Abstract
We study an M/M/m preemptive last-come, first-served queue with impatient customers without priority classes. We focus on the probability of service completion and abandonment as well as the waiting and service times of a unique customer who has the mean service and patience times that are different from those of all other customers in the steady state. The problem is formulated as the first passage times in a combination of two one-dimensional birth-and-death processes each with two absorbing states. We provide explicit expressions in terms of Laplace-Stieltjes transform of the distribution function for the time to service completion or abandonment, which is decomposed into the waiting and service times of the unique customer. A numerical example is presented in order to demonstrate the computation of theoretical formulas.
This work is supported by the Grant-in-Aid for Scientific Research (C) No. 26330354 from the Japan Society for the Promotion of Science (JSPS) in the academic year 2016.
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References
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Takagi, H.: Times to service completion and abandonment in the M/M/\( m \) preemptive LCFS queue with impatient customers. In: QTNA 2016, Proceedings of the 11th International Conference on Queueing Theory and Network Applications, Article No. 2. ACM (2016). ISBN 978-1-4503-4842-3/16/12. https://dx.doi.org/10.1145/3016032.3016036
Takagi, H.: Distribution of the waiting and service time in an M/M/\( m \) preemptive-resume LCFS queue with impatient customers. Int. J. Pure Appl. Math. 116(2), 501–545 (2017). https://dx.doi.org/10.12732/ijpam.v.116i2.21
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Takagi, H. (2017). Waiting and Service Time of a Unique Customer in an M/M/m Preemptive LCFS Queue with Impatient Customers. In: Yue, W., Li, QL., Jin, S., Ma, Z. (eds) Queueing Theory and Network Applications. QTNA 2017. Lecture Notes in Computer Science(), vol 10591. Springer, Cham. https://doi.org/10.1007/978-3-319-68520-5_2
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DOI: https://doi.org/10.1007/978-3-319-68520-5_2
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