Analysis of an MAP/PH/1 queue with flexible group service

• Autorzy: Brugno A. , D’Apice C. , Dudin A. , Manzo R.

Identyfikatory

DOI

10.1515/amcs-2017-0009

• Języki publikacji: EN

Abstrakty

EN

A novel customer batch service discipline for a single server queue is introduced and analyzed. Service to customers is offered in batches of a certain size. If the number of customers in the system at the service completion moment is less than this size, the server does not start the next service until the number of customers in the system reaches this size or a random limitation of the idle time of the server expires, whichever occurs first. Customers arrive according to a Markovian arrival process. An individual customer’s service time has a phase-type distribution. The service time of a batch is defined as the maximum of the individual service times of the customers which form the batch. The dynamics of such a system are described by a multi-dimensional Markov chain. An ergodicity condition for this Markov chain is derived, a stationary probability distribution of the states is computed, and formulas for the main performance measures of the system are provided. The Laplace–Stieltjes transform of the waiting time is obtained. Results are numerically illustrated.

Słowa kluczowe

EN

queueing system; batch service; multirate service; stationary distribution; optimization

PL

system kolejkowy; usługa wsadowa; usługa wielotaryfowa; rozkład stacjonarny

• Wydawca: Oficyna Wydawnicza Uniwersytetu Zielonogórskiego
• Czasopismo: International Journal of Applied Mathematics and Computer Science
• Rocznik: 2017
• Tom: Vol. 27, no. 1
• Strony: 119--131
• Opis fizyczny: Bibliogr. 21 poz., rys., wykr.

Twórcy

autor

Brugno A.

• Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano (SA), Italy

autor

D’Apice C.

• Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano (SA), Italy

autor

Dudin A.

• Department of Applied Mathematics and Computer Science, Belarusian State University, 4, Nezavisimosti Av., Minsk, 220030, Belarus

autor

Manzo R.

• Department of Information Engineering, Electrical Engineering and Applied Mathematics, University of Salerno, Via Giovanni Paolo II, 132, 84084, Fisciano (SA), Italy

Bibliografia

• [1] 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.
• [2] Bailey, N. (1954). On queueing processes with bulk service, Journal of the Royal Statistical Society B 16(1): 80–87.
• [3] Banerjee, A., Gupta, U. and Chakravarthy, S. (2015). Analysis of afinite-buffer bulk-service queue under Markovian arrival process with batch-size-dependent service, Computers and Operations Research 60: 138–149.
• [4] Casale, G., Zhang, E. and Smirn, E. (2010). Trace data characterization and fitting for Markov modeling, Performance Evaluation 67(2): 61–79.
• [5] Chakravarthy, S. (2001). The batch Markovian arrival process: A review and future work, in V.R.E.A. Krishnamoorthy and N. Raju (Eds.), Advances in Probability Theory and Stochastic Processes, Notable Publications Inc., ,Branchburg, NJ, pp. 21–29.
• [6] Chydzinski, A. (2006). Transient analysis of the MMPP/G/1/K queue, Telecommunication Systems 32(4): 247–262.
• [7] Deb, R. and Serfozo, R. (1973). Optimal control of batch service queues, Advances in Applied Probability 5(2): 340–361.
• [8] Downton, F. (1955). Waiting time in bulk service queues, Journal of the Royal Statistical Society B 17(2): 256–261.
• [9] Dudin, A., Manzo, R. and Piscopo, R. (2015). Single server retrial queue with adaptive group admission of customers, Computers and Operations Research 61: 89–99.
• [10] Dudin, A., Lee, M.H. 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.
• [11] 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.
• [12] Heyman, D. and Lucantoni, D. (2003). Modelling multiple IP traffic streams with rate limits, IEEE/ACMTransactions on Networking 11(6): 948–958.
• [13] Kesten, H. and Runnenburg, J. (1956). Priority in Waiting Line Problems, Mathematisch Centrum, Amsterdam.
• [14] 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.
• [15] Klemm, A., Lindermann, C. and Lohmann, M. (2003). Modelling IP traffic using the batch Markovian arrival process, Performance Evaluation 54(2): 149–173.
• [16] Lucatoni, D. (1991). New results on the single server queue with a batch Markovian arrival process, Communication in Statistics: Stochastic Models 7(1): 1–46.
• [17] Mèszáros, A., Papp, J. and Telek, M. (2014). Fitting traffic with discrete canonical phase type distribution and Markov arrival processes, International Journal of Applied Mathematics and Computer Science 24(3): 453–470, DOI: 10.2478/amcs-2014-0034.
• [18] Neuts, M. (1967). A general class of bulk queues with Poisson input, The Annals of Mathematical Statistics 38(3): 759–770.
• [19] Neuts, M. (1981). Matrix-geometric Solutions in Stochastic Models—An Algorithmic Approach, Johns Hopkins University Press, Baltimore, MD.
• [20] Sasikala, S. and Indhira, K. (2016). Bulk service queueing models—a survey, International Journal of Pure and Applied Mathematics 106(6): 43–56.
• [21] van Dantzig, D. (1955). Chaines de markof dans les ensembles abstraits et applications aux processus avec regions absorbantes et au probleme des boucles, Annales de l’Institut Henri Poincar´e 14(3): 145–199.

Uwagi

PL

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).