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On multiserver feedback retrial queues with balking and control retrial rate. (English) Zbl 1102.60079

The paper considers a multiserver retrial queueing system in which primary customers arrive according to a Poisson process. The service facility consists of a finite number of identical servers and service times are exponentially distributed. An arriving customer who finds all servers busy enters the retrial group with probability \(p\) or is lost forever with probability \(1 - p\). If the primary customer finds some servers free, he immediately occupies a server and obtains service. After the customer is served completely, he decides either to join the retrial group again for another service with probability \(1 - \theta \) or to leave the system forever with probability \(\theta \). Every customer in the retrial group conducts a retrial after an exponentially distributed amount of time and is independent of the number of customers applying for service. Upon return from the retrial group, customers who find all servers busy always rejoin the retrial group; this operation continues until they are eventually served. The probability of a repeated attempt during \((t,t + dt)\), given that \(n\) customers are in orbit at time \(t\), is \(n\sigma dt + o(dt)\). This system is analyzed as a quasi-birth-and-death process and a necessary and sufficient condition for stability of the system is discussed. The effects of various parameters on the system performance measures are illustrated numerically. Finally, the optimization of the retrial rate and specific probabilistic descriptors of the system are investigated.

MSC:

60K25 Queueing theory (aspects of probability theory)
90B22 Queues and service in operations research
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References:

[1] Artalejo, J.R. (1999). ”Accessible Bibliography on Retrial Queues.” Mathematical and Computer Modeling 30, 1–6. · Zbl 1009.90001
[2] Artalejo, J.R., A. Gomez-Corral, and M.F. Neuts. (2000). ”Numerical Analysis of Multiserver Retrial Queues Operating Under Full Access Policy.” In G. Latouche and Peter Taylor (eds.), Advances in Algorithmic Methods for Stochastic Models. Notable Publications Inc., New Jersey, USA, pp. 1–19.
[3] Artalejo, J.R., A. Gomez-Corral, and M.F. Neuts. (2001). ”Analysis of Multiserver Queues with Constant Retrial Rate.” Euporean Journal of Operations Research 135, 569–581. · Zbl 0989.90028
[4] Artalejo, J.R. and M.J. Lopez-Herrero. (2000).”On the Single Server Retrial Queue with Balking.” INFOR 38, 35–50.
[5] Artalejo, J.R. and M. Pozo. (2002). ”Numberical Calculation of the Stationary Distribution of the Main Multiserver Retrial Queue.” Annals of Operations Research 116, 41–56. · Zbl 1013.90038
[6] Bright, L. and P.G. Taylor. (1995). ”Calculating the Equilibrium Distribution in Level Dependent Quasi-birth-and-death Processes.” Stochastic Models 11, 495–525. · Zbl 0837.60081
[7] Choi, B.D. and V.G. Kulkarni. (1992). ”Feedback Retrial Queueing Systems.” In U.N. Bhat and I.V. Basawa (eds.), Queueing and Related Models. Oxford University Press, New York, pp. 93–105. · Zbl 0783.60092
[8] Choi, B.D., Y.W. Shin and W.C. Ahn. (1992). ”Retrial Queues with Collision Arising from Unslotted CSMA/CD Protocol.” Queueing Systems 11, 335–356. · Zbl 0762.60088
[9] Choi, B.D. and Y. Chang, (1999). ”Single Server Retrial Queues with Priority Calls.” Mathematical and Computer Modeling 30, 7–32. · Zbl 1042.60533
[10] Choi, B.D., K.H. Rhee, and K.K. Park. (1993). ”The M/G/1 Retrial Queue with Retrial Rate Control Policy.” Probability in the Engineering and Information Sciences 7, 29–46.
[11] Choi, B.D., Y.G. Kim, and Y.W. Lee. (1998). ”The M/M/c Retrial Queue with Geometric Loss and Feedback.” Computer Mathematics and Applications 36, 41–52. · Zbl 0947.90024
[12] Cohen, J.W. (1957). ”Basic Problems of Telephone Traffic Theory and the Influence of Repeated Calls.” Phillips Telecommunictaion Review 18, 49–100.
[13] Falin, G.I. and Templeton, J.G.C. (1997). Retrial Queues. Chapman and Hall, London. · Zbl 0944.60005
[14] Falin, G.I. (1983). ”Calculation of Probability Characteristic of a Multiline System with Repeat Calls.” Mascow University Computational Mathematics and Cybernetics 1, 43–49. · Zbl 0534.90035
[15] Falin, G.I. (1990). ”A Survey of Retrial Queues.” Queueing Systems 7, 127–168. · Zbl 0709.60097
[16] Falin, G.I. and J.R. Artalejo. (1995). ”Approximations for Multiserver Queues with Balking/Retrial Discipline.” OR Spektrum 17, 239–244. · Zbl 0843.90046
[17] Farahmand, K. (1990). ”Single Server Line Queue with Repeated Demands.” Queueing Systems 6, 223–228. · Zbl 0712.60100
[18] Fayolle, G. (1986). ”A Simple Telephone Exchange with Delayed Feedbacks.” In O.J. Boxma, J.W. Cohen, and H.C. Tijms (eds.), Teletraffic analysis and computer performance evaluation. vol.7, Elsevier, Amsterdam, pp. 245–253.
[19] Gomez-Corral, A. and M.F. Ramalhoto. (1999). ”On the Stationary Distribution of a Markovian Process Arising in the Theory of Multiserver Retrial Queueing Systems.” Mathematical and Computer Modeling 30, 141–158. · Zbl 1042.60538
[20] Gomez-Corral, A. and M.F. Ramalhoto. (2000). ”On the Waiting Time Distribution and the Busy Period of a Retrial Queue with Constant Retrial Rate.” Stochastic Modelling and Applications 3(2), 37–47.
[21] Khalil, Z., G. Falin and T. Yang. (1992). ”Some Analytical Results for Congestion in Subscriber Line Modules.” Queueing Systems 10, 381–402. · Zbl 0786.60116
[22] Krishna Kumar, B., S. Pavai Madheswari, and A. Vijayakumar. (2002). ”The M/G/1 retrial queue with feedback and starting failures.” Applied Mathematical Modelling 26, 1057–1075. · Zbl 1018.60088
[23] Kulkarni, V.G. and H.M. Liang. (1997). ”Retrial Queues Revisited.” In J.H. Dshalalow (ed.), Frontiers in Queueing. CRC Press, New York, pp. 19–34. · Zbl 0871.60074
[24] Latouche, G. and V. Ramaswami. (1999). Introduction to Matrix Analytic Methods in Stochastic Modelling. ASA-SIAM, Philadelphia. · Zbl 0922.60001
[25] Neuts, M.F. (1981). Matrix-Geometric Solutions in Stochastic Models -An Algorithmic Approach. John Hopkins University Press, Baltimore, MD. · Zbl 0469.60002
[26] Neuts, M.F. and Rao, B.M. (1990). ”Numerical Investigation of a Multiserver Retrial Model.” Queueing Systems 7, 169–190. · Zbl 0711.60094
[27] Pearce, C.E.M. (1989). ”Extended Continued Fractions, Recurrence Relations and two Diamensional Markov Processes.” Advances in Applied Probability 21, 357–375. · Zbl 0672.60073
[28] Ramalhoto, M.F. and A. Gomez-Corral. (1998). ”Some Decomposition Formulae for M/M/r/r+d Queues and Without Retrials.” Communications in Statistics – Stochastic Models 14(1&2), 123–145. · Zbl 0928.60079
[29] Stepanov, S.N. (1999). ”Markov Models with Retrials: The Calculation of Stationary Performance Measures Based on the Concept of Truncation.” Mathematical and Computer Modeling 30, 207–228. · Zbl 1042.60547
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