# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020101

## A ${BMAP/BMSP/1}$ queue with Markov dependent arrival and Markov dependent service batches

 1 Centre for Research in Mathematics, C.M.S. College, Kottayam-686001, India 2 Department of Mathematics, Union Christian College, Aluva-683102, India

Received  November 2019 Published  May 2020

Fund Project: The first author is supported by UGC, Govt. of India, Emeritus Fellow(EMERITUS 2017-18 GEN 10822(SA-II)) and DST, Indo-Russian project: INT/RUS/RSF/P-15

Batch arrival and batch service queueing systems are of importance in the context of telecommunication networks. None of the work reported so far consider the dependence of consecutive arrival and service batches. Batch Markovian Arrival Process($BMAP$) and Batch Markovian Service Process ($BMSP$) take care of the dependence between successive inter-arrival and service times, respectively. However in real life situations dependence between consecutive arrival and service batch sizes also play an important role. This is to regulate the workload of the server in the context of service and to restrict the arrival batch size when the flow is from the same source. In this paper we study a queueing system with Markov dependent arrival and service batch sizes. The arrival and service batch sizes are assumed to be finite. Further, successive inter-arrival and service time durations are also assumed to be correlated. Specifically, we consider a $BMAP/BMSP/1$ queue with Markov dependent arrival and Markov dependent service batch sizes. The stability of the system is investigated. The steady state probability vectors of the system state and some important performance measures are computed. The Laplace-Stieltjes transform of waiting time and idle time of the server are obtained. Some numerical examples are provided.

Citation: Achyutha Krishnamoorthy, Anu Nuthan Joshua. A ${BMAP/BMSP/1}$ queue with Markov dependent arrival and Markov dependent service batches. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020101
##### References:
 [1] A. D. Banik, Analyzing state-dependent arrival in $GI/BMSP/1/\infty$ queues, Math. Comput. Modelling, 53 (2011), 1229-1246.  doi: 10.1016/j.mcm.2010.12.007.  Google Scholar [2] A. D. Banik, Queueing analysis and optimal control of $BMAP/G^{(a, b)}/1/N$ and $BMAP/MSP^{(a, b)}/1/N$ systems, Comput. Industrial Engineering, 57 (2009), 748-761.  doi: 10.1016/j.cie.2009.02.002.  Google Scholar [3] A. D. Banik, Stationary analysis of a $BMAP/R/1$ queue with $R$-type multiple working vacations, Comm. Statist. Simulation Comput., 46 (2017), 1035-1061.  doi: 10.1080/03610918.2014.990096.  Google Scholar [4] A. D. Banik, M. L. Chaudhry and U. C. Gupta, On the finite buffer queue with renewal input and batch Markovian service process: $GI/BMSP/1/N$, Methodol. Comput. Appl. Probab., 10 (2008), 559-575.  doi: 10.1007/s11009-007-9064-0.  Google Scholar [5] S. Ghosh and A. D. Banik, An algorithmic analysis of the $BMAP/MSP/1$ generalized processor-sharing queue, Comput. Oper. Res., 79 (2017), 1-11.  doi: 10.1016/j.cor.2016.10.001.  Google Scholar [6] S. Ghosh and A. D. Banik, Computing conditional sojourn time of a randomly chosen tagged customer in a $BMAP/MSP/1$ queue under random order service discipline, Ann. Oper. Res., 261 (2018), 185-206.  doi: 10.1007/s10479-017-2534-z.  Google Scholar [7] S. Ghosh and A. D. Banik, Efficient computational analysis of non-exhaustive service vacation queues: $BMAP/R/1/N(\infty)$ under gated-limited discipline, Appl. Math. Model., 68 (2019), 540-562.  doi: 10.1016/j.apm.2018.11.040.  