# Performance Analysis of a Block-Structured Discrete-Time Retrial Queue with State-Dependent Arrivals

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## Abstract

In this paper, we introduce a new discrete block state-dependent arrival (*D*-*BSDA*) distribution which provides fresh insights leading to a successful generalization of the discrete-time Markovian arrival process (*D*-*MAP*). The *D*-*BSDA* distribution is related to structured Markov chains and the method of stages. The consideration of this new discrete-time state-dependent block description gives one the ability of construct new stochastic models. The retrial queue analyzed in this paper gives an example of application of the *D*-*BSDA* distribution to construct more general and sophisticated models. We assume that the primary arrivals and the retrials follow the *D*-*BSDA* description and the service times are of discrete phase-type (*PH*). We study the underlying level dependent Markov chain of *M*/*G*/1-type at the epochs immediately after the slot boundaries. To this end, we employ the UL-type *RG*-factorization which provides an expression for the stationary probabilities. We also perform an analysis of waiting times. Numerical experiments are presented to study the system performance.

## Keywords

Queueing Discrete-time Structured Markov chain Retrial LU-factorizations Waiting time## Notes

### Acknowledgement

The constructive comments of the referees on an earlier version of this paper are greatly appreciated. The authors are also grateful to Jingqi Wang (Tsinghua University) for preparing codes for the numerical implementation. J.R. Artalejo was supported by grant MTM2005-01248 from MEC. The work of Q.L. Li was supported by the National Science Foundation of China under grant No. 10671107, 10871114, 60736028 and the National Grand Fundamental Research 973 Program of China under grant No. 2006CB805901.

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