American Institute of Mathematical Sciences

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doi: 10.3934/jimo.2020135

Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization

Rakesh Nandi 1, , Sujit Kumar Samanta 1,, and  Chesoong Kim 2,

1. 

Department of Mathematics, National Institute of Technology Raipur, Chhattisgarh-492010, India
2. 

Department of Business Administration, Sangji University, Wonju, Kangwon-26339, Republic of Korea

* Corresponding author: Sujit Kumar Samanta

Received  January 2020 Revised  May 2020 Published  August 2020

Fund Project: The third author acknowledges the Sangji University for partial support from the Sangji University research fund 2018

This article studies an infinite buffer single server queueing system under $ N $-policy in which customers arrive according to a discrete-time batch Markovian arrival process. The service times of customers are independent and obey a common general discrete distribution. The idle server begins to serve the customers as soon as the queue size becomes at least $ N $ and serves the customers until the system becomes empty. We determine the queue length distribution at post-departure epoch with the help of roots of the associated characteristic equation of the vector probability generating function. Using the supplementary variable technique, we develop the system of vector difference equations to derive the queue length distribution at random epoch. An analytically simple and computationally efficient approach is also presented to compute the waiting time distribution in the queue of a randomly selected customer of an arrival batch. We also construct an expected linear cost function to determine the optimal value of $ N $ at minimum cost. Some numerical results are demonstrated for different service time distributions through the optimal control parameter to show the key performance measures.

Keywords: Discrete-time batch Markovian arrival process (D-BMAP), N-policy, queueing, roots, waiting time distribution.
Mathematics Subject Classification: Primary: 60K25, 90B22, 68M20, 60K20.
Citation: Rakesh Nandi, Sujit Kumar Samanta, Chesoong Kim. Analysis of $ D $-$ BMAP/G/1 $ queueing system under $ N $-policy and its cost optimization. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020135
References:
[1]

H. K. Aksoy and S. M. Gupta, Near optimal buffer allocation in remanufacturing systems with $N$-policy, Computers & Industrial Engineering, 59 (2010), 496-508.  doi: 10.1016/j.cie.2010.06.004.  Google Scholar

[2]

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

[3]

C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science, 32 (1993), 3-23.   Google Scholar

[4]

J. A. Buzacott and J. G. Shanthikumar, Stochastic models for production control, In Optimization Models and Concepts in Production Management, Gordon and Breach Publishers, 1995,213–255. Google Scholar

[5]

S. R. Chakravarthy, The batch Markovian arrival process: a review and future work, In Advances in Probability Theory and Stochastic Processes, Notable Publications, Inc., New Jersey, USA, 2001, 21–49. Google Scholar

[6]

K. H. Choi and B. K. Yoon, A roots method in $GI/PH/1$ queueing model and its application, Industrial Engineering & Management Systems, 12 (2013), 281-287.  doi: 10.7232/IEMS.2013.12.3.281.  Google Scholar

[7]

G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar

[8]

H. R. Gail, S. L. Hantler and B. A. Taylor, Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains, Advances in Applied Probability, 28 (1996), 114-165.  doi: 10.2307/1427915.  Google Scholar

[9]

T. Hofkens, K. Spaey and C. Blondia, Transient analysis of the $D$-$BMAP/G/1$ queue with an application to the dimensioning of a playout buffer for VBR video, in Networking 2004, Springer, Berlin, Heidelberg, 2004, 1338–1343. doi: 10.1007/978-3-540-24693-0_116.  Google Scholar

[10]

A. J. E. M. Janssen and J. S. H. van Leeuwaarden, Analytic computation schemes for the discrete-time bulk service queue, Queueing Systems, 50 (2005), 141-163.  doi: 10.1007/s11134-005-0402-z.  Google Scholar

[11]

F. C. Jiang, D. C. Huang, C. T. Yang and F. Y. Leu, Lifetime elongation for wireless sensor network using queue-based approaches, The Journal of Supercomputing, 59 (2012), 1312-1335.  doi: 10.1007/s11227-010-0537-5.  Google Scholar

[12]

S. Kasahara, T. Takine, Y. Takahashi and T. Hasegawa, $MAP/G/1$ queues under $N$-policy with and without vacations, Journal of the Operations Research Society of Japan, 39 (1996), 188-212.  doi: 10.15807/jorsj.39.188.  Google Scholar

[13]

A. Kavusturucu and S. M. Gupta, Expansion method for the throughput analysis of open finite manufacturing/queueing networks with $N$-Policy, Computers & Operations Research, 26 (1999), 1267-1292.  doi: 10.1016/S0305-0548(98)00107-5.  Google Scholar

