.007241 0.006043 0.023753
5 0.012534 0.006493 0.012352 0.013954 0.045334
10 0.005820 0.002639 0.005923 0.004720 0.019101
30 0.000272 0.000123 0.000275 0.000218 0.000887
60 0.000002 0.000001 0.000002 0.000002 0.000007
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.268062 0.144522 0.263029 0.324388 1.000000
Table 2.  System-length distribution at random epoch (LAS-DA)
$ n $ $ \pi_1(n) $ $ \pi_2(n) $ $ \pi_3(n) $ $ \pi_4(n) $ $ \mathit{\boldsymbol{\pi }}(n){\bf e} $
0 0.111295 0.065475 0.106245 0.160319 0.443334
1 0.022916 0.010687 0.022938 0.019981 0.076522
2 0.013339 0.006155 0.013392 0.011322 0.044209
3 0.013437 0.006222 0.013490 0.011506 0.044656
4 0.013841 0.006409 0.013895 0.011853 0.045999
5 0.014556 0.006778 0.014600 0.012641 0.048575
10 0.006965 0.003204 0.007013 0.005865 0.023048
30 0.000402 0.000181 0.000410 0.000321 0.001314
60 0.000004 0.000002 0.000004 0.000003 0.000012
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.279037 0.142937 0.275076 0.302949 1.000000
$ L_{q}= 4.110096 $, $ W_{q}\equiv L_{q}/\lambda\overline{g}=16.988397 $
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$ n $ $ \pi_1(n) $ $ \pi_2(n) $ $ \pi_3(n) $ $ \pi_4(n) $ $ \mathit{\boldsymbol{\pi }}(n){\bf e} $
0 0.111295 0.065475 0.106245 0.160319 0.443334
1 0.022916 0.010687 0.022938 0.019981 0.076522
2 0.013339 0.006155 0.013392 0.011322 0.044209
3 0.013437 0.006222 0.013490 0.011506 0.044656
4 0.013841 0.006409 0.013895 0.011853 0.045999
5 0.014556 0.006778 0.014600 0.012641 0.048575
10 0.006965 0.003204 0.007013 0.005865 0.023048
30 0.000402 0.000181 0.000410 0.000321 0.001314
60 0.000004 0.000002 0.000004 0.000003 0.000012
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.279037 0.142937 0.275076 0.302949 1.000000
$ L_{q}= 4.110096 $, $ W_{q}\equiv L_{q}/\lambda\overline{g}=16.988397 $
Table 3.  System-length distribution at intermediate epoch (LAS-DA)
$ n $ $ \pi^{\bullet}_1(n) $ $ \pi^{\bullet}_2(n) $ $ \pi^{\bullet}_3(n) $ $ \pi^{\bullet}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{\bullet}(n){\bf e} $
0 0.103342 0.060768 0.098612 0.148741 0.411462
1 0.025961 0.012561 0.025829 0.024745 0.089096
2 0.013329 0.006179 0.013370 0.011447 0.044325
3 0.013265 0.006142 0.013316 0.011358 0.044081
4 0.013661 0.006326 0.013714 0.011699 0.045400
5 0.016859 0.008165 0.016802 0.016105 0.057931
10 0.007064 0.003265 0.007104 0.006021 0.023455
30 0.000442 0.000202 0.000450 0.000366 0.001460
60 0.000004 0.000002 0.000004 0.000003 0.000013
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.279037 0.142937 0.275076 0.302949 1.000000
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$ n $ $ \pi^{\bullet}_1(n) $ $ \pi^{\bullet}_2(n) $ $ \pi^{\bullet}_3(n) $ $ \pi^{\bullet}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{\bullet}(n){\bf e} $
0 0.103342 0.060768 0.098612 0.148741 0.411462
1 0.025961 0.012561 0.025829 0.024745 0.089096
2 0.013329 0.006179 0.013370 0.011447 0.044325
3 0.013265 0.006142 0.013316 0.011358 0.044081
4 0.013661 0.006326 0.013714 0.011699 0.045400
5 0.016859 0.008165 0.016802 0.016105 0.057931
10 0.007064 0.003265 0.007104 0.006021 0.023455
30 0.000442 0.000202 0.000450 0.000366 0.001460
60 0.000004 0.000002 0.000004 0.000003 0.000013
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.279037 0.142937 0.275076 0.302949 1.000000
Table 4.  System-length distribution at post-departure epoch (LAS-DA)
