# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020124

## On the $BMAP_1, BMAP_2/PH/g, c$ retrial queueing system

 1 School of Mathematical Science, Changsha Normal University, Changsha 410100, Hunan, China 2 School of Mathematics and Statistics, Central South University, Changsha 410083, Hunan, China

* Corresponding author: Jinbiao Wu

Received  December 2019 Revised  April 2020 Published  June 2020

Fund Project: The first author is supported by Provincial Natural Science Foundation of Hunan under Grant 2019JJ50677 and the Program of Hehua Excellent Young Talents of Changsha Normal University. The second author is supported by Provincial Natural Science Foundation of Hunan under Grant 2020JJ4760

In this paper, we consider the BMAP/PH/c retrial queue with two types of customers where the rate of individual repeated attempts from the orbit is modulated according to a Markov Modulated Poisson Process. Using the theory of multi-dimensional asymptotically quasi-Toeplitz Markov chain, we obtain the algorithm for calculating the stationary distribution of the system. Main performance measures are presented. Furthermore, we investigate some optimization problems. The algorithm for determining the optimal number of guard servers and total servers is elaborated. Finally, this queueing system is applied to the cellular wireless network. Numerical results to illustrate the optimization problems and the impact of retrial on performance measures are provided. We find that the performance measures are mainly affected by the two types of customers' arrivals and service patterns, but the retrial rate plays a less crucial role.

Citation: Yi Peng, Jinbiao Wu. On the $BMAP_1, BMAP_2/PH/g, c$ retrial queueing system. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020124
##### References:
 [1] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 30 (1999), 1-6.   Google Scholar [2] J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211.  doi: 10.1007/BF02564721.  Google Scholar [3] L. Breuer, A. Dudin and V. Klimenok, A retrial $BMAP/PH/N$ system, Queueing Systems, 40 (2002), 433-457.  doi: 10.1023/A:1015041602946.  Google Scholar [4] S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Processes, (1999), 21–49. Google Scholar [5] A. Dudin and V. Klimenok, A retrial BMAP/PH/N queueing system with Markov modulated retrials, 2012 2nd Baltic Congress on Future Internet Communications, IEEE, (2012), 246–251. doi: 10.1109/BCFIC.2012.6217953.  Google Scholar [6] A. N. Dudin, G. V. Tsarenkov and V. I. Klimenok, Software "SIRIUS++" for performance evaluation of modern communication networks, Modelling and Simulation 2002. 16th European Simulation Multi-conference, Darmstadt, (2002), 489–493. Google Scholar [7] G. Falin, A survey of retrial queues, Queueing Systems Theory Appl., 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar [8] A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Ltd., Chichester, Halsted Press [John Wiley & Sons, Inc.], New York, 1981,130 pp.  Google Scholar [9] R. Guerin, Queueing-blocking system with two arrival streams and guard channels, IEEE Transations in Communications, 36 (1988), 153-163.  doi: 10.1109/26.2745.  Google Scholar [10] D. Hong and S.S. Rappaport, Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and nonprioritized handoff procedures, IEEE Transactions on Vehicular Technology, 35 (1996), 77-99.   