State probabilities for future periods beginning

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STATE PROBABILITIES FOR FUTURE PERIODS BEGINNING INITIALLY WITH AN ASHLEY’S CUSTOMER 1 5 0.9 1 1 0.2 2 (16.2) 23610_ch16_ptg01_Web.indd 8 01/10/14 6:20 PM
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16-9 and 16.1 Market Share Analysis 2 5 0.1 1 1 0.8 2 (16.3) However, we also know the steady-state probabilities must sum to 1 with 1 1 2 5 1 (16.4) Using equation (16.4) to solve for p 2 and substituting the result in equation (16.2), we obtain 1 5 0.9 1 1 1 5 0.9 1 1 1 2 0.7 1 5 0.3 1 5 1 5 0.2 s 1 2 1 d 0.2 2 0.2 1 0.2 0.2 2 3 Then, using equation (16.4), we can conclude that 2 5 1 2 1 5 1 / 3 . Thus, using equa- tions (16.2) and (16.4), we can solve for the steady-state probabilities directly. You can check for yourself that we could have obtained the same result using equations (16.3) and (16.4). 1 Thus, if we have 1000 customers in the system, the Markov process model tells us that   in the long run, with steady-state probabilities 1 5 2 / 3 and 2 5 1 / 3 , 2 / 3 (1000) 5 667   customers will be Murphy’s and 1 / 3 (1000) 5 333 customers will be Ashley’s. The steady-state probabilities can be interpreted as the market shares for the two stores. Market share information is often quite valuable in decision making. For example, suppose Ashley’s Supermarket is contemplating an advertising campaign to attract more of Murphy’s customers to its store. Let us suppose further that Ashley’s believes this promo- tional strategy will increase the probability of a Murphy’s customer switching to Ashley’s from 0.10 to 0.15. The revised transition probabilities are given in Table 16.4. Can you now compute the steady-state probabilities for Markov processes with two states? Problem 3 provides an application. 1 Even though equations (16.2) and (16.3) provide two equations and two unknowns, we must include equation (16.4) when solving for 1 and 2 to ensure that the sum of steady-state probabilities will equal 1. Current Weekly Next Weekly Shopping Period Shopping Period Murphy’s Foodliner Ashley’s Supermarket Murphy’s Foodliner 0.85 0.15 Ashley’s Supermarket 0.20 0.80 TABLE 16.4 REVISED TRANSITION PROBABILITIES FOR MURPHY’S AND ASHLEY’S GROCERY STORES 23610_ch16_ptg01_Web.indd 9 01/10/14 6:20 PM
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16-10 Chapter 16 Markov Processes Given the new transition probabilities, we can modify equations (16.2) and (16.4) to solve for the new steady-state probabilities or market shares. Thus, we obtain 1 5 0.85 1 1 0.20 2 Substituting 2 5 1 2 1 from equation (16.4), we have 1 5 0.85 1 1 1 5 0.85 1 1 1 2 0.65 1 5 0.35 1 5 1 5 0.20 s 1 2 1 d 0.20 2 0.20 1 0.20 0.20 0.57 and 2 5 1 2 0.57 5 0.43 We see that the proposed promotional strategy will increase Ashley’s market share from 2 5 0.33 to 2 5 0.43. Suppose that the total market consists of 6000 customers per week. The new promotional strategy will increase the number of customers doing their weekly shopping at Ashley’s from 2000 to 2580. If the average weekly pro fi t per customer is $10, the proposed promotional strategy can be expected to increase Ashley’s pro fi ts by $5800 per week. If the cost of the promotional campaign is less than $5800 per week, Ashley should consider implementing the strategy.
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  • Spring '18
  • Markov process, Markov chain, Andrey Markov, Markov decision process

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