Google Scholar [8] C. Kim, V. I. Klimenok and A. N. Dudin, Analysis of unreliable $BMAP/PH/N$ type queue with Markovian flow of breakdowns, Appl. Math. Comput., 314 (2017), 154-172.  doi: 10.1016/j.amc.2017.06.035.  Google Scholar [9] V. Klimenok, A. N. Dudin and K. Samouylov, Computation of moments of queue length in the $BMAP/SM/1$ queue, Oper. Res. Lett., 45 (2017), 467-470.  doi: 10.1016/j.orl.2017.07.003.  Google Scholar [10] V. Klimenok and O. Dudina, Retrial tandem queue with controllable strategy of repeated attempts, Quality Tech. Quantitative Manag., 14 (2017), 74-93.  doi: 10.1080/16843703.2016.1189177.  Google Scholar [11] V. Klimenok, O. Dudina, V. Vishnevsky and K. Samouylov, Retrial tandem queue with $BMAP$ input and semi-Markovian service process, in International Conference on Distributed Computer and Communication Networks, Communications in Computer and Information Science, 700, Springer, Cham, 2017,159–173. doi: 10.1007/978-3-319-66836-9_14.  Google Scholar [12] V. I. Klimenok, A. N. Dudin and K. E. Samouylov, Analysis of the $BMAP/PH/N$ queueing systems with backup servers, Appl. Math. Model., 57 (2018), 64-84.  doi: 10.1016/j.apm.2017.12.024.  Google Scholar [13] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar [14] D. M. Lucantoni, K. S. Meier-Hellstern and M. F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. in Appl. Probab., 22 (1990), 676-705.  doi: 10.2307/1427464.  Google Scholar [15] M. F. Neuts, Versatile Markovian point process, J. Appl. Probab., 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar [16] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences, 2, John Hopkins University Press, Baltimore, MD, 1981.  doi: 10.2307/2287748.  Google Scholar [17] G. Rama, R. Ramshankar, R. Sandhya, V. Sundar and R. Ramanarayanan, $BMAP/M/C$ bulk service queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 33-47.   Google Scholar [18] V. Ramaswami, The $N/G/1$ queue and its detailed analysis, Adv. in Appl. Probab., 12 (1980), 222-261.  doi: 10.2307/1426503.  Google Scholar [19] S. K. Samanta, M. L. Chaudhry and A. Pacheco, Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.  doi: 10.1007/s11009-014-9429-0.  Google Scholar [20] R. Sandhya, V. Sundar, G. Rama, R. Ramshankar and R. Ramanarayanan, $BMAP/BMSP/1$ queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 01-12.   Google Scholar [21] V. M. Vishnevskii and A. N. Dudin, Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks, Autom. Remote Control, 78 (2017), 1361-1403.  doi: 10.1134/S000511791708001X.  Google Scholar

show all references

##### References:
 [1] A. D. Banik, Analyzing state-dependent arrival in $GI/BMSP/1/\infty$ queues, Math. Comput. Modelling, 53 (2011), 1229-1246.  doi: 10.1016/j.mcm.2010.12.007.  Google Scholar [2] A. D. Banik, Queueing analysis and optimal control of $BMAP/G^{(a, b)}/1/N$ and $BMAP/MSP^{(a, b)}/1/N$ systems, Comput. Industrial Engineering, 57 (2009), 748-761.  doi: 10.1016/j.cie.2009.02.002.  Google Scholar [3] A. D. Banik, Stationary analysis of a $BMAP/R/1$ queue with $R$-type multiple working vacations, Comm. Statist. Simulation Comput., 46 (2017), 1035-1061.  doi: 10.1080/03610918.2014.990096.  Google Scholar [4] A. D. Banik, M. L. Chaudhry and U. C. Gupta, On the finite buffer queue with renewal input and batch Markovian service process: $GI/BMSP/1/N$, Methodol. Comput. Appl. Probab., 10 (2008), 559-575.  doi: 10.1007/s11009-007-9064-0.  Google Scholar [5] S. Ghosh and A. D. Banik, An algorithmic analysis of the $BMAP/MSP/1$ generalized processor-sharing queue, Comput. Oper. Res., 79 (2017), 1-11.  doi: 10.1016/j.cor.2016.10.001.  