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Unformatted text preview: Balance equations: Balance equations: In the context of physicalchemistry, it is called the master equation. Flows: The FlowIn = FlowOut 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!...
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 Spring '11
 Zhou
 Demography, Poisson Distribution, Probability theory, Exponential distribution, steady state probabilities

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