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July  2019, 15(3): 1421-1446. doi: 10.3934/jimo.2018102

## Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption

 1 School of Mathematics and Statistics, Sichuan University of Science and Engineering, Zigong 643000, Sichuan, China 2 School of Mathematics and Software Science, Sichuan Normal University, Chengdu 610066, Sichuan, China 3 School of Fundamental Education, Sichuan Normal University, Chengdu 610066, Sichuan, China

* Corresponding author: Shaojun Lan

Received  October 2017 Revised  March 2018 Published  July 2018

This paper is concerned with a discrete-time $Geo/G/1$ retrial queueing system with non-preemptive priority, working vacations and vacation interruption where the service times and retrial times are arbitrarily distributed. If an arriving customer finds the server free, his service commences immediately. Otherwise, he either joins the priority queue with probability $α$, or leaves the service area and enters the retrial group (orbit) with probability $\mathit{\bar{\alpha }}\left( = 1-\alpha \right)$. Customers in the priority queue have non-preemptive priority over those in the orbit. Whenever the system becomes empty, the server takes working vacation during which the server can serve customers at a lower service rate. If there are customers in the system at the epoch of a service completion, the server resumes the normal working level whether the working vacation ends or not (i.e., working vacation interruption occurs). Otherwise, the server proceeds with the vacation. Employing supplementary variable method and generating function technique, we analyze the underlying Markov chain of the considered queueing model, and obtain the stationary distribution of the Markov chain, the generating functions for the number of customers in the priority queue, in the orbit and in the system, as well as some crucial performance measures in steady state. Furthermore, the relation between our discrete-time queue and its continuous-time counterpart is investigated. Finally, some numerical examples are provided to explore the effect of various system parameters on the queueing characteristics.

Citation: Shaojun Lan, Yinghui Tang. Performance analysis of a discrete-time $Geo/G/1$ retrial queue with non-preemptive priority, working vacations and vacation interruption. Journal of Industrial & Management Optimization, 2019, 15 (3) : 1421-1446. doi: 10.3934/jimo.2018102
##### References:
 [1] A. K. Aboul-Hassan, S. I. Rabia and F. A. Taboly, Performance evaluation of a discrete-time $Geo^{[X]}/G/1$ retrial queue with general retrial times, Computers & Mathematics with Applications, 58 (2009), 548-557.  doi: 10.1016/j.camwa.2009.03.101.  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] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar [4] J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.  Google Scholar [5] I. Atencia and P. Moreno, A discrete-time $Geo/G/1$ retrial queue with general retrial times, Queueing Systems, 48 (2004), 5-21.  doi: 10.1023/B:QUES.0000039885.12490.02.  Google Scholar [6] I. Atencia and P. Moreno, A discrete-time $Geo/G/1$ retrial queue with the server subject to starting failures, Annals of Operations Research, 141 (2006), 85-107.  doi: 10.1007/s10479-006-5295-7.  Google Scholar [7] I. Atencia and P. Moreno, A single-server retrial queue with general retrial times and Bernoulli schedule, Applied Mathematics and Computation, 162 (2005), 855-880.  doi: 10.1016/j.amc.2003.12.128.  Google Scholar [8] H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Boston, 1993. doi: 10.1007/978-1-4615-3130-2.  Google Scholar [9] B. D. Choi and J. W. Kim, Discrete-time $Geo_1, Geo_2/G/1$ retrial queueing systems with two types of calls, Computers & Mathematics with Applications, 33 (1997), 79-88.  doi: 10.1016/S0898-1221(97)00078-3.  