MGTOC10 Analysis for Decision Making
Final Exam Aid Sheet
λ e −λ − μt
P[x] = , P[service time ≤ t] = 1 − e
x!
L = λW, L = λW , W = W + 1/μ L = L + λ/μ
q q q , q
(M/M/1) : (FCFS/∞/∞) model
1 1 λ n n
P 0 ∞ = 1 = 1 −μ λ ∞ ∞ λ λ λ2
∑ (λ / μ ) , Pn = μ P0 , Lq = ∑ (n − 1)Pn = ∑ (n − 1)μ 1 − μ = μ (μ − λ )
n = 0 1 − λ / μ n =1 n =1
W q L /q, W = W + 1qμ, L = L + λ/q, P = 1- w = λ/µ 0
(M/M/k) : (FCFS/∞/∞) Model
n
(λ/μ )
P0 0≤n≤k
P = 1 n! k
0 k−1(λ /μ) n (λ /μ)k kμ , P= , L = (λ / μ) (λμ ) P
∑ + n n q (k − 1)!(kμ − λ) 2 0
n=0 n! k! kμ −λ (λ/μ )
P n>k
n−k 0
k!k
k
L = L + λ / μ W = L Lq P = (λ / μ) kμ P
q , λ , W q λ , w k! kμ − λ 0
(M/G/1) : (FCFS/∞/∞)
λ σ +(λ /μ) 2
P 0 1− λ / μ , L q
2(1−λ /μ)
Pw= 1- P 0 λ/µ, L = L + λqμ, W = L /q, W q W + 1/μ q
(λ / μ)
If:(M/D/1) : (FCFS/∞/∞) then L q
2(1− λ / μ)
(M/G/k) : (FCFS/k/∞)
1 (λ / μ) / n!
P n λ
k n n = 0, 1, … k, L = (1 − k ) w = 1/µ
∑ (λ / μ) / n! μ
n=0
(M/M/1) : (FCFS/N/∞)
1−ρ
n ifρ≠1
λ ρP0n

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