[14]

R. G. V. Krishna Reddy, R. Nadarajan and R. Arumuganathan, Analysis of a bulk queue with $N$-policy multiple vacations and setup times, Computers & Operations Research, 25 (1998), 957-967.  doi: 10.1016/S0305-0548(97)00098-1.  Google Scholar

[15]

A. Krishnamoorthy and T. G. Deepak, Modified $N$-policy for $M/G/1$ queues, Computers & Operations Research, 29 (2002), 1611-1620.  doi: 10.1016/S0305-0548(00)00108-8.  Google Scholar

[16]

S. Lan and Y. Tang, An $N$-policy discrete-time $Geo/G/1$ queue with modified multiple server vacations and Bernoulli feedback, RAIRO-Operations Research, 53 (2019), 367-387.  doi: 10.1051/ro/2017027.  Google Scholar

[17]

H. W. Lee, B. Y. Ahn and N. I. Park, Decompositions of the queue length distributions in the $MAP/G/1$ queue under multiple and single vacations with $N$-Policy, Stochastic Models, 17 (2001), 157-190.  doi: 10.1081/STM-100002062.  Google Scholar

[18]

H. W. Lee, S. W. Lee and J. Jongwoo, Using factorization in analyzing $D$-$BMAP/G/1$ queues, Journal of Applied Mathematics and Stochastic Analysis, 2005 (2005), 119-132.  doi: 10.1155/JAMSA.2005.119.  Google Scholar

[19]

S. S. Lee, H. W. Lee, S. H. Yoon and K. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189.   Google Scholar

[20]

H. W. Lee and W. J. Seo, The performance of the $M/G/1$ queue under the dyadic Min($N$, $D$)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758.   Google Scholar

[21]

D. H. Lee and W. S. Yang, The $N$-policy of a discrete time $Geo/G/1$ queue with disasters and its application to wireless sensor networks, Applied Mathematical Modelling, 37 (2013), 9722-9731.  doi: 10.1016/j.apm.2013.05.012.  Google Scholar

[22]

D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Communications in Statistics Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar

[23]

M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar

[24] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981.   Google Scholar
[25]

A. Oblakova, A. Al Hanbali, R. J. Boucherie, J. C. W. V. Ommeren and W. H. M. Zijm, An exact root-free method for the expected queue length for a class of discrete-time queueing systems, Queueing Systems, 92 (2019), 257-292.  doi: 10.1007/s11134-019-09614-1.  Google Scholar

[26]

S. K. Samanta, Waiting-time analysis of $D$-$BMAP/G/1$ queueing system, Annals of Operations Research, 284 (2020), 401-413.  doi: 10.1007/s10479-015-1974-6.  Google Scholar

[27]

C. Sreenivasan, S. R. Chakravarthy and A. Krishnamoorthy, $MAP/PH/1$ queue with working vacations, vacation interruptions and $N$ policy, Applied Mathematical Modelling, 37 (2013), 3879-3893.  doi: 10.1016/j.apm.2012.07.054.  Google Scholar

[28]

K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel 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.   Google Scholar

[29]

T. Y. Wang, T. H. Liu and F. M. Chang, Analysis of a random $N$-policy $Geo/G/1$ queue with the server subject to repairable breakdowns, Journal of Industrial and Production Engineering, 34 (2017), 19-29.   Google Scholar

[30]

M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405.   Google Scholar

[31]

Y. Q. Zhao and L. L. Campbell, Equilibrium probability calculations for a discrete-time bulk queue model, Queueing Systems, 22 (1996), 189-198.  doi: 10.1007/BF01159401.  Google Scholar

[32]

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.   Google Scholar

show all references

References:
[1]

H. K. Aksoy and S. M. Gupta, Near optimal buffer allocation in remanufacturing systems with $N$-policy, Computers & Industrial Engineering, 59 (2010), 496-508.  doi: 10.1016/j.cie.2010.06.004.  Google Scholar

[2]

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

[3]

C. Blondia, A discrete-time batch Markovian arrival process as B-ISDN traffic model, Belgian Journal of Operations Research, Statistics and Computer Science, 32 (1993), 3-23.   Google Scholar

[4]

J. A. Buzacott and J. G. Shanthikumar, Stochastic models for production control, In Optimization Models and Concepts in Production Management, Gordon and Breach Publishers, 1995,213–255. Google Scholar

[5]

S. R. Chakravarthy, The batch Markovian arrival process: a review and future work, In Advances in Probability Theory and Stochastic Processes, Notable Publications, Inc., New Jersey, USA, 2001, 21–49. Google Scholar

[6]