Google Scholar [10] I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Applied Mathematical Modelling, 37 (2013), 1295-1309.  doi: 10.1016/j.apm.2012.04.011.  Google Scholar [11] I. Dimitriou, A two class retrial system with coupled orbit queues, Probability in the Engineering and Informational Sciences, 31 (2017), 139-179.  doi: 10.1017/S0269964816000528.  Google Scholar [12] I. Dimitriou, A queueing model with two types of retrial customers and paired services, Annals of Operations Research, 238 (2016), 123-143.  doi: 10.1007/s10479-015-2059-2.  Google Scholar [13] I. Dimitriou, Analysis of a priority retrial queue with dependent vacation scheme and application to energy saving in wireless communication systems, The Computer Journal, 56 (2013), 1363-1380.   Google Scholar [14] T. V. Do, $M/M/1$ retrial queue with working vacations, Acta Informatica, 47 (2010), 67-75.  doi: 10.1007/s00236-009-0110-y.  Google Scholar [15] B. T. Doshi, Queueing systems with vacation-a survey, Queueing Systems, 1 (1986), 29-66.  doi: 10.1007/BF01149327.  Google Scholar [16] A. Dudin, C. S. Kim, S. Dudin and O. Dudina, Priority retrial queueing model operating in random environment with varying number and reservation of servers, Applied Mathematics and Computation, 269 (2015), 674-690.  doi: 10.1016/j.amc.2015.08.005.  Google Scholar [17] G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar [18] G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar [19] A. Gandhi, V. Gupta, M. Harchol-Balter and M. A. Kozuch, Optimality analysis of energy-performance trade-off for server farm management, Performance Evaluation, 67 (2010), 1155-1171.   Google Scholar [20] S. Gao and J. Wang, On a discrete-time $GI^X/Geo/1/N-G$ queue with randomized working vacations and at most J vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [21] S. Gao and J. Wang, Discrete-time $Geo^X/G/1$ retrial queue with general retrial times, working vacations and vacation interruption, Quality Technology & Quantitative Management, 10 (2013), 495-512.   Google Scholar [22] J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, New York, 1983.  Google Scholar [23] M. Jain, G. C. Sharma and R. Sharma, Maximum entropy approach for discrete-time unreliable server $Geo^X/Geo/1$ queue with working vacation, International Journal of Mathematics in Operational Research, 4 (2012), 56-77.  doi: 10.1504/IJMOR.2012.044473.  Google Scholar [24] M. Jain, A. Bhagat and C. Shekhar, Double orbit finite retrial queues with priority customers and service interruptions, Applied Mathematics and Computation, 253 (2015), 324-344.  doi: 10.1016/j.amc.2014.12.066.  Google Scholar [25] J. C. Ke, C. H. Wu and Z. G. Zhang, Recent developments in vacation queueing models: a short survey, International Journal of Operations Research, 7 (2010), 3-8.   Google Scholar [26] P. V. Laxmi and K. Jyothsna, Finite buffer $GI/Geo/1$ batch servicing queue with multiple working vacations, RAIRO-Operations Research, 48 (2014), 521-543.  doi: 10.1051/ro/2014022.  Google Scholar [27] H. Li and T. Yang, $Geo/G/1$ discrete time retrial queue with Bernoulli schedule, European Journal of Operational Research, 111 (1998), 629-649.  doi: 10.1016/S0377-2217(97)90357-X.  Google Scholar [28] J. Li, N. Tian and W. Liu, The discrete-time $GI/Geo/1$ queue with multiple working vacations, Queueing Systems, 56 (2007), 53-63.  doi: 10.1007/s11134-007-9030-0.  Google Scholar [29] J. Li, Analysis of the discrete-time $Geo/G/1$ working vacation queue and its application to network scheduling, Computers & Industrial Engineering, 65 (2013), 594-604.  doi: 10.1016/j.cie.2013.04.009.  Google Scholar [30] T. Li, Z. Wang and Z. Liu, $Geo/Geo/1$ retrial queue with working vacations and vacation interruption, Journal of Applied Mathematics and Computing, 39 (2012), 131-143.  doi: 10.1007/s12190-011-0516-x.  Google Scholar [31] T. Li, Z. Liu and Z. Wang, $M/M/1$ retrial queue with collisions and working vacation interruption under N-policy, RAIRO-Operations Research, 46 (2012), 355-371.  