Week 10: DTMC Applications Randomized Routing. Network Performance 10-1

Transcription

1 Week 10: DTMC Applications Randomized Routing Network Performance 10-1

2 Random Walk: Probabilistic Routing Random neighbor selection e.g. in ad-hoc/sensor network due to: Scalability: no routing table (e.g. geographical routing) Resilience: node failures, multiple paths Simplicity: easy algorithm Can be modeled as random walk in graph Formulate as a discrete-time Markov chain E.g.: 1 is source, 6 is destination Network Performance 10-2

3 Metrics of interest Can we solve the Markov chain? Not irreducible, so no Metrics of interest: Where is the packet in k hops? Iterative: π [ k + 1] = π[ k] P How many hops are needed for a 90% chance of the packet reaching the destination? Will the packet always get to the destination? What is the average number of hops needed? What is the impact of changing transition probabilities? How would you compute the above analytically? Network Performance 10-3

4 Exercise 1: Gambler s Ruin You go into a casino with $2 in your pocket, and play in a slot-machine. Each play costs $1. In each play, if you win (which happens with probability p), you get back double your money (i.e. $2), whereas if you lose (with probability q = 1-p), you lose your dollar. You stop playing and leave the moment you either reach $4, or when you have lost all your money, whichever happens first. Draw the Discrete Time Markov chain (DTMC) that models your performance at the casino. Does this Markov chain have a steady-state solution? Justify clearly. Compute the probability (in terms of p and q) that you lose all your money. Hint: Let u i denote the chances of losing all your money starting at state i. Compute the probability of losing all your money when p = q = ½. Explain your answer intuitively. Network Performance 10-4

5 Exercise Consider a small organization that sells a software product, and has a corporate intranet consisting of just five web-pages linked as below: Page 1 is the company s main page. It has links to the software download page (page 2) and the customer support page (page 3). Page 2 provides the software product for download. It links back to the main page (page 1). Page 3 is the customer support page, and links to the software download page (page 2), installation support page (pages 4) and tech support page (page 5). Page 4 provides installation support for the software. It links back to the main page (page 1). Page 5 provides technical support for the software (bugs, patches, etc), and has a link back to the software download page (page 2). A search-engine uses the discrete-time Markov chain based PageRank algorithm for ranking these five web-pages, with the following simplifying assumptions: There are no links to any web-pages external to the organization. Each link on a web-page is equally likely to be clicked by the user. A user navigating the corporate web-site is assumed to click links and not explicitly type the URL (i.e., the parameter α of the PageRank algorithm is set to 1; equivalently, the user never restarts the walk on the webgraph). Searching for a word or phrase only returns the top two web-pages that match. Network Performance 10-5

6 Exercise 2 (contd.) Questions: Write the transition probability matrix P for the Markov chain corresponding to the above web-graph. Argue why the Markov chain for the above web-graph is irreducible and aperiodic (and consequenty has a steadystate stationary solution). What are two ways by which you can compute the state probabilities for the above web-graph? Which method would you prefer to employ (using pen and paper, not a computer), and why? Now compute the stationary probabilities for each of the five states. A user searches for a keyword that is present in all five web-pages. Which are the two top-ranked web-pages that are returned by the search engine? Network Performance 10-6

7 Exercise 2 (contd.) Now suppose you are the director of the customer support division, and are free to modify web-pages 3, 4, and 5 in any way you want, but you cannot modify web-pages 1 and 2 (which are managed by another group). Further, you are given the constraints that: (a) you can add additional links in pages 3, 4, and 5 (including links from a page back to itself), but (b) you cannot remove any existing links from these pages. Qualitatively (i.e. using words, not numbers) argue what links you might want to add to improve the rank of page 3 (the main customer support page). Draw the modified web-graph that represents your solution. Write the transition probability matrix P for the modified web-graph. For your modified web-graph from the previous part, compute the stationary state probabilities for all five web-pages. A user now searches for a keyword that is present in all five web-pages. Which are the two top-ranked web-pages that are returned by the search engine? Network Performance 10-7

8 Exercise Consider a network of N = 4 identical hosts interconnected in a ring, as shown in the figure below. One day, a hacker infects host 1 with a network virus that spreads in the network in the direction indicated by the arrows in the figure The virus spreads as follows: every morning, between 2am and 6am, the virus on each infected host has an infection probability p I = 0.2 of spreading to the host adjacent in the clockwise direction, provided the latter host is not already infected. When the system administrator comes in to work at 9am, he is able to identify the hosts that are infected, and attempts to disinfect (i.e., remove the virus from) hosts in the order in which they were infected. Further, he is able to disinfect at most one host per day, with probability p D = 0.5, before he leaves at 5pm. Since the virus always spreads in the clockwise direction and the system administrator disinfects them in the order of infection, note that at any point of time, the infected hosts are contiguous on the ring. Describe the state of the system at midnight each day, and draw the Markov chain that models the spread of the virus. Is this Markov chain irreducible, i.e., is it possible to reach any state from any other state in a finite number of steps? Justify or show a counter-example. From the previous part, would you say that the Markov chain has a stationary solution for the state probability vector? Network Performance 10-8

9 Exercise 3 (contd.) Now suppose the hacker monitors the network between 2am and 6am each day, and if he finds that no host is infected (i.e. the virus is eradicated from the network), reinfects host 1 with probability p R = 0.2. Draw the Markov chain for this revised system. Is this Markov chain irreducible and aperiodic? Justify. On a random day what is the probability that the network is virusfree? Network Performance 10-9