$ n $ $ \pi^{+}_1(n) $ $ \pi^{+}_2(n) $ $ \pi^{+}_3(n) $ $ \pi^{+}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{+}(n){\bf e} $
0 0.032873 0.019458 0.031552 0.047853 0.131736
1 0.019883 0.011780 0.019101 0.028996 0.079761
2 0.019751 0.011709 0.018985 0.028835 0.079280
3 0.020343 0.012060 0.019554 0.029699 0.081656
4 0.020959 0.012425 0.020146 0.030599 0.084130
5 0.011311 0.006712 0.010884 0.016550 0.045457
10 0.010060 0.005977 0.009689 0.014744 0.040469
30 0.000495 0.000297 0.000481 0.000737 0.002009
60 0.000005 0.000003 0.000005 0.000007 0.000020
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
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$ n $ $ \pi^{+}_1(n) $ $ \pi^{+}_2(n) $ $ \pi^{+}_3(n) $ $ \pi^{+}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{+}(n){\bf e} $
0 0.032873 0.019458 0.031552 0.047853 0.131736
1 0.019883 0.011780 0.019101 0.028996 0.079761
2 0.019751 0.011709 0.018985 0.028835 0.079280
3 0.020343 0.012060 0.019554 0.029699 0.081656
4 0.020959 0.012425 0.020146 0.030599 0.084130
5 0.011311 0.006712 0.010884 0.016550 0.045457
10 0.010060 0.005977 0.009689 0.014744 0.040469
30 0.000495 0.000297 0.000481 0.000737 0.002009
60 0.000005 0.000003 0.000005 0.000007 0.000020
150 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
Table 5.  Waiting-time distribution of an arbitrary customer (LAS-DA)
$ k $ $ w_1(k) $ $ w_2(k) $ $ w_3(k) $ $ w_4(k) $ $ {\bf w}(k){\bf e} $
0 0.034508 0.020385 0.033070 0.050094 0.138057
1 0.009993 0.005467 0.009024 0.012961 0.037445
2 0.008841 0.005188 0.008367 0.012412 0.034807
3 0.008718 0.005172 0.008340 0.012581 0.034811
4 0.008830 0.005239 0.008473 0.012846 0.035388
5 0.008691 0.005166 0.008366 0.012724 0.034948
10 0.006267 0.003757 0.006088 0.009341 0.025452
30 0.003144 0.001879 0.003045 0.004660 0.012727
60 0.000421 0.000252 0.000408 0.000625 0.001705
100 0.000033 0.000020 0.000032 0.000049 0.000133
190 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=16.988398 $
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$ k $ $ w_1(k) $ $ w_2(k) $ $ w_3(k) $ $ w_4(k) $ $ {\bf w}(k){\bf e} $
0 0.034508 0.020385 0.033070 0.050094 0.138057
1 0.009993 0.005467 0.009024 0.012961 0.037445
2 0.008841 0.005188 0.008367 0.012412 0.034807
3 0.008718 0.005172 0.008340 0.012581 0.034811
4 0.008830 0.005239 0.008473 0.012846 0.035388
5 0.008691 0.005166 0.008366 0.012724 0.034948
10 0.006267 0.003757 0.006088 0.009341 0.025452
30 0.003144 0.001879 0.003045 0.004660 0.012727
60 0.000421 0.000252 0.000408 0.000625 0.001705
100 0.000033 0.000020 0.000032 0.000049 0.000133
190 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=16.988398 $
Table 6.  System-length distribution at prearrival epoch (EAS)
$ n $ $ \pi^{-}_1(n) $ $ \pi^{-}_2(n) $ $ \pi^{-}_3(n) $ $ \pi^{-}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{-}(n){\bf e} $
0 0.177488 0.104369 0.169315 0.255259 0.706432
1 0.031834 0.014078 0.032731 0.024080 0.102724
2 0.020658 0.009143 0.021320 0.015664 0.066785
3 0.013410 0.005939 0.013880 0.010188 0.043417
4 0.008708 0.003859 0.009033 0.006625 0.028224
5 0.005655 0.002507 0.005877 0.004308 0.018347
10 0.000655 0.000290 0.000683 0.000500 0.002129
15 0.000076 0.000034 0.000079 0.000058 0.000247
20 0.000009 0.000004 0.000009 0.000007 0.000029
25 0.000001 0.000000 0.000001 0.000001 0.000003
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.268240 0.144548 0.263085 0.324127 1.000000
