A vast amount of literature has appeared on vacation queues. In the well-known number- and time-limited vacation policies, the server goes on vacation if the number of customers, respectively, work (time slots) served since the previous vacation reaches a specified value, or if the system becomes empty, whichever occurs first. However, in practice, the server does not always go on vacation when the system is empty if the number of customers/work to be served has not yet reached the specified amount. Therefore, we study modified number- and time-limited vacation policies, where we account for this feature. We complement our recent work on these vacation policies by considering a discrete time, instead of a continuous-time, setting. We therefore adopt a different analysis approach, which enables us to obtain similar as well as new results as compared to our previous work. The results in this paper are valid for a memoryless distribution, but also for distributions with finite support, and a mixture of geometric distributions.
Modified vacation policies Queueing Discrete time Matrix generating functions Decomposition
Mathematics Subject Classification
60K25 68M20 90B22
This is a preview of subscription content, log in to check access.
The authors would like to thank Ivo Adan, Onno Boxma, and Offer Kella, as was well as the reviewers, for their valuable comments on this paper. We believe that these comments have considerably improved the quality of this paper.
Krishnamoorthy, A., Pramod, P., Chakravarthy, S.: Queues with interruptions: a survey. TOP 22, 290–320 (2014)CrossRefGoogle Scholar
Andriansyah, R.: Order-picking workstations for automated warehouses. Ph.D. thesis, Eindhoven University of Technology (2011)Google Scholar
Claeys, D., Adan, I., Boxma, O.: Stochastic bounds for order flow times in parts-to-picker warehouses with remotely located order-picking workstations. Eur. J. Oper. Res. 254(3), 895–906 (2016)CrossRefGoogle Scholar
Wang, H.: A survey of maintenance policies of deteriorating systems. Eur. J. Oper. Res. 139, 469–489 (2002)CrossRefGoogle Scholar
Jardine, A., Tsang, A.: Maintenance, Replacement, and Reliability. Theory and Applications. CRC Press, Boca Raton (2006)Google Scholar
Si, X., Wang, W., Hu, C., Zhou, D.: Remaining useful life estimation—a review on the statistical data driven approaches. Eur. J. Oper. Res. 213, 1–14 (2011)CrossRefGoogle Scholar
Boxma, O., Claeys, D., Gulikers, L., Kella, O.: A queueing system with vacations after N services. Nav. Res. Logist. 62(8), 648–658 (2015)CrossRefGoogle Scholar
Adan, I., Boxma, O., Claeys, D., Kella, O.: A queueing system with vacations after a random amount of work. SIAM J. Appl. Math. 78(3), 1697–1711 (2018)CrossRefGoogle Scholar
Fiems, D., Bruneel, H.: Discrete-time queueing systems with Markovian preemptive vacations. Math. Comput. Model. 57, 782–792 (2013)CrossRefGoogle Scholar
Gail, H., Hantler, S., Taylor, B.: Spectral analysis of M/G/1 and G/M/1 type Markov chains. Adv. Appl. Prob. 28, 114–165 (1996)CrossRefGoogle Scholar
Bruneel, H., Kim, B.: Discrete-Time Models for Communication Systems Including ATM. Kluwer Academic, Boston (1993)CrossRefGoogle Scholar
De Clercq, S., Rogiest, W., Steyaert, B., Bruneel, H.: Stochastic decomposition in discrete-time queues with generalized vacations and applications. J. Ind. Manag. Optim. 8(4), 925–938 (2013)CrossRefGoogle Scholar
Kim, N., Chae, K., Chaudhry, M.: An invariance relation and a unified method to derive stationary queue length distributions. Oper. Res. 52(5), 756–764 (2004)CrossRefGoogle Scholar
Adan, I., van Leeuwaarden, J., Winands, E.: On the application of Rouché’s theorem in queueing theory. Oper. Res. Lett. 34, 355–360 (2006)CrossRefGoogle Scholar
Takagi, H., Leung, K.: Analysis of a discrete-time queueing system with time-limited service. Queueing Syst. 18, 183–197 (1994)CrossRefGoogle Scholar