A single-server queueing system with a marked Markovian arrival process of heterogeneous customers is considered. Type-1 customers have limited preemptive priority over type-2 customers. There is an infinite buffer for type-2 customers and no buffer for type-1 customers. There is also a finite buffer (stock) for consumable additional items (semi-products, half-stocks, etc.) which arrive according to the Markovian arrival process. Service of a customer requires a fixed number of consumable additional items depending on the type of the customer. The service time has a phase-type distribution depending on the type of the customer. Customers in the buffer are impatient and may leave the system without service after an exponentially distributed amount of waiting time. Aiming to minimize the loss probability of type-1 customers and maximize throughput of the system, a threshold strategy of admission to service of type-2 customers is offered. Service of type-2 customer can start only if the server is idle and the number of consumable additional items in the stock exceeds the fixed threshold. Stationary distributions of the system states and the waiting time are computed. In the numerical example, we show some interesting effects and illustrate a possibility of application of the presented results for solution of optimization problems.
If the inline PDF is not rendering correctly, you can download the PDF file here.
Atencia, I. (2014). A discrete-time system with service control and repairs, International Journal of Applied Mathematics and Computer Science 24(3): 471-484, DOI: 10.2478/amcs-2014-0035.
Cardoen, B., Demeulemeester, E. and Beli¨en, J. (2010). Operating room planning and scheduling: A literature review, European Journal of Operational Research 201(3): 921-932.
Chakravarthy, S. (2001). The batchMarkovian arrival process: A review and future work, in A. Krishnamoorthy et al. (Eds.), Advances in Probability Theory and Stochastic Processes, Notable Publications Inc., Branchburg, NJ, pp. 21-29.
Dantzig, D.v. (1955). Chaˆınes de Markof dans les ensembles abstraits et applications aux processus avec r´egions absorbantes et au probl`eme des boucles, Annales de l’Institut Henri Poincar´e 14(3): 145-199.
Dudin, A. and Klimenok, V. (1996). Queueing systems with passive servers, Journal of Applied Mathematics and Stochastic Analysis 9(2): 185-204.
Dudin, A., Lee, M. and Dudin, S. (2016). Optimization of the service strategy in a queueing system with energy harvesting and customers’ impatience, International Journal of Applied Mathematics and Computer Science 26(2): 367-378, DOI: 10.1515/amcs-2016-0026.
Dudina, O., Kim, C. and Dudin, S. (2013). Retrial queuing system with Markovian arrival flow and phase-type service time distribution, Computers & Industrial Engineering 66(2): 360-373.
Gaidamaka, Y., Pechinkin, A., Razumchik, R., Samouylov, K. and Sopin, E. (2014). Analysis of an M/G/1/R queue with batch arrivals and two hysteretic overload control policies, International Journal of Applied Mathematics and Computer Science 24(3): 519-534, DOI: 10.2478/amcs-2014-0038.
Gelenbe, E. (2015). Synchronising energy harvesting and data packets in a wireless sensor, Energies 8(1): 356-369.
He, Q.-M. (1996). Queues with marked customers, Advances in Applied Probability 28(2) : 567-587.
Kesten, H. and Runnenburg, J.T. (1956). Priority inWaiting Line Problems, Mathematisch Centrum, Amsterdam.
Kim, C., A., D., Dudin, S. and Klimenok, V. (2012). Queueing system with batch arrival of customers in sessions, Computers and Industrial Engineering 62(4): 890-897.
Kim, C., Dudin, A., Dudin, S. and Dudina, O. (2014). Analysis of an MMAP/PH1, PH2/N/∞ queueing system operating in a random environment, International Journal of Applied Mathematics and Computer Science 24(3): 485-501, DOI: 10.2478/amcs-2014-0036.
Kim, C., Dudin, S. and Klimenok, V. (2009). The map/ph/1/n queue with flows of customers as model for traffic control in telecommunication networks, Performance Evaluation 66(9): 564-579.
Klimenok, V. and Dudin, A. (2006). Multi-dimensional asymptotically quasi-Toeplitz Markov chains and their application in queueing theory, Queueing Systems 54(4): 245-259.
Krishnamoorthy, A., Benny, B. and Shajin, D. (2016a). A revisit to queueing-inventory system with reservation, cancellation and common life time, OPSEARCH 54(2): 336-350, DOI: 10.1007/s12597-016-0278-1.
Krishnamoorthy, A., Shajin, D. and Lakshmy, B. (2016b). On a queueing-inventory with reservation, cancellation, common life time and retrial, Annals of Operations Research 247(1): 365-389.
Krishnamoorthy, A., Shajin, D. and Lakshmy, B. (2016c). Product form solution for some queueing-inventory supply chain problem, OPSEARCH 53(1): 85-102.
Manzini, R., Heragu, S. and Bozer, Y. (2015). Decision models for the design,optimization and management of warehousing and material handling systems, International Journal of Production Economics 170(C): 711-716.
Neuts, M. (1981). Matrix-Geometric Solutions in Stochastic Models-An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD.
Sharma, V., Mukherji, U., Joseph, V. and Gupta, S. (2010). Optimal energy management policies for energy harvesting sensor nodes, IEEE Transactions on Wireless Communications 9(4): 1326-1336.
Tutuncuoglu, K. and Yener, A. (2012). Optimum transmission policies for battery limited energy harvesting nodes, IEEE Transactions on Wireless Communications 11(3): 1180-1189.
Yang, J. and Ulukus, S. (2012a). Optimal packet scheduling in a multiple access channel with energy harvesting transmitters, Journal of Communications and Networks 14(2): 140-150.
Yang, J. and Ulukus, S. (2012b). Optimal packet scheduling in an energy harvesting communication system, IEEE Transactions on Communications 60(1): 220-230.
Zhao, N. and Lian, Z. (2011). A queueing-inventory system with two classes of customers, International Journal of Production Economics 129(1): 225-231.