Abstract
In this paper, we study the transient behavior of a level dependent single server queuing system with a waiting room of finite size during the busy period. The focus is on the level dependent PH/PH/1/K queue. We derive in closed form the joint transform of the length of the busy period, the number of customers served during the busy period, and the number of losses during the busy period. We differentiate between two types of losses: the overflow losses that are due to a full queue and the losses due to an admission controller. For the M/PH/1/K, M/PH/1/K under a threshold policy, and PH/M/1/K queues, we determine simple expressions for their joint transforms.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Al Hanbali, A. Busy period analysis of the level dependent PH/PH/1/K queue. Queueing Syst 67, 221–249 (2011). https://doi.org/10.1007/s11134-011-9213-6
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DOI: https://doi.org/10.1007/s11134-011-9213-6
Keywords
- PH/PH/1/K queue
- Phase-type distributions
- Level dependent queues
- Busy period
- Transient analysis
- Absorbing Markov chains
- Matrix analytical approach