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Probability II / Continuous-Time Markov Chains and Queueing Theory

Exit Distributions and Exit Times for Continuous-Time Chains

Exit Distributions and Exit Times solved absorption problems for discrete chains via (I−r)−1(I-r)^{-1}. This note lifts the same first-step-analysis technique to continuous time, in two equivalent forms: work with the embedded jump chain (reducing directly to the discrete theory already built) or work directly with the generator matrix QQ, replacing (I−r)−1(I-r)^{-1} with (−R)−1(-R)^{-1} where RR is QQ restricted to the transient states.
Learning objectives. By the end of this note you should be able to:
  • solve an exit-distribution problem via the embedded jump chain;
  • solve an exit-distribution and exit-time problem directly via the generator matrix;
  • interpret (−R)−1(i,j)(-R)^{-1}(i,j) as an expected sojourn time;
  • apply both methods to a queueing example.

In this lesson

  1. Exit distributions via the embedded chain
  2. Exit distributions via the generator matrix
  3. Exit times via the generator matrix

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