homework2_2013_solution
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Problems from Kleinrock’s book (Queueing system Vol.1)
2.5 Consider the homogeneous Markov chain whose state diagram is
(a) Find P, the probability transition matrix.
(b) Under what conditions (if any) will the chain be irreducible and aperiodic?
(c) Solve for the equilibrium probability vector π.
(d) What is the mean recurrence time for state E3 ?
(e) For which values of α and p will we have π1=π2=π3 ?
Solution:
(a) 1000110ppP
(b) 10p且10且p与不同时为1。
(c)
221111,32132131331221321p
p
p
p
p
P
1-α
1
2
3
1-p
p
α
1
因此平稳分布2,2,2pppp
(d) ppTE21333
(e) p
2.6 Consider the discrete-state, discrete-time Markov chain whose transition probability matrix is
given by
41432121P
(a) Find the stationary state probability vector π.
(c) Find the general form for Pn.
Solution:
(a)
52531,2121P
(c)
nnnnnnP4153524153534152524152535252525341001231
11
2.7 Consider a Markov chain with states E0, E1, E2, … and with transition probabilities
jnnjninijnjqpniep0
!
Where p+q=1 (0
(a) Is this chain irreducible? Periodic? Explain.
(b) We wish to find
π
i = equilibrium probability of Ei
write πi in terms of pji and πj for j= 0, 1, 2, ….
Solution:(a)给定的马氏链是不可约,非周期的,因为任意0,jip,并且任意0iip。
(b)jijjip0