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

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.

[1]  Jesús R. Artalejo,et al.  Retrial Queueing Systems , 2008 .

[2]  Edward Chan,et al.  A new method for evaluating the cell loss probability in an atm multiplexer , 2002, Eur. Trans. Telecommun..

[3]  Jesus R. Artalejo,et al.  Cellular mobile networks with repeated calls operating in random environment , 2010, Comput. Oper. Res..

[4]  Valentina Klimenok,et al.  The BMAP/PH/1 retrial queueing system operating in random environment , 2007 .

[5]  Vaidyanathan Ramaswami,et al.  Introduction to Matrix Analytic Methods in Stochastic Modeling , 1999, ASA-SIAM Series on Statistics and Applied Mathematics.

[6]  Bo Li,et al.  A matrix-analytic solution for the DBMAP/PH/1 priority queue , 2006, Queueing Syst. Theory Appl..

[7]  Jesús R. Artalejo,et al.  On the distribution of the number of retrials , 2007 .

[8]  Hideaki Takagi,et al.  Queueing analysis: a foundation of performance evaluation , 1993 .

[9]  Quan-Lin Li,et al.  THE RG-FACTORIZATION IN BLOCK-STRUCTURED MARKOV RENEWAL PROCESSES , 2004 .

[10]  Mohan L. Chaudhry,et al.  On Numerical Computations of Some Discrete-Time Queues , 2000 .

[11]  Marcel F. Neuts,et al.  Matrix-Geometric Solutions in Stochastic Models , 1981 .

[12]  Jesús R. Artalejo,et al.  A SIMULATION STUDY OF A DISCRETE-TIME MULTISERVER RETRIAL QUEUE WITH FINITE POPULATION , 2007 .

[13]  Chris Blondia,et al.  Statistical Multiplexing of VBR Sources: A Matrix-Analytic Approach , 1992, Perform. Evaluation.

[14]  Jesus R. Artalejo,et al.  Algorithmic analysis of the Geo / Geo / c retrial queue , 2007 .

[15]  Quan-Lin Li.  Constructive Computation in Stochastic Models with Applications: The RG-Factorizations , 2010 .

[16]  Jesús R. Artalejo,et al.  Retrial Queueing Systems: A Computational Approach , 2008 .

[17]  L. Allen.  An introduction to stochastic processes with applications to biology , 2003 .

[18]  Raphael Rom,et al.  Multiple Access Protocols: Performance and Analysis , 1990, SIGMETRICS Perform. Evaluation Rev..

[19]  C. Blondia,et al.  Response Time Distribution in a D-MAP/PH/1 Queue with General Customer Impatience , 2005 .

[20]  R. B. Lenin,et al.  Loss Probability of a D-BMAP/PH/1/N Queue , 2006 .

[21]  Herwig Bruneel,et al.  Discrete-time models for communication systems including ATM , 1992 .

[22]  Quanlin Li,et al.  Two Types of RG-Factorizations of Quasi-birth-and-death Processes and Their Applications to Stochastic Integral Functionals , 2004 .

[23]  Attahiru Sule Alfa,et al.  Discrete-time analysis of the GI/G/1 system with Bernoulli retrials: An algorithmic approach , 2006, Ann. Oper. Res..

[24]  Jorma Virtamo,et al.  Performance analysis of finite-source retrial queues operating in random environments , 2007 .

[25]  Ivan Atencia,et al.  A discrete-time Geo[X]/G/1 retrial queue with control of admission , 2005 .

[26]  Hui Li,et al.  On the steady-state queue size distribution of the discrete-timeGeo/G/1 queue with repeated customers , 1995, Queueing Syst. Theory Appl..

[27]  Qing Zhao,et al.  A discrete-time Geo/G/1 retrial queue with starting failures and second optional service , 2007, Comput. Math. Appl..

[28]  Byung Kyu Kim,et al.  A simple eigenvalue method for low-order D-BMAP/G/1 queues , 2005 .

[29]  Raphael Rom,et al.  Multiple Access Protocols: Performance and Analysis , 1990, SIGMETRICS Perform. Evaluation Rev..

[30]  Ivan Atencia,et al.  A single-server G-queue in discrete-time with geometrical arrival and service process , 2005, Perform. Evaluation.