Google Scholar [11] C. S. Kim, V. I. Klimenok and D. S. Orlovsky, The BMAP/PH/N retrial queue with Markovian flow of breakdowns, European Journal of Operational Research, 189 (2008), 1057-1072.  doi: 10.1016/j.ejor.2007.02.053.  Google Scholar [12] C. S. Kim, V. Klimenok, V. Mushko and A. Dudin, The BMAP/PH/N retrial queueing system operating in {M}arkovian random environment, Comput. Oper. Res., 37 (2010), 1228-1237.  doi: 10.1016/j.cor.2009.09.008.  Google Scholar [13] C. S. Kim, V. I. Klimenok and A. N. Dudin, Analysis and optimization of guard channel policy in cellular mobile networks with account of retrials, Comput. Oper. Res., 43 (2014), 181-190.  doi: 10.1016/j.cor.2013.09.005.  Google Scholar [14] A. Klemm, C. Lindemann and M. Lohmann, Modeling IP traffic using the batch Markovian arrival process, Performance Evaluation, 54 (2003), 149-173.  doi: 10.1007/3-540-46029-2_6.  Google Scholar [15] V. I. Klimenok, D. S. Orlovsky and A. N. Dudin, A $BMAP/PH/N$ system with impatient repeated Calls, Asia-Pacific Journal of Operational Research, 24 (2007), 293-312.  doi: 10.1142/S0217595907001310.  Google Scholar [16] V. I. Klimenok and A. N. Dudin, Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems, 54 (2006), 245-259.  doi: 10.1007/s11134-006-0300-z.  Google Scholar [17] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar [18] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar [19] M. Martin and J. R. Artalejo, Analysis of an $M/G/1$ queue with two types of impatient units, Advances in Applied Probability, 27 (1995), 840-861.  doi: 10.2307/1428136.  Google Scholar [20] E. Morozov, A. Rumyantsev, S. Dey and T. G. Deepak, Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking, Performance Evaluation, 134 (2019), 102005. doi: 10.1016/j.peva.2019.102005.  Google Scholar [21] M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar [22] M. F. Neuts, Structured Stochastic Matrices of M/G/1-type and their Applications, Probability: Pure and Applied, 5. Marcel Dekker, Inc., New York, 1989.  Google Scholar [23] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences, 2. Johns Hopkins University Press, Baltimore, Md., 1981.  Google Scholar [24] P. I. Panagoulias, I. D. Moscholios, P. G. Sarigiannidis and M. D. Logothetis, Congestion probabilities in OFDM wireless networks with compound Poisson arrivals, IET Communications, 14 (2020), 674-681.  doi: 10.1049/iet-com.2019.0845.  Google Scholar [25] K. S. Trivedi, S. Dharmaraja and X. M. Ma, Analytic modeling of handoffs in wireless cellular networks, Information Sciences, 148 (2000), 155-166.  doi: 10.1016/S0020-0255(02)00292-X.  Google Scholar [26] T. M. Walingo and F. Takawira, Performance analysis of a connection admission scheme for future networks, IEEE Transactions on Wireless Communications, 14 (2015), 1994-2006.  doi: 10.1109/TWC.2014.2378777.  Google Scholar [27] Y. L. Wu, G. Y. Min and L. T. Yang, Performance analysis of hybrid wireless networks under bursty and correlated traffic, IEEE Transactions on Vehicular Technology, 62 (2013), 449-454.  doi: 10.1109/TVT.2012.2219890.  Google Scholar [28] J. B. Wu, Z. M. Lian and G. Yang, Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment, Applied Mathematical Modelling, 35 (2011), 1184-1193.  doi: 10.1016/j.apm.2010.08.006.  Google Scholar [29] J. B. Wu and Z. T. Lian, Analysis of $M_1, M_2/G/1$ G-queueing system with retrial customers, Nonlinear Analysis: Real World Applications, 14 (2013), 365-382.  doi: 10.1016/j.nonrwa.2012.06.009.  Google Scholar