doi: 10.1051/ro/2012022.  Google Scholar [32] Z. Liu and S. Gao, Discrete-time $Geo_1, Geo^X_2/G_1, G_2/1$ retrial queue with two classes of customers and feedback, Mathematical and Computer Modelling, 53 (2011), 1208-1220.  doi: 10.1016/j.mcm.2010.11.090.  Google Scholar [33] T. Phung-Duc, Retrial queueing models: A survey on theory and applications, in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), World Scientific Publisher, (2017), 1–26. Google Scholar [34] T. Phung-Duc, Single server retrial queues with setup time, Journal of Industrial and Management Optimization, 13 (2017), 1329-1345.  doi: 10.3934/jimo.2016075.  Google Scholar [35] T. Phung-Duc, Exact solutions for $M/M/c/$ Setup queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z.  Google Scholar [36] T. Phung-Duc, W. Rogiest and S. Wittevrongel, Single server retrial queues with speed scaling: Analysis and performance evaluation, Journal of Industrial and Management Optimization, 13 (2017), 1927-1943.  doi: 10.3934/jimo.2017025.  Google Scholar [37] T. Phung-Duc and K. Kawanishi, Impacts of retrials on power-saving policy in data centers, Proceedings of the 11th International Conference on Queueing Theory and Network Applications, 22 (2016), 1-4.  doi: 10.1145/3016032.3016047.  Google Scholar [38] L. R. Ronald, Optimization in Operations Research, Prentice Hall, New Jersey, 1997. Google Scholar [39] L. D. Servi and S. G. Finn, $M/M/1$ queue with working vacations ($M/M/1/WV$), Performance Evaluation, 50 (2002), 41-52.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [40] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar [41] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006.  Google Scholar [42] S. Upadhyaya, Working vacation policy for a discrete-time $Geo^X/Geo/1$ retrial queue, OPSEARCH, 52 (2015), 650-669.  doi: 10.1007/s12597-015-0200-2.  Google Scholar [43] J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, preprint, arXiv: 1801.06993. Google Scholar [44] J. Wang and Q. Zhao, A discrete-time $Geo/G/1$ retrial queue with starting failures and second optional service, Computers & Mathematics with Applications, 53 (2007), 115-127.  doi: 10.1016/j.camwa.2006.10.024.  Google Scholar [45] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Society Press, Los Alamitos, California, 1994. Google Scholar [46] J. Wu, Z. Liu and Y. Peng, A discrete-time $Geo/G/1$ retrial queue with preemptive resume and collisions, Applied Mathematical Modelling, 35 (2011), 837-847.  doi: 10.1016/j.apm.2010.07.039.  Google Scholar [47] D. A. Wu and H. Takagi, $M/G/1$ queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681.   Google Scholar [48] T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 2 (1987), 201-233.  doi: 10.1007/BF01158899.  Google Scholar [49] T. Yang and H. Li, On the steady-state queue size distribution of the discrete-time $Geo/G/1$ queue with repeated customers, Queueing Systems, 21 (1995), 199-215.  doi: 10.1007/BF01158581.  Google Scholar [50] M. Yu, Y. Tang, Y. Fu and L. Pan, $GI/Geom/1/N/MWV$ queue with changeover time and searching for the optimum service rate in working vacation period, Journal of Computational and Applied Mathematics, 235 (2011), 2170-2184.  doi: 10.1016/j.cam.2010.10.013.  Google Scholar

show all references

