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A threshold policy in a Markov-modulated production system with server vacation: the case of continuous and batch supplies

  • Yonit Barron (a1)


We consider a Markov-modulated fluid flow production model under the D-policy, that is, as soon as the storage reaches level 0, the machine becomes idle until the total storage exceeds a predetermined threshold D. Thus, the production process alternates between a busy and an idle machine. During the busy period, the storage decreases linearly due to continuous production and increases due to supply; during the idle period, no production is rendered by the machine and the storage level increases by only supply arrivals. We consider two types of model with different supply process patterns: continuous inflows with linear rates (fluid type), and batch inflows, where the supplies arrive according to a Markov additive process (MAP) and their sizes are independent and have phase-type distributions depending on the type of arrival (MAP type). Four types of cost are considered: a setup cost, a production cost, a penalty cost for an idle machine, and a storage cost. Using tools from multidimensional martingale and hitting time theory, we derive explicit formulae for these cost functionals in the discounted case. Numerical examples, a sensitivity analysis, and insights are provided.


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* Postal address: Department of Industrial Engineering and Management, Ariel University, Ariel 40700, Israel. Email address:


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Aggarwal, V.,Gautam, N.,Kumara, S. R. T. and Greaves, M. (2005).Stochastic fluid flow models for determining optimal switching thresholds.Performance Eval. 59,19–46.
Ahn, S. and Ramaswami, V. (2005).Efficient algorithms for transient analysis of stochastic fluid flow models.J. Appl. Prob. 42,531–549.
Ahn, S.,Badescu, A. L. and Ramaswami, V. (2007).Time dependent analysis of finite buffer fluid flows and risk models with a dividend barrier.Queueing Systems 55,207–222.
Artalejo, J. R. (2001).On the M/G/1 queue with D-policy.Appl. Math. Modelling 25,1055–1069.
Asmussen, S. (2003).Applied Probability and Queues.Springer,New York.
Asmussen, S. and Kella, O. (2000).A multi-dimensional martingale for Markov additive processes and its applications.Adv. Appl. Prob. 32,376–393.
Badescu, A.,Drekic, S. and Landriault, D. (2007).On the analysis of a multi-threshold Markovian risk model.Scand. Actuarial J. 2007,248–260.
Baek, J. W.,Lee, H. W.,Lee, S. W. and Ahn, S. (2011).A Markov-modulated fluid flow queueing model under D-policy.Numer. Linear Algebra Appl. 18,993–1010.
Baek, J. W.,Lee, H. W.,Lee, S. W. and Ahn, S. (2013).A MAP-modulated fluid flow model with multiple vacations.Ann. Operat. Res. 202,19–34.
Baek, J. W.,Lee, H. W.,Lee, S. W. and Ahn, S. (2014).A workload factorization for BMAP/G/1 vacation queues under variable service speed.Operat. Res. Lett. 42,58–63.
Balachandran, K. R. (1973).Control policies for a single server system.Manag. Sci. 19,1013–1018.
Barron, Y. (2016).Performance analysis of a reflected fluid production/inventory model.Math. Methods Operat. Res. 83,1–31.
Barron, Y. (2018).An order-revenue inventory model with returns and sudden obsolescence.Operat. Res. Lett. 46,88–92.
Barron, Y.,Perry, D. and Stadje, W. (2016).A make-to-stock production/inventory model with MAP arrivals and phase-type demands.Ann. Operat. Res. 241,373–409.
Bean, N. G.,O'Reilly, M. M. and Taylor, P. G. (2005).Hitting probabilities and hitting times for stochastic fluid flows.Stoch. Process. Appl. 115,1530–1556.
Bosman, J. W. and NΓΊΓ±ez-Queija, R. (2014).A spectral theory approach for extreme value analysis in a tandem of fluid queues.Queueing Systems 78,121–154.
