Novel Solutions for Closed Queueing Networks with Load-Dependent Stations

Abstract

Load-dependent closed queueing networks are difficult to approximate since their analysis requires to consider statedependent service demands. Standard evaluation techniques, such as mean-value analysis, are not equally efficient in the load-dependent setting, where mean queue-lengths are insufficient alone to recursively determine the model equilibrium performance. As such, novel exact techniques to address this class of models can benefit performance engineering practice by offering alternative trade-offs between accuracy and computational cost. In this paper, we derive novel exact solutions for the normalizing constant of state probabilities in the load-dependent setting. For single-class load-dependent models, we provide an explicit exact formula for the normalizing constant that is valid for models with arbitrary load-dependent rates. From this result, we derive two novel integral forms for the normalizing constant in multiclass load-dependent models, which involve integration in the real and complex domains, leading to novel numerical approximations. The paper also illustrates through experiments the computational gains and accuracy of the obtained expressions.