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Answer .

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Outline • Introduction to Poisson Processes – Definition of arrival process – Definition of renewal process – Definition of Poisson process • Properties of Poisson processes – Inter-arrival time distribution – Waiting time (Arrival Time) distribution – Superposition and decomposition • Non-homogeneous Poisson processes (relaxing stationary) • Compound Poisson processes (relaxing single arrival) • Modulated Poisson processes (relaxing independent)

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N is a random variable

Random Sum IID RVs

Compound Poisson Processes • A stochastic process is said to be a compound Poisson process if – it can be represented as

– is a Poisson process with mean λ

– is a family of i.i.d. random variables that is independent

of

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(Relaxing Single Arrival)

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Proof

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Example

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Solution

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Example (Batch Arrival Process)

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Solution

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Poisson’ moment generating function

Geometric’s moment generating function

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ln

Modulated Poisson Processes • Assume that there are two states, 0 and 1, for a "modulating process."

0 1

• When the state of the modulating process equals 0 then the arrive rate

of customers is given by λ0 , and when it equals 1 then the arrival rate

is λ1 .

• The residence time in a particular modulating state is exponentially

distributed with parameter μ and, after expiration of this time, the

modulating process changes state. • The initial state of the modulating process is randomly selected and is

equally likely to be state 0 or 1.

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arrival rate λ0 arrival rate λ1

(Relaxing Independent Increment)

Modulated Poisson Processes (con’t) • For a given period of time (0, t ), let be a random variable that indicates the total amount of time that the modulating process has been in state 0. Let X(t) be the number of arrivals in (0, t ).

• Then, given , the value of X(t) is distributed as a non-homogeneous

Poisson process and thus • The difficulty in determining the distribution for X(t) is to calculate the

density of . There are some limiting cases that are of interest.

• As μ → 0, the probability that the modulating process makes no transitions

within t seconds converges to 1, and we expect for this case that

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Modulated Poisson Processes (con’t) • As μ → ∞, then the modulating process makes an infinite number of transitions within t seconds, and we expect the modulating process to spend an equal amount of time in each state such that

• Example (Modeling Voice). – A basic feature of speech is that it comprises an alternation of silent periods and non-silent periods. – The arrival rate of packets during a talk spurt period is Poisson

with rate λ1 and silent periods produce a Poisson rate with λ0 ≈ 0. – The duration of times for talk and silent periods are exponentially

distributed with parameters , respectively.

⇒ The model of the arrival stream of packets is given by a modulated

Poisson process.

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Interrupted Poisson Process (IPP)

ON OFF

Poisson process with rate

Stay in ON state for a

period exponentially

distributed with mean 1/

Stay in OFF state for a

period exponentially

distributed with mean 1/

Markov Modulated Poisson Process (MMPP)

Example: 3-state MMPP

Poisson process with

rate 1

1

2

3

Poisson process with

rate 2

Poisson process with

rate 3

p12

p21

p13

p31

p23 p32 Stay in state i for a

period exponentially

distributed with mean

1/i

• Since Equation (4) is satisfied when X is exponentially distributed (for ), it follows that exponential random variable are memoryless.

• Not only is the exponential distribution " memoryless," but it is the

unique continuous distribution possessing this property.

Memoryless Property of the Exponential Distribution • A random variable X is said to be memoryless or without memory, if

(3)

• The condition in Equation (3) is equivalent to

or

(4)

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Example

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Solution

1.

2.

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Example

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Solution

1.

2.

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pdf: λ e−λx

cdf: 1 − e−λx

mean: λ−1

exponential

http://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Cumulative_distribution_functionhttp://en.wikipedia.org/wiki/Expected_value

Minimum of Exponential Random Variables

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Comparison of Two Exponential Random Variables

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pdf: λ e−λx

cdf: 1 − e−λx

mean: λ−1

exponential

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http://en.wikipedia.org/wiki/Probability_density_functionhttp://en.wikipedia.org/wiki/Cumulative_distribution_functionhttp://en.wikipedia.org/wiki/Expected_value

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Example

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Solution

Minimum of R1 and R2

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Maximum of Exponential Random Variables

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Example

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Solution

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Solution

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Example

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Solution

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Example

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Solution

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Example

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Solution

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#

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#

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Solution II for (c)

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Example

Solution

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Solution

1.

2.

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3.

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Queue Server

Queuing System

Queuing Time Service Time

Response Time (or Delay)

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Queuing Theory for Studying Networks

View network as collections of queues FIFO data-structures

Queuing theory provides probabilistic analysis of these queues Examples:

Average length

Average waiting time

Probability queue is at a certain length

Probability a packet will be lost

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Little’s Law

The long-term average number of customers in a stable system N, is equal to the long-term average arrival rate, λ, multiplied by the long-term average time a customer spends in the system, T.

E EN T

Expected number of

customers in the system

Expected time in the system

Arrival rate IN the system

Generality of Little’s Law

Little’s Law is a pretty general result

It does not depend on the arrival process distribution

It does not depend on the service process distribution

It does not depend on the number of servers and buffers in the system.

