Abstract
This paper deals with retrial queueing models having an unlimited/a finite orbit capacity with control retrial policy of a multiprogramming–multiprocessor computer network system. Under the Markovian assumptions and light-traffic condition, the steady-state probabilities of the number of programs in the system and the mean number of programs in the orbit are studied using matrix geometric/analytic methods. The expressions for the Laplace-Stieglitz transforms of the busy period and the waiting time are obtained. The probability generating function for the number of retrials made by a tagged program is also derived. Some interesting performance measures of the system and the various moments of quantities of interest are discussed. Finally, extensive numerical results are illustrated to reveal the impact of the system parameters on the performance measures.
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References
Adiri, I., Hofri, M., & Yadin, M. (1973). A multiprogramming queue. Journal of ACM, 20(4), 589–603.
Allen, A. O. (2006). Probability, statistics and queuing theory with computer applications (2nd ed.). New Delhi, India: Academic Press (An imprint of Elsevier).
Almasi, B., Roszik, J., & Sztrik, J. (2005). Homogeneous finite-source retrial queues with server subject to breakdowns and repairs. Mathematical and Computer Modelling, 42, 673–682.
Artalejo, J. R. (2010). Accessible bibliography on retrial queues: Progress in 2000–2009. Mathematical and Computer Modelling, 51, 1071–1081.
Artalejo, J. R., & Gomez-Corral, A. (2008). Retrial queueing systems: A computational approach. Berlin: Springer.
Artalejo, J. R., Gomez-Corral, A., & Neuts, M. F. (2000). Numerical analysis of multiserver retrial queues operating under a full access policy. In G. Lotouche & P. G. Taylor (Eds.), Advances in algorithmic methods for stochastic models (pp. 1–19). New Jersey: Notable Publications Inc.
Artalejo, J. R., Gomez-Corral, A., & Neuts, M. F. (2001). Analysis of multiserver queues with constant retrial rate. European Journal of Operational Research, 135, 569–581.
Avi-Itzhak, B., & Halfin, S. (1987). Server sharing with a limited number of service positions and symmetric queues. Journal of Applied Probability, 24(4), 990–1000.
Avi-Itzhak, B., & Halfin, S. (1988). Response times in M/M/1 time-sharing schemes with limited number of service positions. Journal of Applied Probability, 25(3), 579–595.
Avi-Itzhak, B., & Heyman, D. P. (1973). Approximate queueing models for multiprogramming computer systems. Operations Research, 21, 1212–1230.
Battestilli, T., & Perros, H. (2003). An introduction to optical burst switching. IEEE Communications Magazine, 41(8), 10–15.
Bertsekas, D., & Gallager, R. (1992). Data networks (2nd ed.). New Jersey: Prentice-Hall International.
Bini, D., & Meini, B. (1995). On cyclic reduction applied to a class of Toeplitz-like matrices arising in queueing problems. In W. J. Stewart (Ed.), Computations with Markov Chains (pp. 21–38). Boston: Kluwer.
Brandwajn, A. (1977). A queueing model of multiprogrammed computer systems under full load conditions. Journal of ACM, 24(2), 222–240.
Chen, Y., Wu, H., Xu, D., & Qiao, C. (2003). Performance analysis of optical burst switched node with deflection routing. Proceedings of IEEE ICC 2003, 2, 1355–1359.
Choi, B. D., Shin, Y. W., & Ahn, W. C. (1992). Retrial queues with collision arising from unslotted CSMA / CD protocol. Queueing Systems, 11, 335–356.
Daduna, H. (1986). Cycle times in two-stage closed queueing networks: Applications to multiprogrammed computer systems with virtual memory. Operations Research, 34(2), 281–288.
Dragieva, V. I. (2013). A finite source retrial queues: Number of retrials. Communications in Statistics-Theory and Method, 42, 812–829.
Elhafsi, E. H., & Molle, M. (2007). On the solution to QBD processes with finite state space. Stochastic Analysis and Applications, 25, 763–779.
