Solutions to Chapter 4 - Communication Networks

Communication Networks (2 nd Edition)
Chapter 4 Solutions
p is directly proportional to the ratio of n to k.
c. How does this p affect the blocking performance of the intermediate crossbar switch
Blocking occurs at the second switches when two of its inputs are to be routed to the same
output. Although the control and routing algorithm of the switch can minimize the chance of this
occurring, the greater the value of p, the greater the chance that such blocking will occur.
d. Supposing that the blocking probability of the intermediate crossbar is small, what is the
proportion of time p′ that a line between the second and third stage is busy
Assuming that the blocking probability of the second stage switches is small, the percentage of
time that the output lines on a switch in the second stage are busy is equal to the percentage of
time that its input lines are busy. This is the case because the second stage switches are
(N/n)x(N/n) components. Thus,
p′ = p = (n/k) 10%
e. For a given input and output line, what is the probability that none of the N/n paths between the input
and output lines are available
In this question N/n should be k, since there are k possible paths from each input to each output
(one different path possible through each second-stage switch). Assuming independence of all
lines, which is not the case generally, for any one of the k paths between a given input and the
output to be unavailable, one of the 2 jumps must be unavailable.
P[path not available] = 1-P[path is available] = 1-(1-p’)(1-p’) ≈ 2p
For none of the paths to be available, all k must be busy. Since we assumed that all are
independent of each other,
P[none of the paths available] = (2p) k
29. Consider the multistage switch in Figure 4.35 with N = 32. Compare the number of crosspoints
required by a nonblocking switch with n = 16, n = 8, n = 4, and n = 2.
Solution:
For any switch to be non-blocking, we require k nb = 2n – 1. The total number of crosspoints is
2Nk + k(N/n) 2 . The resulting number of crosspoints necessary for different values of n is shown
below.
N n k Number of Crosspoints needed
32 16 31 2108
32 8 15 1200
32 4 7 896
32 2 3 960
For a one-stage N x N switch with n = 32, we would require 1032 crosspoints. Thus we see that,
just as was noted in question 26c, for k much less than n, multistage switches can provide good
hardware economy while remaining non-blocking.
30. A multicast connection involves transmitting information from one source user to several
destination users.
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