manufacturing_sol

# manufacturing_sol - Pstat160B Manufacturing management...

This preview shows pages 1–2. Sign up to view the full content.

Pstat160B Winter 2009 Manufacturing management example Solution The exponential property of the inter-arrival of the demand and production give the Markov property, provided that the state space encode the necessary information: quantity in stock and if the machine is on or oﬀ. The states are: 0: 0 in stock (on) 1: 1 ” ” 2: 2 ” ” 3: 3 ” ” 4: 4 ” ” (oﬀ) 5: 3 ” ” (oﬀ) The matrix Q is given by Q = - λ λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 μ - ( λ + μ ) λ 0 0 0 0 0 - μ μ 0 0 μ 0 0 - μ To ﬁnd the limiting probabilities, we solve πQ = 0 or v j π j = X k 6 = j π k q kj , j = 0 , ··· 5 λπ 0 = μπ 1 ( λ + μ ) π 1 = μπ 2 + λπ 0 ( λ + μ ) π 2 = λπ 1 + μπ 3 + μπ 5 ( λ + μ ) π 3 = λπ 2 μπ 4 = λπ 3 μπ 5 = μπ 4 This can easily be solved as π 1 = λ μ π 0 π 2 = λ μ π 1 = ± λ μ ² 2 π 0 π 3 = λ λ + μ π 2 = λ 3 μ 2 ( λ + μ ) π 0 π 4 = λ μ π 3 = λ 4 μ 3

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

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

## This note was uploaded on 01/10/2011 for the course STAT PStat 160b taught by Professor Bonnet during the Spring '10 term at UCSB.

### Page1 / 2

manufacturing_sol - Pstat160B Manufacturing management...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online