Weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain

Our aim is to find sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure of a Markov chain. First, we study properties of the state indicator function and the state occupation measure of a Markov chain. In particular, we establish weak convergence of the state occupation measure under a scaling of the generator matrix. Then, relying on the connection between the state occupation measure and the Dynkin martingale, we provide sufficient conditions for weak convergence of stochastic integrals with respect to the state occupation measure. We apply our results to derive diffusion limits for the Markov-modulated Erlang loss model and the regime-switching Cox–Ingersoll–Ross process.

Sensitivity of mean-field fluctuations in Erlang loss models with randomized routing

In this paper we study a large system of N servers, each with capacity to process at most C simultaneous jobs; an incoming job is routed to a server if it has the lowest occupancy amongst d (out of N) randomly selected servers. A job that is routed to a server with no vacancy is assumed to be blocked and lost. Such randomized policies are referred to JSQ(d) (Join the Shortest Queue out of d) policies. Under the assumption that jobs arrive according to a Poisson process with rate $N\lambda^{(N)}$ where $\lambda^{(N)}=\sigma-\frac{\beta}{\sqrt{N}\,}$ , $\sigma\in\mathbb{R}_+$ and $\beta\in\mathbb{R}$ , we establish functional central limit theorems for the fluctuation process in both the transient and stationary regimes when service time distributions are exponential. In particular, we show that the limit is an Ornstein–Uhlenbeck process whose mean and variance depend on the mean field of the considered model. Using this, we obtain approximations to the blocking probabilities for large N, where we can precisely estimate the accuracy of first-order approximations.

A Queuing Model for Ventilator Capacity Management during the COVID-19 Pandemic

We present a queue model to inform ventilator capacity management under different COVID-19 pandemic scenarios. Our model was used to support ventilator capacity planning during the first wave of the COVID-19 epidemic in British Columbia (BC), Canada. The core of our framework is an extended Erlang loss model, which incorporates COVID-19 case projections, along with the proportion of cases requiring a ventilator, the delay from symptom onset to ventilation, non-COVID-19 ventilator demand, and ventilation time. We implemented our model using discrete event simulation to forecast ventilator utilization. The results predict when capacity would be reached and the rate at which patients would be unable to access a ventilator. We further determined the number of ventilators required to meet a performance indicator target for ventilator access. We applied our model to BC by calibrating to the BC Intensive Care Unit Database and by using local epidemic projections. Epidemic scenarios with and without reduced transmission, due to social distancing and other behavioral changes, were used to link public health interventions to operational impacts on ventilator utilization. The results predict that reduced transmission could potentially avert up to 50 deaths per day by ensuring that ventilator capacity would likely not be reached. Without reduced transmission, an additional 181 ventilators would be required to meet our performance indicator target that 95% of patients can access a ventilator immediately. Our model provides a tool for policy makers to quantify the interplay between public health interventions, necessary critical care resources, and performance indicators for patient access.

Surrogate Modeling for Capacity Planning of Charging Station Equipped With PV and Hydropneumatic Energy Storage

Due to the promising potential for environmental sustain-ability, there has been a significant increase of electric vehicles (EVs) and plug-in hybrid electric vehicles (PHEV) in the market. To support this increasing demand for EVs and PHEVs, challenges related to capacity planning and investment costs of public charging infrastructure must be addressed. Hence, in this paper, a capacity planning problem for EV charging stations is developed and aims to balance current capital investment costs and future operational revenue. The charging station considered in this work is assumed to be equipped with solar photovoltaic panel (PV) and an energy storage system which could be electric battery or the recently invented hydro-pneumatic energy storage (GLIDES, Ground-Level Integrated Diverse Energy Storage) system. A co-optimization model that minimizes investment and operation cost is established to determine the global optimal solution while combining the capacity and operational decision making. The operational decision making considers EV mobility which is modeled as an Erlang-loss system. Meanwhile, stochastic programming is adopted to capture uncertainties from solar radiation and charging demand of the EV fleet. To provide a more general and computationally efficient model, main configuration parameters are sampled in the design space and then fixed in solving the co-optimization model. The model can be used to provide insights for charging station placement in different practical situations. The sampled parameters include: the total number of EV charging slots, the PV area, the maximum capacity of the energy storage system, and daily mean EV arrival number in the Erlang-loss system. Based on the sampled parameter combinations and its responses, black-box mappings are then constructed using surrogate models (RBF, Kriging etc). The effectiveness of proposed surrogate modeling approach is demonstrated in the numerical experiments.