K. H. Choi and B. K. Yoon, A roots method in $GI/PH/1$ queueing model and its application, Industrial Engineering & Management Systems, 12 (2013), 281-287.  doi: 10.7232/IEMS.2013.12.3.281.  Google Scholar

[7]

G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar

[8]

H. R. Gail, S. L. Hantler and B. A. Taylor, Spectral analysis of $M/G/1$ and $G/M/1$ type Markov chains, Advances in Applied Probability, 28 (1996), 114-165.  doi: 10.2307/1427915.  Google Scholar

[9]

T. Hofkens, K. Spaey and C. Blondia, Transient analysis of the $D$-$BMAP/G/1$ queue with an application to the dimensioning of a playout buffer for VBR video, in Networking 2004, Springer, Berlin, Heidelberg, 2004, 1338–1343. doi: 10.1007/978-3-540-24693-0_116.  Google Scholar

[10]

A. J. E. M. Janssen and J. S. H. van Leeuwaarden, Analytic computation schemes for the discrete-time bulk service queue, Queueing Systems, 50 (2005), 141-163.  doi: 10.1007/s11134-005-0402-z.  Google Scholar

[11]

F. C. Jiang, D. C. Huang, C. T. Yang and F. Y. Leu, Lifetime elongation for wireless sensor network using queue-based approaches, The Journal of Supercomputing, 59 (2012), 1312-1335.  doi: 10.1007/s11227-010-0537-5.  Google Scholar

[12]

S. Kasahara, T. Takine, Y. Takahashi and T. Hasegawa, $MAP/G/1$ queues under $N$-policy with and without vacations, Journal of the Operations Research Society of Japan, 39 (1996), 188-212.  doi: 10.15807/jorsj.39.188.  Google Scholar

[13]

A. Kavusturucu and S. M. Gupta, Expansion method for the throughput analysis of open finite manufacturing/queueing networks with $N$-Policy, Computers & Operations Research, 26 (1999), 1267-1292.  doi: 10.1016/S0305-0548(98)00107-5.  Google Scholar

[14]

R. G. V. Krishna Reddy, R. Nadarajan and R. Arumuganathan, Analysis of a bulk queue with $N$-policy multiple vacations and setup times, Computers & Operations Research, 25 (1998), 957-967.  doi: 10.1016/S0305-0548(97)00098-1.  Google Scholar

[15]

A. Krishnamoorthy and T. G. Deepak, Modified $N$-policy for $M/G/1$ queues, Computers & Operations Research, 29 (2002), 1611-1620.  doi: 10.1016/S0305-0548(00)00108-8.  Google Scholar

[16]

S. Lan and Y. Tang, An $N$-policy discrete-time $Geo/G/1$ queue with modified multiple server vacations and Bernoulli feedback, RAIRO-Operations Research, 53 (2019), 367-387.  doi: 10.1051/ro/2017027.  Google Scholar

[17]

H. W. Lee, B. Y. Ahn and N. I. Park, Decompositions of the queue length distributions in the $MAP/G/1$ queue under multiple and single vacations with $N$-Policy, Stochastic Models, 17 (2001), 157-190.  doi: 10.1081/STM-100002062.  Google Scholar

[18]

H. W. Lee, S. W. Lee and J. Jongwoo, Using factorization in analyzing $D$-$BMAP/G/1$ queues, Journal of Applied Mathematics and Stochastic Analysis, 2005 (2005), 119-132.  doi: 10.1155/JAMSA.2005.119.  Google Scholar

[19]

S. S. Lee, H. W. Lee, S. H. Yoon and K. Chae, Batch arrival queue with $N$-policy and single vacation, Computers & Operations Research, 22 (1995), 173-189.   Google Scholar

[20]

H. W. Lee and W. J. Seo, The performance of the $M/G/1$ queue under the dyadic Min($N$, $D$)-policy and its cost optimization, Performance Evaluation, 65 (2008), 742-758.   Google Scholar

[21]

D. H. Lee and W. S. Yang, The $N$-policy of a discrete time $Geo/G/1$ queue with disasters and its application to wireless sensor networks, Applied Mathematical Modelling, 37 (2013), 9722-9731.  doi: 10.1016/j.apm.2013.05.012.  Google Scholar

[22]

D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Communications in Statistics Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar

[23]

M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar

[24] M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981.   Google Scholar
[25]

A. Oblakova, A. Al Hanbali, R. J. Boucherie, J. C. W. V. Ommeren and W. H. M. Zijm, An exact root-free method for the expected queue length for a class of discrete-time queueing systems, Queueing Systems, 92 (2019), 257-292.  doi: 10.1007/s11134-019-09614-1.  Google Scholar

[26]

