Abstract
We study the busy period of a single server queueing system operating in two alternating modes - working and vacation. In the two modes the systems run as an \(M^{X}/G/1\) queue with disasters, but with different parameters. The vacation mode starts once the number of customers drops to zero. It is terminated randomly (when it is not empty) with a transition to the working mode. At such a transition moment all the customers are transferred to the working mode; the service of the customer being served is lost and it starts from scratch in the working mode. Every busy period starts with a batch arrival into an empty system and terminates at the first time that the number of customers drops to zero. The working and the vacation periods are analyzed too. Finally, we apply the results to obtain the probability generating functions of the number of customers in the working, as well as in the vacation periods.
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References
Baba Y (1986) On the \(M^X/G/1\) queue with vacation time. Operation Research Letters. 5(2):93–98
Doshi BT (1986) Queueing systems with vacations - A survey. Queueing System 1:29–66
I. Kleiner, E. Frostig, and D. Perry (2021). A decomposition property for a \(M^X/G/1\) queues with vacation. Indagationes Mathematicae, forthcoming
Levy Y, Yechiali U (1975) Utilization of idle time in \(M/G/1\) queueing system. Management Science. 22(2):202–211
Li J (2013) Analysis of the dicrete-time \(Geo/G/1\) working vacation queue and its application to network scheduling. Computers and Industrial Engineering. 65:594–604
Mytalas GC, Zazanis MA (2015) An \(M^X/G/1\) queueing system with disasters and repairs under a multiple adapted vacation policy. Naval Research Logistics Quarterly. 62(3):171–189
Tian N, Zhang ZG (2006) Vacation Queueing Models Theory and Applications. Springer, New-York
Wolff RW (1989) Stochastic Modeling and the Theory of Queues. Prentice-Hall, Englewood Cliffs, NJ, p 07632
Ye J, Liu L, Jiang T (2016) Analysis of a single-server queue with disasters and repairs under Bernoulli vacation schedule. Systems Science and Information. 4(6):547–559
Yechiali U (2007) Queues with system disasters and impatient customers when system is down. Queueing Systems. 56:195–202
Funding
The research of Esther Frostig is partially funded by ISF (Israel Science Foundation), Grant Grant 1999/18). The research of David Perry is partially funded by ISF (Israel Science Foundation), Grant 3274/19.
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Kleiner, I., Frostig, E. & Perry, D. Busy Periods for Queues Alternating Between Two Modes. Methodol Comput Appl Probab 25, 60 (2023). https://doi.org/10.1007/s11009-023-10037-y
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DOI: https://doi.org/10.1007/s11009-023-10037-y