An Internet Service Provider (ISP) offers three plans, High Speed (H), Medium Speed (M) and Low Speed (L). Each customer has the option of changing plan each month, and using past data they have estimated the proportion of customers who change between plans as follows: 0.16 of those on H change to M, 0.12 change to L. 0.27 of those on M change to H, 0.09 change to L. 0.07 of those on L change to H, 0.27 change to M. All other customers stay on the same plan. (a) Construct a transition matrix A, such that if C = (Hk, MkLk) describes the state at time kthen ck+1 = Ack ab sin (a) f 52 ?х P (b) If at the moment 40 % of customers are on plan H, 30 % on plan M and 30 % on plan L, then what proportion are expected to be on each plan after 1 month? Enter your answer as proportions (not percentages) in the form H,M,L with each number accurate to 2 decimal places. (c) What are the expected proportions after 6 months? (d) Suppose the ISP buys out a competitor and all of the competitors customers are transferred to them and by default are on plan H. These customers can then change to other plans each month. Assuming that they change according to the same probabilities as the existing customers, what will be the proportions of these new customers on each of plan after 7 months. Enter once again in the form H,M,L to two decimal places. 8 R
An Internet Service Provider (ISP) offers three plans, High Speed (H), Medium Speed (M) and Low Speed (L). Each customer has the option of changing plan each month, and using past data they have estimated the proportion of customers who change between plans as follows: 0.16 of those on H change to M, 0.12 change to L. 0.27 of those on M change to H, 0.09 change to L. 0.07 of those on L change to H, 0.27 change to M. All other customers stay on the same plan. (a) Construct a transition matrix A, such that if C = (Hk, MkLk) describes the state at time kthen ck+1 = Ack ab sin (a) f 52 ?х P (b) If at the moment 40 % of customers are on plan H, 30 % on plan M and 30 % on plan L, then what proportion are expected to be on each plan after 1 month? Enter your answer as proportions (not percentages) in the form H,M,L with each number accurate to 2 decimal places. (c) What are the expected proportions after 6 months? (d) Suppose the ISP buys out a competitor and all of the competitors customers are transferred to them and by default are on plan H. These customers can then change to other plans each month. Assuming that they change according to the same probabilities as the existing customers, what will be the proportions of these new customers on each of plan after 7 months. Enter once again in the form H,M,L to two decimal places. 8 R
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:An Internet Service Provider (ISP) offers three plans, High Speed (H), Medium Speed (M) and Low Speed (L). Each customer has the option of changing plan
each month, and using past data they have estimated the proportion of customers who change between plans as follows:
0.16 of those on H change to M, 0.12 change to L.
0.27 of those on M change to H, 0.09 change to L.
0.07 of those on L change to H, 0.27 change to M.
All other customers stay on the same plan.
(a) Construct a transition matrix A, such that if ck = (Hk, MkLk) describes the state at time k then ck+1 = Ack
b sin (a)
f
52
?х
(b) If at the moment 40 % of customers are on plan H, 30 % on plan M and 30 % on plan L, then what proportion are expected to be on each plan after 1
month? Enter your answer as proportions (not percentages) in the form H,M,L with each number accurate to 2 decimal places.
(c) What are the expected proportions after 6 months?
(d) Suppose the ISP buys out a competitor and all of the competitors customers are transferred to them and by default are on plan H. These customers can
then change to other plans each month. Assuming that they change according to the same probabilities as the existing customers, what will be the
proportions of these new customers on each of plan after 7 months. Enter once again in the form H,M,L to two decimal places.
BI
8
R
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