a.leon-garcia_communication_networks.pdf

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| | | Textbook Table of Contentse-Text Main Menu

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Message,packet,cellarrivals

Message,packet,celldepartures

Delay boxMultiplexer

SwitchNetwork

T seconds

Lost orblocked

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A(t) n + 1

n − 1

2

1

0

n

� � �

τ1 τ2 τ3 τn τn + 1t

Arrivalrate

Arrival rate = 1�mean interarrival time

n arrivals 1 1

Time of nth arrival = τ1 + τ2 + + τn�

� �τ1 + τ2 + + τn seconds (τ1 + τ2 + + τn)�n E[τ]= �

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T

D(t)A(t)N(t)

Delay box

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T7

T6

T5

A(t)

D(t)

T4

T3

T2

T1

C1 C2 C3

C3 C4 C5 C6 C7C2C1

C4 C5 C6 C7

Assumesfirst-infirst-out

Arrivals

Departures

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Arrival process

Service time

Queue

Servers

A(t)

B(t)X

D(t)tt

1

2

c

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τi τi + 1

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Arrival process� Service time� Servers� Max occupancy

Interarrival times τM = exponentialD = deterministicG = generalArrival rate:λ = 1/E[τ]

Multiplexer models:Trunking models:User activity:

M/M/1/K, M/M/1, M/G/1, M/D/1M/M/c/c, M/G/c/cM/M/∞, M/G/∞

Service times XM = exponentialD = deterministicG = generalService rate: = 1/E[X]�

1 serverc serversinfinite

K customersunspecified ifunlimited

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N(t) = number in systemNq(t) = number in queueNs(t) = number in service

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

Nq(t)

Ns(t)

1

2

cW

X

T = total delayW = waiting timeX = service time

T = W + X

λ Pb

λ(1−Pb)

� � �λ

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Poissonarrivalsrate λ

Exponentialservice timewith rate �

K − 1 buffer��

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�Δt �Δt �Δt �Δt

λ Δtλ Δtλ Δtλ Δt

1 − λΔt 1 − (λ + �)Δt 1 − (λ + �)Δt 1 − (λ + �)Δt 1 − (λ + �)Δt 1 − (λ + �)Δt

0 1 2 n − 1 n + 1n� � � � � �

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mseparatesystems

Oneconsolidated

systemversus mλ m�

λ

λ

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Poissonarrivalsrate λ

Infinite buffer � = 1/E[X]

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Interarrivals Constant

M�D�1

0

1�2

Erlang

M�Er�1

<1

1�2<, <1

Exponential

M�M�1

1

1

Hyperexponential

M�H�1

>1

>1E[W]�E[WM�M�1]

CX2

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Manylines

Limitednumberof trunks

Poissonarrivals λ λ(1 − Pb)

N(t)

λ PbE[X ] = 1��

1

2

c

� � �

� Blocked calls are cleared from the system; no waiting allowed.

� Performance parameter: Pb � fraction of arrivals that are blocked.

� Pb� P[N(t) � c] �B(c, a) where a �λ��.

� B(c, a) is the Erlang B formula, which is valid for any service time distribution.

B(c, a)�

ac

c!a j

j!j � 0

c

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0 1 2 c − 1 c

�Δt 2�Δt c�Δt

λ Δtλ Δtλ Δt

1 − λ�t 1 − (λ + �)�t 1 − (λ + 2�)�t 1 − (λ + (c − 1)�)�t 1 − c ��t

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a.leon-garcia_communication_networks.pdf

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