UNIVERSITY OF SOUTHERN CALIFORNIA, FALL 2008
1
EE 550: Problem Set
#
6
Due: Wednesday October 29, 2008
I. R
ANDOM
W
ALK ON A
F
INITE
C
ONNECTED
G
RAPH
Consider a particle that makes a random walk over a finite connected graph. Specifically, assume there are
N
nodes of
the graph and
L
links that connect different nodes. If a link connects two nodes of the graph, then we say the nodes are
“neighbors.” Let
d
i
represents the number of neighbors of node
i
, for each
i
∈ {
1
, . . . , N
}
. The amount of time the particle
spends in node
i
is exponentially distributed with rate
v
i
. The particle moves to a new node independently and uniformly over
all neighboring nodes (so that if there are
3
neighbors it chooses with probability
1
/
3
for each neighbor). We assume the graph
is
connected
so that there is a path from every node to every other node. Compute the steady state probabilities
p
i
for being
in node
i
for
i
∈ {
1
, . . . , N
}
, and conclude that this system can be viewed as a reversible Markov chain. (Hint: First
guess
that the probabilities
p
i
will be proportional to the amount of time spent in node
i
before transitioning, and also proportional
to the number of neighbor nodes. Then verify this guess).
II. W
AVELENGTH
C
ONTINUITY
C
ONSTRAINT
Consider the optical network with three links and four sessions, shown below. The network operates under the
wavelength
continuity constraint
, which requires a single active session to use the same wavelength on all of its links. For this problem, we
assume that each of the four sessions is either ON or OFF, so that it is not possible to have more than one call on the network
from a given session
i
. For each session
i
, calls arrive as a Poisson process with rate
λ
i
and have independent exponential
holding times with rate
μ
i
.
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 Fall '08
 Neely
 Probability theory, wavelength continuity constraint

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