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Order processing time
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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Hint
Write one equation for the expected remaining steps from each non-final state.
Solution
- Let and be the expected steps to reach Done from Queued and from Working.
- Then and .
- The second equation gives , so .
- The first equation gives .
- So , and the expected number of steps is .
Answer: