lec20 - Balance equations Balance equations In the context...

Info iconThis preview shows pages 1–5. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Balance equations: Balance equations: In the context of physical-chemistry, it is called the master equation. Flows: The Flow-In = Flow-Out Principle provides us with the means to derive equations between the steady state probabilities. State 0: μ 1 p 1 = λ p i.e. p 1 = λ μ 1 p . 1 Balance equations (cont’d) State 1: μ 1 p 1 + λ 1 p 1 = λ p + μ 2 p 2 i.e. p 2 = λ 1 μ 2 p 1 = λ λ 1 μ 1 μ 2 p . State 2: μ 2 p 2 + λ 2 p 2 = λ p + μ 3 p 3 i.e. p 3 = λ 2 μ 3 p 2 = λ λ 1 λ 2 μ 1 μ 2 μ 3 p . 2 for state k we get: p k = λ λ 1 λ 2 · ... · λ k- 1 μ 1 μ 2 μ 3 · ... · μ k p . ♥ ok, so now we know all the steady state probabilities depending on p . But what use has that, if we don’t know p ? ♣ Here, we need another trick: we know, that the steady state probabilities are the density function for the state X . Their sum must therefore be 1! ♥ Hence: 1 = p + p 1 + p 2 + ... = p parenleftbigg 1 + λ μ 1 + λ λ 1 μ 1 μ 2 + ... parenrightbigg bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright := S 3 Balance equations (cont’d) ♥ If this sum S converges, we get p = S- 1 . If it doesn’t converge, we know that we don’t have any steady state probabilities, i.e. the B+D process never reaches an equilibrium. The analysis of S is crucial!...
View Full Document

{[ snackBarMessage ]}

Page1 / 11

lec20 - Balance equations Balance equations In the context...

This preview shows document pages 1 - 5. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online