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$ n $ $ \pi^{-}_1(n) $ $ \pi^{-}_2(n) $ $ \pi^{-}_3(n) $ $ \pi^{-}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{-}(n){\bf e} $
0 0.177488 0.104369 0.169315 0.255259 0.706432
1 0.031834 0.014078 0.032731 0.024080 0.102724
2 0.020658 0.009143 0.021320 0.015664 0.066785
3 0.013410 0.005939 0.013880 0.010188 0.043417
4 0.008708 0.003859 0.009033 0.006625 0.028224
5 0.005655 0.002507 0.005877 0.004308 0.018347
10 0.000655 0.000290 0.000683 0.000500 0.002129
15 0.000076 0.000034 0.000079 0.000058 0.000247
20 0.000009 0.000004 0.000009 0.000007 0.000029
25 0.000001 0.000000 0.000001 0.000001 0.000003
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.268240 0.144548 0.263085 0.324127 1.000000
Table 7.  System-length distribution at random epoch (EAS)
$ n $ $ \pi_1(n) $ $ \pi_2(n) $ $ \pi_3(n) $ $ \pi_4(n) $ $ \mathit{\boldsymbol{\pi }}(n){\bf e} $
0 0.133257 0.077530 0.125951 0.188221 0.524958
1 0.051790 0.022946 0.051871 0.039576 0.166183
2 0.033578 0.014882 0.033952 0.025651 0.108062
3 0.021782 0.009656 0.022187 0.016637 0.070262
4 0.014135 0.006268 0.014481 0.010796 0.045680
5 0.009176 0.004069 0.009443 0.007009 0.029697
10 0.001061 0.000471 0.001104 0.000811 0.003446
15 0.000123 0.000055 0.000128 0.000094 0.000400
20 0.000014 0.000006 0.000015 0.000011 0.000046
25 0.000002 0.000001 0.000002 0.000001 0.000005
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.280718 0.142893 0.275506 0.300882 1.000000
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$ n $ $ \pi_1(n) $ $ \pi_2(n) $ $ \pi_3(n) $ $ \pi_4(n) $ $ \mathit{\boldsymbol{\pi }}(n){\bf e} $
0 0.133257 0.077530 0.125951 0.188221 0.524958
1 0.051790 0.022946 0.051871 0.039576 0.166183
2 0.033578 0.014882 0.033952 0.025651 0.108062
3 0.021782 0.009656 0.022187 0.016637 0.070262
4 0.014135 0.006268 0.014481 0.010796 0.045680
5 0.009176 0.004069 0.009443 0.007009 0.029697
10 0.001061 0.000471 0.001104 0.000811 0.003446
15 0.000123 0.000055 0.000128 0.000094 0.000400
20 0.000014 0.000006 0.000015 0.000011 0.000046
25 0.000002 0.000001 0.000002 0.000001 0.000005
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.280718 0.142893 0.275506 0.300882 1.000000
Table 8.  System-length distribution at outside observer's epoch (EAS)
$ n $ $ \pi^{\circ}_1(n) $ $ \pi^{\circ}_2(n) $ $ \pi^{\circ}_3(n) $ $ \pi^{\circ}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{\circ}(n){\bf e} $
0 0.111612 0.064802 0.105303 0.157092 0.438808
1 0.058730 0.027593 0.058203 0.052204 0.196731
2 0.038411 0.017807 0.038510 0.032991 0.127719
3 0.025082 0.011509 0.025373 0.020975 0.082940
4 0.016359 0.007448 0.016665 0.013400 0.053872
5 0.010660 0.004824 0.010920 0.008593 0.034997
10 0.001243 0.000555 0.001292 0.000964 0.004055
15 0.000144 0.000064 0.000151 0.000111 0.000470
20 0.000017 0.000007 0.000017 0.000013 0.000054
25 0.000002 0.000001 0.000002 0.000001 0.000006
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.280718 0.142893 0.275506 0.300882 1.000000
$ L_{q}=1.040384 $, $ W_{q}\equiv L_{q}/\lambda\overline{g}=4.265574 $
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$ n $ $ \pi^{\circ}_1(n) $ $ \pi^{\circ}_2(n) $ $ \pi^{\circ}_3(n) $ $ \pi^{\circ}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{\circ}(n){\bf e} $
0 0.111612 0.064802 0.105303 0.157092 0.438808
1 0.058730 0.027593 0.058203 0.052204 0.196731
2 0.038411 0.017807 0.038510 0.032991 0.127719
3 0.025082 0.011509 0.025373 0.020975 0.082940
