Outline Application examples Google page rank algorithm Aloha protocol Virtual circuit with window flow control Store-and-Forward packet-switched network Interactive system with infinite servers 1
Example1: Google Page Rank Algorithm Search engine of Google Page rank algorithm is the key Rank the page in a proper order show the most relevant page in top 10 search results Question: How to rank the page? Count # of backlinks to this page, or citation counting Problem: different backlink should have different weight Trick: create 1000 pages linking to my page A backlink from Yahoo should be given more importance Google s solution: define page rank recursively 2
Page rank algorithm Recursively define the page rank A page has high rank if the sum of the ranks of its backlinks is high Similar to the Markov chain j i ij i 1 P ij is the joint matrix among pages. P is huge and spare The rank of a page can be viewed as its limiting probability in a Markov chain n P 3
Page rank algorithm Google s page rank algorithm (simple case) Create a DTMC transition diagram where there is one state for each page and there is an arrow from state i to state j if page i has a link to page j If page i has k>0 outgoing links, then set the probability on each outgoing arrow from state i to be 1/k Solve the DTMC to determine the limiting probability. Pages are then ranked based on their limiting probabilities (higher probability first) 4
Page rank algorithm An example of 3 pages 1/2 N 1/2 1/2 1/2 A M 1 Solve the balance equation and obtain π A = π N =2/5, π M =1/5 Page A and N are more important than page M 5
Page rank algorithm Dead End or Spider Trap 1/2 N 1/2 1/2 1/2 A M 1 Solve the balance equation and obtain π A = π N =0, π M =1 Actually, page M is anti-social and has not link to anyone else It should be given less importance in Internet 6
Page rank algorithm Google s solution to dead ends and spider traps Tax 0.7*1/2+0.1 0.7*1/2+0.1 N 0.7*1/2+0.1 0.7*1/2+0.1 A M 0.8 Tax each page some fraction of its importance and distribute the tax equally among all pages. E.g., apply a 30% tax on each page and distribute 30%/n=10% to each page. Solve the balance equation and obtain π A =0.19, π N =0.26, π M =0.55 7
Page rank algorithm Modifications to tax policy Not equally distribute the tax back to pages Distribute only among predecessors of the page Practical implementation issues How to solve the balance equation? P is very huge and spare Google did a lot of efforts on this issue Not solve equation take powers of P, calculate P t Spare matrix calculation, partition matrix, parallel computation, 8
Example2: Aloha Protocol Analysis Aloha is the progenitor of Ethernet protocol Link layer protocol Aloha: 1970, N. Abramson, Hawaii Univ., radio network Ethernet : CSMA/CD CSMA/CD is developed based on Aloha CSMA/CA is used in WiFi 9
Slotted Aloha protocol Divide the time into discrete time slots Each slot can be used to transmit a packet Allow only one transmission at one slot, otherwise, collision and need retransmission # of independent hosts is m, each host transmit a new packet with probability p, p<1/m If collide, retransmit at every following slot with probability q until succeed, q<p Aloha is a Decentralized Protocol 10
Markov chain to model Aloha State: # of packets need retransmission Transition probability P 0, j k 2 k, j m P (1 p) kq(1 q) kk, 1 kk, kk, 1 k, j k 1 P m q p p kq q p k m 1 k 1 m (1 ) (1 ) (1 (1 ) )(1 ) P mp p q m 1 k (1 ) (1 (1 ) ) m j m j Pk, k j p (1 p), j 2,..., m j P 0, j k m 11
Property of Aloha Markov chain Is this chain aperiodic and irreducible? Yes. Does Aloha work well? Or is the chain ergodic? Not well The probability of transiting to a lower state is k 1 (k) k 1 back k, j k, k 1 j 0 P P P kq(1 q) (1 p) When k, this probability 0, which means the # of retransmission always grows. Markov chain is transient, grows to infinity m 12
Property of Aloha Markov chain The expected # of transmissions during a slot E[N]=mp+kq How to set q so as to make the chain ergodic? mp+kq<1 q<(1-mp)/k E.g., q=1/k n, q=1/β k Exponential backoff scheme in Ethernet, CSMA/CD and CSMA/CA But, it will make the queue longer, large delay 13
Example 3: Applications to Computer Networks Closed Network Model
Virtual circuit with window flow control Closed Network Model with Multiple Classes Suitable for modeling virtual circuit (VC) with window flow control Data sources/sinks are modeled explicitly 15
Model of a VC with Window Flow Control 16
Model of a VC with Window Flow Control Source, sink, channels are modeled by FCFS single server facilities ACK (acknowledgement) delay is modeled by an IS (infinite server) facility When source server is not idle, each customer represents an authorization to send one data packet on VC Data packets are individually acknowledged 17
Model of a VC with Window Flow Control A customer entering the source queue represents the reception of an ACK by the source N is the window size When all N data packets are unacknowledged, the source is idle; this models the situation that the window is closed No loss of packets due to error or buffer overflow 18
Closed Network Model Each VC is modeled by a closed class No. of customers belonging to each class is constant Channels visited by each class are defined by the routing algorithm 19
Example 20
Model Assumptions 21
Analytic Results 22
Example 4: Applications to Computer Networks Open Network Model
Store-and-Forward Packet- Switched Network 24
Assumptions for the Classical Model 25
Assumptions for the Classical Model 26
Notation and Definition 27
Open Network Model with Multiple Classes for simplicity, only odd numbered channels are shown 28
Open Network Model with Multiple Classes Suitable for datagram networks Analysis is based on the approach for Markovian queuing networks provide results for end-to-end delay of each class results are useful for network design Capacity assignment Optimal routing Topological design 29
Analytic Results 30
Analytic Results 31
Analytic Results 32
Shortest-Path Routing Example 33
Shortest-Path Routing Example 34
Shortest-Path Routing Example 35
End-to-End Delay Results 36
Alternate Routing Example 37
Alternate Routing Example 38
End-to-End Delay Results 39
Typical Behavior 40
Observations 41
Example 5: Interactive system model with infinite servers Thinking stage Computing stage To model the multiple terminals access to a central computer 42
Relative Arrival Rates 43
Server Utilization 44
Upper Bound Utilization 45
Upper Bound Utilization 46
Lower Bound Mean Response Time 47
Lower Bound Mean Response Time 48
Heavy Load Approximation 49
Interactive System Model Analytic Results 50
Interactive System Model Analytic Results 51