Boxma, O. J.,Schlegel, S. and Yechiali, U. (2002).A note on an M/G/1 queue with a waiting server, timer, and vacations. In Analytic Methods in Applied Probability (Amer. Math. Soc. Transl. Ser. 2 207),American Mathematical Society,Providence, RI, pp. 25–35.
Breuer, L. (2010).A quintuple law for Markov additive processes with phase-type jumps.J. Appl. Prob. 47,441–458.
Chae, K. C. and Park, Y. (2001).The queue length distribution for the M/G/1 queue under the D-policy.J. Appl. Prob. 38,278–279.
Chang, S. H.,Takine, T.,Chae, K. C. and Lee, H. W. (2002).A unified queue length formula for BMAP/G/1 queue with generalized vacations.Stoch. Models 18,369–386.
Choudhury, G. (2005).An M/G/1 queueing system with two phase service under D-policy.Internat. J. Inform. Manag. Sci. 16,1–17.
Doshi, B. T. (1986).Queueing systems with vacations–a survey.Queueing Systems 1,29–66.
Dshalalow, J. H. (1998).Queueing processes in bulk systems under the D-policy.J. Appl. Prob. 35,976–989.
Efrosinin, D. and Winkler, A. (2011).Queueing system with a constant retrial rate, non-reliable server and threshold-based recovery.European J. Operat. Res. 210,594–605.
Guo, P. and Hassin, R. (2011).Strategic behavior and social optimization in Markovian vacation queues.Operat. Res. 59,986–997.
Gupta, U. C. and Sikdar, K. (2006).Computing queue length distributions in MAP/G/1/N queue under single and multiple vacation.Appl. Math. Comput. 174,1498–1525.
Jain, M. and Bhagat, A. (2012).Finite population retrial queueing model with threshold recovery, geometric arrivals and impatient customers.J. Inform. Operat. Manag. 3,162–165.
Ke, J.-C.,Wu, C.-H. and Zhang, Z. G. (2010).Recent developments in vacation queueing models: a short survey.Internat. J. Operat. Res. 7,3–8.
Kella, O. (1989).The threshold policy in the M/G/1 queue with server vacations.Naval Res. Logistics 36,111–123.
Lee, H. W. and Baek, J. W. (2005).BMAP/G/1 queue under D-policy: queue length analysis.Stoch. Models 21,485–505.
Lee, H. W. and Song, K. S. (2004).Queue length analysis of MAP/G/1 queue under D-policy.Stoch. Models 20,363–380.
Lee, H. W.,Park, N. I. and Jeon, J. (2002).Application of the factorization property to the analysis of production systems with a non-renewal input, bilevel threshold control and maintenance. In Matrix-Analytic Methods (Adelaide, 2002), eds G. Latouche and P. Taylor,World Scientific,River Edge, NJ, pp. 219–236.
Lee, H. W.,Cheon, S. H.,Lee, E. Y. and Chae, K. C. (2004).Workload and waiting time analyses of MAP/G/1 queue under D-policy.Queueing Systems 48,421–443.
Levy, Y. and Yechiali, U. (1975).Utilization of idle time in an M/G/1 queueing system.Manag. Sci. 22,139–260.
Liu, R. and Deng, Z. (2014).The steady-state system size distribution for a modified D-policy GEO/G/1 queueing system.Math. Prob. Eng. 2014, 10pp.
Malhotra, R.,Mandjes, M. R. H.,Scheinhardt, W. and van den Berg, J. L. (2009).A feedback fluid queue with two congestion control thresholds.Math. Methods Operat. Res. 70,149–169.
Mao, B.-W.,Wang, F.-W. and Tian, N. S. (2010).Fluid model driven by an M/M/1 queue with exponential vacation. In Information Technology and Computer Science, 2010 Second International Conference on IEEE, pp. 539–542.
Tian, N. and Zhang, Z. G. (2006).Vacation Queueing Models: Theory and Applications.Springer,New York.
Vijayashree, K. V. and Anjuka, A. (2016).Fluid queue driven by an M/M/1 queue subject to Bernoulli-schedule-controlled vacation and vacation interruption.Adv. Operat. Res. 2016, 11pp.
Yang, D.-Y.,Yen, C.-H. and Chiang, Y.-C. (2013).Numerical analysis for time-dependent machine repair model with threshold recovery policy and server vacations. In Proc. Internat. Multiconference of Engineers and Computer Scientists, Vol. II, pp. 1117–1120.
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Advances in Applied Probability
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  • EISSN: 1475-6064
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