Dynamic Control for Nonstationary Queueing Networks

Nonstationary queueing networks are notoriously difficult to analyze and control. One reason is that steady state analysis and techniques are not useful since the model parameters in practice are not constant and depend on time. In this work, we analyze two optimal control problems for nonstationary Jackson networks with abandonment where our main goal is to optimally control the number of servers in the network. The first control problem approximates the stochastic dynamics of the Jackson network with a deterministic fluid model and optimizes with respect to the fluid model. In this case, we prove that the optimal solution is bang-bang and prove an asymptotic optimality result, which shows the fluid model optimal solution is asymptotically optimal for the scaled stochastic control problem. Our second approach exploits a Gaussian infinite server approximation for the queue length process and optimizes with respect to the number of servers. Unlike the fluid model’s bang-bang solution, the infinite server approximation yields an square root staffing formula that depends on the cost to revenue ratio at each station. This proves the optimality of the square root staffing formula in a network setting.