Elen 602 lecture 8
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Review of Last lecture HDLC, PPP TDM, FDM Today’s lecture Wavelength Division Multiplexing Statistical Multiplexing Preliminary Queuing theory Reading -- Chapter 4.3, 5.5.1, Appendix A.1 - A.3. ELEN 602 Lecture 8. HeaderData payload. Statistical Multiplexing. Input lines. A.

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ELEN 602 Lecture 8

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Elen 602 lecture 8

Review of Last lecture

HDLC, PPP

TDM, FDM

Today’s lecture

Wavelength Division Multiplexing

Statistical Multiplexing

Preliminary Queuing theory

Reading -- Chapter 4.3, 5.5.1, Appendix A.1 - A.3

ELEN 602 Lecture 8


Elen 602 lecture 8

HeaderData payload

Statistical Multiplexing

Input lines

A

Output line

B

Buffer

C


Elen 602 lecture 8

Dedicated versus Shared Lines

(a)

Dedicated Lines

A1

A2

B1

B2

C1

C2

(b)

Shared Line

B2

C2

A2

A1

C1

B1


Elen 602 lecture 8

Number of Packets in System

(a)

Dedicated Lines

A1

A2

B1

B2

C1

C2

(b)

Shared Line

B2

C2

A2

A1

C1

B1

(c)

N(t)


Tdm fdm wdm multiplexing

In TDM, FDM, and WDM link capacity is subdivided into m portions

A packet of length L takes L/(C/m) = Lm/C time

Resources are allocated to individual streams

some streams may have empty queues while others may have long queues

Delay behavior dependent on individual stream arrival

Resources could be wasted

Statistical multiplexing -- no resource wastage

smaller delays, but larger delay variance

In TDM/FDM/WDM -- no need for packet headers

less overhead, simpler packet processing

TDM/FDM/WDM Multiplexing


Elen 602 lecture 8

Network Delay Analysis

Delay Box:

Multiplexer

Switch

Network

Message,

Packet,

Cell

Arrivals

Message,

Packet,

Cell

Departures

T seconds

Lost or

Blocked


Elen 602 lecture 8

Arrival Rates and Interarrival Times

n+1

A(t)

n

n-1

•••

2

1

t

2

n

1

n+1

0

3

Time of nth arrival = 1 + 2 + . . . + n

n arrivals

1

Arrival

Rate

1

=

=

E[]

1 + 2 + . . . + nseconds

(1+2 +...+n)/n

Arrival Rate = 1 / mean interarrival time


Elen 602 lecture 8

Little’s Theorem

T

A(t)

D(t)

Delay Box

N(t)


Little s theorem

N = T

N = Average Number of packets in the system

 = Packet Arrival rate

T = Average Service Delay per packet

Larger the service delay (queuing delay +service time), larger the number of waiting (or buffered) packets

Higher the arrival rate, larger the number of buffered packets

Little’s Theorem


Elen 602 lecture 8

Arrivals and Departures in a FIFO System

A(t)

T7

Assumes

first-in

first-out

T6

T5

T4

D(t)

T3

T2

T1

Arrivals

C1

C2

C3

C4

C5

C6

C7

C1

C2

C3

C4

C5

C6

C7

Departures


Elen 602 lecture 8

Exponentail interarrival

Probability density

e-t

0

t


Elen 602 lecture 8

Queuing Model Classification

Arrival Process / Service Time / Servers / Max Occupancy

Interarrival times 

M = exponential

D = deterministic

G = general

Arrival Rate:

E[ ]

Service times X

M = exponential

D = deterministic G = general

Service Rate:

E[X]

K customers

unspecified if

unlimited

1 server

c servers

infinite

Multiplexer Models: M/M/1/K, M/M/1, M/G/1, M/D/1

Trunking Models: M/M/c/c, M/G/c/c

User Activity: M/M/, M/G/ 


Elen 602 lecture 8

Queuing System Variables

N(t) = number in system

N(t) = Nq(t) + Ns(t)

Nq(t) = number in queue

Ns(t)

Nq(t)

Ns(t) = number in service

1

Pb)

2

T = total delay

c

W

X

W = waiting time

Pb

T = W + X

X = service time


Elen 602 lecture 8

Exponential service

time with rate 

K-1 buffer

Poisson arrivals

rate 

M/M/1K Queue


Elen 602 lecture 8

A Markov State transition diagram

1 - (t

1 - (t

1 - (t

1 - (t

1 - t

1 - (t

t

t

t

t

n

2

n-1

0

1

n+1

t

t

t

t


Elen 602 lecture 8

Average Packet Delay vs. Load M/M/1/10

Finite buffer multiplexer

Normalizedaveragedelay

Load


Elen 602 lecture 8

Packet loss probability vs. Load M/M/1/10

Loss probability

Load


Elen 602 lecture 8

Average Delay with infinite Buffers

M/M/1

Normalizedaveragedelay

M/D/1

Load


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