Design and Implement a Driver Delivery Payroll Service

You are building a backend service to track delivery drivers' work intervals and compute wages. Implement the following functions and specify system details.

Functions to Implement

1. create_driver(base_rate)
   • Registers a driver with a base hourly rate.
2. record_delivery(driver_id, start_time, end_time)
   • Logs a delivery interval for a driver.
3. get_total_cost()
   • Returns the total cost across all drivers as the sum over deliveries of (driver_base_rate × delivery_duration).
4. pay_out_to_time(time)
   • Settles/marks wages as paid for all applicable deliveries up to the given timestamp.
5. remaining_cost_to_pay_out()
   • Returns the total unsettled wages after payouts.

Clarifications and Constraints

• Time handling
  • Use UTC timestamps in seconds.
  • Intervals are half-open: [start_time, end_time), so duration = end_time − start_time.
  • Validate start_time < end_time.
• Money and units
  • base_rate is hourly (e.g., dollars/hour). Convert correctly when multiplying by seconds.
• Overlaps/back-to-back
  • Back-to-back delivery intervals for the same driver (end_time == next start_time) are allowed.
  • Specify how overlapping intervals for the same driver are handled.
• Edits/cancellations and invalid inputs
  • Define how updates and cancellations are handled, especially after payouts.
• Idempotency
  • Define guarantees for repeated API calls (e.g., retries).
• Data models and persistence
  • Specify storage schemas/maps and how payout state is stored.
• Performance
  • Provide expected time/space complexity for each function.

Provide and justify your design and implementation choices for the above.