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### Probability Review

### Mobility Models for Mobile Computing

### Probability Review

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

Problem 1

- We roll a regular dice and we toss a coin as many times as the roll indicates.
- What is the probability that we get no tails?
- What is the probability that we get 3 heads?

- Using the same process, we are asked to get at least 6 tails in total.
- If we don’t get 6 tails with the first roll, we roll again and repeat as many times as needed to get at least 6 tails.
- What is the probability that we need more than 1 rolls to get 6 tails?

Problem 2

- A process has to send 1000 bytes over a wireless link using a stop-and-wait protocol. The payload size per packet is 10 bytes. The bit-wise error probability is 10-3.
- Find the expected number of total packet transmissions needed to transfer the 1000 bytes.

The Markov Process

- Discrete time Markov Chain: is a process that changes states at discrete times (steps). Each change in the state is an event.
- The probability of the next state depends solely on the current state, and not on any of the past states:
- Markov Process’s versatility
- can be used in building mobility models, data traffic models, call pattern models etc.

Transition Matrix

- Putting all the transition probabilities together, we get a transition probability matrix:
- The sum of probabilities across each row has to be 1.

Stationary Probabilities

- We denote Pn,n+1 as P(n). If for each step n the transition probability matrix does not change, then we denote all matrices P(n) as P.
- The transition probabilities after 2 steps are given by PP=P2; transition probabilities after 3 steps by P3, etc.
- Usually, limnPn exists, and shows the probabilities of being in a state after a really long period of time.
- Stationary probabilities are independent of the initial state

Numeric Example

- No matter where you start, after a long period of time, the probability of being in a state will not depend on your initial state.

r

r

1

1

q

q

q

1

2

3

4

5

p

p

p

Markov Random Walk- A Random Walk Is a subcase of Markov chains.
- The transition probabilitymatrix looks like this:

Markov Random Walk

- Hitting probability (gambler’s ruin) ui=Pr{XT=0|Xo=i}
- Mean hitting time (soujourn time) vi=E[T|Xo=k]

D

Example Problem- A mouse walks equally likely between adjacent rooms in the following maze:
- Once it reaches room D, it finds food and stays there indefinitely.
- What is the probability that it reaches room D within 5 steps, starting in A?
- Is there a probability it will never reach room D?

0.99

LOW

HIGH

0.9

0.1

Example Problem 2- A variable bit rate (VBR) audio stream follows a bursty traffic model described by the following two-state Markov chain:
- In the LOW state it has a CBR of 50Kbps and in the HIGH it has a CBR of 150Kbps.
- What is the average bit-rate of the stream?

Example Problem 2 (cont’d)

- We find the stationary probabilities
- Using the stationary probabilities we find the average bitrate:

Continuous time Markov chains

- Simple variation of the time-discrete markov chain
- Self-transitions, i.e. (n,n) transitions, are replaced with the PDF of the pause.
- At any state, the process remains for a random time under exponential distribution

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

Mobility models play an important role in various layers

Location management

Routing

Location-aware applications

Mobility models are used in three phases in developing a LM scheme

At assumptions phase

At performance analysis phase

At simulation phase

Analytical vs simulation mobility models

two types of mobility models:

Aggregate (analytical)

per-user (simulation)

Two granularity domains of mobility models:

Discrete space

Continuous space

Aggregate mobility models

flow models

per-user mobility models

based on a stochastic modeling of velocity.

random way-point (RWP)

Angular RWP

Habitual models

Trip-based mobility model

Markovian models

Variations

Time-dependent behavior (temporal dependency)

Location-dependent behavior (spatial dependency)

Mobility modelsRandom Waypoint Process (WRP)

- First appeared in the publication of DSR[1996]
- Basic configuration
- Choose a random target point (waypoint) distributed (usually uniformly) over some area
- Move from current location to the target waypoint using a constant velocity

0

2

vo

vo

vo

1

3

RWP variants/extensions

- Non-uniform distribution of waypoint selection
- Uniform distribution makes the nodes spend more time around the center

- Pause times
- Select a random or a constant time to wait at each destination

- Random velocities
- Choose a random velocity and move from current location to the target waypoint using this velocity

0

2

vo

v1

v2

1

3

Random Turn Process

- Choose a random direction (way) distributed (usually uniformly) over [0, 2π]
- Choose a random velocity and move from current location over specified direction for certain time or distance.

