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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77<br />
dp i3 (t)<br />
dt<br />
dp i4 (t)<br />
dt<br />
dp i5 (t)<br />
dt<br />
= 2p i2 (t) − 4p i3 (t)<br />
= −2p 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 −5t<br />
⎛<br />
∞ ∑<br />
n=0<br />
Q n (5t)n<br />
n!<br />
4/5 1/5 0 0 0<br />
2/5 1/5 2/5 0 0<br />
1/5 0 1/5 0 3/5<br />
0 0 0 3/5 2/5<br />
0 0 0 1 0<br />
⎞<br />
⎟<br />
⎠<br />
(c) (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 />
= 2p i2 (t)+p i3 (t)<br />
= −4p i2 (t)<br />
= 2p i2 (t) − 4p i3 (t)<br />
= −2p i4 (t)<br />
= 3p i3 (t)+2p i4 (t)<br />
(ii)<br />
where<br />
Q =<br />
⎜<br />
⎝<br />
P(t) =e −4t<br />
⎛<br />
∞ ∑<br />
n=0<br />
Q n (4t)n<br />
n!<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 1/2 1/2<br />
0 0 0 0 1<br />
⎞<br />
⎟<br />
⎠