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Order processing time

ProbabilityMedium

An order moves between states Queued, Working and Done once per step. From Queued: stay Queued with probability , go to Working with probability , go to Done with probability . From Working: go back to Queued with probability , stay Working with probability , go to Done with probability . Done is final. An order starts Queued. What is the expected number of steps until it is Done? Give a fraction a/b in lowest terms.

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A whole number, a decimal, a fraction like 3/8 or a percent.

Hint

Write one equation for the expected remaining steps from each non-final state.

Solution
  1. Let and be the expected steps to reach Done from Queued and from Working.
  2. Then and .
  3. The second equation gives , so .
  4. The first equation gives .
  5. So , and the expected number of steps is .

Answer: