Tools for Computer Systems Analysis Homework

Tools for Computer Systems Analysis
Homework 8 – Due December 29, 2014
December 8, 2014
Problem 1. Following Example 2,
1. Show that Y ∗ (s) =
1−X ∗ (s)
sEX .
2. Using (1), show that
E[Y k ] =
E[X k+1 ]
(k + 1)EX
Problem 2. For a renewal process, let A(t) be the age at time t. Prove that if µ < ∞, then
with probability 1,
A(t) t→∞
−−−→ 0
t
Problem 3. Consider a half-duplex transmission regime with two transmitters A and B.
A and B share the channel bu transmitting alternately, forming A-period and B-period of
transmissions. Consider Xi - length of the i-th transmission period, and random process
{Xn , n > 0}. Xn are independent. A-period is uniform, U [2, 4]. B-period is uniform, U [1, 3].
1. Is {Xn , n > 0} Markovian? Is it a renewal process? Explain.
2. A message randomly arrives to be transmitted by transmitter A. Find the pdf or PDF
of the waiting time until the transmission starts.
3. Find and draw f (X A ), F (X A ), f (XrA ), F (XrA ) where X A is the length of A-period and
XrA is the excess life of A-period.
4. Consider a message to be transmitted by A arriving during an A-period. Find the
probability it has at least 1 second of transmission time.
1

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