# ELEN 602 Lecture 8 - PowerPoint PPT Presentation

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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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#### Presentation Transcript

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

Statistical Multiplexing

Input lines

A

Output line

B

Buffer

C

Dedicated versus Shared Lines

(a)

Dedicated Lines

A1

A2

B1

B2

C1

C2

(b)

Shared Line

B2

C2

A2

A1

C1

B1

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)

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

### TDM/FDM/WDM Multiplexing

Network Delay Analysis

Delay Box:

Multiplexer

Switch

Network

Message,

Packet,

Cell

Arrivals

Message,

Packet,

Cell

Departures

T seconds

Lost or

Blocked

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

Little’s Theorem

T

A(t)

D(t)

Delay Box

N(t)

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

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

Exponentail interarrival

Probability density

e-t

0

t

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/ 

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

Exponential service

time with rate 

K-1 buffer

Poisson arrivals

rate 

M/M/1K Queue

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

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

Finite buffer multiplexer

Normalizedaveragedelay

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

Loss probability