Google Scholar [6] S. Ghosh and A. D. Banik, Computing conditional sojourn time of a randomly chosen tagged customer in a $BMAP/MSP/1$ queue under random order service discipline, Ann. Oper. Res., 261 (2018), 185-206.  doi: 10.1007/s10479-017-2534-z.  Google Scholar [7] S. Ghosh and A. D. Banik, Efficient computational analysis of non-exhaustive service vacation queues: $BMAP/R/1/N(\infty)$ under gated-limited discipline, Appl. Math. Model., 68 (2019), 540-562.  doi: 10.1016/j.apm.2018.11.040.  Google Scholar [8] C. Kim, V. I. Klimenok and A. N. Dudin, Analysis of unreliable $BMAP/PH/N$ type queue with Markovian flow of breakdowns, Appl. Math. Comput., 314 (2017), 154-172.  doi: 10.1016/j.amc.2017.06.035.  Google Scholar [9] V. Klimenok, A. N. Dudin and K. Samouylov, Computation of moments of queue length in the $BMAP/SM/1$ queue, Oper. Res. Lett., 45 (2017), 467-470.  doi: 10.1016/j.orl.2017.07.003.  Google Scholar [10] V. Klimenok and O. Dudina, Retrial tandem queue with controllable strategy of repeated attempts, Quality Tech. Quantitative Manag., 14 (2017), 74-93.  doi: 10.1080/16843703.2016.1189177.  Google Scholar [11] V. Klimenok, O. Dudina, V. Vishnevsky and K. Samouylov, Retrial tandem queue with $BMAP$ input and semi-Markovian service process, in International Conference on Distributed Computer and Communication Networks, Communications in Computer and Information Science, 700, Springer, Cham, 2017,159–173. doi: 10.1007/978-3-319-66836-9_14.  Google Scholar [12] V. I. Klimenok, A. N. Dudin and K. E. Samouylov, Analysis of the $BMAP/PH/N$ queueing systems with backup servers, Appl. Math. Model., 57 (2018), 64-84.  doi: 10.1016/j.apm.2017.12.024.  Google Scholar [13] D. M. Lucantoni, New results on the single server queue with a batch Markovian arrival process, Comm. Statist. Stochastic Models, 7 (1991), 1-46.  doi: 10.1080/15326349108807174.  Google Scholar [14] D. M. Lucantoni, K. S. Meier-Hellstern and M. F. Neuts, A single-server queue with server vacations and a class of non-renewal arrival processes, Adv. in Appl. Probab., 22 (1990), 676-705.  doi: 10.2307/1427464.  Google Scholar [15] M. F. Neuts, Versatile Markovian point process, J. Appl. Probab., 16 (1979), 764-779.  doi: 10.2307/3213143.  Google Scholar [16] M. F. Neuts, Matrix-Geometric Solutions in Stochastic Models. An Algorithmic Approach, Johns Hopkins Series in the Mathematical Sciences, 2, John Hopkins University Press, Baltimore, MD, 1981.  doi: 10.2307/2287748.  Google Scholar [17] G. Rama, R. Ramshankar, R. Sandhya, V. Sundar and R. Ramanarayanan, $BMAP/M/C$ bulk service queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 33-47.   Google Scholar [18] V. Ramaswami, The $N/G/1$ queue and its detailed analysis, Adv. in Appl. Probab., 12 (1980), 222-261.  doi: 10.2307/1426503.  Google Scholar [19] S. K. Samanta, M. L. Chaudhry and A. Pacheco, Analysis of $BMAP/MSP/1$ queue, Methodol. Comput. Appl. Probab., 18 (2016), 419-440.  doi: 10.1007/s11009-014-9429-0.  Google Scholar [20] R. Sandhya, V. Sundar, G. Rama, R. Ramshankar and R. Ramanarayanan, $BMAP/BMSP/1$ queue with randomly varying environment, IOSR J. Engineering, 5 (2015), 01-12.   Google Scholar [21] V. M. Vishnevskii and A. N. Dudin, Queueing systems with correlated arrival flows and their applications to modeling telecommunication networks, Autom. Remote Control, 78 (2017), 1361-1403.  doi: 10.1134/S000511791708001X.  Google Scholar
Transition rate submatrices within level 0
