CPU
0
I/O 1
I/O m
I/O 2
μ
1
μ
1
μ
1
μ
2
μ
2
μ
2
3
2
1
0
λ
very fast M/M/1
Homework 8 – Student Name: Son Tran – Student ID: 200723576
Class: Data Networks – Spring 2008
3.61
We have
=
i 0mpi
=
1
.
The arrival rate at the CPU is λ/
p
0
and the arrival rate at the
i
th
I/O port is λ
p
i
/
p
0
.
By Jackson’s Theorem, we have
,
,…,
= =
(  )
Pn0 n1
nm i 0mρini 1 ρi
Where
=
ρ λμ0p0
, and
=
ρi λpiμip0
for
i
> 0
The tandem system is as follows
The arrival rate is λ. The service rate for CPU 0 is μ
0
p
0
and for I/O
i
th
(
i
= 1,.
.,m) is μ
i
p
0
/
p
i
.
3.64
a) The state is determined by the number of customers at node 1, when having customers at node
1, the deperture rate from node 1 is
μ
1
, each departure causes the state to decrease as the figure.
When customers come to node 2, the departure rate from node 2 is
μ
2
, each departure causes the
state to increase.
b) The steady state probability at state
i
is
p
i
and
p
i
=
p
i1
ρ
, where
ρ
=
μ
2
/
μ
1
.
So
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 Spring '08
 Choi
 Markov chain, Queueing theory, data networks, arrival rate

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