We consider a Markov decision process for an queue that is controlled by batches of negative customers. More specifically, we derive conditions that imply threshold-type optimal policies, under either the total discounted cost criterion or the average cost criterion. The performance analysis of the model when it operates under a given threshold-type policy is also studied. We prove a stability condition and a complete stochastic comparison characterization for models operating under different thresholds. Exact and asymptotic results concerning the computation of the stationary distribution of the model are also derived.
Mots clés : queueing, Markov decision processes, negative customers, stationary distribution, stochastic comparison
@article{RO_2004__38_2_121_0,
author = {Artalejo, Jesus R. and Economou, Antonis},
title = {Optimal control and performance analysis of an $M^{X}/M/1$ queue with batches of negative customers},
journal = {RAIRO - Operations Research - Recherche Op\'erationnelle},
pages = {121--151},
publisher = {EDP-Sciences},
volume = {38},
number = {2},
year = {2004},
doi = {10.1051/ro:2004016},
zbl = {1092.90013},
mrnumber = {2081834},
language = {en},
url = {http://www.numdam.org/item/RO_2004__38_2_121_0/}
}
Artalejo, Jesus R.; Economou, Antonis. Optimal control and performance analysis of an $M^{X}/M/1$ queue with batches of negative customers. RAIRO - Operations Research - Recherche Opérationnelle, Tome 38 (2004) no. 2, pp. 121-151. doi : 10.1051/ro:2004016. http://www.numdam.org/item/RO_2004__38_2_121_0/
[1] , G-networks: A versatile approach for work removal in queueing networks. Eur. J. Oper. Res. 126 (2000) 233-249. | MR 1785793 | Zbl 0971.90007
[2] , Dynamic Programming, Deterministic and Stochastic Models. Prentice-Hall, Englewood Cliffs, New Jersey (1987). | MR 896902 | Zbl 0649.93001
[3] , Optimal control of bulk queues with compound Poisson arrivals and batch service. Opsearch 21 (1984) 227-245. | MR 777833 | Zbl 0552.60092
[4] and, Optimal control of batch service queues. Adv. Appl. Prob. 5 (1973) 340-361. | MR 341657 | Zbl 0264.60066
[5] , and, Queueing Networks: Customers, Signals and Product Form Solutions. Wiley, Chichester (1999). | Zbl 0936.90010
[6] , On the control of a compound immigration process through total catastrophes. Eur. J. Oper. Res. 147 (2003) 522-529. | MR 1965253 | Zbl 1026.90091
[7] and, Computation of the stationary distribution of the queue size in an queueing system with variable service rate. J. Appl. Prob. 17 (1980) 515-522. | MR 568961 | Zbl 0428.60093
[8] , Random neural networks with negative and positive signals and product-form solutions. Neural Comput. 1 (1989) 502-510.
[9] , Product-form queueing networks with negative and positive customers. J. Appl. Prob. 28 (1991) 656-663. | MR 1123837 | Zbl 0741.60091
[10] , G-networks with signals and batch removal. Probab. Eng. Inf. Sci. 7 (1993) 335-342.
[11] , and, Queues with negative arrivals. J. Appl. Prob. 28 (1991) 245-250. | MR 1090463 | Zbl 0744.60110
[12] and, Stability of product form G-networks. Probab. Eng. Inf. Sci. 6 (1992) 271-276. | Zbl 1134.60396
[13] and, Introduction to Queueing Networks. Wiley, Chichester (1998). | MR 874339 | Zbl 0654.60079
[14] and, The M/G/1 queue with negative customers. Adv. Appl. Prob. 28 (1996) 540-566. | MR 1387890 | Zbl 0861.60088
[15] and, Discrete-time Markov Control Processes. Springer, New York (1996). | MR 1363487 | Zbl 0840.93001
[16] , Optimal control of a truncated general immigration process through total catastrophes. J. Appl. Prob. 36 (1999) 461-472. | MR 1724812 | Zbl 0955.90147
[17] , Characterization of the optimal policy for the control of a simple immigration process through total catastrophes. Oper. Res. Letters 24 (1999) 245-248. | MR 1719920 | Zbl 0954.90063
[18] , and, An system with set-up time for server replacement. Transactions of the Institute of Electronics, Information and Communication Engineers J74-A-10 (1991) 1586-1593.
[19] and, An vacation model with two service modes. Prob. Eng. Inform. Sci. 9 (1994) 355-374. | MR 1365266
[20] and, Optimal control of an queue with two service modes. Eur. J. Oper. Res. 113 (1999) 610-619. | Zbl 0947.90028
[21] , Markov Decision Processes. Wiley, New York (1994). | MR 1270015 | Zbl 0829.90134
[22] , Applied Probability Models with Optimization Applications. Holden-Day Inc., San Francisco (1970). | MR 264792 | Zbl 0213.19101
[23] , Introduction to Stochastic Dynamic Programming. Academic Press, New York (1983). | MR 749232 | Zbl 0567.90065
[24] , Stochastic Dynamic Programming and the Control of Queueing Systems. Wiley, New York (1999). | MR 1645435 | Zbl 0997.93503
[25] , and, Mean drifts and the non-ergodicity of Markov chains. Oper. Res. 31 (1983) 783-789. | MR 720515 | Zbl 0525.60072
[26] , An equivalence between continuous and discrete time Markov decision processes. Oper. Res. 27 (1979) 616-620. | MR 533923 | Zbl 0413.90079
[27] , Comparison Methods for Queues and Other Stochastic Models. Wiley, Chichester (1983). | MR 754339 | Zbl 0536.60085
[28] , Control of the service process in a queueing system. Eur. J. Oper. Res. 23 (1986) 141-158. | MR 825606 | Zbl 0583.60092
[29] , A First Course in Stochastic Models. Wiley, Chichester (2003). | MR 2190630 | Zbl 1088.60002
[30] , and, Analysis of M/G/1 stochastic clearing systems. Stochastic Anal. Appl. 20 (2002) 1083-1100. | MR 1938256 | Zbl 1019.60087
