Compare Fixed and Variable Train Gaps Without a Timetable

# Compare Fixed and Variable Train Gaps Without a Timetable Compare two train services for a commuter who arrives without consulting a timetable. Service A has a fixed 10-minute gap. Service B repeats gaps of 5, 10, and 15 minutes. Assuming the commuter's arrival time is uniformly distributed over a long operating cycle, compute the expected waiting time for each service and explain which is preferable under this criterion. ### Constraints & Assumptions - Train dwell time is negligible and service repeats indefinitely. - The commuter arrives independently of the schedule. - For Service B, each 5-, 10-, and 15-minute interval occurs once per cycle. - Preference is based only on expected wait, not variance, crowding, or missed connections. ### Clarifying Questions to Ask - Are the variable gaps a deterministic repeating schedule or independent random draws? - Is the commuter's arrival independent of train arrivals? - Should the comparison include worst-case waiting time as well as expectation? ```hint Compare identical mean gaps carefully Both services average ten minutes between trains; explain why that fact alone may or may not determine a random commuter's wait. ``` ```hint Check the extremes Relate the short and long gaps to both the best-case and worst-case wait before calculating an expectation. ``` ### What a Strong Answer Covers - The residual-wait calculation for a fixed interval. - Length-biased sampling of variable intervals, also known as the inspection paradox. - Correct weighting by time spent in each interval rather than by interval count. - A conclusion that names the assumptions under which it holds. ### Follow-up Questions - How do the wait-time variances compare? - What changes if the 5-, 10-, and 15-minute gaps are chosen with unequal frequencies?