 $\rm{ From}$ $\rm{To}$ $\rm{ Rate}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p, n_1, n_2 = 0(k))$ $\textbf I_s \otimes D_0$ $(0, p, n_1, n_2)$ $(0, p, n_1, n_2)$ $S_0 \oplus D_0$ $(0, p, n_1, n_2)$ $(0, p + m_1, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1}D_{c}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p + m_1 - k, m_1, k)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p+m_1, m_1, 0(k))$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(0, p, n_1, n_2 )$ $(0, p, n_1, m_2 = 0(k))$ $q_{n_2 k} S_d \otimes \textbf I_r$ $(0, p, n_1, n_2)$ $(0, p-m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$
 $\rm{ From}$ $\rm{To}$ $\rm{ Rate}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p, n_1, n_2 = 0(k))$ $\textbf I_s \otimes D_0$ $(0, p, n_1, n_2)$ $(0, p, n_1, n_2)$ $S_0 \oplus D_0$ $(0, p, n_1, n_2)$ $(0, p + m_1, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1}D_{c}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p + m_1 - k, m_1, k)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(0, p, n_1, n_2 = 0(k))$ $(0, p+m_1, m_1, 0(k))$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(0, p, n_1, n_2 )$ $(0, p, n_1, m_2 = 0(k))$ $q_{n_2 k} S_d \otimes \textbf I_r$ $(0, p, n_1, n_2)$ $(0, p-m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$
Transition rate submatrices except those within level 0
 $\rm{ From}$ $\rm{To}$ $\rm{ Rate}$ $(0, p, n_1 , n_2)$ $(1, p + m_1- q, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1}D_{c}$ $(1, p, n_1, n_2)$ $(0, q + p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$ $(l, p, n_1, n_2)$ $(l+1, p + m_1 - q, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(l, p, n_1, n_2)$ $(l, p, n_1, n_2)$ $S_0 \oplus D_0$ $(l, p, n_1, n_2)$ $(l, p + m_1, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(l, p, n_1, n_2)$ $(l, p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$ $(l, p, n_1, n_2)$ $(l-1, q + p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$
 $\rm{ From}$ $\rm{To}$ $\rm{ Rate}$ $(0, p, n_1 , n_2)$ $(1, p + m_1- q, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1}D_{c}$ $(1, p, n_1, n_2)$ $(0, q + p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$ $(l, p, n_1, n_2)$ $(l+1, p + m_1 - q, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(l, p, n_1, n_2)$ $(l, p, n_1, n_2)$ $S_0 \oplus D_0$ $(l, p, n_1, n_2)$ $(l, p + m_1, m_1, n_2)$ $\textbf I_s \otimes p_{n_1 m_1} D_{c}$ $(l, p, n_1, n_2)$ $(l, p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$ $(l, p, n_1, n_2)$ $(l-1, q + p- m_2, n_1, m_2)$ $q_{n_2 m_2} S_d \otimes \textbf I_r$
Expected queue length under various arrival and service processes
 NCA (2a) NCA (2b) ZCA PCA (3a) PCA (3b) NCS(2b) ${4.5898}$ ${5.7659}$ ${3.1824}$ ${42.5934}$ ${179.1428}$ NCS(2a) ${3.3006}$ ${4.3856}$ ${2.1585}$ ${41.1375}$ ${177.7154}$ ZCS ${1.1070}$ ${1.1905}$ ${0.9351}$ ${8.1323}$ ${40.2739}$ PCS ${40.5835}$ ${42.1713}$ ${32.6996}$ ${82.5983}$ ${214.3443}$
 NCA (2a) NCA (2b) ZCA PCA (3a) PCA (3b) NCS(2b) ${4.5898}$ ${5.7659}$ ${3.1824}$ ${42.5934}$ ${179.1428}$ NCS(2a) ${3.3006}$ ${4.3856}$ ${2.1585}$ ${41.1375}$ ${177.7154}$ ZCS ${1.1070}$ ${1.1905}$ ${0.9351}$ ${8.1323}$ ${40.2739}$ PCS ${40.5835}$ ${42.1713}$ ${32.6996}$ ${82.5983}$ ${214.3443}$
Expected idle time in $(0, 0, 1, 0(1), 1, 1), (0, 0, 1, 0(2), 1, 1), (0, 0, 1, 0(3), 1, 1)$ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes
 L(1, 1) M(1, 1) N(1, 1) L(1, 2) M(1, 2) N(1, 2) L(1, 3) M(1, 3) N(1, 3) ZCS NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ ZCS NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ ZCS ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ ZCS PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$ PCS NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ PCS NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ PCS ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ PCS PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$ NCS(2a) NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ NCS(2a) NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ NCS(2a) ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ NCS(2a) PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$
