# American Institute of Mathematical Sciences

November  2021, 17(6): 3373-3391. 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  November 2021 Early access  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 and Management Optimization, 2021, 17 (6) : 3373-3391. doi: 10.3934/jimo.2020124
##### References:
 [1] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 30 (1999), 1-6. [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. [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. [4] S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Processes, (1999), 21–49. [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. [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. [7] G. Falin, A survey of retrial queues, Queueing Systems Theory Appl., 7 (1990), 127-167.  doi: 10.1007/BF01158472. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143. [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. [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. [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. [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. [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. [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. [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. [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.

show all references

##### References:
 [1] J. R. Artalejo, Accessible bibliography on retrial queues, Mathematical and Computer Modelling, 30 (1999), 1-6. [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. [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. [4] S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Processes, (1999), 21–49. [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. [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. [7] G. Falin, A survey of retrial queues, Queueing Systems Theory Appl., 7 (1990), 127-167.  doi: 10.1007/BF01158472. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [21] M. F. Neuts, A versatile Markovian point process, Journal of Applied Probability, 16 (1979), 764-779.  doi: 10.2307/3213143. [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. [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. [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. [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. [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. [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. [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. [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.
$L_{orb}$ 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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