S. K. Samanta, Waiting-time analysis of $D$-$BMAP/G/1$ queueing system, Annals of Operations Research, 284 (2020), 401-413.  doi: 10.1007/s10479-015-1974-6.  Google Scholar

[27]

C. Sreenivasan, S. R. Chakravarthy and A. Krishnamoorthy, $MAP/PH/1$ queue with working vacations, vacation interruptions and $N$ policy, Applied Mathematical Modelling, 37 (2013), 3879-3893.  doi: 10.1016/j.apm.2012.07.054.  Google Scholar

[28]

K. D. Turck, S. D. Vuyst, D. Fiems, H. Bruneel 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.   Google Scholar

[29]

T. Y. Wang, T. H. Liu and F. M. Chang, Analysis of a random $N$-policy $Geo/G/1$ queue with the server subject to repairable breakdowns, Journal of Industrial and Production Engineering, 34 (2017), 19-29.   Google Scholar

[30]

M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405.   Google Scholar

[31]

Y. Q. Zhao and L. L. Campbell, Equilibrium probability calculations for a discrete-time bulk queue model, Queueing Systems, 22 (1996), 189-198.  doi: 10.1007/BF01159401.  Google Scholar

[32]

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.   Google Scholar

Figure 1.  Various time epochs in LAS-DA
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Figure 2.  Queue length distribution at random epoch in idle mode
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Figure 3.  Queue length distribution at random epoch in busy mode
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Figure 4.  Waiting time distribution
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Figure 5.  Mean queue length versus threshold value $ N $ for different coefficients of correlation in the D-BMAPs
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Figure 6.  Cost per unit time over different service pattern and $ N $-policy
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Table 1.  Queue length distribution at post-departure epoch
$ n $ $ \pi^{+}_{1}(n) $ $ \pi^{+}_{2}(n) $ $ \pi^{+}_{3}(n) $ $ \pi^{+}_{4}(n) $ $ \pi^{+}_{5}(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^{+}(n){\bf e} $
0 0.00114837 0.00255731 0.00122101 0.00057458 0.00030733 0.00580861
1 0.00196732 0.00413389 0.00211277 0.00132765 0.00067006 0.01021170
2 0.00268813 0.00542042 0.00287623 0.00214745 0.00100214 0.01413437
3 0.00336532 0.00676685 0.00362099 0.00302599 0.00135476 0.01813391
4 0.00407581 0.00806720 0.00442181 0.00388853 0.00172228 0.02217563
5 0.00480560 0.00931208 0.00519387 0.00478410 0.00209565 0.02619131
6 0.00545231 0.01038279 0.00590375 0.00577743 0.00247998 0.02999627
7 0.00550115 0.01037107 0.00599819 0.00616987 0.00258839 0.03062867
10 0.00518961 0.00969826 0.00573383 0.00601163 0.00253004 0.02916336
20 0.00355502 0.00663985 0.00392615 0.00413473 0.00174204 0.01999780
50 0.00113199 0.00211426 0.00125016 0.00131658 0.00055470 0.00636770
100 0.00016808 0.00031392 0.00018562 0.00019549 0.00008236 0.00094547
200 0.00000371 0.00000692 0.00000409 0.00000431 0.00000182 0.00002084
300 0.00000008 0.00000015 0.00000009 0.00000010 0.00000004 0.00000046
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17884285 0.33699076 0.19680812 0.20180982 0.08554844 1.00000000
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$ n $ $ \pi^{+}_{1}(n) $ $ \pi^{+}_{2}(n) $ $ \pi^{+}_{3}(n) $ $ \pi^{+}_{4}(n) $ $ \pi^{+}_{5}(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^{+}(n){\bf e} $
0 0.00114837 0.00255731 0.00122101 0.00057458 0.00030733 0.00580861
1 0.00196732 0.00413389 0.00211277 0.00132765 0.00067006 0.01021170
2 0.00268813 0.00542042 0.00287623 0.00214745 0.00100214 0.01413437
3 0.00336532 0.00676685 0.00362099 0.00302599 0.00135476 0.01813391
4 0.00407581 0.00806720 0.00442181 0.00388853 0.00172228 0.02217563
5 0.00480560 0.00931208 0.00519387 0.00478410 0.00209565 0.02619131
6 0.00545231 0.01038279 0.00590375 0.00577743 0.00247998 0.02999627
7 0.00550115 0.01037107 0.00599819 0.00616987 0.00258839 0.03062867
10 0.00518961 0.00969826 0.00573383 0.00601163 0.00253004 0.02916336
20 0.00355502 0.00663985 0.00392615 0.00413473 0.00174204 0.01999780
50 0.00113199 0.00211426 0.00125016 0.00131658 0.00055470 0.00636770
100 0.00016808 0.00031392 0.00018562 0.00019549 0.00008236 0.00094547
200 0.00000371 0.00000692 0.00000409 0.00000431 0.00000182 0.00002084
300 0.00000008 0.00000015 0.00000009 0.00000010 0.00000004 0.00000046
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17884285 0.33699076 0.19680812 0.20180982 0.08554844 1.00000000
Table 2.  Queue length distribution at random epoch
$ n $ $ \omega_{1}(n) $ $ \omega_{2}(n) $ $ \omega_{3}(n) $ $ \omega_{4}(n) $ $ \omega_{5}(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}(n){\bf e} $
0 0.00222695 0.00529393 0.00242011 0.00090775 0.00048977 0.01133850
1 0.00122073 0.00248229 0.00141283 0.00079891 0.00049111 0.00640588
2 0.00105468 0.00179357 0.00112890 0.00103029 0.00047010 0.00547755
3 0.00089757 0.00190533 0.00096004 0.00118332 0.00048134 0.00542760
4 0.00096295 0.00185514 0.00111938 0.00116282 0.00051063 0.00561091
5 0.00111788 0.00198211 0.00120152 0.00116430 0.00051719 0.00598300
6 0.00106199 0.00200132 0.00116720 0.00115114 0.00049567 0.00587732
$ \pi_{1}(n) $ $ \pi_{2}(n) $ $ \pi_{3}(n) $ $ \pi_{4}(n) $ $ \pi_{5}(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}(n){\bf e} $
0 0.00162452 0.00345778 0.00174092 0.00103506 0.00052768 0.00838598
1 0.00234181 0.00477333 0.00250811 0.00179593 0.00085359 0.01227277
2 0.00300104 0.00603923 0.00322414 0.00261562 0.00118356 0.01606358
3 0.00366851 0.00729361 0.00397075 0.00344304 0.00152948 0.01990539
4 0.00435868 0.00849806 0.00471593 0.00428708 0.00188379 0.02374353
5 0.00500101 0.00957287 0.00541216 0.00520452 0.00224699 0.02743755
10 0.00486647 0.00905994 0.00535865 0.00563201 0.00238020 0.02729727
20 0.00330525 0.00617334 0.00365030 0.00384423 0.00161965 0.01859277
50 0.00105246 0.00196571 0.00116233 0.00122408 0.00051573 0.00592030
100 0.00015627 0.00029187 0.00017258 0.00018175 0.00007658 0.00087904
200 0.00000345 0.00000643 0.00000380 0.00000401 0.00000169 0.00001938
300 0.00000008 0.00000014 0.00000008 0.00000009 0.00000003 0.00000042
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17895473 0.33796971 0.19708476 0.20065459 0.08533621 1.00000000
$ L_{q}=27.32669405 $, $ W_{q}\equiv L_{q}/\lambda^{\ast}=42.16910717 $
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$ n $ $ \omega_{1}(n) $ $ \omega_{2}(n) $ $ \omega_{3}(n) $ $ \omega_{4}(n) $ $ \omega_{5}(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}(n){\bf e} $
0 0.00222695 0.00529393 0.00242011 0.00090775 0.00048977 0.01133850
1 0.00122073 0.00248229 0.00141283 0.00079891 0.00049111 0.00640588
2 0.00105468 0.00179357 0.00112890 0.00103029 0.00047010 0.00547755
3 0.00089757 0.00190533 0.00096004 0.00118332 0.00048134 0.00542760
4 0.00096295 0.00185514 0.00111938 0.00116282 0.00051063 0.00561091
5 0.00111788 0.00198211 0.00120152 0.00116430 0.00051719 0.00598300
6 0.00106199 0.00200132 0.00116720 0.00115114 0.00049567 0.00587732
$ \pi_{1}(n) $ $ \pi_{2}(n) $ $ \pi_{3}(n) $ $ \pi_{4}(n) $ $ \pi_{5}(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}(n){\bf e} $
0 0.00162452 0.00345778 0.00174092 0.00103506 0.00052768 0.00838598
1 0.00234181 0.00477333 0.00250811 0.00179593 0.00085359 0.01227277
2 0.00300104 0.00603923 0.00322414 0.00261562 0.00118356 0.01606358
3 0.00366851 0.00729361 0.00397075 0.00344304 0.00152948 0.01990539
4 0.00435868 0.00849806 0.00471593 0.00428708 0.00188379 0.02374353
5 0.00500101 0.00957287 0.00541216 0.00520452 0.00224699 0.02743755
10 0.00486647 0.00905994 0.00535865 0.00563201 0.00238020 0.02729727
20 0.00330525 0.00617334 0.00365030 0.00384423 0.00161965 0.01859277
50 0.00105246 0.00196571 0.00116233 0.00122408 0.00051573 0.00592030
100 0.00015627 0.00029187 0.00017258 0.00018175 0.00007658 0.00087904
200 0.00000345 0.00000643 0.00000380 0.00000401 0.00000169 0.00001938
300 0.00000008 0.00000014 0.00000008 0.00000009 0.00000003 0.00000042
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17895473 0.33796971 0.19708476 0.20065459 0.08533621 1.00000000
$ L_{q}=27.32669405 $, $ W_{q}\equiv L_{q}/\lambda^{\ast}=42.16910717 $
Table 3.  Queue length distribution at prearrival epoch
$ n $ $ \omega_{1}^-(n) $ $ \omega_{2}^-(n) $ $ \omega_{3}^-(n) $ $ \omega_{4}^-(n) $ $ \omega_{5}^-(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}^-(n){\bf e} $
0 0.00174343 0.00277470 0.00221558 0.00262957 0.00132854 0.01069181
1 0.00097452 0.00160317 0.00122966 0.00163056 0.00074979 0.00618770
2 0.00083109 0.00141140 0.00103647 0.00156583 0.00062849 0.00547327
3 0.00547327 0.00133665 0.00103247 0.00165829 0.00060400 0.00546386
4 0.00085171 0.00139904 0.00105950 0.00167026 0.00063105 0.00561157
5 0.00090939 0.00152759 0.00113284 0.00173509 0.00068207 0.00598698
6 0.00089418 0.00148113 0.00111396 0.00171177 0.00066572 0.00586676
$ \pi_{1}^-(n) $ $ \pi_{2}^-(n) $ $ \pi_{3}^-(n) $ $ \pi_{4}^-(n) $ $ \pi_{5}^-(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^-(n){\bf e} $
0 0.00128465 0.00209323 0.00162002 0.00213136 0.00097544 0.00810471
1 0.00187638 0.00307986 0.00235698 0.00326222 0.00141681 0.01199226
2 0.00245421 0.00402943 0.00307416 0.00440903 0.00184105 0.01580788
3 0.00303820 0.00499321 0.00379932 0.00556828 0.00227196 0.01967097
4 0.00362127 0.00597046 0.00452247 0.00673302 0.00270420 0.02355141
5 0.00418155 0.00691160 0.00521393 0.00790838 0.00311484 0.02733030
10 0.00415110 0.00687971 0.00516200 0.00810034 0.00307831 0.02737146
20 0.00282716 0.00468171 0.00351552 0.00552180 0.00209539 0.01864157
50 0.00090022 0.00149075 0.00111941 0.00175825 0.00066721 0.00593585
100 0.00013366 0.00022135 0.00016621 0.00026106 0.00009907 0.00088135
200 0.00000295 0.00000488 0.00000366 0.00000576 0.00000218 0.00001943
300 0.00000006 0.00000010 0.00000008 0.00000012 0.00000005 0.00000042
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.15214569 0.25164805 0.18938631 0.29385469 0.11296526 1.00000000
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$ n $ $ \omega_{1}^-(n) $ $ \omega_{2}^-(n) $ $ \omega_{3}^-(n) $ $ \omega_{4}^-(n) $ $ \omega_{5}^-(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}^-(n){\bf e} $
0 0.00174343 0.00277470 0.00221558 0.00262957 0.00132854 0.01069181
1 0.00097452 0.00160317 0.00122966 0.00163056 0.00074979 0.00618770
2 0.00083109 0.00141140 0.00103647 0.00156583 0.00062849 0.00547327
3 0.00547327 0.00133665 0.00103247 0.00165829 0.00060400 0.00546386
4 0.00085171 0.00139904 0.00105950 0.00167026 0.00063105 0.00561157
5 0.00090939 0.00152759 0.00113284 0.00173509 0.00068207 0.00598698
6 0.00089418 0.00148113 0.00111396 0.00171177 0.00066572 0.00586676
$ \pi_{1}^-(n) $ $ \pi_{2}^-(n) $ $ \pi_{3}^-(n) $ $ \pi_{4}^-(n) $ $ \pi_{5}^-(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^-(n){\bf e} $
