c2

Question 1

Probability distributions, continuous vs. discrete

Know what it is and what the differences are

Answer 1

Random number from a range

Continuous probability distribution:

• Can have any real value
• Examples are time-based (waiting time, processing time)

Discrete:

• Can only have integer values
• Number of things (jobs, calls, failures, etc.)

Question 2

PDF/CDF, density functions

Know what it is

Answer 2

Pick a random number x between A and B (uniform distribution)
PDF: 1/b-a when x is between
- 0, otherwise

CDF: For a given x, 0 for x < a
(x - a) / (b - a), for x is between a and b
1 for x > b

Question 3

Moments, mean, variance, standard deviation, SCV

Know what it is

Answer 3

Moments: Mean, variance, StD, etc.

Mean: Average

Variance: Squared deviation of value from the mean

Standard deviation: The variance helps determine the data’s spread size when compared to the mean value.

SCV: Squared Coefficient of variation: The ratio of the standard deviation to the mean

Question 4

Exponential distribution, memoryless property

Know what it is and understand the concept of memoryless

Answer 4

Exponential distribution: the probability distribution of the time between events in a Poisson point process

Any number generated from a exponential distribution is not based on any numbers that were generated before it. It does not care about the “history”. This is the memoryless property.

Question 5

Poisson process, superposition and thinning property

Know what it is and its importance to model random events

Answer 5

Poisson process:
a model for a series of discrete events where the average time between events is known
And the series is random (exponentially distributed). The time between events is memory-less (independent)

Superposition property:
Two poisson processes with rates lambda 1 and lambda 2, added together is a poisson process with rate lambda 1+2

Thinning property:
Given a poisson process with rate lambda, removing arrivals with a probability p, gives a poisson process with rate lambda(1-p)

Question 6

Relation Poisson process and Poisson distribution

Know what their relation is

Answer 6

If events occur according to a Poisson process with rate lambda, and we consider N number of events in an interval in that process of length T.
Then, N has a Poisson distribution with parameter lambda*T

Question 7

Continuous and discrete-time Markov chains, balance
equations, equilibrium distribution
Understand the concept and know the difference, know how
to specify transition rates, and formulate balance equations

Answer 7

Continuous-time markov chain:

• Transition, arrival rate: lambda
• departure rate: mu * number of calls

Discrete-time Markov chain:

• Transition, arrival rate: lambda
• departure rate: mu * number of calls

Balance equations are built from the outgoing and incoming rates:
rate in = rate out

equilibrium distribution:

Question 8

Erlang-B formula, insensitivity property

Understand what it is, and what the insensitivity property is

Answer 8

Erlang-B formula uses the probability of being in a state where an added call would be blocked.
Insensitivity property states that this formula holds even if the duration beta is not exponentially distributed. (Only the average is needed)

Question 9

PASTA property

Understand the concept and the intuition behind it

Answer 9

When arrivals are random, the time average will be seen,

Whereas arrival that are not random, may see a value that varies largely from the mean.

Question 10

Reliability, bathtub curve

What it is, and be able to make simple calculations

Answer 10

Reliability of a component is its ability to function correctly over a specified period of time
X = lifetime of system
R(t) = Probability that lifetime is larger than t

Bathtub curve:

• Failure rate over time
• As things grow older, the failure rate may go up
• When things are new, the may perform badly and as such their failure rate may be high.
• When they are in the middle, the failure rate is constant

Calculations: