Two independent repair techs work on two machines. You care about (i) when the first machine comes back, (ii) which tech finishes first, (iii) when both are done. Exponential clocks make these questions algebraic instead of simulation-only.
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Lesson close · Stochastic Processes (Advanced)
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Sequence
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Equivalently, pdf on .
Then
Scaling: if then () is exponential with rate (mean becomes ).
For ,
Sketch: .
Reading: given survival to , remaining life is again , independent of age. This is the modeling reason people use exponentials for “no wear-out” waiting times — and also why the model is wrong when aging matters.
Let , independent.
Sketch: .
Sketch: integrate joint density over , or condition on and use memoryless/ competing-rate intuition: instantaneous hazard of “1 finishes” is vs .
Always , so
Solutions: soon add
Interactive · Competing exponential clocks
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