0 0.00128465 0.00209323 0.00162002 0.00213136 0.00097544 0.00810471
1 0.00187638 0.00307986 0.00235698 0.00326222 0.00141681 0.01199226
2 0.00245421 0.00402943 0.00307416 0.00440903 0.00184105 0.01580788
3 0.00303820 0.00499321 0.00379932 0.00556828 0.00227196 0.01967097
4 0.00362127 0.00597046 0.00452247 0.00673302 0.00270420 0.02355141
5 0.00418155 0.00691160 0.00521393 0.00790838 0.00311484 0.02733030
10 0.00415110 0.00687971 0.00516200 0.00810034 0.00307831 0.02737146
20 0.00282716 0.00468171 0.00351552 0.00552180 0.00209539 0.01864157
50 0.00090022 0.00149075 0.00111941 0.00175825 0.00066721 0.00593585
100 0.00013366 0.00022135 0.00016621 0.00026106 0.00009907 0.00088135
200 0.00000295 0.00000488 0.00000366 0.00000576 0.00000218 0.00001943
300 0.00000006 0.00000010 0.00000008 0.00000012 0.00000005 0.00000042
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.15214569 0.25164805 0.18938631 0.29385469 0.11296526 1.00000000
Table 4.  Queue length distribution at intermediate epoch
$ n $ $ \omega_{1}^\bullet(n) $ $ \omega_{2}^\bullet(n) $ $ \omega_{3}^\bullet(n) $ $ \omega_{4}^\bullet(n) $ $ \omega_{5}^\bullet(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}^\bullet(n){\bf e} $
0 0.00148277 0.00363672 0.00162886 0.00053541 0.00029061 0.00757437
1 0.00122073 0.00248229 0.00141283 0.00079891 0.00049111 0.00640588
2 0.00105468 0.00179357 0.00112890 0.00103029 0.00047010 0.00547755
3 0.00089757 0.00190533 0.00096004 0.00118332 0.00048134 0.00542760
4 0.00096295 0.00185514 0.00111938 0.00116282 0.00051063 0.00561091
5 0.00111788 0.00198211 0.00120152 0.00116430 0.00051719 0.00598300
6 0.00106199 0.00200132 0.00116720 0.00115114 0.00049567 0.00587732
$ \pi_{1}^\bullet(n) $ $ \pi_{2}^\bullet(n) $ $ \pi_{3}^\bullet(n) $ $ \pi_{4}^\bullet(n) $ $ \pi_{5}^\bullet(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^\bullet(n){\bf e} $
0 0.00109382 0.00243612 0.00116304 0.00054705 0.00029262 0.00553266
1 0.00187471 0.00393963 0.00201337 0.00126468 0.00063839 0.00973078
2 0.00256220 0.00516670 0.00274151 0.00204630 0.00095506 0.01347178
3 0.00320810 0.00645095 0.00345180 0.00288409 0.00129132 0.01728624
4 0.00388575 0.00769135 0.00421561 0.00370673 0.00164183 0.02114127
5 0.00458192 0.00887902 0.00495214 0.00456082 0.00199793 0.02497183
10 0.00498603 0.00928331 0.00549189 0.00575755 0.00243227 0.02795104
20 0.00339147 0.00633438 0.00374553 0.00394451 0.00166190 0.01907779
50 0.00107991 0.00201699 0.00119265 0.00125601 0.00052918 0.00607475
100 0.00016034 0.00029948 0.00017708 0.00018649 0.00007857 0.00090197
200 0.00000353 0.00000660 0.00000390 0.00000411 0.00000173 0.00001989
300 0.00000008 0.00000015 0.00000009 0.00000009 0.00000003 0.00000044
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17895473 0.33796971 0.19708476 0.20065459 0.08533621 1.00000000
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$ n $ $ \omega_{1}^\bullet(n) $ $ \omega_{2}^\bullet(n) $ $ \omega_{3}^\bullet(n) $ $ \omega_{4}^\bullet(n) $ $ \omega_{5}^\bullet(n) $ $ {\bf{\it\Large \pmb{\mathsf{ ω}}}}^\bullet(n){\bf e} $
0 0.00148277 0.00363672 0.00162886 0.00053541 0.00029061 0.00757437
1 0.00122073 0.00248229 0.00141283 0.00079891 0.00049111 0.00640588
2 0.00105468 0.00179357 0.00112890 0.00103029 0.00047010 0.00547755
3 0.00089757 0.00190533 0.00096004 0.00118332 0.00048134 0.00542760
4 0.00096295 0.00185514 0.00111938 0.00116282 0.00051063 0.00561091
5 0.00111788 0.00198211 0.00120152 0.00116430 0.00051719 0.00598300
6 0.00106199 0.00200132 0.00116720 0.00115114 0.00049567 0.00587732
$ \pi_{1}^\bullet(n) $ $ \pi_{2}^\bullet(n) $ $ \pi_{3}^\bullet(n) $ $ \pi_{4}^\bullet(n) $ $ \pi_{5}^\bullet(n) $ $ {\bf{ \pmb{\mathsf{ π}}}}^\bullet(n){\bf e} $
