Katz, Stoica F04

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

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Outline

Motivations Timing diagrams Metrics Evaluation techniques

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Motivations

Understanding network behavior Improving protocols Verifying correctness of implementation Detecting faults Monitor service level agreements Choosing provides Billing

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Outline

Motivations Timing diagrams Metrics Evaluation techniques

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Timing Diagrams

Sending one packet Queueing Switching

- Store and forward

- Cut-through

Fluid view

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Definitions

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)

T seconds

P bits

Bandwidth: R bpsPropagation delay: T sec

time

Transmission time = P/RT

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 KbyteR = 1 Gbps100 Km, fiber => T = 500 usec P/R = 8 usec

T

P/Rtime

time

T

P/R

P = 1 KbyteR = 100 Mbps1 Km, fiber => T = 5 usec P/R = 80 usec

T >> P/R

T << P/R

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Queueing

The queue has Q bits when packet arrives packet has to wait for the queue to drain before being transmitted

P bits

time

P/RT

Q bits

Queueing delay = Q/R

Capacity = R bpsPropagation delay = T sec

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Queueing Example

P = 1 Kbit; R = 1 Mbps P/R = 1 ms

Packet arrivalTime (ms)

Time

Delay for packet that arrives at time t, d(t) = Q(t)/R + P/RDelay for packet that arrives at time t, d(t) = Q(t)/R + P/RQ(t)

1 Kb

P bits Q bits

0 0.5 1 7 7.5

0.5 Kb

1.5 Kb

2 Kb

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

A packet is stored (enqueued) before being forwarded (sent)

Sender Receiver

10 Mbps 5 Mbps 100 Mbps 10 Mbps

time

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Store and Forward: Multiple Packet Example

Sender Receiver

10 Mbps 5 Mbps 100 Mbps 10 Mbps

time

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Switching: Cut-Through

A packet starts being forwarded (sent) as soon as its header is received

Sender Receiver

R1 = 10 Mbps R2 = 10 Mbps

time

Header

What happens if R2 > R1 ?

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Fluid Flow System

Packets are serviced bit-by-bit as they arrive

a(t)

e(t)

Q(t)

t

Rate or Queue size

Q(t)a(t) – arrival rate e(t) – departure rate

Q(t) = queueing size at time t

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Outline

Motivations Timing diagrams Metrics

• Throughput

• Delay

Evaluation techniques

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Throughput

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

Connection: - Send W bits (window size)

- Wait for ACKs

- Repeat

Assume the round-trip-time is RTT seconds

Throughput = W/RTT bps

Numerical example:- W = 64 Kbytes

- RTT = 200 ms

- Throughput = W/T = 2.6 Mbps

time

Source Destination

RTT

RTT

RTT

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Throughput: Fluctuations

Throughput may vary over time

max

min

mean

Throughput

Time

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Delay

Delay (Latency) of bit (packet, file) from A to B- The time required for bit (packet, file) to go from A to B

Jitter- Variability in delay

Round-Trip Time (RTT)- Two-way delay from sender to receiver and back

Bandwidth-Delay product- Product of bandwidth and delay “storage” capacity of

network

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Delay: Illustration 1

at point 2at point 1

Latest bit seenby time t

Delay

Sender Receiver

1 2

time

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Delay: Illustration 2

Sender Receiver

1 2

2

1Packet arrival times at 1

Packet arrival times at 2Delay

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Little’s Theorem

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?

systema(t) – arrival rate

d(i) = delay of packet i

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

Latest bit seenby time t

Sender Receiver

1 2

time

x(t)

d(i) = delay of packet ix(t) = number of packets in transit (in the system) at time t

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

Latest bit seenby time t

Sender Receiver

1 2

time

x(t)

T

Average occupancy = S/T

d(i) = delay of packet ix(t) = number of packets in transit (in the system) at time t

S= area

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Little’s Theorem

Latest bit seenby time t

Sender Receiver

1 2

time

x(t)

S(N)

Pd(N-1)

S(N-1)

T

S = S(1) + S(2) + … + S(N) = P*(d(1) + d(2) + … + d(N))

d(i) = delay of packet ix(t) = number of packets in transit (in the system) at time t

S= area

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Average occupancy

Average arrival time Average delay

Little’s Theorem

Latest bit seenby time t

Sender Receiver

1 2

time

x(t)

S(N)

Pd(N-1)

S(N-1)

TS/T = (P*(d(1) + d(2) + … + d(N)))/T = ((P*N)/T) * ((d(1) + d(2) + … + d(N))/N)

d(i) = delay of packet ix(t) = number of packets in transit (in the system) at time t

S= area

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Little’s Theorem

Latest bit seenby time t

Sender Receiver

1 2

time

x(t)

S(N)

Average occupancy = (average arrival rate) x (average delay)

S= area

a(i)d(N-1)

S(N-1)

T

d(i) = delay of packet ix(t) = number of packets in transit (in the system) at time t

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Outline

Motivations Timing diagrams Metrics Evaluation techniques

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Evaluation Techniques

Measurements- gather data from a real network

- e.g., ping www.berkeley.edu

- realistic, specific Simulations: run a program that pretends to be a real

network- e.g., NS network simulator, Nachos OS simulator

Models, analysis- write some equations from which we can derive conclusions

- general, may not be realistic Usually use combination of methods

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Analysis

Example: M/M/1 Queue

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

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

a s

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Simulation

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 eventY(n) = state after event nT(n) = time when even n occurs[Y(n+1), T(n+1)] = g(Y(n), T(n), E(n))

Output analysis: estimates, confidence intervals

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Evaluation: Putting Everything Together

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)

Reality Model

HypothesisConclusion

Abstraction

Plausibility

Derivation Tractability

Realism

Prediction