    doi: 10.3934/jimo.2020135

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

 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.

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:
  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  A. S. Alfa, Applied Discrete-Time Queues, 2$^nd$ edition, Springer, New York, 2016. doi: 10.1007/978-1-4939-3420-1.  Google Scholar  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  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  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  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  G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar  M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981. Google Scholar  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  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  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  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  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  M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405.   Google Scholar  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  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:
  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  A. S. Alfa, Applied Discrete-Time Queues, 2$^nd$ edition, Springer, New York, 2016. doi: 10.1007/978-1-4939-3420-1.  Google Scholar  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  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  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  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  G. L. Curry and R. M. Feldman, Manufacturing Systems Modeling and Analysis, Springer, Berlin, Heidelberg, 2009. Google Scholar  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  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  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  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  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  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  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  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  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  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  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  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  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  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  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  M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar  M. F. Neuts, Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach,, John Hopkins University Press, Baltimore, MD, 1981. Google Scholar  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  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  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  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  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  M. Yadin and P. Naor, Queueing systems with a removable service station, Operational Research Quarterly, 14 (1963), 393-405.   Google Scholar  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  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 Mean queue length versus threshold value $N$ for different coefficients of correlation in the D-BMAPs
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
 $n$ $\pi^{+}_{1}(n)$ $\pi^{+}_{2}(n)$ $\pi^{+}_{3}(n)$ $\pi^{+}_{4}(n)$ $\pi^{+}_{5}(n)$ ${\bf{ \pmb{\mathsf{ π}}}}^{+}(n){\bf e}$ 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
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 $L_{q}=27.32669405$, $W_{q}\equiv L_{q}/\lambda^{\ast}=42.16910717$
 $n$ $\omega_{1}(n)$ $\omega_{2}(n)$ $\omega_{3}(n)$ $\omega_{4}(n)$ $\omega_{5}(n)$ ${\bf{\it\Large \pmb{\mathsf{ ω}}}}(n){\bf e}$ 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 $L_{q}=27.32669405$, $W_{q}\equiv L_{q}/\lambda^{\ast}=42.16910717$
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
 $n$ $\omega_{1}^-(n)$ $\omega_{2}^-(n)$ $\omega_{3}^-(n)$ $\omega_{4}^-(n)$ $\omega_{5}^-(n)$ ${\bf{\it\Large \pmb{\mathsf{ ω}}}}^-(n){\bf e}$ 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
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
 $n$ $\omega_{1}^\bullet(n)$ $\omega_{2}^\bullet(n)$ $\omega_{3}^\bullet(n)$ $\omega_{4}^\bullet(n)$ $\omega_{5}^\bullet(n)$ ${\bf{\it\Large \pmb{\mathsf{ ω}}}}^\bullet(n){\bf e}$ 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
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 $W_{q}\equiv \sum_{k=1}^{\infty}k{\bf w}(k){\bf e}=42.16910716$
 $k$ $w_{1}(k)$ $w_{2}(k)$ $w_{3}(k)$ $w_{4}(k)$ $w_{5}(k)$ ${\bf w}(k){\bf e}$ 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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