EECS 122:
Introduction to Computer Networks
Performance Modeling
Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California, Berkeley
Berkeley, CA 94720-1776
Katz, Stoica F04
Outline
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Motivations
Timing diagrams
Metrics
Evaluation techniques
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Motivations
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Understanding network behavior
Improving protocols
Verifying correctness of implementation
Detecting faults
Monitor service level agreements
Choosing provides
Billing
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Outline
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Motivations
Timing diagrams
Metrics
Evaluation techniques
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Timing Diagrams
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Sending one packet
Queueing
Switching
- Store and forward
- Cut-through
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Fluid view
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Definitions
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Link bandwidth (capacity): maximum rate (in bps)
at which the sender can send data along the link
Propagation delay: time it takes the signal to
travel from source to destination
Packet transmission time: time it takes the
sender to transmit all bits of the packet
Queuing delay: time the packet need to wait
before being transmitted because the queue was
not empty when it arrived
Processing Time: time it takes a router/switch to
process the packet header, manage memory, etc
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Sending One Packet
R bits per second (bps)
Bandwidth: R bps
Propagation delay: T sec
T seconds
P bits
Transmission time = P/R
T
time
Propagation delay =T = Length/speed
1/speed = 3.3 usec in free space
4 usec in copper
5 usec in fiber
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Sending one Packet: Examples
P = 1 Kbyte
R = 1 Gbps
100 Km, fiber =>
T = 500 usec
P/R = 8 usec
T >> P/R
T
P/R
time
T << P/R
T
P = 1 Kbyte
R = 100 Mbps
1 Km, fiber =>
T = 5 usec
P/R = 80 usec
P/R
time
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Queueing
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The queue has Q bits when packet arrives  packet has to
wait for the queue to drain before being transmitted
Capacity = R bps
Propagation delay = T sec
P bits
Q bits
Queueing delay = Q/R
T
P/R
time
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Queueing Example
P bits
Q bits
P = 1 Kbit; R = 1 Mbps  P/R = 1 ms
Packet arrival
Q(t)
Time (ms)
0 0.5 1
7
7.5
Delay for packet that arrives at time t, d(t) = Q(t)/R + P/R
2 Kb
1.5 Kb
1 Kb
0.5 Kb
Time
packet 1, d(0) = 1ms
packet 2, d(0.5) = 1.5ms
packet 3, d(1) = 2ms
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Switching: Store and Forward
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A packet is stored (enqueued) before being forwarded
(sent)
10 Mbps
Sender
5 Mbps
100 Mbps
10 Mbps
Receiver
time
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Store and Forward: Multiple Packet
Example
10 Mbps
Sender
5 Mbps
100 Mbps
10 Mbps
Receiver
time
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Switching: Cut-Through
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A packet starts being forwarded (sent) as soon as its
header is received
R1 = 10 Mbps
R2 = 10 Mbps
Receiver
Sender
Header
time
What happens if R2 > R1 ?
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Fluid Flow System
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Packets are serviced bit-by-bit as they arrive
Q(t) = queueing size at time t
a(t) – arrival rate
Q(t)
e(t) – departure rate
Rate or
Queue size
a(t)
e(t)
Q(t)
t
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Outline
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Motivations
Timing diagrams
Metrics
• Throughput
• Delay
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Evaluation techniques
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Throughput
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Throughput of a connection or link = total number of bits
successfully transmitted during some period [t, t + T)
divided by T
Link utilization = throughput of the link / link rate
Bit rate units: 1Kbps = 103bps, 1Mbps = 106bps, 1 Gbps
= 109bps [For memory: 1 Kbyte = 210 bytes = 1024 bytes]
- Some rates are expressed in packets per second (pps) 
relevant for routers/switches where the bottleneck is the header
processing
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Example: Windows Based Flow Control
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Source
Connection:
- Send W bits (window size)
- Wait for ACKs
- Repeat
RTT
RTT
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Assume the round-triptime is RTT seconds
Throughput = W/RTT bps
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Numerical example:
RTT
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- W = 64 Kbytes
- RTT = 200 ms
- Throughput = W/T = 2.6
Mbps
Destination
time
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Throughput: Fluctuations
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Throughput may vary over time
Throughput
max
mean
min
Time
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Delay
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Delay (Latency) of bit (packet, file) from A to B
- The time required for bit (packet, file) to go from A to B
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Jitter
- Variability in delay
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Round-Trip Time (RTT)
- Two-way delay from sender to receiver and back
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Bandwidth-Delay product
- Product of bandwidth and delay  “storage” capacity of
network
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Delay: Illustration 1
1
Sender
2
Receiver
Latest bit seen
by time t
at point 2
at point 1
Delay
time
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Delay: Illustration 2
1
Sender
2
Receiver
Packet arrival times at 1
1
2
Delay
Packet arrival times at 2
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Little’s Theorem
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Assume a system (e.g., a queue) at which packets
arrive at rate a(t)
Let d(i) be the delay of packet i , i.e., time packet i
spends in the system
What is the average number of packets in the
system?
