store and forward: packets
moveonehop at a time
Nodereceives completepacket before
forwarding
Bandwidth division into “pieces”
Dedicated allocation
Resource reservation
Introduction
159
Packet Switching: Statistical Multiplexing
Sequenceof A & B packets does not havefixed pattern, bandwidth shared on
demand
➨
statistical multiplexing
.
TDM: each host gets same slot in revolving TDM frame.
A
B
C
100 Mb/s
Ethernet
1.5 Mb/s
D
E
statistical multiplexing
queueof packets
waiting for output
link
Introduction
160
Packetswitching: storeandforward
takes L/R seconds to transmit
(push out) packet of L bits on
to link at R bps
storeand forward:
entire
packet must
arrive at router
before it can be transmitted on
next link
delay = 3L/R (assuming zero
propagation delay)
Example:
L = 7.5 Mbits
R = 1.5 Mbps
transmission delay = 15 sec
R
R
R
L
more on delay shortly …
Introduction
161
Packet switching versus circuit switching
1 Mb/s link
each user:
100 kb/s when “active”
active 10% of time
circuitswitching:
10 users
packet switching:
with 35 users, probability > 10
active at sametimeis less
than .0004
Packet switching allows moreusers to usenetwork!
N users
1 Mbps link
Q: how did weget value 0.0004?
Use binomial distribution …
Introduction
162
Probability Background:
variables, stats and distributions
Discreterandomvariables:
whereE[X] is theexpected (or mean) value
2
nd
moment:
∑
2200
=
=
k
k
X
1
]
Pr[
∑
2200
=
=
k
k
X
k
X
E
]
Pr[
]
[
∑
2200
=
=
k
k
X
k
X
E
]
Pr[
]
[
2
2
Introduction
163
Continuous random variables:
whereF[x] is thecumulativedistribution, f(y) is the
probability density function,
F[
∞
]=0, F[
∞
]=1
Variance:
Var[X]=E[(XE[X])
2
]=E[X
2
](E[X])
2
Standard deviation
∫
∞

=
≤
=
x
dy
y
f
x
X
x
F
)
(
]
Pr[
]
[
]
[
X
Var
x
=
σ
Introduction
164

Bernoulli experiment:

probability of success p, failure q=1p

Geometric distribution:

X is the number of (independent identically distributed i.i.d.)
Bernoulli experiments to get success

Pr[X=k]=q
k1
p (1
st
k1 failures then success)

E(X)=
Σ
kPr[X=k]=1/p

p=0.1, E(X)=1/p=10
Introduction
165
Introduction
166

Binomial distribution:
•
x
is the number of successes in
n
Bernoulli experiments/trials
•
E[X]=np
(
29
(
29
!
)!
(
!
,
)
(
k
k
n
n
k
X
P
n
k
k
k
n
n
k
p
q

=
=
=

Introduction
167

Exponential distribution:
F[x]=1e

λ
x
, f(x)=
λ
e

λ
x
, Pr[X>x]=1F[x]=e

λ
x
, E[X]=1/
λ
Introduction
168
Poisson Distribution:
Pr[X=k]= (
λ
k
/k!) e

λ
,E[X]=Var[X]=
λ
Used in queuing theory:

Pr[k items arriving in T interval]= ((
λ
T)
k
/k!) e

λ
T
,

Expected number of items to arrive in T=
λ
T, where
λ
is the rate of
arrival
Introduction
169

Poisson processes areused in M/M/1 and M/D/1
queuing models

Interarrival times Ta

Pr[Ta<t]=1e

λ
t
, E[Ta]=1/
λ
, is exponentially distributed

good for modeling human generated actions

phone call arrivals

call duration

telnet/ftp session arrivals
Introduction
170
Packet switching versus circuit switching
great for bursty data
resource sharing (scalable!)
simpler, no call setup, morerobust (rerouting)
excessive congestion:
packet delay and loss
Without admission control: protocols needed for reliabledata
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 Spring '09
 Helmy
 Trigraph, State highways in Mississippi, ne twork, Ne tworks, ne de tworks