 L(1, 1) M(1, 1) N(1, 1) L(1, 2) M(1, 2) N(1, 2) L(1, 3) M(1, 3) N(1, 3) ZCS NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ ZCS NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ ZCS ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ ZCS PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$ PCS NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ PCS NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ PCS ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ PCS PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$ NCS(2a) NCA(2a) ${0.0200}$ ${1.5725}$ ${0.7685}$ ${0}$ ${0}$ ${0}$ ${1.9756}$ ${0.4426}$ ${2.5034}$ NCS(2a) NCA(2b) ${0.2199}$ ${1.2456}$ ${1.1507}$ ${0.2274}$ ${0.1551}$ ${0.2979}$ ${1.4797}$ ${0.8040}$ ${1.7725}$ NCS(2a) ZCA ${1.0333}$ ${1.8600}$ ${2.7280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${1.8098}$ ${2.2052}$ ${2.3602}$ ${1.8682}$ ${2.3471}$ ${2.5799}$ ${2.5189}$ ${4.2899}$ ${6.0942}$ NCS(2a) PCA(3b) ${1.9756}$ ${1.5247}$ ${5.1267}$ ${0}$ ${0}$ ${0}$ ${0.0200}$ ${0.0514}$ ${0.0965}$
Expected idle time in $(0, 0, 1, 0(1), 1, 2), (0, 0, 1, 0(2), 1, 2), (0, 0, 1, 0(3), 1, 2)$ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes
 L(2, 1) M(2, 1) N(2, 1) L(2, 2) M(2, 2) N(2, 2) L(2, 3) M(2, 3) N(2, 3) ZCS NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ ZCS NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ ZCS ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ ZCS PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$ PCS NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ PCS NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ PCS ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ PCS PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$ NCS(2a) NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ NCS(2a) NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ NCS(2a) ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ NCS(2a) PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$
 L(2, 1) M(2, 1) N(2, 1) L(2, 2) M(2, 2) N(2, 2) L(2, 3) M(2, 3) N(2, 3) ZCS NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ ZCS NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ ZCS ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ ZCS PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$ PCS NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ PCS NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ PCS ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ PCS PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$ NCS(2a) NCA(2a) ${0.0100}$ ${0.7881}$ ${0.3977}$ ${0}$ ${0}$ ${0}$ ${0.9878}$ ${0.2292}$ ${1.8764}$ NCS(2a) NCA(2b) ${0.0567}$ ${0.5166}$ ${0.5367}$ ${0.0351}$ ${0.0530}$ ${0.2016}$ ${0.5964}$ ${0.3021}$ ${1.2421}$ NCS(2a) ZCA ${0.5333}$ ${1.3600}$ ${2.2280}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${1.9276}$ ${2.5820}$ ${2.8420}$ ${1.9727}$ ${2.6957}$ ${3.0442}$ ${1.4113}$ ${3.6018}$ ${5.8677}$ NCS(2a) PCA(3b) ${0.9878}$ ${2.5447}$ ${4.1548}$ ${0}$ ${0}$ ${0}$ ${0.0100}$ ${0.0336}$ ${0.0707}$
Expected idle time in $(0, 0, 1, 0(1), 1, 3), (0, 0, 1, 0(2), 1, 3), (0, 0, 1, 0(3), 1, 3)$ respectively and arrival phase changes to 1, 2, 3 respectively at the end of idle time under various arrival and service processes
 L(3, 1) M(3, 1) N(3, 1) L(3, 2) M(3, 2) N(3, 2) L(3, 3) M(3, 3) N(3, 3) ZCS NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ ZCS NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ ZCS ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ ZCS PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$ PCS NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ PCS NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ PCS ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ PCS PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$ NCS(2a) NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ NCS(2a) NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ NCS(2a) ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ NCS(2a) PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$
 L(3, 1) M(3, 1) N(3, 1) L(3, 2) M(3, 2) N(3, 2) L(3, 3) M(3, 3) N(3, 3) ZCS NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ ZCS NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ ZCS ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ZCS PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ ZCS PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$ PCS NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ PCS NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ PCS ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ PCS PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ PCS PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$ NCS(2a) NCA(2a) ${0.0044}$ ${0.0168}$ ${1.2523}$ ${0}$ ${0}$ ${0}$ ${0}$ ${1.5682}$ ${0.7559}$ NCS(2a) NCA(2b) ${0.0483}$ ${0.1888}$ ${1.0304}$ ${0.0032}$ ${0.1781}$ ${0.1684}$ ${0.0088}$ ${1.1688}$ ${0.9376}$ NCS(2a) ZCA ${0.2000}$ ${1.0267}$ ${1.8947}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ ${0}$ NCS(2a) PCA(3a) ${0.0176}$ ${0.1166}$ ${0.2426}$ ${0.0214}$ ${0.1252}$ ${0.2598}$ ${0.0891}$ ${0.2358}$ ${0.4843}$ NCS(2a) PCA(3b) ${0}$ ${0.0159}$ ${0.0449}$ ${0}$ ${0}$ ${0}$ ${0.0044}$ ${0.0080}$ ${0.0120}$
$A_1(j, j')$ and $B_1(j, j')$
 $A (1, 1)$ $A (1, 2)$ $A(2, 1)$ $A(2, 2)$ $B(1, 1)$ B(1, 2) $B(2, 1)$ $B(2, 2)$ ZCS NCA(2a) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS NCA(2b) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS ZCA ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS PCA(3a) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS PCA(3b) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ PCS NCA(2a) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS NCA(2b) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS ZCA ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS PCA(3a) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS PCA(3b) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ NCS(2a) NCA(2a) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) NCA(2b) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) ZCA ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) PCA(3a) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) PCA(3b) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$
 $A (1, 1)$ $A (1, 2)$ $A(2, 1)$ $A(2, 2)$ $B(1, 1)$ B(1, 2) $B(2, 1)$ $B(2, 2)$ ZCS NCA(2a) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS NCA(2b) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS ZCA ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS PCA(3a) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ ZCS PCA(3b) ${0}$ ${0.3000}$ ${0}$ ${0.1000}$ ${0}$ ${0.3400}$ ${0}$ ${0.1150}$ PCS NCA(2a) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS NCA(2b) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS ZCA ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS PCA(3a) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ PCS PCA(3b) ${0.0838}$ ${0.0093}$ ${0.0784}$ ${2.0463}$ ${0.0963}$ ${0.0110}$ ${0.0974}$ ${2.3432}$ NCS(2a) NCA(2a) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) NCA(2b) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) ZCA ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) PCA(3a) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$ NCS(2a) PCA(3b) ${0.3631}$ ${0.1072}$ ${0.1397}$ ${0.0389}$ ${0.4172}$ ${0.1227}$ ${0.1618}$ ${0.0453}$
$A_2(j, j')$ and $B_2(j, j')$
 $A 2(1, 1)$ $A2 (1, 2)$ $A2(2, 1)$ $A2(2, 2)$ $B2(1, 1)$ B2(1, 2) $B2(2, 1)$ $B2(2, 2)$ ZCS NCA(2a) ${0}$ ${1.4212}$ ${0}$ ${0.4737}$ ${0}$ ${0.4599}$ ${0}$ ${0.1717}$ ZCS NCA(2b) ${0}$ ${1.1870}$ ${0}$ ${0.3957}$ ${0}$ ${0.5642}$ ${0}$ ${0.2033}$ ZCS ZCA ${0}$ ${1.9840}$ ${0}$ ${0.6613}$ ${0}$ ${1.5335}$ ${0}$ ${0.5363}$ ZCS PCA(3a) ${0}$ ${2.5277}$ ${0}$ ${0.8426}$ ${0}$ ${2.3911}$ ${0}$ ${0.8288}$ ZCS PCA(3b) ${0}$ ${3.7649}$ ${0}$ ${1.2550}$ ${0}$ ${2.9230}$ ${0}$ ${1.0221}$ PCS NCA(2a) ${0.3972}$ ${0.0441}$ ${0.3716}$ ${9.6942}$ ${0.1426}$ ${0.0197}$ ${0.2146}$ ${3.4026}$ PCS NCA(2b) ${0.3317}$ ${0.0368}$ ${0.3103}$ ${8.0964}$ ${0.1693}$ ${0.0220}$ ${0.2257}$ ${4.0690}$ PCS ZCA ${0.5544}$ ${0.0615}$ ${0.5187}$ ${13.5328}$ ${0.4477}$ ${0.0550}$ ${0.5298}$ ${10.8229}$ PCS PCA(3a) ${0.7063}$ ${0.0784}$ ${0.6608}$ ${17.2411}$ ${0.6924}$ ${0.0835}$ ${0.7879}$ ${16.7681}$ PCS PCA(3b) ${1.0521}$ ${0.1168}$ ${0.9843}$ ${25.6805}$ ${0.8533}$ ${0.1048}$ ${1.0087}$ ${20.6271}$ NCS(2a) NCA(2a) ${1.7200}$ ${0.5080}$ ${0.6618}$ ${0.1843}$ ${0.6196}$ ${0.1777}$ ${0.2529}$ ${0.0732}$ NCS(2a) NCA(2b) ${1.4365}$ ${0.4243}$ ${0.5528}$ ${0.1539}$ ${0.7351}$ ${0.2128}$ ${0.2949}$ ${0.0844}$ NCS(2a) ZCA ${2.4011}$ ${0.7091}$ ${0.9239}$ ${0.2573}$ ${1.9419}$ ${0.5664}$ ${0.7671}$ ${0.2174}$ NCS(2a) PCA(3a) ${3.0590}$ ${0.9035}$ ${1.1771}$ ${0.3278}$ ${3.0025}$ ${0.8777}$ ${1.1804}$ ${0.3335}$ NCS(2a) PCA(3b) ${4.5564}$ ${1.3457}$ ${1.7533}$ ${0.4882}$ ${3.7009}$ ${1.0794}$ ${1.4617}$ ${0.4143}$
 $A 2(1, 1)$ $A2 (1, 2)$ $A2(2, 1)$ $A2(2, 2)$ $B2(1, 1)$ B2(1, 2) $B2(2, 1)$ $B2(2, 2)$ ZCS NCA(2a) ${0}$ ${1.4212}$ ${0}$ ${0.4737}$ ${0}$ ${0.4599}$ ${0}$ ${0.1717}$ ZCS NCA(2b) ${0}$ ${1.1870}$ ${0}$ ${0.3957}$ ${0}$ ${0.5642}$ ${0}$ ${0.2033}$ ZCS ZCA ${0}$ ${1.9840}$ ${0}$ ${0.6613}$ ${0}$ ${1.5335}$ ${0}$ ${0.5363}$ ZCS PCA(3a) ${0}$ ${2.5277}$ ${0}$ ${0.8426}$ ${0}$ ${2.3911}$ ${0}$ ${0.8288}$ ZCS PCA(3b) ${0}$ ${3.7649}$ ${0}$ ${1.2550}$ ${0}$ ${2.9230}$ ${0}$ ${1.0221}$ PCS NCA(2a) ${0.3972}$ ${0.0441}$ ${0.3716}$ ${9.6942}$ ${0.1426}$ ${0.0197}$ ${0.2146}$ ${3.4026}$ PCS NCA(2b) ${0.3317}$ ${0.0368}$ ${0.3103}$ ${8.0964}$ ${0.1693}$ ${0.0220}$ ${0.2257}$ ${4.0690}$ PCS ZCA ${0.5544}$ ${0.0615}$ ${0.5187}$ ${13.5328}$ ${0.4477}$ ${0.0550}$ ${0.5298}$ ${10.8229}$ PCS PCA(3a) ${0.7063}$ ${0.0784}$ ${0.6608}$ ${17.2411}$ ${0.6924}$ ${0.0835}$ ${0.7879}$ ${16.7681}$ PCS PCA(3b) ${1.0521}$ ${0.1168}$ ${0.9843}$ ${25.6805}$ ${0.8533}$ ${0.1048}$ ${1.0087}$ ${20.6271}$ NCS(2a) NCA(2a) ${1.7200}$ ${0.5080}$ ${0.6618}$ ${0.1843}$ ${0.6196}$ ${0.1777}$ ${0.2529}$ ${0.0732}$ NCS(2a) NCA(2b) ${1.4365}$ ${0.4243}$ ${0.5528}$ ${0.1539}$ ${0.7351}$ ${0.2128}$ ${0.2949}$ ${0.0844}$ NCS(2a) ZCA ${2.4011}$ ${0.7091}$ ${0.9239}$ ${0.2573}$ ${1.9419}$ ${0.5664}$ ${0.7671}$ ${0.2174}$ NCS(2a) PCA(3a) ${3.0590}$ ${0.9035}$ ${1.1771}$ ${0.3278}$ ${3.0025}$ ${0.8777}$ ${1.1804}$ ${0.3335}$ NCS(2a) PCA(3b) ${4.5564}$ ${1.3457}$ ${1.7533}$ ${0.4882}$ ${3.7009}$ ${1.0794}$ ${1.4617}$ ${0.4143}$
