Abstract
In this paper, we consider an MMPP/M/1/1 retrial queue where incoming fresh calls arrive at the server according to a Markov modulated Poisson process (MMPP). Upon arrival, an incoming call either occupies the server if it is idle or joins a virtual waiting room called orbit if the server is busy. From the orbit, incoming calls retry to occupy the server in an exponentially distributed time and behave the same as a fresh incoming call. After an exponentially distributed idle time, the server makes an outgoing call whose duration is also exponentially distributed but with a different parameter from that of incoming calls. Our contribution is to derive the first order (law of large numbers) and the second order (central limit theorem) asymptotics for the distribution of the number of calls in the orbit under the condition that the retrial rate is extremely low. The asymptotic results are used to obtain the Gaussian approximation for the distribution of the number of calls in the orbit. Our result generalizes earlier results where Poisson input was assumed.
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References
Arrar N. K, Djellab N. V, Baillon J. B (2012). On the asymptotic behaviour of M/G/1 retrial queues with batch arrivals and impatience phenomenon. Mathematical and Computer Modelling 55: 654–665.
Artalejo J. R, Gomez-Corral A (2008). Retrial Queueing Systems: A Computational Approach. Springer, Berlin.
Artalejo J. R, Phung-Duc T (2012). Markovian retrial queues with two way communication. Journal of Industrial and Management Optimization 8: 781–806.
Artalejo J. R, Phung-Duc T (2013). Single server retrial queues with two way communication. Applied Mathematical Modelling 37(4): 1811–1822.
Bhulai S, Koole G (2003). Aqueueing model for Call Blending in Call Centers. IEEE Transactions on Automatic Control 48: 1434–1438.
Blom J, De Turck K, Mandjes M (2015). Analysis of Markovmodulated infinite-server queues in the central-limit regime. Probability in the Engineering and Informational Sciences 29: 433–459.
Choi B. D, Choi K. B, Lee Y. W (1995). M/G/1 Retrial queueing systems with two types of calls and finite capacity. Queueing Systems 19: 215–229.
Deslauriers A, L’Ecuyer P, Pichitlamken J, Ingolfsson A, Avramidis A.N (2007). Markov chain models of a telephone call center with call blending. Computers and Operations Research 34: 1616–1645.
Falin G. I (1979). Model of coupled switching in presence of recurrent calls. Engineering Cybernetics Review 17: 53–59.
Falin G. I, Templeton J. G. C (1997). Retrial Queues. Chapman and Hall, London, 1997.
Fedorova E (2015). The second order asymptotic analysis under heavy load condition for retrial queueing system MMPP/M/1. Information Technologies and Mathematical Modelling -Queueing Theory and Applications. Communications in Computer and Information Science 564: 344–357. Springer, Cham.
Nazarov A, Phung-Duc T, Paul S (2017) Heavy outgoing call asymptotics for MMPP/M/1/1 retrial queue with two-way communication. Information Technologies and Mathematical Modelling. Queueing Theory and Applications CCIS 800: 28–41.
Sakurai H, Phung-Duc T (2016). Scaling limits for single server retrial queues with two-way communication. Annals of Operations Research 247: 229–256.
Shin Y. W (2016) Stability of MAP/PH/c/K queue with customer retrials and server vacations. Bulletin of the Korean Mathematical Society 53: 985–1004.
Tran-Gia P, Mandjes M (1997). Modeling of customer retrial phenomenon in cellular mobile networks IEEE Journal on Selected Areas in Communications 15: 1406–1414.
Phung-Duc T, Kawanishi K (2014). An efficient method for performance analysis of blended call centers with redial. Asia-Pacific Journal of Operational Research 31(2): 665–716.
Phung-Duc T, Rogiest W, Takahashi Y, Bruneel H (2016) Retrial queues with balanced call blending: Analysis of single-server and multiserver Model. Annals of Operations Research 239: 429–449.
Acknowledgements
The reported study was funded by RFBR according to the research project 18-01-00277. The research of TP was partially supported by University of Tsukuba Basic Research Support Program Type A.
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Nazarov Anantoly is a full professor at the Institute of Applied Mathematics and Computer Science in Tomsk State University, Russia. He is a Head of Department of Probability Theory and Mathematical Statistics. His research interests are in the field of queuing theory, applied probability analysis and mathematical modeling. He is a leader of Tomsk science school on queueing theory.
Tuan Phung-Duc is an associate professor at University of Tsukuba. He received a Ph.D. in Informatics from Kyoto University in 2011. He is currently on the editorial board of six international journals and is a guest editor of three special issues of annals of operations research. Dr. Phung-Duc received the Research Encourage Award from The Operations Research Society of Japan in 2013. His research interests include stochastic modelling, performance analysis and stochastic models.
Svetlana Paul obtained her degree in 2008 in Tomsk State University, Russia. Since 2009 she works as an associate professor of Department of Probability Theory and Mathematical Statistics of Tomsk State University. She is the member of research group in the field of queueing theory in Tomsk State University.
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Nazarov, A., Phung-Duc, T. & Paul, S. Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input. J. Syst. Sci. Syst. Eng. 28, 181–193 (2019). https://doi.org/10.1007/s11518-018-5404-6
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DOI: https://doi.org/10.1007/s11518-018-5404-6