Applies to any system in equilibrium, as long as nothing in black box is

creating or destroying tasks

E EN T

Queueing

Network λ

Aggregate

Arrival rate

Mean number tasks in system = mean arrival rate x mean response time

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Characteristics of queuing systems

Arrival Process The distribution that determines how the tasks arrives in the

system.

Service Process The distribution that determines the task processing time

Number of Servers Total number of servers available to process the tasks

Specification of Queueing Systems

Arrival/Departure

Customer arrival and service stochastic models

Structural Parameters

Number of servers: What is the number of servers?

Storage capacity: are buffer finite or infinite?

Operating policies

Customer class differentiation

are all customers treated the same or do some have priority over others?

Scheduling/Queueing policies

which customer is served next

Admission policies

which/when customers are admitted

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Kendall Notation A/B/m(/K/N/X)

To specify a queue, we use the Kendall Notation.

The first three parameters are typically used, unless specified A: Inter arrival distribution

B: Service time distribution

m: Number of servers (1, 2,… ∞)

K: Storage Capacity (1, 2,… ∞, infinite if not specified)

N: Population Size (1, 2,… ∞, infinite if not specified)

X: Service Discipline (FCFS/FIFO/RSS)

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Kendall Notation of Queueing System

A/B/m/K/N/X

Arrival Process

• M: Markovian

• D: Deterministic

• Er: Erlang

• G: General

Service Process

• M: Markovian

• D: Deterministic

• Er: Erlang

• G: General

Number of servers

m=1,2,…

Storage Capacity

K= 1,2,…

(if ∞ then it is omitted)

Number of customers

N= 1,2,…

(for closed networks, otherwise

it is omitted)

Service Discipline

FIFO, LIFO, Round Robin, …

CS352 Fall,2005 113

Distributions

M: stands for "Markovian", implying exponential distribution for service times or inter-arrival times.

D: Deterministic (e.g. fixed constant)

Ek: Erlang with parameter k http://en.wikipedia.org/wiki/Erlang_distribution

Hk: Hyper-exponential with parameter k

G: General (anything)

http://en.wikipedia.org/wiki/Erlang_distribution

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Kendall Notation Examples

M/M/1 Queue Poisson arrivals (exponential inter-arrival), and exponential service,

1 server, infinite capacity and population, FCFS (FIFO)

the simplest ‘realistic’ queue

M/M/m Queue Same, but m servers

M/D/1 Queue Poisson arrivals and CONSTANT service times, 1 server, infinite

capacity and population, FIFO.

G/G/3/20/1500/SPF General arrival and service distributions, 3 servers, 17 queues (20-

3), 1500 total jobs, Shortest Packet First

Performance Measures of Interest

We are interested in steady state behavior

Even though it is possible to pursue transient results, it is a significantly more difficult task.

E[S]: average system (response) time (average time spent in the system)

E[W]: average waiting time (average time spent waiting in queue(s))

E[X]: average queue length

E[U]: average utilization (fraction of time that the resources are being used)

E[R]: average throughput (rate that customers leave the system)

E[L]: average customer loss (rate that customers are lost or probability that a customer is lost)

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Queue

• A memoryless, Poisson process always gives an exponential distribution for inter-arrival or inter-service times

• However there are cases where the service process times are not exponentially distributed

• Ex.1 A packet switching system may handle only two different types of packet, one with 100 bytes, and one with 2,000 bytes. The big packets will take longer to serve (transmit)

Service time, t

F(t)

A dumbbell distribution: The service times of the small packets would cluster around a low value, and there would be a cluster at a longer time for the larger packets

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Queue (con’t)

• Ex.2 A server (computer) has to perform different operations on packets, depending on what type of packet it is

─ Possible operations are encrypting, processing for an on-line game, and simple transmission

• Ex.3 Within an ATM switch, where all packets (or“cells”) are the same size, and thus take the same time to transmit

• In these cases, the service time distribution is said to be “general“, and we describe the queue as M/G/1

─ Customer arrival: Poisson with rate λ

─ Service times: i.i.d. general distribution G, independent to the arrival distribution

Busy Period of a Queue

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• Sk : The time until k additional customers have arrived

─ Sk has Erlang distribution with parameters (k, λ)

• Y1, Y2, … : The sequence of service times

• The busy period will last a time t and consist of n services iff

• Sk · Y1+Y2+…Yk, k = 1,…n-1

• Y1+Y2+…Yn = t

• There are n-1 arrivals in (0, t)

(total n customers are served)

0

S1

Y1 Y2 Yk

Sk S2

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0

S1

Y1 Y2 Yn-1

Sn-1 S2

t

Sn

Yn

Busy Period

idle period

(1) (2)

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(1)

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Y1 Yn-1 Y2 Yn

0 t

…

∵ Y1, Y2, … Yn are of i.i.d. general distribution G

∴

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∵ The arrival process is independent of the service times

∴

(2)

Gn: n-fold convolution

of G with itself

convolution of x() an h()

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