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Compare Fixed and Variable Train Gaps Without a Timetable

DeshawSoftware Engineer · Technical Screen
HardSoftware EngineerTechnical ScreenStatistics & Math

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Compare two train services for a commuter who arrives without consulting a timetable. Explain the assumptions and derivation clearly, check edge cases, and show how the result changes when those assumptions no longer hold.

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?

Hints

Hint 1

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 2

Check the extremes: Relate the short and long gaps to both the best-case and worst-case wait before calculating an expectation.

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?