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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103<br />
(c) For the M/M/s/c with s ≤ c<br />
There<strong>for</strong>e,<br />
⎧<br />
⎪⎨<br />
d j =<br />
⎪⎩<br />
(λ/µ) j<br />
j!<br />
, j =0, 1,...,s<br />
(λ/µ) j<br />
,<br />
s! s j−s j = s +1,...,c<br />
0, j = c +1,c+2,...<br />
<strong>and</strong><br />
(λ/µ) j<br />
π j = π 0 d j = π 0 , j =0, 1,...,s<br />
j!<br />
(λ/µ) j<br />
π j = π 0 d j = π 0<br />
s! s = π (λ/µ) s<br />
j−s 0<br />
s!<br />
( ) j−s<br />
λ<br />
sµ<br />
= π s ρ j−s , j = s +1,s+2,...,c<br />
Thus,<br />
Pr{L = j | L ≤ s} =<br />
=<br />
=<br />
Pr{L = j}<br />
Pr{L ≤ s}<br />
π 0 (λ/µ) j<br />
j!<br />
∑ si=0<br />
π 0<br />
(λ/µ) i<br />
i!<br />
(λ/µ) j /j!<br />
∑ si=0<br />
(λ/µ) i /i!<br />
Also,<br />
Pr{L = s + j | L ≥ s} =<br />
=<br />
=<br />
Pr{L = s + j}<br />
Pr{L ≥ s}<br />
π s ρ j<br />
∑ ci=s<br />
π s ρ i−s<br />
ρ j<br />
∑ c−s<br />
i=0 ρ i<br />
15. No answer provided.