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doi: 10.3934/jimo.2019085

A two-priority single server retrial queue with additional items

Dhanya Shajin 1, , A. N. Dudin 2,3,, , Olga Dudina 2,3, and  A. Krishnamoorthy 4,

1. 

Department of Mathematics, S. N. College, Chempazhanthy, Trivandrum, Kerala-695587, India
2. 

Belarusian State University, 4, Nezavisimosti Ave., Minsk, 220030, Belarus
3. 

Peoples Friendship University of Russia, 6 Miklukho-Maklaya St, Moscow, 117198, Russia
4. 

Department of Mathematics, CMS College, Kottayam-686001, India

* Corresponding author: A. N. Dudin

Received  October 2018 Revised  March 2019 Published  July 2019

Fund Project: The first author is supported by Kerala State Council for Science, Technology & Environment: 001-07/PDF/2016/KSCSTE in Department of Mathematics, CMS College, Kottayam-686001, India.
The second author is supported by "RUDN University Program 5-100".
The fourth author is supported by UGC No.F.6-6/2017-18/EMERITUS-2017-18-GEN-10822 (SA-Ⅱ) and DST project INT/RUS/RSF/P-15.

In this paper, we study a priority queueing-inventory problem with two types of customers. Arrival of customers follows Marked Markovian arrival process and service times have phase-type distribution with parameters depending on the type of customer in service. For service of each type of customer, a certain number of additional items are needed. High priority customers do not have waiting space and so leave the system when on their arrival a priority 1 customer is in service or the number of available additional items is less than the required threshold. Preemptive priority is assumed. Type 2 customers, encountering a busy server or idle with the number of available additional items less than a threshold, go to an orbit of infinite capacity to retry for service. The customers in orbit are non-persistent: if on retrial the server is found to be busy/idle with the number of additional items less than the threshold, this customer abandons the system with certain probability. Such a system represents an accurate enough model of many real-world systems, including wireless sensor networks and system of cognitive radio with energy harvesting and healthcare systems. The probability distribution of the system states is computed, using which several of the characteristics are derived. A detailed numerical study of the system, including the analysis of the influence of the threshold, is performed.

Keywords: Markovian arrival process (MAP), marked Markovian arrival process (MMAP), phase-type distribution, additional items, retrial queue.
Mathematics Subject Classification: Primary: 60K25, 90B05; Secondary: 68M20, 90B22.
Citation: Dhanya Shajin, A. N. Dudin, Olga Dudina, A. Krishnamoorthy. A two-priority single server retrial queue with additional items. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2019085
References:
[1]

A. Arafa, T. Tong, M. Fu, S. Ulukus and W. Chen, Delay minimal policies in energy harvesting communication systems, IEEE Transactions on Communications, 66 (2018), 2918-2930. Google Scholar

[2]

J. R. Artalejo and A. Gomez Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. Google Scholar

[3]

J. Baek, O. Dudina and C. Kim, Queueing system with heterogeneous impatient customers and consumable additional items, International Journal of Applied Mathematics and Computer Science, 27 (2017), 367-384. doi: 10.1515/amcs-2017-0026. Google Scholar

[4]

L. Boutarfa and N. Djellab, On the performance of the $M_1, M_2/G_1, G_2/1$ retrial queue with pre-emptive resume policy, Yugoslav Journal of Operations Research, 25 (2015), 153-164. doi: 10.2298/YJOR130217001B. Google Scholar

[5]

B. Cardoen, E. Demeulemeester and J. Belien, Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932. doi: 10.1016/j.ejor.2012.11.029. Google Scholar

[6]

S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Process: Notable Publications (eds. A. Krishnamoorthy, et al.), New Jersey, 2001, 21–49.Google Scholar

[7]

B. D. Choi and Y. Chang, Single server retrial queues with priority customers, Mathematical and Computer Modelling, 30 (1999), 7-32. doi: 10.1016/S0895-7177(99)00129-6. Google Scholar

[8]

