Assignment 2Solution(Prob) PDF

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The Open University of Hong Kong STAT S315F Applied Probability Models for Investment Assignment 2 Solution Survival Analysis (20 Marks) Queuing (20 Marks) (a) The state space is the set of all non-negative integers. Transition Rate i→i+ 1 λ (i= 0) i→i− 1 μ (i= 1) (b) The traffic intensity isρ=λμ Th...

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The Open University of Hong Kong STAT S315F Applied Probability Models for Investment Assignment 2 Solution

1. Survival Analysis (20 Marks)

2. Queuing (20 Marks) (a) The state space is the set of all non-negative integers. Transition Rate i→ i+1 λ (i = 0) i→ i−1 µ (i = 1) (b) The traffic intensity is ρ = The condition is ρ < 1

λ µ

(c) πx (t) = λπx−1 (t) + µπx+1 (t) − (λ + µ)πx (t) π0 (t) = µπ1 (t) − λπ0 (t) ∂π0 (t) = µπ1 − λπ0 = 0 π1 = µλ π0 = ρπ0 ∂πx (t) = λπx−1 + µπx+1 − (λ + µ)πx = 0 (λ+µ)πx −λπx−1 πx+1 = µ πx+1 = (ρ + 1)πx − ρπx−1 π2 = (ρ + 1)π1 − ρπ0 π2 = ρ2 π0 Likewise, π3 = ρ3 π0 πn = ρn π0 π0 + π1 + π2 + · · · + πn = 1 π0 (1 + ρ + · · · + ρn ) = 1 1 π0 ( 1−ρ )=1 π0 = 1 − ρ πn = (1 − ρ)ρn (d) Idle Period =

1 3

(e) P2 = (1 − ρ)ρ2 =

4 27

(f) Mean queue time = (g) K = 2 P (X = 2) =

1 ε−λ

=6

1 9

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3. Brownian Motion (20 Marks)

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