Estimating the ambulance pool

Estimating the pool size from total weekly ambulance hours

The model has a fixed maximum number of ambulances available throughout the simulation.

The available data provide the total number of ambulance-hours per week. One ambulance available continuously for a full week provides:

24 \times 7 = 168\text{ ambulance hours per week}

Therefore, if the service provides 50,400 ambulance-hours per week:

\frac{50,400\text{ ambulance hours per week}}{168\text{ hours per week}} = 300 \text{ ambulances}

The model therefore uses an ambulance pool of 300 ambulances.

This does not mean there are 300 different ambulances on each day. It means that, across a week, the service provides the same total capacity as 300 ambulances each being available continuously for the whole week.

Reducing the pool for sickness abstraction

Absence due to sickness reduces the ambulance capacity from the planned capacity.

The sickness data contains the total absence-hours for individual staff members per day. As the majority of resources require two staff members, these hours are divided by two to estimate the equivalent ambulance-hours unavailable.

For example, suppose the mean daily abstraction due to sickness was 240 hours. One ambulance available continuously for a day provides 24 ambulance hours, so:

\frac{240 \text{ ambulance-hours unavailable per day}}{24 \text{ hours per day}} = 10 \text{ ambulances}

This is equivalent to 10 ambulances being continuously unavailable. This value is either:

• Calculated once, as a mean, and subtracted once from the fixed ambulance pool size, or-
• Sampled per day from a distribution of unavailability, and the ambulance pool size is then changed each day to match.

Reducing the pool for multi-resource incidents.

The model assigns one ambulance to each incident. However, some incidents require more than one ambulance.

The first ambulance is represented directly in the simulation. Any second, third, or later ambulances attending the same incident are not explicitly represented. Their work still uses ambulance capacity, so the model reduces the pool to account for it.

For example, suppose there are ten Category 1 incidents on an average day, each incident uses two ambulances, and the mean ambulance job duration is 100 minutes. The model already represents one ambulance per incident, so the omitted resources per incident are:

2 - 1 = 1 \text{ additional ambulance}

The total omitted workload is:

10 \text{ incidents} \times 1 \text{ additional ambulance} \times 100 \text{ minutes} = 1000 \text{ additional ambulance minutes per day}

One ambulance continuously occupied for a complete day accounts for:

1 \times 24 \times 7 = 1440 \text{ minutes per day}

Therefore, the omitted workload is equivalent to:

\frac{1000 \text{ ambulance minutes per day}}{1440 \text{ minutes per day}} = 0.694 \text{ ambulances}

So those extra Category 1 responses consume the same amount of capacity as 0.694 ambulances being continuously occupied throughout the day.

When to update the reductions

The sickness reduction should be updated if the scenario changes the assumed level of sickness absence.

The multi-resource reduction should be updated only when the scenario changes the amount of work done by second and subsequent ambulances.

Scenario change	Update sickness reduction?	Update multi-resource reduction?
Increase or decrease overall ambulance pool size	No	No
Change in average levels of sickness	Yes	No
Increase incident demand by 5%	No	Yes, increase by 5%
Change incident demand by response category	No	Yes, recalculate by category
Change ambulance job durations	No	Yes
Change the number of ambulances attending incidents	No	Yes

Just changing the pool size will not itself create more sickness or multi-resource incidents, so the reduction remains unchanged. However, if other factors also change (e.g., if you assumed sickness should be a proportion related to pool size, or if incident demand also changed with pool size), then amendments are required.