A Fast Algorithm for Multiserver Queueing Systems with Setup Times and Power-saving Modes

Multi-server queueing models with setup times have been extensively investigated due to their applications in data centers, which have rapidly expanded in recent years. These models employ an ON-OFF policy to reduce energy consumption for idle servers, though this can increase delay due to setup processes. To minimize the drawback of the ON-OFF policy, a setup queueing model with a power-saving policy was recently proposed in [9], enabling servers in a power-saving mode to process incoming jobs while simultaneously setting up to a normal mode. Our paper considers the same model but adopts a generating function approach rather than the matrix geometric method as in [9]. Our approach provides exact expressions for the joint stationary queue length distribution, generating functions, and factorial moments of any order. By exploiting the special structure of the Markov chain, the generating function approach achieves significant reductions in computational complexity compared to existing methods. In addition, this paper presents numerical experiments to evaluate the performance-energy trade-off using an existing speed-based cost function.