# Transmission Errors in System A

See the attached file.

2. System A consists of a single ring with 100 stations, one per repeater. System B consists of four 25 stations rings linked by a bridge. If the probability of a link failure is, a repeater failure is, and a bridge failure is, derive an expression for parts (a) to (e). 50 points

a) Probability of failure of system A.

b) Probability of complete failure of system B.

c) Probability that a particular station will find the network unavailable, for systems A and B.

d) Probability that any two stations selected at random will be unable to communicate for systems A and B.

e) Compare values of parts (a) and (b) for .

2. We send a frame of 256 bits. If the probability that a bit changes in transmission is 0.001 and each bit is independent.

a. What is the probability that exactly 6 bits change?

b. What is the probability of no bits changing?

3. If a Signal with a power level of 40 mW is inserted onto a transmission line and the measured power some distance away is 10 mW, calculate the loss of signal strength is decibels.

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#### Solution Preview

2. Probability of change of a bit, p = 0.001

a.

Probability that exactly 6 bits change: P(6) = ...

#### Solution Summary

In transmission, possibilities of changes are estimated in the solution.

Probability distribution and intervals

2. A system uses one of two networks to send binary information, where only one of those networks is used to send all bits of a single block of 5 bits. When network 1 is used the probability of an error in one bit is 0.1 and when network 2 is used the probability of error in one bit is 0.3. The probability of an error in using network 1 to send a block is 0.6. Different blocks may be sent via one of the two different networks based on the same probability.

a. Find the probability of receiving a block of 5 bits with at least two errors.

Hint: {at least 2} = {2 or more} = {more than 1} = {Not 0 or 1} = {2,3,4,5} This is clearly the correct definition of {at least 2} for this problem.

b. If a block of 5 bits was received with at least two errors, what is the probability that network 2 was used?

Hint: All 5 bits go through the SAME network. (Think about tossing the same coin 5 times).

In order to avoid error propagation, assume that the answer to (a) is 0.4 (this is not the correct answer) and proceed to do part (c)

c. Blocks received with at least two errors are retransmitted. Each re-transmission may be sent via either channel. What is average number of times we have RE-transmit a block until it is received with less than 2 errors?

Hint: re-transmission needs to include {X - 1}

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3. Voice and data calls arrive at a switch following a Poisson model, but different arrival rates of 6 per second for data, and 2 per second for voice.

a. Find the total arrival rate of calls to the switch.

b. Find the probability that in an interval of length 0.2 seconds, zero voice calls and exactly one data call arrive at the switch.

c. Find the probability that in 0.5 seconds exactly 4 calls (of any kind) arrive at the switch.

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4. A system is composed of three links as shown. The links fail independently of each other, with failure probabilities show on each link. Find the failure probability of the system.

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