Patient Choice & Queuing Model¶

The NHRA model uses a Wardrop Equilibrium approach to model endogenous demand at the Emergency Department (ED). Patients choose between the ED and General Practice (GP) based on relative utility.

1. Mathematical Formulation¶

The utility $U$ for a patient choosing an interface is defined as:

\[ U_{ED} = U_{base, ED} - \left( \frac{W_{ED}}{60} \times V_{time} \right) \]

\[ U_{GP} = -( \frac{W_{GP}}{60} \times V_{time} ) - C_{OOP} \]

Where:

$W_{ED}$: Wait time in ED (minutes), derived from system occupancy.
$W_{GP}$: Fixed wait time for a GP appointment.
$V_{time}$: Value of patient time ($/hour).
$C_{OOP}$: Out-of-pocket cost for GP.

2. Queuing Dynamics (M/M/s)¶

The ED wait time $W_{ED}$ is approximated using an M/M/s queuing model:

\[ W_{ED} = \frac{\rho^{\sqrt{2(s+1)}-1}}{s(1-\rho)} \times 60 \]

Where:

$\rho$: System utilization (arrival rate / service rate).
$s$: Number of servers (staffed beds).

3. Equilibrium Selection¶

We solve for the probability of choosing ED ($P_{ED}$) using a Logit choice model:

\[ P_{ED} = \frac{e^{\lambda U_{ED}}}{e^{\lambda U_{ED}} + e^{\lambda U_{GP}} + e^{\lambda U_{outside}}} \]

The system iterates until demand $D_{ED} = D_{total} \times P_{ED}$ stabilizes.

4. Evidence Grounding¶

Time Value: Grounded in ABS hourly wage data.
Wait Times: Sourced from AIHW MyHospitals performance reports.
Costs: Sourced from Medicare MBS data.