Buffer Overflow Duration in a Processing System with a Flexible Server Vacation

Among the technological challenges of today, the issue of reducing energy consumption is of particular importance. This problem concerns service systems, data transmission networks, communication solutions and production systems. On the other hand, however, the aim is to constantly improve the quality of customer service, which, of course, naturally entails the risk of increased energy consumption. In the article, we propose a queue-motivated service system with bounded accumulation buffer capacity in which a solution supporting power consumption based on a single vacation policy is installed. Simultaneously, the vacation policy is flexible in the sense that the vacation period can be interrupted if the number of waiting messages (jobs, clients, etc.) reaches a predefined level. A closed form of the tail cumulative distribution function of the buffer overflow duration conditioned by the initial buffer state is found. The analytic approach is based on integral equations and linear algebra. Numerical study of the sensitivity of this stochastic characteristic on main system parameters is attached as well. More precisely, the impact of the initial buffer state, the arrival rate, the mean vacation duration and the value of the threshold level interrupting the ongoing vacation is investigated separately.