d(i) = delay of packet i
a(t) – arrival rate
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system
Intuition:
- Assume arrival rate is a = 1 packet per second and the delay
of each packet is s = 5 seconds
- What is the average number of packets in the system?
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Little’s Theorem
1
2
Latest bit seen
by time t
d(i) = delay of packet i
x(t) = number of packets
in transit (in the system)
at time t
Sender
Receiver
x(t)
time
T
What is the system occupancy, i.e., average
number of packets in transit between 1 and 2 ?
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Little’s Theorem
1
2
Latest bit seen
by time t
d(i) = delay of packet i
x(t) = number of packets
in transit (in the system)
at time t
Sender
Receiver
x(t)
S= area
time
T
Average occupancy = S/T
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Little’s Theorem
1
2
Latest bit seen
by time t
d(i) = delay of packet i
x(t) = number of packets
in transit (in the system)
at time t
Sender
Receiver
S(N)
S(N-1)
P
d(N-1)
x(t)
S= area
time
T
S = S(1) + S(2) + … + S(N) = P*(d(1) + d(2) + … + d(N))
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Little’s Theorem
1
2
Latest bit seen
by time t
d(i) = delay of packet i
x(t) = number of packets
in transit (in the system)
at time t
Sender
Receiver
S(N)
S(N-1)
P
d(N-1)
x(t)
S= area
time
T
Average
S/T = (P*(d(1) + d(2) + … + d(N)))/T
occupancy
= ((P*N)/T) * ((d(1) + d(2) + … + d(N))/N)
Average arrival time
Average delay
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Little’s Theorem
1
Latest bit seen
by time t
d(i) = delay of packet i
x(t) = number of packets
in transit (in the system)
at time t
2
Sender
Receiver
S(N)
a(i)
S(N-1)
d(N-1)
x(t)
S= area
time
T
Average occupancy = (average arrival rate) x (average delay)
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Outline
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Motivations
Timing diagrams
Metrics
Evaluation techniques
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Evaluation Techniques
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Measurements
- gather data from a real network
- e.g., ping www.berkeley.edu
- realistic, specific
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Simulations: run a program that pretends to be a real
network
- e.g., NS network simulator, Nachos OS simulator
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Models, analysis
- write some equations from which we can derive conclusions
- general, may not be realistic
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Usually use combination of methods
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Analysis
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Example: M/M/1 Queue
a
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s
Arrivals are Poisson with rate a
Service times are exponentially distributed with
mean 1/s
- s - rate at which packets depart from a full queue
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Average delay per packet:
T = 1/(s – a) = (1/a)/(1 – u), where u = a/s = utilization
Numerical example, 1/a = 1ms; u = 80%  Q = 5ms
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Simulation
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Model of traffic
Model of routers, links
Simulation:
- Time driven:
X(t) = state at time t
X(t+1) = f(X(t), event at time t)
- Event driven:
E(n) = n-th event
Y(n) = state after event n
T(n) = time when even n occurs
[Y(n+1), T(n+1)] = g(Y(n), T(n), E(n))
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Output analysis: estimates, confidence intervals
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Evaluation: Putting Everything Together
Abstraction
Reality
Model
Plausibility
Derivation
Tractability
Prediction
Conclusion
Hypothesis
Realism
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Usually favor plausibility, tractability over realism
- Better to have a few realistic conclusions than none (could not
derive) or many conclusions that no one believes (not
plausible)
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