 [1] Yi Peng, Jinbiao Wu. Analysis of a batch arrival retrial queue with impatient customers subject to the server disasters. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2243-2264. doi: 10.3934/jimo.2020067 [2] Wenjuan Zhao, Shunfu Jin, Wuyi Yue. A stochastic model and social optimization of a blockchain system based on a general limited batch service queue. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1845-1861. doi: 10.3934/jimo.2020049 [3] Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040 [4] Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393 [5] Jan Rychtář, Dewey T. Taylor. Moran process and Wright-Fisher process favor low variability. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3491-3504. doi: 10.3934/dcdsb.2020242 [6] Zhenquan Zhang, Meiling Chen, Jiajun Zhang, Tianshou Zhou. Analysis of non-Markovian effects in generalized birth-death models. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3717-3735. doi: 10.3934/dcdsb.2020254 [7] Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021048 [8] Alexander Tolstonogov. BV solutions of a convex sweeping process with a composed perturbation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021012 [9] José Antonio Carrillo, Martin Parisot, Zuzanna Szymańska. Mathematical modelling of collagen fibres rearrangement during the tendon healing process. Kinetic & Related Models, 2021, 14 (2) : 283-301. doi: 10.3934/krm.2021005 [10] Xu Zhang, Zhanglin Peng, Qiang Zhang, Xiaoan Tang, Panos M. Pardalos. Identifying and determining crowdsourcing service strategies: An empirical study on a crowdsourcing platform in China. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021045 [11] Peng Tong, Xiaogang Ma. Design of differentiated warranty coverage that considers usage rate and service option of consumers under 2D warranty policy. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1577-1591. doi: 10.3934/jimo.2020035 [12] Jinsen Guo, Yongwu Zhou, Baixun Li. The optimal pricing and service strategies of a dual-channel retailer under free riding. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021056 [13] Jun Tu, Zijiao Sun, Min Huang. Supply chain coordination considering e-tailer's promotion effort and logistics provider's service effort. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021062 [14] Kehan Si, Zhenda Xu, Ka Fai Cedric Yiu, Xun Li. Open-loop solvability for mean-field stochastic linear quadratic optimal control problems of Markov regime-switching system. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021074 [15] Hai-Yang Jin, Zhi-An Wang. The Keller-Segel system with logistic growth and signal-dependent motility. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3023-3041. doi: 10.3934/dcdsb.2020218 [16] Paul Deuring. Spatial asymptotics of mild solutions to the time-dependent Oseen system. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021044 [17] Qian Cao, Yongli Cai, Yong Luo. Nonconstant positive solutions to the ratio-dependent predator-prey system with prey-taxis in one dimension. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021095 [18] Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 [19] Liqiang Jin, Yanqing Liu, Yanyan Yin, Kok Lay Teo, Fei Liu. Design of probabilistic $l_2-l_\infty$ filter for uncertain Markov jump systems with partial information of the transition probabilities. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021070 [20] Omer Gursoy, Kamal Adli Mehr, Nail Akar. Steady-state and first passage time distributions for waiting times in the $MAP/M/s+G$ queueing model with generally distributed patience times. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021078

2019 Impact Factor: 1.366

## Tools

Article outline

Figures and Tables