Falin, G. I. (1990). A survey of retrial queue. Queueing Systems, 7, 127–167.
Falin, G. I., & Artalejo, J. R. (1998). A finite source retrial queue. European Journal of Operational Research, 108, 409–424.
Falin, G. I., & Templeton, J. G. C. (1997). Retrial queues. London: Chapman and Hall.
Fayolle, G. (1986). A simple telephone exchange with delayed feedbacks. In O. J. Boxma, J. W. Cohen, & H. C. Tijms (Eds.), Teletraffic analysis and computer performance evaluation (Vol. 7, pp. 245–253). Amsterdam: Elsevier.
Gaver, D. P. (1967). Probability models for multiprogramming computer systems. Journal of ACM, 14(3), 423–438.
Gaver, D. P., & Humfeld, G. (1976). Multitype multiprogramming models. Acta Informatica, 7, 111–121.
Gaver, D. P., & Shedler, G. S. (1973). Processor utilization in multiprogramming systems via diffusion approximations. Operations Research, 21(2), 569–576.
Gelenbe, E., & Mitrani, I. (2010). Analysis and synthesis of computer systems (2nd ed.). London: Imperial College Press.
Hofri, M. (1978). A generating-function analysis of multiprogramming queues. International Journal of Computer Information Sciences, 7(2), 121–155.
Kameda, H. (1986). Effects of job loading policies for multiprogramming systems in processing a job stream. ACM Transactions on Computer Systems, 4(1), 71–106.
Konheim, A. G., & Reiser, M. (1976). A Queuing model with finite waiting room and blocking. Journal of ACM, 23(2), 328–341.
Konheim, A. G., & Reiser, M. (1978). Finite capacity queuing systems with applications in computer modeling. SIAM Journal of Computing, 7(2), 210–229.
Krishna Kumar, B., & Raja, J. (2006). On multiserver feedback retrial queues with balking and control retrial rate. Annals of Operations Research, 141(1), 211–232.
Kulkarni, V. G., & Liang, H. M. (1997). Retrial queues revisited. In J. H. Dshalalow (Ed.), Frontiers in queueing: Models and applications in science and engineering (pp. 19–34). New York, Boca Raton: CRC Press.
Latouche, G. (1981). Algorithmic analysis of a multiprogramming-multiprocessor computer system. Journal of ACM, 28(4), 662–679.
Latouche, G., & Ramaswami, V. (1999). Introduction to matrix analytic method in stochastic modelling. Philadelphia: ASA-SIAM.
Lewis, P. V. W., & Shedler, G. S. (1971). A cycle-queue model of system overhead in multiprogrammed computer systems. Journal of ACM, 18(2), 199–220.
Neuts, M. F. (1981). Matrix-geometric solutions in stochastic models: An algorithmic approach. Baltimore: John Hopkins University Press.
Ramalhoto, M. F., & Gomez-Corral, A. (1998). Some decomposition formulae for M/M/r/r+d queues with constant retrial rate. Stochastic Models, 14, 123–145.
Rege, K. M., & Sengupta, B. (1985). Sojourn time distribution in a multiprogrammed computer system. AT & T Tech Journal, 64, 1077–1090.
Verma, S., Chaskar, H., & Ravikanth, R. (2000). Optical burst switching: A viable solution for terabit IP backbone. IEEE Network, 14(6), 48–53.
Wong, E. W. M., Andrew, L. L. H., Cui, T., Moran, B., Zalesky, A., Tucker, R. S., et al. (2009). Towards a buffer less optical internet. Journal of Lightwave Technology, 27, 2817–2833.
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The authors would like to thank the anonymous reviewers and the guest editors for their valuable comments and suggestions to improve the quality of this paper.
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Kumar, B.K., Thanikachalam, A., Kanakasabapathi, V. et al. Performance analysis of a multiprogramming–multiprocessor retrial queueing system with orderly reattempts. Ann Oper Res 247, 319–364 (2016). https://doi.org/10.1007/s10479-015-2005-3
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DOI: https://doi.org/10.1007/s10479-015-2005-3