E&CE 710: Assignment #2

Prof. X. Shen

Email: xshen@bbcr.uwaterloo.ca

Office: EIT 4155

Office Hours: Tu 9:30AM - 11:00AM

(Available by appointment)

Prof. X. Shen E&CE 710 : Assignment #2

Problem 1

A compressed video sequence can be represented by the following autoregressive model equation:

λ(n) = aλ(n− 1) + bω(n), |a| < 1

where ω(n) is a stationary Gaussian white noise (i.i.d.) process with unit variance and average value E(ω) =

η. Show the average bit rate is given by

E(λ) = bη/(1− a), |a| < 1

while the autocovariance function has the exponential decay form of the following equation:

C(n) = σ2λ(a)

n, n ≥ 0

where σ2λ = C(0) = E(λ2)− E2(λ) = b2/(1− a2).

Problem 2

Find the shortest path tree from every node to node 1 in the graph using the Bellman-Ford algorithm.

Problem 3

The number shown next to each link of the network is the probability of the link failing during the lifetime of

a virtual circuit from node A to node B. It is assumed that links fail independently of each other. Find the

most reliable path from A to B, i.e., the path for which the probability that all its links stay intact during

the virtual circuits lifetime is maximal. What is this probability?

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Prof. X. Shen E&CE 710 : Assignment #2

Problem 4

A multiple minisource model is used to model five video sources multiplexed at an access buffer. Each source

generates, on the average, 4.5 Mbits/sec. The standard deviation is σ = 2 Mbits/sec. The autocovariance

time constant is 2.5 sec. The source output is converted to 45-octet cells before transmission to the buffer.

The buffer in turn transmits into the network at a rate of 80,000 cells/sec. Use 20 minisources to model each

video source. Apply fluid analysis to obtain the asymptotic survivor function:

G(x) ∼ Amρme−kx

(Hint: Am = 2.0294× 109)

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