4 0.016359 0.007448 0.016665 0.013400 0.053872
5 0.010660 0.004824 0.010920 0.008593 0.034997
10 0.001243 0.000555 0.001292 0.000964 0.004055
15 0.000144 0.000064 0.000151 0.000111 0.000470
20 0.000017 0.000007 0.000017 0.000013 0.000054
25 0.000002 0.000001 0.000002 0.000001 0.000006
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.280718 0.142893 0.275506 0.300882 1.000000
$ L_{q}=1.040384 $, $ W_{q}\equiv L_{q}/\lambda\overline{g}=4.265574 $
Table 9.  System-length distribution at post-departure epoch (EAS)
$ n $ $ \pi^{+}_1(n) $ $ \pi^{+}_2(n) $ $ \pi^{+}_3(n) $ $ \pi^{+}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{+}(n){\bf e} $
0 0.088744 0.052185 0.084658 0.127630 0.353216
1 0.056838 0.033672 0.054596 0.082865 0.227970
2 0.036522 0.021762 0.035273 0.053821 0.147378
3 0.023528 0.014083 0.022821 0.034965 0.095397
4 0.015187 0.009123 0.014781 0.022720 0.061811
5 0.009819 0.005914 0.009581 0.014765 0.040079
10 0.001124 0.000682 0.001104 0.001712 0.004622
15 0.000130 0.000079 0.000128 0.000199 0.000535
20 0.000015 0.000009 0.000015 0.000023 0.000062
25 0.000002 0.000001 0.000002 0.000002 0.000007
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
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$ n $ $ \pi^{+}_1(n) $ $ \pi^{+}_2(n) $ $ \pi^{+}_3(n) $ $ \pi^{+}_4(n) $ $ \mathit{\boldsymbol{\pi }}^{+}(n){\bf e} $
0 0.088744 0.052185 0.084658 0.127630 0.353216
1 0.056838 0.033672 0.054596 0.082865 0.227970
2 0.036522 0.021762 0.035273 0.053821 0.147378
3 0.023528 0.014083 0.022821 0.034965 0.095397
4 0.015187 0.009123 0.014781 0.022720 0.061811
5 0.009819 0.005914 0.009581 0.014765 0.040079
10 0.001124 0.000682 0.001104 0.001712 0.004622
15 0.000130 0.000079 0.000128 0.000199 0.000535
20 0.000015 0.000009 0.000015 0.000023 0.000062
25 0.000002 0.000001 0.000002 0.000002 0.000007
50 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
Table 10.  Waiting-time distribution of an arbitrary customer (EAS)
$ k $ $ w_1(k) $ $ w_2(k) $ $ w_3(k) $ $ w_4(k) $ $ {\bf w}(k){\bf e} $
0 0.088744 0.052185 0.084658 0.127630 0.353216
1 0.028093 0.015539 0.025592 0.037079 0.106303
2 0.020283 0.012108 0.019471 0.029323 0.081185
3 0.016689 0.010130 0.016294 0.025037 0.068151
4 0.014155 0.008612 0.013897 0.021520 0.058184
5 0.012056 0.007334 0.011861 0.018415 0.049667
10 0.005375 0.003268 0.005294 0.008231 0.022168
20 0.001064 0.000647 0.001048 0.001629 0.004388
30 0.000211 0.000128 0.000207 0.000323 0.000869
50 0.000008 0.000005 0.000008 0.000013 0.000034
80 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=4.265574 $
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$ k $ $ w_1(k) $ $ w_2(k) $ $ w_3(k) $ $ w_4(k) $ $ {\bf w}(k){\bf e} $
0 0.088744 0.052185 0.084658 0.127630 0.353216
1 0.028093 0.015539 0.025592 0.037079 0.106303
2 0.020283 0.012108 0.019471 0.029323 0.081185
3 0.016689 0.010130 0.016294 0.025037 0.068151
4 0.014155 0.008612 0.013897 0.021520 0.058184
5 0.012056 0.007334 0.011861 0.018415 0.049667
10 0.005375 0.003268 0.005294 0.008231 0.022168
20 0.001064 0.000647 0.001048 0.001629 0.004388
30 0.000211 0.000128 0.000207 0.000323 0.000869
50 0.000008 0.000005 0.000008 0.000013 0.000034
80 0.000000 0.000000 0.000000 0.000000 0.000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
Sum 0.248718 0.147678 0.239427 0.364177 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=4.265574 $
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