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2011, 1(4): 713-725. doi: 10.3934/naco.2011.1.713

## On the stationary LCFS-PR single-server queue: A characterization via stochastic intensity

 1 Department of Mathematical and Computing Sciences, Tokyo Institute of Technology, Tokyo 152-8552, Japan

Received  June 2011 Revised  July 2011 Published  November 2011

We consider a stationary single-server queue with preemptive-resume last-come, first-served (LCFS-PR) queueing discipline. The LCFS-PR single-server queue has some interesting properties and has been studied in the literature. In this paper, we generalize the previous results such that the input process to the queue is given as a general stationary marked point process and derive some formulas concerning the joint distribution of queue length and remaining service times of respective customers in the system at arbitrary time instances as well as at arrival instances. The tool for derivation is the Palm-martingale calculus; that is, the connection between the notion of Palm probability and that of stochastic intensity.
Citation: Naoto Miyoshi. On the stationary LCFS-PR single-server queue: A characterization via stochastic intensity. Numerical Algebra, Control and Optimization, 2011, 1 (4) : 713-725. doi: 10.3934/naco.2011.1.713
##### References:
 [1] F. Baccelli and P. Brémaud, "Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences,'' 2nd edition, Springer-Verlag, Berlin, 2003. [2] V. E. Beneš, On queues with Poisson arrivals, Ann. Math. Statist., 28 (1957), 670-677. doi: 10.1214/aoms/1177706878. [3] P. Brémaud, "Point Processes and Queues: Martingale Dynamics,'' Springer-Verlag, New York, 1981. [4] P. Brémaud, Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability, Queueing Systems Theory Appl., 5 (1989), 99-111. doi: 10.1007/BF01149188. [5] R. B. Cooper and S. C. Niu, Beneš's formula for M/G/1-FIFO ''explained'' by preemptive-resume LIFO, J. Appl. Probab., 23 (1986), 550-554. doi: 10.2307/3214199. [6] D. Fakinos, The G/G/1 queueing system with a particular queue discipline, J. Roy. Statist. Soc. Ser. B, 43 (1981), 190-196. [7] D. Fakinos, The single-server queue with service depending on queue size with the preemptive-resume last-come-first-served queue discipline, J. Appl. Probab., 24 (1987), 758-767. doi: 10.2307/3214105. [8] F. P. Kelly, The departure process from a queueing system, Math. Proc. Cambridge Philos. Soc., 80 (1976), 283-285. doi: 10.1017/S0305004100052919. [9] R. M. Loynes, The stability of a queue with non-independent interarrival and service times, Proc. Cambridge Philos. Soc., 58 (1962), 497-520. doi: 10.1017/S0305004100036781. [10] B. Melamed and W. Whitt, On arrivals that see time averages: A martingale approach, J. Appl. Probab., 27 (1990), 376-384. doi: 10.2307/3214656. [11] N. Miyoshi, On the stationary workload distribution of work-conserving single-server queues: A general formula via stochastic intensity, J. Appl. Probab., 38 (2001), 793-798. doi: 10.1239/jap/1005091044. [12] F. Papangelou, Integrability of expected increments of point processes and a related random change of scale, Trans. Amer. Math. Soc., 165 (1972), 483-506. doi: 10.1090/S0002-9947-1972-0314102-9. [13] T. Takine, Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain, Queueing Syst., 42 (2002), 131-151. doi: 10.1023/A:1020152920794. [14] G. Yamazaki, The GI/G/1 queue with last-come-first-served, Ann. Inst. Statist. Math., 34 (1982), 599-604. doi: 10.1007/BF02481057. [15] G. Yamazaki, Invariance relations of GI/G/1 queueing systems with preemptive-resume last-come-first-served queue discipline, J. Oper. Res. Soc. Japan, 27 (1984), 338-347.

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##### References:
 [1] F. Baccelli and P. Brémaud, "Elements of Queueing Theory: Palm-Martingale Calculus and Stochastic Recurrences,'' 2nd edition, Springer-Verlag, Berlin, 2003. [2] V. E. Beneš, On queues with Poisson arrivals, Ann. Math. Statist., 28 (1957), 670-677. doi: 10.1214/aoms/1177706878. [3] P. Brémaud, "Point Processes and Queues: Martingale Dynamics,'' Springer-Verlag, New York, 1981. [4] P. Brémaud, Characteristics of queueing systems observed at events and the connection between stochastic intensity and Palm probability, Queueing Systems Theory Appl., 5 (1989), 99-111. doi: 10.1007/BF01149188. [5] R. B. Cooper and S. C. Niu, Beneš's formula for M/G/1-FIFO ''explained'' by preemptive-resume LIFO, J. Appl. Probab., 23 (1986), 550-554. doi: 10.2307/3214199. [6] D. Fakinos, The G/G/1 queueing system with a particular queue discipline, J. Roy. Statist. Soc. Ser. B, 43 (1981), 190-196. [7] D. Fakinos, The single-server queue with service depending on queue size with the preemptive-resume last-come-first-served queue discipline, J. Appl. Probab., 24 (1987), 758-767. doi: 10.2307/3214105. [8] F. P. Kelly, The departure process from a queueing system, Math. Proc. Cambridge Philos. Soc., 80 (1976), 283-285. doi: 10.1017/S0305004100052919. [9] R. M. Loynes, The stability of a queue with non-independent interarrival and service times, Proc. Cambridge Philos. Soc., 58 (1962), 497-520. doi: 10.1017/S0305004100036781. [10] B. Melamed and W. Whitt, On arrivals that see time averages: A martingale approach, J. Appl. Probab., 27 (1990), 376-384. doi: 10.2307/3214656. [11] N. Miyoshi, On the stationary workload distribution of work-conserving single-server queues: A general formula via stochastic intensity, J. Appl. Probab., 38 (2001), 793-798. doi: 10.1239/jap/1005091044. [12] F. Papangelou, Integrability of expected increments of point processes and a related random change of scale, Trans. Amer. Math. Soc., 165 (1972), 483-506. doi: 10.1090/S0002-9947-1972-0314102-9. [13] T. Takine, Matrix product-form solution for an LCFS-PR single-server queue with multiple arrival streams governed by a Markov chain, Queueing Syst., 42 (2002), 131-151. doi: 10.1023/A:1020152920794. [14] G. Yamazaki, The GI/G/1 queue with last-come-first-served, Ann. Inst. Statist. Math., 34 (1982), 599-604. doi: 10.1007/BF02481057. [15] G. Yamazaki, Invariance relations of GI/G/1 queueing systems with preemptive-resume last-come-first-served queue discipline, J. Oper. Res. Soc. Japan, 27 (1984), 338-347.
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