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finalexamformulas

# finalexamformulas - Formula Sheet Markov Chains...

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Formula Sheet Markov Chains Chapman–Kolmogorov Equations: p ( n ) ij = M X k =0 p ( m ) ik p ( n - m ) kj for any m < n . Steady-State Probabilities: π j = M X i =0 π i p ij j = 0 , 1 , . . . , M M X j =0 π j = 1 Queueing Theory M/M/ 1 M/M/s M/G/ 1 P n = C n P 0 = C n ˆ X n =0 C n ! - 1 C n = λ μ n = ρ n ( λ/μ ) n n ! , for n = 1 , 2 , . . . , s ( λ/μ ) n s ! s n - s , for n = s + 1 , s + 2 , . . . λ μ n = ρ n P 0 = 1 - ρ " s - 1 X n =0 ( λ/μ ) n n ! + ( λ/μ ) s s ! 1 1 - λ/ ( ) # - 1 1 - ρ L q = λ 2 μ
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