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##### References:
 [1] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 30 (1999), 1-6.   Google Scholar [2] J. R. Artalejo, A classified bibliography of research on retrial queues: Progress in 1990-1999, Top, 7 (1999), 187-211.  doi: 10.1007/BF02564721.  Google Scholar [3] L. Breuer, A. Dudin and V. Klimenok, A retrial $BMAP/PH/N$ system, Queueing Systems, 40 (2002), 433-457.  doi: 10.1023/A:1015041602946.  Google Scholar [4] S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Processes, (1999), 21–49. Google Scholar [5] A. Dudin and V. Klimenok, A retrial BMAP/PH/N queueing system with Markov modulated retrials, 2012 2nd Baltic Congress on Future Internet Communications, IEEE, (2012), 246–251. doi: 10.1109/BCFIC.2012.6217953.  Google Scholar [6] A. N. Dudin, G. V. Tsarenkov and V. I. Klimenok, Software "SIRIUS++" for performance evaluation of modern communication networks, Modelling and Simulation 2002. 16th European Simulation Multi-conference, Darmstadt, (2002), 489–493. Google Scholar [7] G. Falin, A survey of retrial queues, Queueing Systems Theory Appl., 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar [8] A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Horwood Ltd., Chichester, Halsted Press [John Wiley & Sons, Inc.], New York, 1981,130 pp.  Google Scholar [9] R. Guerin, Queueing-blocking system with two arrival streams and guard channels, IEEE Transations in Communications, 36 (1988), 153-163.  doi: 10.1109/26.2745.  Google Scholar [10] D. Hong and S.S. Rappaport, Traffic model and performance analysis for cellular mobile radio telephone systems with prioritized and nonprioritized handoff procedures, IEEE Transactions on Vehicular Technology, 35 (1996), 77-99.   Google Scholar [11] C. S. Kim, V. I. Klimenok and D. S. Orlovsky, The BMAP/PH/N retrial queue with Markovian flow of breakdowns, European Journal of Operational Research, 189 (2008), 1057-1072.  doi: 10.1016/j.ejor.2007.02.053.  Google Scholar [12] C. S. Kim, V. Klimenok, V. Mushko and A. Dudin, The BMAP/PH/N retrial queueing system operating in {M}arkovian random environment, Comput. Oper. Res., 37 (2010), 1228-1237.  doi: 10.1016/j.cor.2009.09.008.  Google Scholar [13] C. S. Kim, V. I. Klimenok and A. N. Dudin, Analysis and optimization of guard channel policy in cellular mobile networks with account of retrials, Comput. Oper. Res., 43 (2014), 181-190.  doi: 10.1016/j.cor.2013.09.005.  Google Scholar [14] A. Klemm, C. Lindemann and M. Lohmann, Modeling IP traffic using the batch Markovian arrival process, Performance Evaluation, 54 (2003), 149-173.  doi: 10.1007/3-540-46029-2_6.  Google Scholar [15] V. I. Klimenok, D. S. Orlovsky and A. N. Dudin, A $BMAP/PH/N$ system with impatient repeated Calls, Asia-Pacific Journal of Operational Research, 24 (2007), 293-312.  doi: 10.1142/S0217595907001310.  Google Scholar [16] V. I. Klimenok and A. N. Dudin, Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems, 54 (2006), 245-259.  doi: 10.1007/s11134-006-0300-z.  Google Scholar [17] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, American Statistical Association, Alexandria, VA, 1999. doi: 10.1137/1.9780898719734.  Google Scholar [18] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar [19] M. Martin and J. R. Artalejo, Analysis of an $M/G/1$ queue with two types of impatient units, Advances in Applied Probability, 27 (1995), 840-861.  doi: 10.2307/1428136.  Google Scholar [20] E. Morozov, A. Rumyantsev, S. Dey and T. G. Deepak, Performance analysis and stability of multiclass orbit queue with constant retrial rates and balking, Performance Evaluation, 134 (2019), 102005. doi: 10.1016/j.peva.2019.102005.  Google Scholar [21] M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar [22] M. F. Neuts, Structured Stochastic Matrices of M/G/1-type and their Applications, Probability: Pure and Applied, 5. Marcel Dekker, Inc., New York, 1989.  Google Scholar [23] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models: An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences, 2. Johns Hopkins University Press, Baltimore, Md., 1981.  Google Scholar [24] P. I. Panagoulias, I. D. Moscholios, P. G. Sarigiannidis and M. D. Logothetis, Congestion probabilities in OFDM wireless networks with compound Poisson arrivals, IET Communications, 14 (2020), 674-681.  doi: 10.1049/iet-com.2019.0845.  Google Scholar [25] K. S. Trivedi, S. Dharmaraja and X. M. Ma, Analytic modeling of handoffs in wireless cellular networks, Information Sciences, 148 (2000), 155-166.  doi: 10.1016/S0020-0255(02)00292-X.  