Question

 

There are 3 different Redbox locations in the Grandville area located at Meijer, Walgreens, and CVS. If you rent a DVD from Redbox, you must return it to any Redbox location the next day.

If you get your DVD from the Meijer location, there is a 50% chance that it will be returned back to Meijer, 20% to Walgreens, and 30% to CVS.

If you get your DVD from the Walgreens location, there is a 40% chance that it will be returned to Meijer, 50% back to Walgreens, and 10% to CVS.

If you get your DVD from the CVS location, there is a 20% chance that it will be returned back to Meijer, 20% back to Walgreens, and 60% back to CVS.

Use Excel and create a transition matrix for this problem. Make sure to label the states.

Suppose that each of the 3 locations starts out with 100 DVD’s. What is the initial-state matrix  S₀?

Find S₃ and S₁₀ for this system.

How many DVD’s will be at Walgreens after 3 days?

How many DVD’s will be at CVS after 10 days?

In the long run, how many DVD’s will be at each location? Use the equation S x T = S to find the stationary matrix.

Answer 1

Meijer Walgreens CVS      

Initial State Matrix S0 = | 100    100         100 |

Probability Transition Matrix, T =

S3 = S0*T³ = | 110.7    86.1    103.2 |

S10 = S0*T¹⁰ = | 110.2    85.7    104.1 |

Steady State (stationar) matrix S = S*T

or S*(T-1) = 0

In matrix terminology, 1 is equivalent to Identity Matrix, I

Therefore, S*(T - I) = 0

T - I =

Let steady state matix, S = | x     y    z |

Solving, the equation, S(T - I) = 0 and S0 = 300 for x, y and z.

S0 = 300 is represented by equation (6) below. To understand this, recall that each location has 100 DVDs in the initial state. Their combined total is 300 DVDs. This quantity will remain same even in steady state.

Therefore, Steady state matix, S = | 110.2     85.7    104.1 |

Number of DVDs at each location,

Meijer = 110.2 ~ 110

Walgreens = 85.7 ~ 86

CVS = 104.1 ~ 104