The Role of Modeling in Designing Communication Networks

R. SrikantECE/CSL

University of Illinois

A Puzzle

• $5 Reward for the first correct answer• Honor code: Please don’t answer if you have

heard this puzzle before• There are 10 people in a room.• Each one is wearing a white hat or a black hat. • Each person can see the color of other

people’s hats but not her own.

The Puzzle• There is at least one white hat and one black

hat in the room.• If everyone correctly guesses the color of their

hats, then they all win a $1 million each.• They are not allowed to talk or explicitly signal

each other once they are in the room, but they can agree on a strategy ahead of time.

• Is there a way to win the million dollars?• Now to the talk….

What is a Model?

What is a Model?

• Mathematical models– M/M/1 model of a buffer– Collision/SINR models for wireless networks– Optimization models of resource allocation– Differential equation models for TCP

• Do these models capture reality? Should they capture all details of reality?

Reality vs. Utility of Math Models

• “In applying mathematics to subjects such as physics or statistics we make tentative assumptions about the real world which we know are false but which we believe may be useful nonetheless.” George E. P. Box, Science and Statistics. Journal of the American Statistical Association, 71:791–799, 1976.

• More succinctly, “Essentially all models are wrong, but some are useful.” 1987 (George Box and Norman Draper)

Three Themes

• Models are useful to obtain insight

• Models are useful to push the envelope

• Theory versus experiments

Example: Buffering

• Packets arrive at the rate 1 every two time slots into a router buffer

• Router can process 1/2 packet every time slot

• Queue length=1; Delay=1; Is this a useful model?

Example: Buffering

• A stochastic model:

Is this model accurate?

• It is unlikely that any arrival process at any computer or communication system consists of i.i.d. Bernoulli arrivals

• Yet, the average queue size in all buffers have such a qualitative behavior

• More refinements possible: what is the impact of variance, correlation, etc.?

Is this model accurate?

• It is unlikely that any arrival process at any computer or communication system consists of i.i.d. Bernoulli arrivals

• Yet, the average queue size in all buffers have such a qualitative behavior

• More refinements possible: what is the impact of variance, correlation, etc.?

• Conclusion: The model is wrong, but it is useful, as George Box said!

What did we learn from the model?(And a word of caution)

• Randomness in the arrival/service processes is crucial to understanding the queueing behavior of a system

• An old model: but still useful, in discussing buffer sizing at the core vs. access routers in the Internet (will come back to this later)

• Deterministic models are also useful: to understand if the system is stable (arrival rate < service rate implies stability)– The usefulness of a model depends on what you want to

understand

Another Example• Statistical Multiplexing: Motivation for moving

from circuit-switched networks to packet-switched networks

• Can admit more than 100 sources

• Law of large numbers: about 75 sources ON simultaneously

• More precise estimates of packet loss using large deviations

What did we learn from this model?

• Randomness can be exploited• Instead of assuming the worst-case (150

packets arriving all at once), the law of large numbers and large deviations allows us to conclude that we can serve 150 sources using a link with capacity 100

• Clearly ON-OFF modeling is crude; but not difficult to refine it, but it will still be wrong!

• Models may be wrong, but can be useful

Three Themes

• Models are useful to obtain insight

• Models are useful to push the envelope

• Theory versus experiments

Pushing the Envelope

• Example: Downlink scheduling

A Simple Average-Case Model

• Each channel is ON half the time.

• Need to give each user 1/3 of the available capacity.

• Therefore, the throughput per user is 1/6.

• Certainly would capture the reality of some naïve scheduling algorithms. But it doesn’t ask or answer the question, can we do better?

A More Refined Model

• Suppose transmitter knows channel conditions

• Transmit to an ON channel• P(all channels are OFF)=1/8• Throughput=7/8• Throughput/user=7/24• Better than 1/6 with the

previous algorithm!

Burstiness

• What if the channel is more bursty?

• For example, to make easy comparisons, let’s say that the mean rate is the same

• With a single user, more burstiness is only bad; get the same throughput with worse delay

Multi-user Diversity

• A much more bursty channel• ON less frequently, but is

able to transmit more packets when ON

• The mean number of packets that can be transmitted per channel is still 1/2

Multi-user Diversity

• Again, transmit to one of the ON channels

• P(all channels are OFF)=0.93=0.729

• Throughput=0.271*5=1.355• Throughput/user=1.355/3 =0.45

> 7/24=0.29• Burstiness may be good, it may

be worth using some resources to get channel state information!

Heterogeneity and QoS

• Different users may need different data rates: some >0.45, some <0.45

• The channel conditions for different users may be different

• QoS may be important, not just throughput

MaxWeight Scheduling

• Associate a weight with each user

• Transmit to a user with max weight

• Weight can be chosen for fairness, to maximize throughput, to reflect QoS requirements, etc.

3G Wireless Networks

• Employ some form of channel-aware scheduling for data

• A huge success story in translating theory to practice

• Moral of the story: Modeling doesn’t just help us understand real systems, but can lead to better technology; can help push the envelope

Three Themes

• Models are useful to obtain insight

• Models are useful to push the envelope

• Theory versus experiments

Theory and Practice

• “In theory, there is no difference between theory and practice. In practice, there is.” Jan L.A. van de Snepscheut or Yogi Berra?

• How does one translate theory to good practice? How do we know that the model we start with is relevant to practice?

• Answer: Experiments/System Building

Comm Networks: A Vibrant Field

• New applications/technology emerge: a close connection between academia, system developers and the industry– Opportunistic scheduling– High-speed TCP variants– BitTorrent and other filesharing systems– P2P Streaming– Switching fabrics for routers– TCP buffer sizing– …….

Modeling and Experimentation

• The two are inextricably conjoined– It is said that the Nobel Prize in Physics is rarely

awarded for purely theoretical work, it should be experimentally validated

– Back to channel-aware scheduling: in practice, there are many considerations that are not modeled; very short flows, transient flows, sometimes strict delay constraints• Start with theoretical weights and modify them using

experimentation; modeling is still the key, the starting point

An Example

• How experiments affect our perception of protocol design?

• Early 1990s: Long-range dependent nature of packet arrival processes in the Internet

• Implications: packet arrival times are correlated over long intervals of time; very different from Poisson process modeling– Queueing models of such systems predict very large

queue lengths; may lead to excessively large, unacceptable delays

A Refined Viewpoint• What is the cause of long-range correlations?

– File sizes are heavy-tailed; most files are very short; but a few large files contribute to most of the traffic in the Internet

• An opportunity– Control the rate at which large files transmit data into the network;

Improve the performance of short files dramatically, without sacrificing the performance of large files

• More experiments– At the core routers, perhaps the arrival rate is indeed Poisson; flows

are access constrained– May be can get away with very small buffers

Summary

• Mathematical models are useful to understand reality– Abstraction is part of the game

• Mathematical models are sometimes necessary to made radical improvements to existing technology– Again, abstraction is the key

• Theory and Experimentation are complementary, and need each other

Back to the Puzzle

• Solution: – Each person observes the number of black hats

and white hats. – Each person waits for T seconds, where T is the

smaller of the two numbers she observes; then, announces that her hat is the same color as the hats with the fewer number

– Everyone else knows their color now

An Example

• Suppose there are 6 White hats and 4 Black hats in the room– Each B observes 6W and 3B– Each W observes 5W and 4B

• All Bs will announce their hat colors after 3 seconds• All Ws, then know their hat color• Called the Hat Puzzle in Wikipedia• Research is not about modeling or experiments: it

is about out-of-the-box thinking