Google Scholar [26] T. M. Walingo and F. Takawira, Performance analysis of a connection admission scheme for future networks, IEEE Transactions on Wireless Communications, 14 (2015), 1994-2006.  doi: 10.1109/TWC.2014.2378777.  Google Scholar [27] Y. L. Wu, G. Y. Min and L. T. Yang, Performance analysis of hybrid wireless networks under bursty and correlated traffic, IEEE Transactions on Vehicular Technology, 62 (2013), 449-454.  doi: 10.1109/TVT.2012.2219890.  Google Scholar [28] J. B. Wu, Z. M. Lian and G. Yang, Analysis of the finite source MAP/PH/N retrial G-queue operating in a random environment, Applied Mathematical Modelling, 35 (2011), 1184-1193.  doi: 10.1016/j.apm.2010.08.006.  Google Scholar [29] J. B. Wu and Z. T. Lian, Analysis of $M_1, M_2/G/1$ G-queueing system with retrial customers, Nonlinear Analysis: Real World Applications, 14 (2013), 365-382.  doi: 10.1016/j.nonrwa.2012.06.009.  Google Scholar
$L_{orb}$ as function of the parameter $g$ with $(c, \lambda_o, \lambda_h) = (10, 2, 2)$
Dependence of the blocking probability for originating calls on the value $g$ with $(c, \lambda_o, \lambda_h) = (10, 2, 2)$
Dependence of the blocking probability for handoff calls on the value $g$ with $(c, \lambda_o, \lambda_h) = (10, 2, 2)$
The stationary join distribution of the system with $(c, g, \lambda_o, \lambda_h, \lambda_r) = (8, 6, 2, 2, 2)$
 $i$ $\backslash$ $b$ 0 1 2 3 4 5 6 7 8 $sum$ 0 0.0467 0.1413 0.2136 0.2148 0.1604 0.0923 0.0376 0.0016 0.0001 0.9084 1 0.0001 0.0006 0.0020 0.0047 0.0091 0.0149 0.0208 0.0013 0.0001 0.0536 2 0.0000 0.0000 0.0002 0.0008 0.0023 0.0054 0.0109 0.0008 0.0000 0.0206 3 0.0000 0.0000 0.0000 0.0002 0.0007 0.0022 0.0055 0.0005 0.0000 0.0092 4 0.0000 0.0000 0.0000 0.0001 0.0002 0.0009 0.0028 0.0003 0.0000 0.0043 5 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0014 0.0001 0.0000 0.0020 1.0e-003 $\times$ 6 0.0000 0.0001 0.0007 0.0055 0.0334 0.1661 0.6986 0.0667 0.0051 0.9761 1.0e-003 $\times$ 7 0.0000 0.0000 0.0002 0.0019 0.0130 0.0731 0.3474 0.0337 0.0027 0.4721 1.0e-003$\times$ 8 0.0000 0.0000 0.0001 0.0007 0.0052 0.0325 0.1724 0.0169 0.0014 0.2291 1.0e-004$\times$ 9 0.0000 0.0000 0.0002 0.0025 0.0211 0.1460 0.8533 0.0846 0.0069 1.1145 1.0e-004$\times$ 10 0.0000 0.0000 0.0001 0.0009 0.0087 0.0660 0.4217 0.0421 0.0035 0.5429 $sum$ 0.0468 0.1419 0.2158 0.2206 0.1728 0.1162 0.0800 0.0047 0.0002 0.999
 $i$ $\backslash$ $b$ 0 1 2 3 4 5 6 7 8 $sum$ 0 0.0467 0.1413 0.2136 0.2148 0.1604 0.0923 0.0376 0.0016 0.0001 0.9084 1 0.0001 0.0006 0.0020 0.0047 0.0091 0.0149 0.0208 0.0013 0.0001 0.0536 2 0.0000 0.0000 0.0002 0.0008 0.0023 0.0054 0.0109 0.0008 0.0000 0.0206 3 0.0000 0.0000 0.0000 0.0002 0.0007 0.0022 0.0055 0.0005 0.0000 0.0092 4 0.0000 0.0000 0.0000 0.0001 0.0002 0.0009 0.0028 0.0003 0.0000 0.0043 5 0.0000 0.0000 0.0000 0.0000 0.0001 0.0004 0.0014 0.0001 0.0000 0.0020 1.0e-003 $\times$ 6 0.0000 0.0001 0.0007 0.0055 0.0334 0.1661 0.6986 0.0667 0.0051 0.9761 1.0e-003 $\times$ 7 0.0000 0.0000 0.0002 0.0019 0.0130 0.0731 0.3474 0.0337 0.0027 0.4721 1.0e-003$\times$ 8 0.0000 0.0000 0.0001 0.0007 0.0052 0.0325 0.1724 0.0169 0.0014 0.2291 1.0e-004$\times$ 9 0.0000 0.0000 0.0002 0.0025 0.0211 0.1460 0.8533 0.0846 0.0069 1.1145 1.0e-004$\times$ 10 0.0000 0.0000 0.0001 0.0009 0.0087 0.0660 0.4217 0.0421 0.0035 0.5429 $sum$ 0.0468 0.1419 0.2158 0.2206 0.1728 0.1162 0.0800 0.0047 0.0002 0.999
The the optimal value $g^*$ for the optimization problem (I) with $(c, \lambda_o, p_0) = (20, 10, 0.0001)$
 $\lambda_r$ $\backslash$ $\lambda_h$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 1 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13 10 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13 20 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13
 $\lambda_r$ $\backslash$ $\lambda_h$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 1 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13 10 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13 20 18 18 18 17 17 17 16 16 16 16 15 15 15 14 14 13
The optimal value $c^*$ for the optimization problem (II) with $(\lambda_r, p_1, p_2) = (10, 0.001, 0.0001)$
 $\lambda_h$ $\backslash$ $\lambda_o$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 1 8 11 14 16 18 20 22 24 26 28 29 31 33 35 37 45 5 10 13 15 17 19 21 23 25 27 29 30 32 34 36 38 46 10 13 15 17 19 21 23 25 27 29 31 32 34 36 37 39 47
 $\lambda_h$ $\backslash$ $\lambda_o$ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 20 1 8 11 14 16 18 20 22 24 26 28 29 31 33 35 37 45 5 10 13 15 17 19 21 23 25 27 29 30 32 34 36 38 46 10 13 15 17 19 21 23 25 27 29 31 32 34 36 37 39 47
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