Practice problems. Practice problems. Example. Grocery store example 2 dairies. Creamwood Cheesedale. Next week This week Creamwood 1 Cheesedale 2

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1 Practice problems Grocery store example dairies Next week This week Creamwood Cheesedale Creamwood Cheesedale Example Practice problems Probability of purchasing Cheesedale in Week 4:.7(.7)() +.7()(.6) + (.4)() + (.6)(.6) =.47

2 Practice problems Find the steady state probabilities: p P.4 = = =7 p + p +.4 p P = = =.43 p + p +.4 Practice problems New-Fangled Soft Drink Company To From Red Pop Super Cola Red Pop Super Cola Practice problems New-Fangled Soft Drink Company Find the long-run market share (steady state probabilities): p P =. = = p + p. +. p. P = = = p + p. +.

3 Practice problems New-Fangled Soft Drink Company Practice problems New-Fangled Soft Drink Company Original market shares are 5% each (based on steady states) To From Red Pop Super Cola Red Pop Super Cola Practice problems New-Fangled Soft Drink Company Re-compute steady state probabilities: p P = = =.6 p + p. + p. P = = =.4 p + p. + 3

4 Markov Processes First Passage Time: This tells how many transitions are needed in order to leave one state (state i) and go to a second state (state j) for the first time. This is not a probability!! µ ij = expected first passage time from state i to state j # transitions it takes to get from i to j Markov Processes µ ij = p ij + Σ p ik ( + µ kj ) transition from i to j is made in step with prob. p ij -step transition prob. of going from i to k step from i to k; an additional exptd. value of µ kj steps to get to j for first time Markov Processes First passage time µ ij = p ij + Σ p ik ( + µ kj ) µ ij = p ij + Σ p ik. + Σ p ik (µ kj ) k=j k=j must = µ ij =. + Σ p ik (µ kj ) Number of transitions from i to j 4

5 p / ij Markov Processes First passage time = the probability that absorbing state j is eventually reached when starting in transient state i p / ij = p ij + Σ p ik (p / kj ) k=j Probability of prob. of direct transition going to state j from i to k prob. of eventually getting to j from k Markov Processes What is an absorbing state? Good question! 3 Different Types of States. Recurring State state that you can move into and out of. Transient State state that once you leave, you cannot get back to 3. Absorbing State state that once you get into, you cannot get out of Trustworthy Car Co. (TCC) To (j) (i) From TCC car Non-TCC car First passage time TCC car.8.6 Non-TCC..4 If a customer is a Trustworthy customer, how long will it be before he/she switches? (need to find first passage times) 5

6 Trustworthy Car Co. (TCC) µ ij =. + Σ p ik (µ kj ) (a) µ = + p µ = +.8 µ.µ = µ =/. = 5 µ = + p µ µ - p µ = µ ( - p ) = µ = - p = = = Trustworthy Car Co. (TCC) If a customer is not a Trustworthy customer, how long will it be before he/she switches back to Trustworthy? µ ij =. + Σ p ik (µ kj ) (b) µ = + p µ - p µ = = = =.67 Next week s leader This week s leader A B C First Passage Time For a 3-state setting 3 TV Stations A B C

7 First Passage Time For a 3-state setting a) If Station B is currently the leader, how long will it be before Station A becomes leader? µ ij =. + Σ p ik (µ kj ) µ BA =. + p BB µ BA + p BC µ CA µ CA =. + p CC µ CA + p CB µ BA Solve simultaneous equations First Passage Time For a 3-state setting µ BA =. +.7 µ BA +. µ CA µ CA =. +.6 µ CA + µ BA µ BA -. µ CA =.4µ CA - µ BA = µ BA = 87 µ CA = 7.85 Expected Recurrence Time Expected Recurrence Time: Once you leave a state, how long it takes to get back to that state ~ µ AA = = = 6 weeks P A.67 µ BB = = = weeks P B µ CC = = ~ = 3 weeks P C 3 7

8 Salesman receives $4 per system sold Estimated cost per housecall = $ To From New New prosp. Interested 3 Sale lost 4 Sale made Int. 3 Lost 4 Made. Salesman receives $4 per system sold Est. cost per housecall = $ To From New New prosp. Interested 3 Sale lost 4 Sale made Int. 3 Lost 4 Made. 5 Decision.75 To From New New prosp. Interested 3 Sale lost 4 Sale made Int. 3 Lost 4 Made. 5 Decision.75 8

9 µ 5 = + p µ 5 + p µ 5 µ 5 = + p µ 5 + p µ 5 µ 5 = + µ 5 µ 5 = + µ 5 µ 5 =.667 µ 5 = 3 How long it takes the customer to make a decision p ij = prob. that absorbing state j is eventually reached when start in transient state i p ij = p ij + Σ p ik p kj k=j Salesman receives $4 per system sold Est. cost per housecall = $ To From New New prosp. Interested 3 Sale lost 4 Sale made Int. 3 Lost 4 Made. 9

10 p 4= p 4 + p p 4 + p p 4 + p 3 p 34 p 4= p 4 + p p 4 + p p 4 + p 3 p 34 p 4=. + p 4 p 4= + p 4 p 4= 3 Prob of sale made starting w/new p 4=.67 Prob of sale made starting w/interest Expected profit = (profit/sale)(prob of making sale) - (cost/housecall)(exptd. # of calls) New: EP = $4 (3) - $ (.667) = $4 Interest: EP = $4 (.67) - $ (3) = $3 µ 5 = + p µ 5 + p µ 5 µ 5 = + p µ 5 + p µ 5 µ 5 = + µ 5 µ 5 = + µ 5 µ 5 =.667 µ 5 = 3 How long it takes the customer to make a decision