CS-438 COMPUTER SYSTEM MODELING WK5LEC9-10.pdf

CS-438
COMPUTER SYSTEMS MODELING
Spring Semester 2023
Batch: 2019
(WEEK # 5: LECTURE # 09 - 10)
FAKHRA AFTAB
LECTURER
DEPARTMENT OF COMPUTER & INFORMATION SYSTEMS ENGINEERING
NED UNIVERSITY OF ENGINEERING & TECHNOLOGY
1

REVIEW OF PROBABILITY
THEORY
Chapter # 2
Prepared by: Ms. Fakhra Aftab (Lecturer, CISD, NEDUET)

Definitions
1) Random Experiment
A random experiment is a process whose outcome is not known in
advance but for which the set of all possible individual outcomes is
known.
2) Trial
Single performance of a random experiment is called a trial.
3) Sample Space
The set of all possible outcomes of a random experiment is called its
sample space, usually denoted by S.

Definitions (Cont’d)
4) Event
An event is a subset of sample space.
5) Probability
Probability of an event A, denoted P(A), is defined as:
𝑃(𝐴)=
|𝐴|
|𝑆|
6) Independent Trials
If an experiment involves a sequence of independent but identical
stages, we say that we have a sequence of independent trials.

7) Bernoulli Trials
In the special case where there are only two possible outcomes in each
trial, we say that we have a sequence of independent Bernoulli trials.
8) Random Variable
A random variable X is defined as:
X: S → ℝ
Discrete random variable assumes finite or countably infinite values,
whereas, a continuous random variable can assume infinite values.

Example:
Consider two Bernoulli trials.
Then, S = {HH, HM, MH, MM}.
Let’s define a random variable X giving number of hits in the two trials.
s P (s) X(s)
HH ¼ 2
HM ¼ 1
MH ¼ 1
MM ¼ 0

9) Probability Mass Function
Probability mass function (pmf) of a discrete random variable X is
defined as:
From the basic axioms of probability,

Continuing with the example mentioned under random variable, what
will be the pmf of X? i.e. p(0), p(1) and p(2)?
• p(0) = ¼
• P(1) = ½
• P(2) = ¼
s P (s) X(s)
HH ¼ 2
HM ¼ 1
MH ¼ 1
MM ¼ 0

10) Probability Distribution Function
Probability distribution function FX(t), also known as cumulative
distribution function (CDF) of a discrete random variable X is defined as:
It gives the probability of X acquiring a value less than or equal to ‘t’ in its
range.

Commonly used Discrete Probability Distributions
Poisson Distribution
• This models occurrences of an event, generally regarded as successes
(or failures, depending upon the context) in a given time duration.
• The Poisson pmf is given by:
• The parameter α is related to time duration t as follows:
α = λt
where, λ is interpreted as the rate of occurring successes.

Commonly used Discrete Probability Distributions
• When n is large and p is small, the Poisson pmf (distribution) can also
be used as a convenient approximation to the binomial pmf
(distribution).
• As a rule of thumb, we use it when n ≥ 20 and p ≤ 0.05

Example
Queries to a database server arrive at a rate of 12/hour. Calculate the
probability that:
a) Exactly six queries will arrive in next 30 min?
b) Three or more queries will arrive in next 15 min?
c) Two, three or four queries will arrive in next 5 minutes?
Solution:
α = λt , here λ = 12/hour
α = 12t

Part a)
t = 0.5 hour
Therefore α = 6.
P[X=6] =
ⅇ−𝛼𝛼𝑘
𝑘!
=
ⅇ−666
6!
=0.1606
Part b)
α = 12x1/4 = 3
P[X≥3] = 1 – P[X < 3]
= 1 – σ𝑘=0
2 ⅇ−𝛼𝛼𝑘
𝑘!
= 1 – [ⅇ−𝛼
{1 +
3
1!
+
32
2!
}]
= 0.5768
Part c)
α = 12x5/60 = 1
P[X=2] + P[X=3] + P[X=4] = 0.2606

Exponential Distribution (Continuous Distribution)
• It is used to model the time elapsed between events.
• If the number of arrivals at a service facility during a specified period
follows Poisson distribution, then,
• automatically, the distribution of the time interval between successive arrivals
must follow the negative exponential (or, simply, exponential) distribution.
• Specifically, if λ is the rate at which Poisson events occur, then the
distribution of time between successive arrivals, t, f(t) = λe-λt, t > 0
• The mean and variance of the exponential distribution are:
𝐸 𝑡 =
1
𝜆
𝑣𝑎𝑟 𝑡 =
1
𝜆

• The mean E{t} is consistent with the definition of λ.
• If λ is the rate at which events occur, then 1/λ is the average time
interval between successive events.
Applications:
• The exponential distribution is one of the widely used continuous
distributions.
• In most queuing situations, the arrival of customers occurs in a totally
random fashion.
• Exponential distribution is widely used to model time in different
applications e.g. inter arrival time of customers entering a system,
lifetime of hardware components, waiting and service times in a
queuing system.

Rare Events
• When two events are extremely unlikely to occur simultaneously or
within a very short period of time, they are called rare or Poissonian
events, as they are modeled using Poisson distribution.
• Job arrivals to a system, telephone calls, e-mail messages, network
breakdowns, virus attacks, software errors are examples of rare
events.

Inter-arrival Times of Rare Events are Exponential
• Let N be a Poisson random variable denoting number of customers
arriving to a system in the interval (0, t]. Hence,
• Let T be a random variable denoting interarrival time of customers.
Then,
• where the parameter α = λt
• which shows that interarrival times are exponentially distributed
when arrivals occur according to Poisson distribution

Memoryless Property
• Let T be a random variable denoting service time of a server.
• Suppose that a job currently with the server has already consumed
service time t.
• We are interested in the probability that the job will stay with the
server for additional time s; i.e. we wish to calculate the conditional
probability P[T > t + s | T > t]

Memoryless Property
• Exponential distribution exhibits a unique property, known as memoryless
property. Let us first establish this property using mathematics, followed
by its intuitive explanation.
• Memoryless property for exponentially distributed lifetimes would mean
that the probability of a component surviving for some additional time
would be independent of how long it has been operating. This means that
component shows no sign of aging.
• This is independent of t which means that the probability of spending an
additional service time does not depend on the time that has already been
spent with the server.
• Past has no bearing on future and hence the name memoryless property.

This memoryless
property is not for
students! It is ONLY
for exponential
distribution!

CS-438 COMPUTER SYSTEM MODELING WK5LEC9-10.pdf

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