##### References:
 [1] A. K. Aboul-Hassan, S. I. Rabia and F. A. Taboly, Performance evaluation of a discrete-time $Geo^{[X]}/G/1$ retrial queue with general retrial times, Computers & Mathematics with Applications, 58 (2009), 548-557.  doi: 10.1016/j.camwa.2009.03.101.  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] J. R. Artalejo, Accessible bibliography on retrial queues: Progress in 2000-2009, Mathematical and Computer Modelling, 51 (2010), 1071-1081.  doi: 10.1016/j.mcm.2009.12.011.  Google Scholar [4] J. R. Artalejo and A. Gómez-Corral, Retrial Queueing Systems: A Computational Approach, Springer, Berlin, 2008. doi: 10.1007/978-3-540-78725-9.  Google Scholar [5] I. Atencia and P. Moreno, A discrete-time $Geo/G/1$ retrial queue with general retrial times, Queueing Systems, 48 (2004), 5-21.  doi: 10.1023/B:QUES.0000039885.12490.02.  Google Scholar [6] I. Atencia and P. Moreno, A discrete-time $Geo/G/1$ retrial queue with the server subject to starting failures, Annals of Operations Research, 141 (2006), 85-107.  doi: 10.1007/s10479-006-5295-7.  Google Scholar [7] I. Atencia and P. Moreno, A single-server retrial queue with general retrial times and Bernoulli schedule, Applied Mathematics and Computation, 162 (2005), 855-880.  doi: 10.1016/j.amc.2003.12.128.  Google Scholar [8] H. Bruneel and B. G. Kim, Discrete-Time Models for Communication Systems Including ATM, Kluwer Academic Publishers, Boston, 1993. doi: 10.1007/978-1-4615-3130-2.  Google Scholar [9] B. D. Choi and J. W. Kim, Discrete-time $Geo_1, Geo_2/G/1$ retrial queueing systems with two types of calls, Computers & Mathematics with Applications, 33 (1997), 79-88.  doi: 10.1016/S0898-1221(97)00078-3.  Google Scholar [10] I. Dimitriou, A mixed priority retrial queue with negative arrivals, unreliable server and multiple vacations, Applied Mathematical Modelling, 37 (2013), 1295-1309.  doi: 10.1016/j.apm.2012.04.011.  Google Scholar [11] I. Dimitriou, A two class retrial system with coupled orbit queues, Probability in the Engineering and Informational Sciences, 31 (2017), 139-179.  doi: 10.1017/S0269964816000528.  Google Scholar [12] I. Dimitriou, A queueing model with two types of retrial customers and paired services, Annals of Operations Research, 238 (2016), 123-143.  doi: 10.1007/s10479-015-2059-2.  Google Scholar [13] I. Dimitriou, Analysis of a priority retrial queue with dependent vacation scheme and application to energy saving in wireless communication systems, The Computer Journal, 56 (2013), 1363-1380.   Google Scholar [14] T. V. Do, $M/M/1$ retrial queue with working vacations, Acta Informatica, 47 (2010), 67-75.  doi: 10.1007/s00236-009-0110-y.  Google Scholar [15] B. T. Doshi, Queueing systems with vacation-a survey, Queueing Systems, 1 (1986), 29-66.  doi: 10.1007/BF01149327.  Google Scholar [16] A. Dudin, C. S. Kim, S. Dudin and O. Dudina, Priority retrial queueing model operating in random environment with varying number and reservation of servers, Applied Mathematics and Computation, 269 (2015), 674-690.  doi: 10.1016/j.amc.2015.08.005.  Google Scholar [17] G. I. Falin, A survey of retrial queues, Queueing Systems, 7 (1990), 127-167.  doi: 10.1007/BF01158472.  Google Scholar [18] G. I. Falin and J. G. C. Templeton, Retrial Queues, Chapman & Hall, London, 1997. Google Scholar [19] A. Gandhi, V. Gupta, M. Harchol-Balter and M. A. Kozuch, Optimality analysis of energy-performance trade-off for server farm management, Performance Evaluation, 67 (2010), 1155-1171.   Google Scholar [20] S. Gao and J. Wang, On a discrete-time $GI^X/Geo/1/N-G$ queue with randomized working vacations and at most J vacations, Journal of Industrial and Management Optimization, 11 (2015), 779-806.  doi: 10.3934/jimo.2015.11.779.  Google Scholar [21] S. Gao and J. Wang, Discrete-time $Geo^X/G/1$ retrial queue with general retrial times, working vacations and vacation interruption, Quality Technology & Quantitative Management, 10 (2013), 495-512.   Google Scholar [22] J. J. Hunter, Mathematical Techniques of Applied Probability, Vol. 2, Discrete Time Models: Techniques and Applications, Academic Press, New York, 1983.  Google Scholar [23] M. Jain, G. C. Sharma and R. Sharma, Maximum entropy approach for discrete-time unreliable server $Geo^X/Geo/1$ queue with working vacation, International Journal of Mathematics in Operational Research, 4 (2012), 56-77.  doi: 10.1504/IJMOR.2012.044473.  