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Statistics Compute a normalized solution

2.21.Compute a normalized solution to the balance equations for the DTMC in
Computational Problem 2.20 (a). When possible, compute:
1. the limiting distribution;
2. the stationary distribution;
3. the occupancy distribution.
2.22.Do Computational Problem2.21 for Computational Problem2.20 (b).
2.23.Do Computational Problem2.21 for Computational Problem2.20 (c).
2.24.Do Computational Problem2.21 for Computational Problem2.20 (d).
2.25.Consider the DTMC of Computational Problem2.5. Compute:
1. the long-run fraction of the time that the buffer is full;
2. the expected number of packets in the buffer in the long run.
2.26.Consider Computational Problem 2.7. Compute the expected number of em-ployees in each grade in steady state.
2.27.Consider the weather model of Conceptual Problem 2.12 . Compute the long-run fraction of days that are rainy.
2.28.Consider the weather model of Conceptual Problem 2.13 . Compute the long-run fraction of days that are sunny.
2.29.What fraction of the time does the coffee addict of Computational Problem
2.4consume brand A coffee?
2.30.Consider the machine descr ibed in Conceptual Problem 2.8. What is the long-run fraction of the time that this machine is up? (Assume p D :90: )
2.31.Consider the manufacturing model of Example2.11 . Compute the expected
number of components in bins A and B in steady state.
2.32.Consider the stock market model of Example 2.7. What fraction of the time
does the chief financial officer have to interfere in the stock market to control the
price of the stock?
2.33.Consider the single-machine production system of Conceptual Problem 2.10 .
Compute the expected number of items processed by the machine in 10 minutes,
assuming that the bin is empty and the machine is idle to begin with. (Assume
p D :95: )
2.34.Do Computational Problem 2.33 for the production system of Conceptual
Problem 2.11 . (Assume p D :95: )
2.35.Which one of the two production systems described in Conceptual Problems
2.10 and 2.11 has a higher per minute rate of production in steady state?
2.9 Problems 57
2.36.Consider the three-machine workshop described in Conceptual Problem2.3.
Suppose each working machine produces revenue of $500 per day, while repairs
cost $300 per day per machine. What is the net rate of revenue per day in steady
state? ( Hint : Can we consider the problem with one machine to obtain the answer
for three machines?)
2.37.Consider the inventory system of Example 2.27 . Compute the long-run ex-pected cost per day of operating this system.
2.38.Consider the manufacturing system of Example2.11 . Compute the expected
number of assemblies produced per hour in steady state.
2.39.(Computational Problem 2.38 continued). What will be the increase in the
production rate (in number of assemblies per hour) if we provide bins of capacity 3
to the two machines in Example 2.11 ?
2.40.Compute the long-run expected number of packets transmitted per unit time
by the data switch of Example 2.12 . How is this connected to the packet-loss rate
computed in Example 2.29 ?
2.41.Consider the brand-switching model of Computational Problem 2.4. Suppose
the per pound cost of coffee is $6, $8, and $15 for brands A, B, and C, respectively.
Assuming Mr. Smith consumes one pound of coffee per week, what is his long-run
expected coffee expense per week?
2.42.Compute the expected time to go from state 1 to 4 in the DTMCs of Compu-tational Problems2.20 (a) and (c).
2.43.Compute the expected time to go from state 1 to 4 in the DTMCs of Compu-tational Problems2.20 (b) and (d).
2.44.Consider the selection pro cedure of Conceptual Problem 2.17 . Suppose the
mean lifetime of Brand A light bulbs is 1, while that of Brand B light bulbs is 1.25.
Compute the expected number of experiments done before the selection procedure
ends. ( Hint : Use the Gambler’s ruin model of Example 2.33 .)
2.45.Consider the DTMC model of the data switch described in Example 2.12 .
Suppose the buffer is full to begin with. Compute the expected amount of time
(counted in number of time slots) before the buffer becomes empty.
2.46.Do Computational Problem2.45 for the data buffer described in Computa-tional Problem 2.5.
2.47.Consider the manufacturing model of Example2.11 . Compute the expected
time (in hours) before one of the two machines is shut down, assuming that both
bins are empty at time 0.
58 2 Discrete-Time Markov Models
Case Study Problems. You may use the Matlab program of Section2.8 to solve
the following problems.
2.48.Suppose Passport has decided not to employ the services of WMB. However,
this has generated discussion within the company about whether it should terminate
accounts earlier. Let T m ( 1 m 7 ) be the policy of terminating the account as
soon as it misses m payments in a row. Which policy should Passport follow?
2.49.Consider the current policy Pc
. One of the managers wants to see if it would
help to alert the customers of their impending account termination in a more dire
form by a phone call when the customer has missed six minimum payments in a
row. This will cost a dollar per call. The m anager estimates that this will decrease
the missed payment probability from the current p 6 D :329 to :250 . Is this policy
cost-effective?
2.50.The company has observed over the past year that the downturn in the econ-omy has increased the bankruptcy rate by 50%. In this changed environment, should
Passport engage the services of WMB? When should it turn over the accounts to
WMB?
2.51.Passport has been approached by another collection agency, which is willing
to work with no annual service contract fee. However, it pays only 60% of the out-standing balance of any account turned over to them. Is this option better than hiring
WMB?
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