Abstract
In this paper, we consider an M/G/1 queueing system with two phases of heterogeneous service and a finite number of immediate Bernoulli feedbacks. All arriving customers are provided with the same type of service in the first phase. In the second phase, the customer has to choose from one of the several optional services which are available in the system. After having completed both phases of service, the customer is allowed to make an immediate feedback. The feedback service also consists of two phases. In the feedback, the first phase of service is of the same type as in the previous service. However, in the second phase, the customer may be permitted to choose an optional service different from the one chosen earlier. This feedback scheme is different from the usual Bernoulli feedback scheme. In the earlier papers on the usual Bernoulli feedback, the customer returns to the tail of the queue and waits patiently for his next round of service. In our system, if the customer desires to make a feedback, the customer immediately proceeds for a second round of service after completion of his first round , that is he is his own successor. The customer is allowed to make a finite number of such feedbacks before departing from the system. Our motivation for considering this model comes from our experience in banking transactions through the ATM. We obtain the probability generating function of the system size distribution in the steady state. Some useful performance measures are also obtained. Numerical examples are presented to illustrate the influence of the various parameters involved on the performance of the system.
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Al-Jararha, J., Madan, K.C.: An M/G/1 queue with second optional service with general service time distribution. Int. J. Inf. Manag. Sci. 14(3), 47–56 (2003)
Bonald, T., Prouti‘ere, A.: On performance bounds for the integration of elastic and adaptive streaming flows. In: Proceedings of Association of Computing Machinery Sigmetrics/Performance Joint International Conference on Measurement and Modeling of Computer Systems, pp. 235–245 (2004)
Chandrasekaran, V.M.: Transient and reliability analysis of M/G/1 feedback queue subject to catastrophes, server failures and repairs. Int. J. Pure Appl. Math. 77(5), 605–625 (2012)
Choudhury, G.: Some aspects of an M/G/1 queueing system with optional second service. Top 11, 141–150 (2003)
Choudhury, G.: An M/G/1 queueing system with two phase service under D-Policy. Int. J. Inf. Manag. Sci. 16(4), 1–17 (2005)
Choudhury, G., Madhuchanda P.: A two phase queueing system with Bernoulli feedback. Int. J. Inf. Manag. Sci. 16(1), 35–52 (2005)
Choudhury, G., Madhuchanda P.: A batch arrival queue with a second optional ervice channel under N-Policy. Stoch. Anal. Appl. 24, 1–21 (2006)
Disney, R.L., McNickle, D.C., Simon, B.: The M/G/1 queue with instantaneous Bernoulli feedback. Nav. Res. Logist. 27, 635–644 (1980)
Disney, R.L., Konig, D., Schmidt, V.: Stationary queue-length and waiting-time distributions in single-server feedback queues. Adv. Appl. Probab. 16, 437–446 (1984)
Disney, R.L., Konig, D.: Queueing networks: a survey of their random process. SIAM Rev. 27, 335–403 (1985)
Fontana, B., Berzosa, C.D.: M/G/1 queue with N-priorities and feedback: joint queue-length distributions and response time distribution for any particular sequence. In: Akiyama, W. (ed.) Teletraffic Issues in Advanced Information Society, ITC-11, pp. 452–458. Elsevier, North-Holland (1985)
Harchol-Balter, M., Schroeder, B., Bansal, N., Agrawal, M.: Sizebased scheduling to improve web performance. ACM Trans. Comput. Syst. 21(2), 207–233 (2003)
Jain, M., Bhargava, C.: Unreliable server M/G/1 queueing system with Bernoulli feedback, repeated attempts modied vacation, phase repair and discouragement. JKAU Eng. Sci. 20(2), 45–77 (2009). A.D. / 1430 A.H.
Kalyanaraman, R., Renganathan, N.: A single server instantaneous Bernoulli feedback queue with multiple vacation. Optim. J. Math. Program. Oper. Res. 1, 87–96 (1996)
Kleinrock, L.: Queueing Systems, vol. II. Computer Applications. Wiley, New York (1976)
Krishna Kumar, B., Arivudainambi D., Vijayakumar A.: An M/G/1 queue with unreliable server and no waiting capacity. Inf. Manag. Sci. 13(2), 35–50 (2002)
Krishna Kumar, B., Arivudainambi, D., Krishnamoorthy, A.: Some results on a generalized M/G/1 feedback queue with negative customers. Ann. Oper. Res. 143, 277–296 (2006)
Li, J., Wang, J.: An M/G/1 retrial queue with second multi optional service, feedback and unreliable server. Appl. Math. J. Chinese Univ. Set. B 21(3), 252–262 (2006)
Madan, K.C.: An M/G/1 queue with second optional service. Queueing Syst. 34, 37–46 (2000)
Madan, K.C., Abu-Dayyeh, W., Saleh, M.F.: An M/G/1 queue with second optional service and Bernoulli schedule server vacations. Syst. Sci. 28, 51–62 (2002)
Medhi, J.: A single server Poisson input queue with a second optional channel. Queueing Syst. 42, 239–242 (2002)
Nunez-Queija, R.: Processor sharing models for integrated services networks, Ph.D. Thesis, Eindhoven University of Technology (2000)
Roberts, J.W.: Engineering for quality of service. In: Park, K., Willinger, W. (eds.) Self-Similar Network Traffic and Performance Evaluation, 401420. Wiley, New York (2000)
Shahkar, G.H., Badamchizadeh, A.: On \(M/(G1,...,Gk )/V/1(BS)\). Far. East. J. Theor. Stat. 20(2), 151–162 (2006)
Salehirad, M.R., Badamchizadeh, A.: On the multi-phase M/G/1 queueing system with random feedback. CEJOR 17, 131–139 (2009)
Takacs, L.: A single server queue with feedback. Bell Sys. Tech. J. 42, 505–519 (1963)
Thangaraj, V.: Santhakumaran, A.: A queue with a pair of instantaneous independent Bernoulli feedback processes. Optim. J. Math. Program. Oper. Res. 27(3), 259–281 (1993)
Thangaraj, V.: Santhakumaran, A.: Sojourn times in queues with a pair of instantaneous independent Bernoulli feedback. Optim. J. Math. Program. Oper. Res. 29(3), 281–294 (1994)
Thangaraj, V., Vanitha, S.: A two phase M/G/1 feedback queue with multiple server vacation. Stoch. Anal. Appl. 27, 1231–1245 (2009)
Thangaraj, V., Vanitha, S.: M/G/1 queue with two-stage heterogeneous service compulsory server vacation and random breakdowns. Int. J. Contemp. Math. Sci. 5(7), 307–322 (2010)
Wang, J.: An M/G/1 queue with second optional service and server breakdowns. Int. J Comput. Math. Appl. 47, 1713–1723 (2004)
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Kalidass, K., Kasturi, R. A two phase service M/G/1 queue with a finite number of immediate Bernoulli feedbacks. OPSEARCH 51, 201–218 (2014). https://doi.org/10.1007/s12597-013-0136-3
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DOI: https://doi.org/10.1007/s12597-013-0136-3