SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
SOLUTIONS MANUAL for Stochastic Modeling: Analysis and ...
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76 CHAPTER 7. CONTINUOUS-TIME PROCESSES<br />
(c) (i)<br />
⎛<br />
P =<br />
⎜<br />
⎝<br />
1 0 0 0 0<br />
1/2 0 1/2 0 0<br />
1/4 0 0 0 3/4<br />
0 0 0 0 1<br />
0 0 0 0 1<br />
⎞<br />
⎟<br />
⎠<br />
(ii) T = {2, 3, 4} R 1 = {1}, R 2 = {5}<br />
(iii) π R1 =1 π R2 =1<br />
3. (a) (i) For i =1, 2,...,5<br />
dp i1 (t)<br />
dt<br />
dp i2 (t)<br />
dt<br />
dp i3 (t)<br />
dt<br />
dp i4 (t)<br />
dt<br />
dp i5 (t)<br />
dt<br />
= −p i1 (t)+2p i2 (t)+p i3 (t)<br />
= p i1 (t) − 5p i2 (t)+p i4 (t)<br />
= 2p i2 (t) − 4p i3 (t)<br />
= p i2 (t) − 6p i4 (t)+5p i5 (t)<br />
= 3p i3 (t)+2p i4 (t) − 5p i5 (t)<br />
(ii)<br />
where<br />
Q =<br />
⎜<br />
⎝<br />
P(t) =e −6t<br />
⎛<br />
∞ ∑<br />
n=0<br />
Q n (6t)n<br />
n!<br />
5/6 1/6 0 0 0<br />
1/3 1/6 1/3 1/6 0<br />
1/6 0 1/3 0 1/2<br />
1/2 1/6 0 0 1/3<br />
0 0 0 5/6 1/6<br />
⎞<br />
⎟<br />
⎠<br />
(b) (i) For i =1, 2,...,5<br />
dp i1 (t)<br />
dt<br />
dp i2 (t)<br />
dt<br />
= −p i1 (t)+2p i2 (t)+p i3 (t)<br />
= p i1 (t) − 4p i2 (t)