A batch arrival retrial queue with starting failures, feedback and admission control

Abstract

This paper is concerned with the analysis of a feedback M^[X]/G/1 retrial queue with starting failures and general retrial times. In a batch, each individual customer is subject to a control admission policy upon arrival. If the server is idle, one of the customers admitted to the system may start its service and the rest joins the retrial group, whereas all the admitted customers go to the retrial group when the server is unavailable upon arrival. An arriving customer (primary or retrial) must turn-on the server, which takes negligible time. If the server is started successfully (with a certain probability), the customer gets service immediately. Otherwise, the repair for the server commences immediately and the customer must leave for the orbit and make a retrial at a later time. It is assumed that the customers who find the server unavailable are queued in the orbit in accordance with an FCFS discipline and only the customer at the head of the queue is allowed for access to the server. The Markov chain underlying the considered queueing system is studied and the necessary and sufficient condition for the system to be stable is presented. Explicit formulae for the stationary distribution and some performance measures of the system in steady-state are obtained. Finally, some numerical examples are presented to illustrate the influence of the parameters on several performance characteristics.

Keywords

Batch arrival / feedback / repair / retrial queue / starting failure

References

[1] Artalejo J.R.. Analysis of an M/G/1 queue with constant repeated attempts and server vacations. Computers and Operations Research, 1997, 24: 493-504.

[2] Artalejo J.R.. A classified bibliography of research on retrial queues: progress in 1990–1999. Top, 1999, 7(2): 187-211.

[3] Artalejo J.R.. Accessible bibliography on retrial queues. Mathematical and Computer Modelling, 1999, 30: 1-6.

[4] Artalejo J.R., Atencia I.. On the single server retrial queue with batch arrivals. Sankhya, 2004, 66: 140-158.

[5] Artalejo J.R., Atencia I., Moreno P.. A discrete-time Geo^[X]/G/1 retrial queue with control of admission. Applied Mathematical Modelling, 2005, 29: 1100-1120.

[6] Artalejo J.R., Gomez-Corral A.. Retrial Queueing Systems: A Computational Approach, 2008, Berlin: Springer.

[7] Bianchi G.. Performance analysis of the IEEE 802.11 distributed coordination function. IEEE Journal on Selected Areas in Communications, 2000, 18: 535-547.

[8] Choi B.D., Shin Y.W., Ahn W.C.. Retrial queues with collision arising from unslotted CSMA/CD protocol. Queueing Systems, 1992, 11: 335-356.

[9] Choi B.D., Rhee K.H., Park K.K.. The M/G/1 retrial queue with retrial rate control policy. Probability in Engineering and Informational Sciences, 1993, 7: 26-46.

[10] Choi B.D., Kulkarni V.G.. Bhat U.N., Basawa I.V.. Feedback retrial queueing systems. Stochastic Model Related Fields, 1992, New York: Oxford University Press 93-105.

[11] Choi B.D., Kim Y.C., Lee Y.W.. The M/M/C retrial queue with geometric loss and feedback. Computers and Mathematics with Applications, 1998, 36: 41-52.

[12] Cooper R.B.. Introduction to Queueing Theory, 1981, New York: North-Holland

[13] Falin G.I.. A survey of retrial queues. Queueing Systems, 1990, 7: 127-167.

[14] Falin G.I., Templeton J.G.C.. Retrial Queues, 1997, London: Chapman & Hall.

[15] Farahmand K.. Single line queue with repeated demands. Queueing Systems, 1990, 6: 223-228.

[16] Fayolle G.. Boxma O.J., Cohen J.W., Tijms M.C.. A simple telephone exchange with delayed feedbacks. Teletraffic Analysis and Computer Performance Evaluation, 1986, Amsterdam: North-Holland 245-253.

[17] Gomez-Corral A.. Stochastic analysis of a single server retrial queue with general retrial times. Naval Resrarch Logistics, 1999, 46: 561-581.

[18] Ke J.-C., Chang F.-M.. Modified vacation policy for M/G/1 retrial queue with balking and feedback. Computers and Industrial Engineering, 2009, 57: 433-443.

[19] Krishna Kumar B., Pavai Madheswari S., Vijayakumar A.. The M/G/1 retrial queue with feedback and starting failures. Applied Mathematical Modelling, 2002, 26: 1057-1075.

[20] Kulkarni V.G., Choi B.D.. Retrial queues with server subject to breakdowns and repairs. Queueing Systems, 1990, 7: 191-208.

[21] Kulkarni V.G., Liang H.M.. Dshalalow J.H.. Retrial queues revisted. Frontiers in Queueing, 1997, Boca Raton: CRC Press 19-34.

[22] Li H., Zhao Y.Q.. A retrial queue with a constant retrial rate server downs and impatient customers. Stochastic Models, 2005, 21: 531-550.

[23] Gomez-Corral M., Artalejo J.R.. Analysis of an M/G/1 queue with two types of impatient units. Advances in Applied Probability, 1995, 27: 840-861.

[24] Sennott L.I., Humblet P.A., Tweedi R.L.. Mean drifts and the non-ergodicity of Markovchains. Operations Research, 1983, 31: 783-789.

[25] Wang J., Cao J., Li Q.. Reliability analysis of the retrial queue with server breakdowns and repairs. Queueing Systems, 2001, 38: 363-380.

[26] Wang J.. Reliability analysis of M/G/1 queues with general retrial times and server breakdowns. Progress in Natural Science, 2006, 5: 464-473.

[27] Wang J., Zhao Q.. Discrete-time retrial queue with general retrial times and starting failures. Mathematical and Computer Modelling, 2007, 45: 853-863.

[28] Wang J., Li J.. A single server retrial queue with general retrial times and two-phase service. Journal of Systems Science and Complexity, 2009, 22: 291-302.

[29] Yang T., Li H.. On the steady-state queue size distribution of the discrete-time Geo/G/1 queue with repeated customers. Queueing Systems, 1995, 21: 199-215.

[30] Yang T., Templeton J.G.C.. A survey on retrial queues. Queueing Systems, 1987, 2: 201-233.