Slow Retrial Asymptotics for a Single Server Queue with Two-Way Communication and Markov Modulated Poisson Input

  • Anatoly Nazarov
  • Tuan Phung-DucEmail author
  • Svetlana PaulEmail author


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.


Retrial queueing system incoming calls and outgoing calls MMPP-process asymptotic analysis method gaussian approximation 


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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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© Systems Engineering Society of China and Springer-Verlag GmbH Germany 2019

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and Computer ScienceNational Research Tomsk State University634050Russian Federation
  2. 2.Faculty of Engineering Information and SystemsUniversity of TsukubaIbarakiJapan

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