Resource planning/allocation: How many service technicians does it take?

Assumptions/factors:

- Available working hours: 8AM - 5PM Monday - Friday, with one hour off for lunch (time not fixed), and two 15-minute breaks (times not fixed).

- Scope of work includes receiving a dispatch (ticket), contacting and visiting user, verifying/clarifying issue, troubleshooting to isolate problem area/cause, repairing or replacing defective items as most appropriate.

- They must successfully resolve 90% of all events within 4 business hours, which can span two business days (example: 1 hour Monday, 3 hours Tuesday). The 4 hours is measured between receipt of dispatch (ticket) and problem/event resolution.

- Historically average (mean) event working time duration: 1.42 hours, with a range of 0.5 hours to 4.0 (or more) hours. Typically less than 2% (assume 2%) require more than 4 business hours to resolve. Assume normal distribution of event durations.

- Assume all times stated above include logistics/comms/prep/cleanup, etc. - no additional tasks remain. Assume no travel or related issues (already at same site as event).

- Average (mean) number of events per month: 3204. Average (mean) number of working days per month: 20.9. Average (mean) number of events per day = 153.3.

- Assume equal competency, skills, and tools by all technicians - no other know variables.

Request:

A formula that accurately predicts minimum number of technicians needed to staff to guarantee 90% of all resolutions will occur within the 4-hour business hour window, and how to account for the following changes:

a. Average event working time (example: increases from 1.42 to 2.5 hours per event)

b. Changes in service level (the 90% or the time window) affect the calculation. For example, if the service level changed from 90% to 95%, or the resolution time requirement changed from 4 hours to 8 hours, how should I account for those changes when recomputing staffing requirements?

c. Changes in available working hours; for example, if work hours change from 8AM - 5PM to 9AM to 5PM, how do I account for that in the staffing model?