Google Scholar [24] M. Jain, A. Bhagat and C. Shekhar, Double orbit finite retrial queues with priority customers and service interruptions, Applied Mathematics and Computation, 253 (2015), 324-344.  doi: 10.1016/j.amc.2014.12.066.  Google Scholar [25] J. C. Ke, C. H. Wu and Z. G. Zhang, Recent developments in vacation queueing models: a short survey, International Journal of Operations Research, 7 (2010), 3-8.   Google Scholar [26] P. V. Laxmi and K. Jyothsna, Finite buffer $GI/Geo/1$ batch servicing queue with multiple working vacations, RAIRO-Operations Research, 48 (2014), 521-543.  doi: 10.1051/ro/2014022.  Google Scholar [27] H. Li and T. Yang, $Geo/G/1$ discrete time retrial queue with Bernoulli schedule, European Journal of Operational Research, 111 (1998), 629-649.  doi: 10.1016/S0377-2217(97)90357-X.  Google Scholar [28] J. Li, N. Tian and W. Liu, The discrete-time $GI/Geo/1$ queue with multiple working vacations, Queueing Systems, 56 (2007), 53-63.  doi: 10.1007/s11134-007-9030-0.  Google Scholar [29] J. Li, Analysis of the discrete-time $Geo/G/1$ working vacation queue and its application to network scheduling, Computers & Industrial Engineering, 65 (2013), 594-604.  doi: 10.1016/j.cie.2013.04.009.  Google Scholar [30] T. Li, Z. Wang and Z. Liu, $Geo/Geo/1$ retrial queue with working vacations and vacation interruption, Journal of Applied Mathematics and Computing, 39 (2012), 131-143.  doi: 10.1007/s12190-011-0516-x.  Google Scholar [31] T. Li, Z. Liu and Z. Wang, $M/M/1$ retrial queue with collisions and working vacation interruption under N-policy, RAIRO-Operations Research, 46 (2012), 355-371.  doi: 10.1051/ro/2012022.  Google Scholar [32] Z. Liu and S. Gao, Discrete-time $Geo_1, Geo^X_2/G_1, G_2/1$ retrial queue with two classes of customers and feedback, Mathematical and Computer Modelling, 53 (2011), 1208-1220.  doi: 10.1016/j.mcm.2010.11.090.  Google Scholar [33] T. Phung-Duc, Retrial queueing models: A survey on theory and applications, in Stochastic Operations Research in Business and Industry (eds. T. Dohi, K. Ano and S. Kasahara), World Scientific Publisher, (2017), 1–26. Google Scholar [34] T. Phung-Duc, Single server retrial queues with setup time, Journal of Industrial and Management Optimization, 13 (2017), 1329-1345.  doi: 10.3934/jimo.2016075.  Google Scholar [35] T. Phung-Duc, Exact solutions for $M/M/c/$ Setup queues, Telecommunication Systems, 64 (2017), 309-324.  doi: 10.1007/s11235-016-0177-z.  Google Scholar [36] T. Phung-Duc, W. Rogiest and S. Wittevrongel, Single server retrial queues with speed scaling: Analysis and performance evaluation, Journal of Industrial and Management Optimization, 13 (2017), 1927-1943.  doi: 10.3934/jimo.2017025.  Google Scholar [37] T. Phung-Duc and K. Kawanishi, Impacts of retrials on power-saving policy in data centers, Proceedings of the 11th International Conference on Queueing Theory and Network Applications, 22 (2016), 1-4.  doi: 10.1145/3016032.3016047.  Google Scholar [38] L. R. Ronald, Optimization in Operations Research, Prentice Hall, New Jersey, 1997. Google Scholar [39] L. D. Servi and S. G. Finn, $M/M/1$ queue with working vacations ($M/M/1/WV$), Performance Evaluation, 50 (2002), 41-52.  doi: 10.1016/S0166-5316(02)00057-3.  Google Scholar [40] H. Takagi, Queueing Analysis: A Foundation of Performance Evaluation, North-Holland Publishing Co., Amsterdam, 1993.  Google Scholar [41] N. Tian and Z. G. Zhang, Vacation Queueing Models: Theory and Applications, Springer, New York, 2006.  Google Scholar [42] S. Upadhyaya, Working vacation policy for a discrete-time $Geo^X/Geo/1$ retrial queue, OPSEARCH, 52 (2015), 650-669.  doi: 10.1007/s12597-015-0200-2.  Google Scholar [43] J. Walraevens, D. Claeys and T. Phung-Duc, Asymptotics of queue length distributions in priority retrial queues, preprint, arXiv: 1801.06993. Google Scholar [44] J. Wang and Q. Zhao, A discrete-time $Geo/G/1$ retrial queue with starting failures and second optional service, Computers & Mathematics with Applications, 53 (2007), 115-127.  