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A repairable Geo X/G/1 retrial queue with Bernoulli feedback and impatient customers

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Abstract

This paper deals with a discrete-time batch arrival retrial queue with the server subject to starting failures. Different from standard batch arrival retrial queues with starting failures, we assume that each customer after service either immediately returns to the orbit for another service with probability θ or leaves the system forever with probability 1 − θ (0 ≤ θ < 1). On the other hand, if the server is started unsuccessfully by a customer (external or repeated), the server is sent to repair immediately and the customer either joins the orbit with probability q or leaves the system forever with probability 1 − q (0 ≤ q < 1). Firstly, we introduce an embedded Markov chain and obtain the necessary and sufficient condition for ergodicity of this embedded Markov chain. Secondly, we derive the steady-state joint distribution of the server state and the number of customers in the system/orbit at arbitrary time. We also derive a stochastic decomposition law. In the special case of individual arrivals, we develop recursive formulae for calculating the steady-state distribution of the orbit size. Besides, we investigate the relation between our discrete-time system and its continuous counterpart. Finally, some numerical examples show the influence of the parameters on the mean orbit size.

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Authors and Affiliations

  1. Department of Mathematics, Fuyang Normal College, Fuyang, 236037, China

    Shan Gao

  2. School of Mathematics and Statistics, Central South University, Changsha, 410075, China

    Zai-ming Liu

Authors
  1. Shan Gao
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  2. Zai-ming Liu
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Correspondence to Shan Gao.

Additional information

Supported by the National Natural Science Foundation of China (Nos. 11171019, 11171179, and 11271373), Program for New Century Excellent Talents in University (No. NCET-11-0568) and the Fundamental Research Funds for the Central Universities (No. 2011JBZ012), the Tianyuan Fund for Mathematics (Nos. 11226200 and 11226251) and Program for Science Research of Fuyang Normal College (2013FSKJ01ZD).

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Gao, S., Liu, Zm. A repairable Geo X/G/1 retrial queue with Bernoulli feedback and impatient customers. Acta Math. Appl. Sin. Engl. Ser. 30, 205–222 (2014). https://doi.org/10.1007/s10255-014-0278-y

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  • DOI: https://doi.org/10.1007/s10255-014-0278-y

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