Abstract
This paper considers the equilibrium balking behavior of customers in a single-server Markovian queue with variable vacation and vacation interruption, where the server can switch across four states: vacation, working vacation, idle period, and busy period. Once the queue becomes empty, the server commences a working vacation and slows down its service rate. However, this period may be interrupted anytime by the vacation interruption. Upon the completion of a working vacation, the server takes a vacation in a probability-based manner and stops service if the system is empty. The system stays idle after a vacation until a new customer arrives. The comparisons between the equilibrium balking strategy of customers and the optimal expected social benefit per time unit for each type of queue are elucidated and the inconsistency between the individual optimization and the social optimization is revealed. Moreover, the sensitivity of the expected social benefit and the equilibrium threshold with respect to the several parameters as well as diverse precision levels is illustrated through numerical examples in a competitive cloud environment.
Highlights
Queueing economics has received increasing attention in recent decades
This paper introduces a variable vacation and vacation interruption policy in the classical M/M/1 queue with the arrival rate λ and the normal service rate μ
Once a working vacation ends, the server immediately switches to a regular busy period if the queue is non-empty, otherwise, the system enters a vacation period with probability p(0 < p ≤ 1), or enters an idle state with probability 1 − p, and the vacation time follows an exponential distribution with parameter θ2
Summary
Queueing economics has received increasing attention in recent decades. It was firstly introduced by Naor [12], who proved that the decisions of individual customers often deviate from the whole interests in that system. Zhang et al [24] illustrated four equilibrium strategies and provided the optimal thresholds in a multiple working vacation queue. Tian and Wang [20] combined multiple classical vacations with working vacation and made the economic analysis of this model in the unobservable cases. Despite these considerable advantages, the modern management system is still far from a sensitive response. If the buffer has a waiting cloud user request, the virtual machine immediately changes from the semi dormancy state to the wake-up state and provides the normal service for cloud users.
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