doi: 10.1016/j.camwa.2006.10.024.  Google Scholar [45] M. E. Woodward, Communication and Computer Networks: Modelling with Discrete-Time Queues, IEEE Computer Society Press, Los Alamitos, California, 1994. Google Scholar [46] J. Wu, Z. Liu and Y. Peng, A discrete-time $Geo/G/1$ retrial queue with preemptive resume and collisions, Applied Mathematical Modelling, 35 (2011), 837-847.  doi: 10.1016/j.apm.2010.07.039.  Google Scholar [47] D. A. Wu and H. Takagi, $M/G/1$ queue with multiple working vacations, Performance Evaluation, 63 (2006), 654-681.   Google Scholar [48] T. Yang and J. G. C. Templeton, A survey on retrial queues, Queueing Systems, 2 (1987), 201-233.  doi: 10.1007/BF01158899.  Google Scholar [49] T. Yang and H. Li, On the steady-state queue size distribution of the discrete-time $Geo/G/1$ queue with repeated customers, Queueing Systems, 21 (1995), 199-215.  doi: 10.1007/BF01158581.  Google Scholar [50] M. Yu, Y. Tang, Y. Fu and L. Pan, $GI/Geom/1/N/MWV$ queue with changeover time and searching for the optimum service rate in working vacation period, Journal of Computational and Applied Mathematics, 235 (2011), 2170-2184.  doi: 10.1016/j.cam.2010.10.013.  Google Scholar
Various time epochs in an early arrival system (EAS)
The effect of $\alpha$ on $p(0,0,0)$ for different values of $r$, $\chi _b = 0.7$, $\chi _v = 0.4$, $\lambda = 0.2$, $\theta = 0.3$
The effect of $\chi _v$ on $p(0,0,0)$ for different values of $\theta$, $\chi _b = 0.7$, $\alpha = 0.5$, $\lambda = 0.2$, $r = 0.35$
The effect of $\alpha$ on $E\left[ {L_1 } \right]$ for different values of $\lambda$, $\chi _b = 0.7$, $\chi _v = 0.4$, $r = 0.35$, $\theta = 0.3$
The effect of $\alpha$ on $E\left[ {L_2 } \right]$ for different values of $\lambda$, $\chi _b = 0.7$, $\chi _v = 0.4$, $r = 0.35$, $\theta = 0.3$
The effect of $\chi _v$ on $E[L]$ for different values of $\theta$, $\chi _b = 0.7$, $\alpha = 0.5$, $\lambda = 0.2$, $r = 0.35$
$E[L]$ versus $\chi _v$ and $\lambda$, $\chi _b = 0.7$, $\alpha = 0.5$, $\theta = 0.3$, $r = 0.35$
$E[L]$ versus $r$ and $\lambda$, $\chi _b = 0.7$, $\chi _v = 0.4$, $\alpha = 0.5$, $\theta = 0.3$
The effect of $\alpha$ on the expected operating cost per unit time
The parabolic method in searching for the optimum solution
 No. of iterations 0 1 2 3 4 5 $\alpha^{(l)}$ 0.55000 0.60000 0.61667 0.61712 0.61734 0.61735 $\alpha^{(m)}$ 0.60000 0.61667 0.61712 0.61734 0.61735 0.61735 $\alpha^{(r)}$ 0.65000 0.65000 0.65000 0.65000 0.65000 0.65000 $TC\left( {\alpha^{(l)} } \right)$ 443.08917 443.06412 443.06226 443.06226 443.06226 443.06226 $TC\left( {\alpha^{(m)} } \right)$ 443.06412 443.06226 443.06226 443.06226 443.06226 443.06226 $TC\left( {\alpha^{(r)} } \right)$ 443.06912 443.06912 443.06912 443.06912 443.06912 443.06912 $\alpha^\ast$ 0.61667 0.61712 0.61734 0.61735 0.61735 0.61735 $TC\left( {\alpha^\ast } \right)$ 443.06226 443.06226 443.06226 443.06226 443.06226 443.06226 Tolerance 1.66744×10$^{-2}$ 4.40785×10$^{-4}$ 2.27516×10$^{-4}$ 8.98158×10$^{-6}$ 3.20017×10$^{-6}$ 1.62729×10$^{-7}$
 No. of iterations 0 1 2 3 4 5 $\alpha^{(l)}$ 0.55000 0.60000 0.61667 0.61712 0.61734 0.61735 $\alpha^{(m)}$ 0.60000 0.61667 0.61712 0.61734 0.61735 0.61735 $\alpha^{(r)}$ 0.65000 0.65000 0.65000 0.65000 0.65000 0.65000 $TC\left( {\alpha^{(l)} } \right)$ 443.08917 443.06412 443.06226 443.06226 443.06226 443.06226 $TC\left( {\alpha^{(m)} } \right)$ 443.06412 443.06226 443.06226 443.06226 443.06226 443.06226 $TC\left( {\alpha^{(r)} } \right)$ 443.06912 443.06912 443.06912 443.06912 443.06912 443.06912 $\alpha^\ast$ 0.61667 0.61712 0.61734 0.61735 0.61735 0.61735 $TC\left( {\alpha^\ast } \right)$ 443.06226 443.06226 443.06226 443.06226 443.06226 443.06226 Tolerance 1.66744×10$^{-2}$ 4.40785×10$^{-4}$ 2.27516×10$^{-4}$ 8.98158×10$^{-6}$ 3.20017×10$^{-6}$ 1.62729×10$^{-7}$
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