0

1

2

v1

3

v2

2

v3

1

3

Markovian:Random Graph Mobility Process

- A random walk mobility model Gu for a user u:
- Nodes represent regions
- Weighted directed edge
- an edge r1-> r2 has weight: Pr[ user will mover to r2 when it leaves r1]

- Pr[path p in Gu] = product of probabilities of all edges in p.
- In general, Gu has cycles

P(n,m)(n,m+1)

P(n,m)(n+1,m)

Grid-based markovian mobility model(2-d random walk)- Based on a time-continuous, space-discrete 2-D markov chain
- Can be used to model urban mobility

P(n,m)(n,m+1)

Hex-based markovian mobility model- Variation of the 2-d model
- Can be used in cellular systems

Georgios VarsamopoulosSchool of Computing and InformaticsArizona State University

Basic concepts and terminology

- Basic concepts by example: dice rolling
- (Intuitively) coin tossing and dice rolling are Stochastic Processes
- The outcome of a dice rolling is an Event
- The values of the outcomes can be expressed as a Random Variable, which assumes a new value with each event.
- The likelihood of an event (or instance) of a random variable is expressed as a real number in [0,1].
- Dice: Random variable
- instances or values: 1,2,3,4,5,6
- Events: Dice=1, Dice=2, Dice=3, Dice=4, Dice=5, Dice=6
- Pr{Dice=1}=0.1666
- Pr{Dice=2}=0.1666

- The outcome of the dice is independent of any previous rolls (given that the constitution of the dice is not altered)

Basic Probability Properties

- The sum of probabilities of all possible values of a random variable is exactly 1.
- Pr{Coin=HEADS}+Pr{Coin=TAILS} = 1

- The probability of the union of two events of the same variable is the sum of the separate probabilities of the events
- Pr{ Dice=1 Dice=2 } = Pr{Dice=1}+{Dice=2} = 1/3

Properties of two or more random variables

- The tossing of two or more coins (such that each does not touch any of the other(s) ) simulateously is called (statistically) independent processes (so are the related variables and events).
- The probability of the intersection of two or more independent events is the product of the separate probabilities:
- Pr{ Coin1=HEADS Coin2=HEADS } =Pr{ Coin1=HEADS} Pr{Coin2=HEADS}

Conditional Probabilities

- The likelihood of an event can change if the knowledge and occurrence of another event exists.
- Notation:
- Usually we use conditional probabilities like this:

Conditional Probability Example

- Problem statement (Dice picking):
- There are 3 identical sacks with colored dice. Sack A has 1 red and 4 green dice, sack B has 2 red and 3 green dice and sack C has 3 red and 2 green dice. We choose 1 sack equally randomly and pull a dice while blind-folded. What is the probability that we chose a red dice?

- Thinking:
- If we pick sack A, there is 0.2 probability that we get a red dice
- If we pick sack B, there is 0.4 probability that we get a red dice
- If we pick sack C, there is 0.6 probability that we get a red dice
- For each sack, there is 0.333 probability that we choose that sack

- Result ( R stands for “picking red”):

Moments and expected values

- mth moment:
- Expected value is the 1st moment: X =E[X]
- mth central moment:
- 2nd central moment is called variance (Var[X],2X)

Geometric Distribution

- Based on random variables that can have 2 outcomes (A and A): Pr{A}, Pr{A}=1-Pr{A}
- Expresses the probability of number of trials needed to obtain the event A.
- Example: what is the probability that we need k dice rolls in order to obtain a 6?

- Formula: (for the dice example, p=1/6)

Geometric distribution example

- A wireless network protocol uses a stop-and-wait transmission policy. Each packet has a probability pE of being corrupted or lost.
- What is the probability that the protocol will need 3 transmissions to send a packet successfully?
- Solution: 3 transmissions is 2 failures and 1 success, therefore:

- What is the average number of transmissions needed per packet?

Binomial Distribution

- Expresses the probability of some events occurring out of a larger set of events
- Example: we roll the dice n times. What is the probability that we get k 6’s?

- Formula: (for the dice example, p=1/6)

Binomial Distribution Example

- Every packet has n bits. There is a probability pΒthat a bit gets corrupted.
- What is the probability that a packet has exactly 1 corrupted bit?
- What is the probability that a packet is not corrupted?
- What is the probability that a packet is corrupted?

Continuous Distributions

- Continuous Distributions have:
- Probability Density function
- Probability Distribution function:

Continuous Distributions

- Normal Distribution:
- Uniform Distribution:
- Exponential Distribution:

Poisson Distribution

- Poisson Distribution is the limit of Binomial when n and p 0, while the product np is constant. In such a case, assessing or using the probability of a specific event makes little sense, yet it makes sense to use the “rate” of occurrence (λ=np).
- Example: if the average number of falling stars observed every night is λ then what is the probability that we observe k stars fall in a night?

- Formula:

Poisson Distribution Example

- In a bank branch, a rechargeable Bluetooth device sends a “ping” packet to a computer every time a customer enters the door.
- Customers arrive with Poisson distribution of customers per day.
- The Bluetooth device has a battery capacity of m Joules. Every packet takes n Joules, therefore the device can send u=m/n packets before it runs out of battery.
- Assuming that the device starts fully charged in the morning, what is the probability that it runs out of energy by the end of the day?

Poisson Process

- It is an extension to the Poisson distribution in time
- It expresses the number of events in a period of time given the rate of occurrence.
- Formula
- A Poisson process with parameter has expected value and variance .
- The time intervals between two events of a Poisson process have an exponential distribution (with mean 1/).

Poisson dist. example problem

- An interrupt service unit takes to sec to service an interrupt before it can accept a new one. Interrupt arrivals follow a Poisson process with an average of interrupts/sec.
- What is the probability that an interrupt is lost?
- An interrupt is lost if 2 or more interrupts arrive within a period of to sec.

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