1. DTMC Applications Ranking Web Pages & Slotted ALOHA
TELE4642: Week11

2. Outline Apply the theory of discrete time Markov chains:
Google’s ranking of web-pages
What page is the user most likely searching for?
Formulate web-graph as a Markov chain
Does steady-state exist?
Does a user randomly walk the web-graph?
Can search results be improved further?
Slotted ALOHA medium access control protocol
Is the protocol stable for large number of nodes?
How should the retransmission probability be chosen?
Network Performance

3. Ranking of Web-pages
Problem: how should a search engine rank web-pages?
Idea: rank pages based on number of in-links (citations)
Weakness: not all in-links are equal
Google’s idea: a page has high rank if the sum of the ranks of its in-link pages is high
Formulate moves between web-pages as Markov chain
Solve to obtain steady-state probability of each state
State probability is proportional to importance of page
Example with three web-pages: N M A
Network Performance

4. Markov Model of the Web 2 3 1 5 4
Issue 1: how to choose transition probabilities?
Assumption: each link is equally likely to be clicked
Can accommodate non-uniform probability if such information available
Issue 2: some rows are zero (dead ends)
Assumption: on reaching dead-end restart at any state
r is an Nx1 column vector whose i-th row is non-zero for dead-end nodes
v is an Nx1 column vector whose entries add to 1
could all be 1/N (uniform)
could be different from uniform (i.e. personalized)
Network Performance

5. Markov Model of the Web (contd.)5 1 4 2 3
Issue 3: Transition probability matrix may still be non-stationary
Solution: inter-connect all nodes:
where u is an Nx1 column vector with all entries 1
α is a number between 0 and 1 (“tax” on “importance”)
For and :
The very sparse initial matrix now becomes the dense matrix
Network Performance

6. Computing the page rank
Issue 4: Computing involves solving billion+ equations!
Instead take powers of
Iterative procedure:
No matrix multiplication, work with only one vector
Multiplication with sparse matrix P, dense matrix not formed
Convergence depends on parameter α
What should α be set at?
Small α allows faster convergence (why?)
Large α preserves better the true nature of the web-graph (why?)
Brin and Page [Google] claim that α=0.85 works well
only 50 to 100 iterations are required for convergence
Network Performance

7. Discussion 2 3 1 5 4 Basic idea: Random walk on the web-graph
The more often you visit a node,
the more “popular” the page
Does your model of the walk path match real user behavior?
Instead of connecting every node to every other node (“tax”), create a dummy node to which all other nodes are connected and that connects to all nodes; this alters the true web-graph less.
At dead-end, user often hits the “back” button; so bias the transition probability towards predecessor pages.
How to increase the ranking of your web-page?
Create replicas of your page?
Create many “dummy” web-pages that point to your page?
Make your web-pages link to each other?
Further reading:
“The PageRank Citation Ranking: Bringing Order to the Web”, 1999
“Random Walks with Back Buttons”, 2000
“Deeper inside PageRank”, 2004
Network Performance

8. Slotted Aloha N nodes, time-slotted system, equal-size packets
Probability of new packet arrival in a slot to any given node is pa and the new packet is transmitted immediately
Collision happens if more than one node transmits in the same slot; detected by all nodes at end of slot
If collision, each backlogged node retries in every slot with probability pr until successful transmission
No queueing: new arrivals to a backlogged node are dropped
Network Performance

9. Slotted Aloha: Markov chain
State: number of backlogged nodes m = 0,…,N
Probability that i backlogged nodes transmit in a slot is
Probability that j non-backlogged nodes transmit in a slot is
Markov chain:
Network Performance

10. Slotted Aloha: Efficiency
Probability of successful transmission in state m:
For small pa and pr , and using for small x:
Let be the transmission attempt rate in state m, the throughput (successful transmissions per slot) is
Throughput maximized at G(m)=1
Max. throughput = 1/e = 36%
Network Performance

11. Slotted Aloha: Instability
Does slotted Aloha work when N is large?
Given you are in state m, what is the probability of moving backwards (i.e. state < m)?
Stated another way, when the number of backlogged nodes is large enough, the average attempt rate G(m) becomes > 1
i.e. there are excessive collisions and state keeps growing
Potential solution: ensure the attempt rate G(m) < 1
How? make the retransmission probability dependent on state
E.g.: exponential backoff:
Price for making retransmission probability too small: large delay
Network Performance