0 0.00109382 0.00243612 0.00116304 0.00054705 0.00029262 0.00553266
1 0.00187471 0.00393963 0.00201337 0.00126468 0.00063839 0.00973078
2 0.00256220 0.00516670 0.00274151 0.00204630 0.00095506 0.01347178
3 0.00320810 0.00645095 0.00345180 0.00288409 0.00129132 0.01728624
4 0.00388575 0.00769135 0.00421561 0.00370673 0.00164183 0.02114127
5 0.00458192 0.00887902 0.00495214 0.00456082 0.00199793 0.02497183
10 0.00498603 0.00928331 0.00549189 0.00575755 0.00243227 0.02795104
20 0.00339147 0.00633438 0.00374553 0.00394451 0.00166190 0.01907779
50 0.00107991 0.00201699 0.00119265 0.00125601 0.00052918 0.00607475
100 0.00016034 0.00029948 0.00017708 0.00018649 0.00007857 0.00090197
200 0.00000353 0.00000660 0.00000390 0.00000411 0.00000173 0.00001989
300 0.00000008 0.00000015 0.00000009 0.00000009 0.00000003 0.00000044
500 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.17895473 0.33796971 0.19708476 0.20065459 0.08533621 1.00000000
Table 5.  Waiting time distribution
$ k $ $ w_{1}(k) $ $ w_{2}(k) $ $ w_{3}(k) $ $ w_{4}(k) $ $ w_{5}(k) $ $ {\bf w}(k){\bf e} $
0 0.00047483 0.00077372 0.00059879 0.00078790 0.00036054 0.00299578
1 0.00076314 0.00119978 0.00097247 0.00140311 0.00061681 0.00495532
2 0.00110386 0.00172634 0.00140867 0.00210776 0.00091559 0.00726221
3 0.00147097 0.00227929 0.00190462 0.00286252 0.00123500 0.00975240
4 0.00185631 0.00287564 0.00242983 0.00368458 0.00157512 0.01242148
5 0.00226328 0.00349641 0.00297861 0.00455765 0.00193035 0.01522630
10 0.00342239 0.00526449 0.00449032 0.00694631 0.00290268 0.02302618
20 0.00253361 0.00388041 0.00331762 0.00511929 0.00214814 0.01699908
50 0.00115231 0.00176460 0.00150943 0.00233341 0.00097752 0.00773727
100 0.00031795 0.00048690 0.00041649 0.00064384 0.00026972 0.00213489
200 0.00002421 0.00003707 0.00003171 0.00004902 0.00002053 0.00016254
300 0.00000184 0.00000282 0.00000241 0.00000373 0.00000156 0.00001237
500 0.00000001 0.00000002 0.00000001 0.00000002 0.00000001 0.00000007
1000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.14901642 0.22858763 0.19513341 0.30097685 0.12628569 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=42.16910716 $
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$ k $ $ w_{1}(k) $ $ w_{2}(k) $ $ w_{3}(k) $ $ w_{4}(k) $ $ w_{5}(k) $ $ {\bf w}(k){\bf e} $
0 0.00047483 0.00077372 0.00059879 0.00078790 0.00036054 0.00299578
1 0.00076314 0.00119978 0.00097247 0.00140311 0.00061681 0.00495532
2 0.00110386 0.00172634 0.00140867 0.00210776 0.00091559 0.00726221
3 0.00147097 0.00227929 0.00190462 0.00286252 0.00123500 0.00975240
4 0.00185631 0.00287564 0.00242983 0.00368458 0.00157512 0.01242148
5 0.00226328 0.00349641 0.00297861 0.00455765 0.00193035 0.01522630
10 0.00342239 0.00526449 0.00449032 0.00694631 0.00290268 0.02302618
20 0.00253361 0.00388041 0.00331762 0.00511929 0.00214814 0.01699908
50 0.00115231 0.00176460 0.00150943 0.00233341 0.00097752 0.00773727
100 0.00031795 0.00048690 0.00041649 0.00064384 0.00026972 0.00213489
200 0.00002421 0.00003707 0.00003171 0.00004902 0.00002053 0.00016254
300 0.00000184 0.00000282 0.00000241 0.00000373 0.00000156 0.00001237
500 0.00000001 0.00000002 0.00000001 0.00000002 0.00000001 0.00000007
1000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000 0.00000000
$\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$ $\vdots$
sum 0.14901642 0.22858763 0.19513341 0.30097685 0.12628569 1.000000
$ W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=42.16910716 $
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