A Non-Preemptive Priority Queueing System with a Single Server Serving Two Queues M/G/1 and M/D/1 with Optional Server Vacations Based on Exhaustive Service of the Priority Units
Kailash C. Madan
DOI: 10.4236/am.2011.26106   PDF    HTML     6,375 Downloads   12,922 Views   Citations


We study a vacation queueing system with a single server simultaneously dealing with an M/G/1 and an M/D/1 queue. Two classes of units, priority and non-priority, arrive at the system in two independent Poisson streams. Under a non-preemptive priority rule, the server provides a general service to the priority units and a deterministic service to the non-priority units. We further assume that the server may take a vacation of random length just after serving the last priority unit present in the system. We obtain steady state queue size distribution at a random epoch. Corresponding results for some special cases, including the known results of the M/G/1 and the M/D/1 queues, have been derived.

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K. Madan, "A Non-Preemptive Priority Queueing System with a Single Server Serving Two Queues M/G/1 and M/D/1 with Optional Server Vacations Based on Exhaustive Service of the Priority Units," Applied Mathematics, Vol. 2 No. 6, 2011, pp. 791-799. doi: 10.4236/am.2011.26106.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] A. Cobham, “Priority Assignments in Waiting Line Problems,” Operions Research, Vol. 2, No. 1, 1954, pp. 70-76. doi:10.1287/opre.2.1.70
[2] T. E. Phipps, “Machine Repair as a Priority Waiting Line Problem,” Operations Research, Vol. 4, No. 1, 1956, pp. 76-85. doi:10.1287/opre.4.1.76
[3] L. E. Schrage, “The Queue M/G/1 with Feedback to Lower Priority Queues,” Management Science, Vol. 13, No. 7, 1967, pp. 466-474. doi:10.1287/mnsc.13.7.466
[4] N. K. Jaiswal, “Priority Queues,” Academic Press, New York, 1968.
[5] K. C. Madan, “A Priority Queueing System with Service Interruptions,” Statistica Neerlandica, Vol. 27, No. 3, 1973, pp. 115-123. doi:10.1111/j.1467-9574.1973.tb00217.x
[6] B. Simon, “Priorty Queues with Feedback,” Journal of the Association for Computing Machinery, Vol. 31, No. 1, 1984, pp. 134-149.
[7] H. Takagi, “Vacation and Priority Systems,” Queueing Analysis, Vol. 1, Amsterdam, 1991.
[8] B. D. Choi, and Y. Chang, “Single Server Retrial Queues with Priority Calls,” Mathematical and Computer Modeling Vol. 30, No. 3-4, 1999, pp. 7-32. doi:10.1016/S0895-7177(99)00129-6
[9] K. C. Madan and W. Abu-Dayyeah, “On a Combination of M/G/1 and M/D/1 Queues in Non-Preemptive Priority Queueing System,” Far East Journal of Theoretical Statistics, Vol. 10, No. 2, 2003, pp. 133-146.
[10] U. N. Bhat, “Elements of Applied Stochastic Processes,” Wiley, New York, 1972.
[11] Y. Levy and U. Yechiali, “Utilization of Idle Time in an M/G/1 Queueing System,” Management Science, Vol. 22, No. 2, 1975, pp. 202-211. doi:10.1287/mnsc.22.2.202
[12] L. Kleinrock, “Queueing Systems, Vol. 2, Computer Applications,” Wiley, New York, 1976.
[13] J. W. Cohen, “The Single Server Queue,” 2nd Edition, North-Holland, Amsterdam, 1982.
[14] T. T. Lee, “M/G/1/N Queue with Vacation Times and Exhaustive Service Discipline,” Operations Research, Vol. 32, No. 4, 1984, pp. 774-786. doi:10.1287/opre.32.4.774
[15] D. Gross and C. M. Harris, “Fundamentals of Queueing Theory,” 2nd Edition, Wiley, New York, 1985.
[16] D. R. Cox and H. D. Miller, “The Theory of Stochastic Processes,” Chapman and Hall, London, 1994.
[17] H. C. Tijms, “Stochastic Models: An Algorithmic Approach,” Wiley, New York, 1994.
[18] T. Yang and H. Li, “The M/G/1 Retrial Queue with the Server Subject to Starting Failures,” Queueing Systems, Vol. 16, No. 1-2, 1994, pp. 83-96. doi:10.1007/BF01158950
[19] B. D. Bunday, “Basic Queueing Theory,” 2nd Edition, Edward Arnold, Melbourne, 1995.
[20] K. C. Madan, “An M/G/1 Queue with Optional Deterministic Server Vacations,” Metron, Vol. 57, No. 3-4, 1999, pp. 83-95.
[21] K. C. Madan, “An M/G/1 Queue with Second Optional Service,” Queueing Systems, Vol. 34, No. 1-4, 2000, pp. 37-46. doi:10.1023/A:1019144716929

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