E. De Cuypere, K. De Turck and D. Fiems, A queueing model of an energy harvesting sensor node with data buffering, Telecommunication Systems, 67 (2018), 281-295. Google Scholar

[9]

I. Dimitrious, Analysis of a priority retrial queue with dependent vacation scheme and application to power saving in wireless communication systems, The Computer Journal, 56 (2015), 1363-1380. Google Scholar

[10]

A. N. Dudin and V. I. Klimenok, Queueing systems with passive servers, International Journal of Stochastic Analysis, 9 (1996), 185-204. doi: 10.1155/S1048953396000184. Google Scholar

[11]

A. N. Dudin, M. H. Lee, O. Dudina and S. K. Lee, Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system, IEEE Transactions on Communications, 65 (2017), 186-199. Google Scholar

[12]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.Google Scholar

[13]

G. I. Falin, J. R. Artalejo and M. Martin, On the single server retrial queue with priority customers, Queueing Systems, 14 (1993), 439-455. doi: 10.1007/BF01158878. Google Scholar

[14]

S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251. Google Scholar

[15]

Q. M. He, Queues with marked customers, Advances in Applied Probability, (1996), 567–587. doi: 10.2307/1428072. Google Scholar

[16]

C. Kim, S. Dudin and V. Klimenok, The $MAP/PH/1/N$ queue with flows of customers as model for traffic control in telecommunication networks, Performance Evaluation, 66 (2009), 564-579. Google Scholar

[17]

C. Kim, S. Dudin and V. Klimenok, Queueing system with batch arrival of customers in sessions, Computers and Industrial Engineering, 62 (2012), 890-897. Google Scholar

[18]

J. Kim and B. Kim, A survey of retrial queueing systems, Annals of Operations Research, 247 (2016), 3-36. doi: 10.1007/s10479-015-2038-7. Google Scholar

[19]

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

[20]

A. Krishnamoorthy, B. Binitha and D. Shajin, A revisit to queueing-inventory system with reservation, cancellation and common life time, OPSEARCH, 54 (2017), 336-350. Google Scholar

[21]

A. Krishnamoorthy, D. Shajin and B. Lakshmy, On a queueing-inventory with reservation, cancellation, common life time and retrial, Annals of Operations Research, (2016), 1–25. doi: 10.1007/s10479-015-1849-x. Google Scholar

[22]

A. Krishnamoorthy, D. Shajin and B. Lakshmy, Product form solution for some queueing-inventory supply chain problem, OPSEARCH, 53 (2016), 85-102. doi: 10.1007/s12597-015-0215-8. Google Scholar

[23]

C. Langaris and E. Moutzoukis, A retrial queue with structured batch arrivals, priorities and server vacations, Queueing Systems, 20 (1995), 341-368. doi: 10.1007/BF01245324. Google Scholar

[24]

M. Liu, M. Feng and C. Y. Wong, Flexible service policies for a Markov inventory system with two demand classes, International Journal of Production Economics, 151 (2014), 180-185. Google Scholar

[25]

K. Patil, K. De Turck and D. Fiems, A two-queue model for optimising the value of information in energy-harvesting sensor networks, Performance Evaluation, 119 (2018), 27-42. doi: 10.1016/j.orl.2018.04.002. Google Scholar

[26]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, (to appear) in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), WSP, 2017.Google Scholar

[27]

J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, Performance Evaluation, 127 (2018), 235-252. Google Scholar

[28]

Y. Wang, X. Tang and T. Wang, A unified QoS and security provisioning framework for wiretap cognitive radio networks: A statistical queueing analysis approach, IEEE Transactions on Wireless Communications, 18 (2019), 1548-1565. doi: 10.1109/TWC.2019.2893381. Google Scholar

[29]

N. Zhao and Z. Lian, A queueing-inventory system with two classes of customers, International Journal of Production Economics, 129 (2011), 225-231. Google Scholar

show all references

References:
[1]

A. Arafa, T. Tong, M. Fu, S. Ulukus and W. Chen, Delay minimal policies in energy harvesting communication systems, IEEE Transactions on Communications, 66 (2018), 2918-2930. Google Scholar

[2]

J. R. Artalejo and A. Gomez Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9. Google Scholar

[3]

J. Baek, O. Dudina and C. Kim, Queueing system with heterogeneous impatient customers and consumable additional items, International Journal of Applied Mathematics and Computer Science, 27 (2017), 367-384. doi: 10.1515/amcs-2017-0026. Google Scholar

[4]

L. Boutarfa and N. Djellab, On the performance of the $M_1, M_2/G_1, G_2/1$ retrial queue with pre-emptive resume policy, Yugoslav Journal of Operations Research, 25 (2015), 153-164. doi: 10.2298/YJOR130217001B. Google Scholar

[5]

B. Cardoen, E. Demeulemeester and J. Belien, Operating room planning and scheduling: A literature review, European Journal of Operational Research, 201 (2010), 921-932. doi: 10.1016/j.ejor.2012.11.029. Google Scholar

[6]

S. R. Chakravarthy, The batch Markovian arrival process: A review and future work, Advances in Probability Theory and Stochastic Process: Notable Publications (eds. A. Krishnamoorthy, et al.), New Jersey, 2001, 21–49.Google Scholar

[7]

B. D. Choi and Y. Chang, Single server retrial queues with priority customers, Mathematical and Computer Modelling, 30 (1999), 7-32. doi: 10.1016/S0895-7177(99)00129-6. Google Scholar

[8]

E. De Cuypere, K. De Turck and D. Fiems, A queueing model of an energy harvesting sensor node with data buffering, Telecommunication Systems, 67 (2018), 281-295. Google Scholar

[9]

I. Dimitrious, Analysis of a priority retrial queue with dependent vacation scheme and application to power saving in wireless communication systems, The Computer Journal, 56 (2015), 1363-1380. Google Scholar

[10]

A. N. Dudin and V. I. Klimenok, Queueing systems with passive servers, International Journal of Stochastic Analysis, 9 (1996), 185-204. doi: 10.1155/S1048953396000184. Google Scholar

[11]

A. N. Dudin, M. H. Lee, O. Dudina and S. K. Lee, Analysis of priority retrial queue with many types of customers and servers reservation as a model of cognitive radio system, IEEE Transactions on Communications, 65 (2017), 186-199. Google Scholar

[12]

G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997.Google Scholar

[13]

G. I. Falin, J. R. Artalejo and M. Martin, On the single server retrial queue with priority customers, Queueing Systems, 14 (1993), 439-455. doi: 10.1007/BF01158878. Google Scholar

[14]

S. Gao, A preemptive priority retrial queue with two classes of customers and general retrial times, Operational Research, 15 (2015), 233-251. Google Scholar

[15]

Q. M. He, Queues with marked customers, Advances in Applied Probability, (1996), 567–587. doi: 10.2307/1428072. Google Scholar

[16]

C. Kim, S. Dudin and V. Klimenok, The $MAP/PH/1/N$ queue with flows of customers as model for traffic control in telecommunication networks, Performance Evaluation, 66 (2009), 564-579. Google Scholar

[17]

C. Kim, S. Dudin and V. Klimenok, Queueing system with batch arrival of customers in sessions, Computers and Industrial Engineering, 62 (2012), 890-897. Google Scholar

[18]

J. Kim and B. Kim, A survey of retrial queueing systems, Annals of Operations Research, 247 (2016), 3-36. doi: 10.1007/s10479-015-2038-7. Google Scholar

[19]

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

[20]

A. Krishnamoorthy, B. Binitha and D. Shajin, A revisit to queueing-inventory system with reservation, cancellation and common life time, OPSEARCH, 54 (2017), 336-350. Google Scholar

[21]

A. Krishnamoorthy, D. Shajin and B. Lakshmy, On a queueing-inventory with reservation, cancellation, common life time and retrial, Annals of Operations Research, (2016), 1–25. doi: 10.1007/s10479-015-1849-x. Google Scholar

[22]

A. Krishnamoorthy, D. Shajin and B. Lakshmy, Product form solution for some queueing-inventory supply chain problem, OPSEARCH, 53 (2016), 85-102. doi: 10.1007/s12597-015-0215-8. Google Scholar

[23]

C. Langaris and E. Moutzoukis, A retrial queue with structured batch arrivals, priorities and server vacations, Queueing Systems, 20 (1995), 341-368. doi: 10.1007/BF01245324. Google Scholar

[24]

M. Liu, M. Feng and C. Y. Wong, Flexible service policies for a Markov inventory system with two demand classes, International Journal of Production Economics, 151 (2014), 180-185. Google Scholar

[25]

K. Patil, K. De Turck and D. Fiems, A two-queue model for optimising the value of information in energy-harvesting sensor networks, Performance Evaluation, 119 (2018), 27-42. doi: 10.1016/j.orl.2018.04.002. Google Scholar

[26]

T. Phung-Duc, Retrial queueing models: A survey on theory and applications, (to appear) in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), WSP, 2017.Google Scholar

[27]

J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, Performance Evaluation, 127 (2018), 235-252. Google Scholar

[28]

Y. Wang, X. Tang and T. Wang, A unified QoS and security provisioning framework for wiretap cognitive radio networks: A statistical queueing analysis approach, IEEE Transactions on Wireless Communications, 18 (2019), 1548-1565. doi: 10.1109/TWC.2019.2893381. Google Scholar

[29]

N. Zhao and Z. Lian, A queueing-inventory system with two classes of customers, International Journal of Production Economics, 129 (2011), 225-231. Google Scholar

Figure A.  Picture representation of the model
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Figure 1.  Dependence of average number of customers in the orbit $ N_{O} $ and average number of additional items in the stock $ N_{item} $ on $ N $
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Figure 2.  Dependence of probability that an arbitrary type 1 customer will be lost $ p_{1}^{loss} $ and probability that an arbitrary additional item will be lost $ p_{item}^{loss} $ on $ N $
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Figure 3.  Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer $ p_{1}^{busy\ loss} $ and probability of an arbitrary arriving type 1 customer loss due to lack of additional items $ p_{1}^{lack\ loss} $ on $ N $
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Figure 4.  Dependence of average number of customers in the orbit $ N_{O} $ and average number of additional items in the stock $ N_{item} $ on $ q $
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Figure 5.  Dependence of probability that an arbitrary type 1 customer will be lost $ p_{1}^{loss} $ and probability that an arbitrary additional item will be lost $ p_{item}^{loss} $ on $ q $
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Figure 6.  Dependence of probability of an arbitrary arriving type 1 customer loss because the server is busy with type 1 customer $ p_{1}^{busy\ loss} $ and probability of an arbitrary arriving type 1 customer loss due to lack of additional items $ p_{1}^{lack\ loss} $ on $ q $
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Figure 7.  Dependence of $ N_O $ on $ \gamma $
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Table 1.  Dependence of $ N_O $ and Nitem on N for q = 0.2
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
2 0.541687 1.367441 1.684218 1.271213
4 0.549554 1.412635 1.703399 1.305689
6 0.5646816 1.518695 1.755786 1.403333
8 0.581552 1.588845 1.784279 1.466440
10 0.60139 1.626319 1.805897 1.509774
12 0.628908 1.656662 1.827878 1.551554
14 0.669292 1.682243 1.849624 1.590796
16 0.732318 1.709136 1.874669 1.631378
18 0.831686 1.742311 1.904638 1.674188
20 1.044837 1.802246 1.949755 1.728375
(A) Dependence of NO
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
2 15.265487 15.01827 12.685517 7.343982
4 15.360819 15.39104 13.30253 8.478718
6 15.489897 15.661518 13.504709 8.945458
8 15.707897 16.001991 13.742329 9.415827
10 16.008232 16.420983 13.98464 10.00213
12 16.376142 16.857811 14.170774 10.51655
14 16.821438 17.324321 14.475216 11.21957
16 17.34789 17.797588 14.7440311 11.92004
18 17.91796 18.255182 15.1044063 12.71894
20 18.51922 18.665273 15.5122786 13.58646
(B) Dependence of Nitem
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$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
2 0.541687 1.367441 1.684218 1.271213
4 0.549554 1.412635 1.703399 1.305689
6 0.5646816 1.518695 1.755786 1.403333
8 0.581552 1.588845 1.784279 1.466440
10 0.60139 1.626319 1.805897 1.509774
12 0.628908 1.656662 1.827878 1.551554
14 0.669292 1.682243 1.849624 1.590796
16 0.732318 1.709136 1.874669 1.631378
18 0.831686 1.742311 1.904638 1.674188
20 1.044837 1.802246 1.949755 1.728375
(A) Dependence of NO
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0}$
2 15.265487 15.01827 12.685517 7.343982
4 15.360819 15.39104 13.30253 8.478718
6 15.489897 15.661518 13.504709 8.945458
8 15.707897 16.001991 13.742329 9.415827
10 16.008232 16.420983 13.98464 10.00213
12 16.376142 16.857811 14.170774 10.51655
14 16.821438 17.324321 14.475216 11.21957
16 17.34789 17.797588 14.7440311 11.92004
18 17.91796 18.255182 15.1044063 12.71894
20 18.51922 18.665273 15.5122786 13.58646
(B) Dependence of Nitem
Table 2.  Dependence of $ p_1^{loss} $ and pitemloss on N for q = 0.2
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.075438 0.594164 0.621250 0.584869
4 0.069154 0.528047 0.612127 0.574752
6 0.053754 0.343713 0.526146 0.420849
8 0.044331 0.230872 0.488863 0.334311
10 0.041304 0.184474 0.466582 0.291888
12 0.039596 0.154385 0.442873 0.251298
14 0.039017 0.138246 0.426295 0.224901
16 0.038708 0.127744 0.407572 0.199159
18 0.038589 0.12157 0.392962 0.179236
20 0.038535 0.117919 0.37687 0.16082
(A) Dependence of ploss of p1loss
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.218984 0.596466 0.720285 0.570702
4 0.22019 0.599877 0.725534 0.58051
6 0.222009 0.60253 0.727949 0.584911
8 0.225624 0.60626 0.731018 0.590137
10 0.23144 0.611019 0.734444 0.596847
12 0.23998 0.616327 0.737665 0.6033
14 0.25284 0.62257 0.742163 0.61164
16 0.272944 0.630337 0.747311 0.620533
18 0.304933 0.641239 0.754921 0.631276
20 0.373814 0.662226 0.767272 0.645899
(B) Dependence of ploss of pitemloss
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$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.075438 0.594164 0.621250 0.584869
4 0.069154 0.528047 0.612127 0.574752
6 0.053754 0.343713 0.526146 0.420849
8 0.044331 0.230872 0.488863 0.334311
10 0.041304 0.184474 0.466582 0.291888
12 0.039596 0.154385 0.442873 0.251298
14 0.039017 0.138246 0.426295 0.224901
16 0.038708 0.127744 0.407572 0.199159
18 0.038589 0.12157 0.392962 0.179236
20 0.038535 0.117919 0.37687 0.16082
(A) Dependence of ploss of p1loss
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.218984 0.596466 0.720285 0.570702
4 0.22019 0.599877 0.725534 0.58051
6 0.222009 0.60253 0.727949 0.584911
8 0.225624 0.60626 0.731018 0.590137
10 0.23144 0.611019 0.734444 0.596847
12 0.23998 0.616327 0.737665 0.6033
14 0.25284 0.62257 0.742163 0.61164
16 0.272944 0.630337 0.747311 0.620533
18 0.304933 0.641239 0.754921 0.631276
20 0.373814 0.662226 0.767272 0.645899
(B) Dependence of ploss of pitemloss
Table 3.  Dependence of $ p_1^{busy\ loss} $and p1lack loss on N for q = 0.2
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.036982 0.045292 0.042889 0.016605
4 0.037234 0.054023 0.043846 0.01701
6 0.03785 0.07846 0.054843 0.023166
8 0.038227 0.093433 0.059606 0.026628
10 0.038347 0.099601 0.062507 0.028324
12 0.038416 0.103602 0.06565 0.029948
14 0.038439 0.105748 0.06781 0.031004
16 0.038452 0.107144 0.07029 0.032034
18 0.038456 0.107965 0.07221 0.032831
20 0.038459 0.10845 0.07433 0.033567
(A) Dependence of p1busy loss
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.038456 0.548872 0.578361 0.568264
4 0.03192 0.474025 0.568281 0.557742
6 0.015904 0.265254 0.471303 0.397683
8 0.006105 0.13744 0.429257 0.307683
10 0.002956 0.084873 0.404075 0.263563
12 0.00118 0.050783 0.377223 0.22135
14 5.77E-4 0.032498 0.358485 0.193897
16 2.56E-4 0.020599 0.337281 0.167125
18 1.33E-4 0.013605 0.320755 0.146405
20 7.63E-5 0.009469 0.302539 0.127253
(B) Dependence of p1lack loss
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$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.036982 0.045292 0.042889 0.016605
4 0.037234 0.054023 0.043846 0.01701
6 0.03785 0.07846 0.054843 0.023166
8 0.038227 0.093433 0.059606 0.026628
10 0.038347 0.099601 0.062507 0.028324
12 0.038416 0.103602 0.06565 0.029948
14 0.038439 0.105748 0.06781 0.031004
16 0.038452 0.107144 0.07029 0.032034
18 0.038456 0.107965 0.07221 0.032831
20 0.038459 0.10845 0.07433 0.033567
(A) Dependence of p1busy loss
$ N $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0}$
2 0.038456 0.548872 0.578361 0.568264
4 0.03192 0.474025 0.568281 0.557742
6 0.015904 0.265254 0.471303 0.397683
8 0.006105 0.13744 0.429257 0.307683
10 0.002956 0.084873 0.404075 0.263563
12 0.00118 0.050783 0.377223 0.22135
14 5.77E-4 0.032498 0.358485 0.193897
16 2.56E-4 0.020599 0.337281 0.167125
18 1.33E-4 0.013605 0.320755 0.146405
20 7.63E-5 0.009469 0.302539 0.127253
(B) Dependence of p1lack loss
Table 4.  Dependence of $ N_O $ and Nitem on q for N = 4
$ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.758933 2.592361 3.261523 2.397430
0.2 0.549554 1.412635 1.703399 1.305689
0.3 0.441755 0.987633 1.162907 0.911481
0.4 0.372968 0.765297 0.886588 0.705013
0.5 0.324288 0.627350 0.718082 0.576937
0.6 0.287627 0.532879 0.604271 0.489305
0.7 0.258835 0.463871 0.522087 0.425367
0.8 0.235527 0.411117 0.459869 0.376551
0.9 0.216218 0.369402 0.411081 0.337998
1 0.199928 0.335543 0.371769 0.306743
(A) Dependence of NO
$ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
0.1 14.631235 15.054343 13.047518 8.294399
0.2 15.360819 15.391041 13.302530 8.478718
0.3 15.760671 15.548513 13.408673 8.583818
0.4 16.021155 15.649013 13.470799 8.656368
0.5 16.206952 15.721749 13.512953 8.710798
0.6 16.347252 15.778002 13.544014 8.753684
0.7 16.457467 15.823323 13.568133 8.788605
0.8 16.546609 15.860874 13.587549 8.817730
0.9 16.620347 15.892632 13.603596 8.842468
1 16.682445 15.919918 13.617128 8.863788
(B) Dependence of Nitem
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$ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.758933 2.592361 3.261523 2.397430
0.2 0.549554 1.412635 1.703399 1.305689
0.3 0.441755 0.987633 1.162907 0.911481
0.4 0.372968 0.765297 0.886588 0.705013
0.5 0.324288 0.627350 0.718082 0.576937
0.6 0.287627 0.532879 0.604271 0.489305
0.7 0.258835 0.463871 0.522087 0.425367
0.8 0.235527 0.411117 0.459869 0.376551
0.9 0.216218 0.369402 0.411081 0.337998
1 0.199928 0.335543 0.371769 0.306743
(A) Dependence of NO
$ q $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+ MMAP^{0} $
0.1 14.631235 15.054343 13.047518 8.294399
0.2 15.360819 15.391041 13.302530 8.478718
0.3 15.760671 15.548513 13.408673 8.583818
0.4 16.021155 15.649013 13.470799 8.656368
0.5 16.206952 15.721749 13.512953 8.710798
0.6 16.347252 15.778002 13.544014 8.753684
0.7 16.457467 15.823323 13.568133 8.788605
0.8 16.546609 15.860874 13.587549 8.817730
0.9 16.620347 15.892632 13.603596 8.842468
1 16.682445 15.919918 13.617128 8.863788
(B) Dependence of Nitem
Table 5.  Dependence of $ p_1^{loss} $ and pitemloss on q for N = 4
$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.086069 0.557286 0.619555 0.588222
0.2 0.069154 0.528047 0.612127 0.574752
0.3 0.061659 0.506329 0.607792 0.566495
0.4 0.057424 0.489551 0.604719 0.560590
0.5 0.054707 0.476222 0.602361 0.556081
0.6 0.052819 0.465381 0.600469 0.552495
0.7 0.051434 0.456390 0.598905 0.549560
0.8 0.050375 0.448809 0.597586 0.547105
0.9 0.049540 0.442329 0.596454 0.545017
1 0.048865 0.436726 0.595472 0.543215
(A) Dependence of p1loss
$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.187902 0.582932 0.711872 0.560124
0.2 0.220190 0.599877 0.725534 0.580510
0.3 0.241291 0.606780 0.731699 0.590660
0.4 0.256630 0.610762 0.735528 0.597193
0.5 0.268448 0.613446 0.738236 0.601898
0.6 0.277902 0.615420 0.740290 0.605506
0.7 0.285673 0.616952 0.741918 0.608389
0.8 0.292192 0.618187 0.743249 0.610759
0.9 0.297748 0.619209 0.744361 0.612748
1 0.302546 0.620072 0.745308 0.614446
(B) Dependence of pitemloss
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$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.086069 0.557286 0.619555 0.588222
0.2 0.069154 0.528047 0.612127 0.574752
0.3 0.061659 0.506329 0.607792 0.566495
0.4 0.057424 0.489551 0.604719 0.560590
0.5 0.054707 0.476222 0.602361 0.556081
0.6 0.052819 0.465381 0.600469 0.552495
0.7 0.051434 0.456390 0.598905 0.549560
0.8 0.050375 0.448809 0.597586 0.547105
0.9 0.049540 0.442329 0.596454 0.545017
1 0.048865 0.436726 0.595472 0.543215
(A) Dependence of p1loss
$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.187902 0.582932 0.711872 0.560124
0.2 0.220190 0.599877 0.725534 0.580510
0.3 0.241291 0.606780 0.731699 0.590660
0.4 0.256630 0.610762 0.735528 0.597193
0.5 0.268448 0.613446 0.738236 0.601898
0.6 0.277902 0.615420 0.740290 0.605506
0.7 0.285673 0.616952 0.741918 0.608389
0.8 0.292192 0.618187 0.743249 0.610759
0.9 0.297748 0.619209 0.744361 0.612748
1 0.302546 0.620072 0.745308 0.614446
(B) Dependence of pitemloss
Table 6.  Dependence of $ p_1^{busy\ loss} $ and p1lack loss on q for N = 4
$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.036557 0.050253 0.043100 0.016471
0.2 0.037234 0.054023 0.043846 0.017010
0.3 0.037534 0.056868 0.044325 0.017340
0.4 0.037703 0.059078 0.044682 0.017576
0.5 0.037812 0.060838 0.044963 0.017757
0.6 0.037887 0.062272 0.045192 0.017900
0.7 0.037943 0.063462 0.045384 0.018018
0.8 0.037985 0.064466 0.045547 0.018116
0.9 0.038018 0.065324 0.045688 0.018199
1.0 0.038045 0.066067 0.045811 0.018271
(A) Dependence of p1busy loss
0.1 0.049511 0.507033 0.576455 0.571751
0.2 0.031920 0.474025 0.568281 0.557742
0.3 0.024125 0.449461 0.563466 0.549155
0.4 0.019721 0.430472 0.560037 0.543014
0.5 0.016895 0.415384 0.557398 0.538324
0.6 0.014932 0.403110 0.555277 0.534595
0.7 0.013491 0.392928 0.553521 0.531543
0.8 0.012390 0.384343 0.552038 0.528990
0.9 0.011521 0.377005 0.550766 0.526817
1.0 0.010820 0.370659 0.549661 0.524944
(B) Dependence of p1lack loss
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$ q $ $ MAP^{0}+MMAP^{0} $ $ MAP^{0}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0.4}$ $ MAP^{0.4}+ MMAP^{0} $
0.1 0.036557 0.050253 0.043100 0.016471
0.2 0.037234 0.054023 0.043846 0.017010
0.3 0.037534 0.056868 0.044325 0.017340
0.4 0.037703 0.059078 0.044682 0.017576
0.5 0.037812 0.060838 0.044963 0.017757
0.6 0.037887 0.062272 0.045192 0.017900
0.7 0.037943 0.063462 0.045384 0.018018
0.8 0.037985 0.064466 0.045547 0.018116
0.9 0.038018 0.065324 0.045688 0.018199
1.0 0.038045 0.066067 0.045811 0.018271
(A) Dependence of p1busy loss
0.1 0.049511 0.507033 0.576455 0.571751
0.2 0.031920 0.474025 0.568281 0.557742
0.3 0.024125 0.449461 0.563466 0.549155
0.4 0.019721 0.430472 0.560037 0.543014
0.5 0.016895 0.415384 0.557398 0.538324
0.6 0.014932 0.403110 0.555277 0.534595
0.7 0.013491 0.392928 0.553521 0.531543
0.8 0.012390 0.384343 0.552038 0.528990
0.9 0.011521 0.377005 0.550766 0.526817
1.0 0.010820 0.370659 0.549661 0.524944
(B) Dependence of p1lack loss
Table 7.  Dependence of $ N_O $ on $ \gamma $ and $ q $ for $ N = 4 $
$ q $ $ \gamma $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0} $
0.1 1.5 0.959861 3.382764 4.300035 3.145578
0.2 0.75 1.328719 3.583343 4.430688 3.384015
0.3 0.5 1.628854 3.746986 4.534785 3.574155
0.4 0.375 1.881123 3.887829 4.623184 3.731486
0.5 0.3 2.098026 4.012896 4.700989 3.865063
0.6 0.25 2.287694 4.126279 4.771088 3.980639
0.7 0.2143 2.455731 4.230599 4.835292 4.082103
0.8 0.1875 2.606171 4.327642 4.89482 4.172209
0.9 0.1667 2.742017 4.418685 4.950534 4.252983
1 0.1500 2.865570 4.504674 5.003066 4.325962
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$ q $ $ \gamma $ $ MAP^{0}+ MMAP^{0}$ $ MAP^{0}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0.4} $ $ MAP^{0.4}+MMAP^{0} $
0.1 1.5 0.959861 3.382764 4.300035 3.145578
0.2 0.75 1.328719 3.583343 4.430688 3.384015
0.3 0.5 1.628854 3.746986 4.534785 3.574155
0.4 0.375 1.881123 3.887829 4.623184 3.731486
0.5 0.3 2.098026 4.012896 4.700989 3.865063
0.6 0.25 2.287694 4.126279 4.771088 3.980639
0.7 0.2143 2.455731 4.230599 4.835292 4.082103
0.8 0.1875 2.606171 4.327642 4.89482 4.172209
0.9 0.1667 2.742017 4.418685 4.950534 4.252983
1 0.1500 2.865570 4.504674 5.003066 4.325962
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