Homework Assignment 1

Suppose a personal telephone call (i.e., a call involving people who have a personal relationship) is unexpectedly terminated.
1. What informal protocol do you use to reestablish communications?
2. "Young lovers" (e.g., two people who have just started dating) probably behave differently from you. What protocol do "young lovers" use?
3. Does the young lovers protocol (YLP) you described above have any flaws? If so, what are the flaws?
Describe the Drive-Thru Protocol (DTP) used by customers and cashiers at McDonalds (or a similar "fast food restaurant").
A roulette board typically looks something like the following:
1. You can bet on red or black by placing your chip(s) on the appropriately colored square at the bottom of the board. You win if the ball lands in any of the eighteen numbers with that color, in which case a $1 bet returns your $1 plus $1 more. What is the expected value of a $1 bet on red?
2. You can also bet on specific pairs of two numbers by placing your chip(s) on the line between the two numbers. For example, one can bet on 1,2 by placing your chip(s) on the line between the 1 and 2. This is a kind of "inside bet" sometimes called a "split bet". You win if the ball lands in any of the two numbers. Assuming the expected value is the same as for a color bet, how much do you win on a "split bet"?
There are currently several popular (digital) television resolutions in the U.S. A standard-definition TV has 480x640 pixels and uses interlacing (i.e., it receives half of the 480 rows at a time, alternating the odd and even rows). A 720i TV has 720x1280 pixels and uses interlacing. A 720p TV has 720x1280 pixels and receives all of the rows at one time. A 1080i TV has 1080x1960 pixels and uses interlacing. Finally, a 1080p TV has 1080x1960 pixels and receives all of the rows at one time.
Why might a movie streaming service (like Netflix or Vudu) want to know the distribution of television resolutions? How would they use the data?
Suppose people's use of search engines can be described as follows: Probability of Changing Search Engines

Next Search

Bing Google

Bing 0.40 0.60

Previous Search

Google 0.20 0.80

Further, suppose that today all of the population uses Bing (i.e., Bing's market share is 100%).
1. What search engine "will" people use tomorrow? That is, what "will" the market share be for each search engine tomorrow? (Show your work.)
  Hint: This problem involves a Markov chain, as discussed in the reading. Though the reading does not contain a "worked example", you should be able to figure out how to do the calculations. To get started, let $B_1$ and $G_1$ denote the market shares for Bing and Google on day 1. Further, let $P\{\mbox{B to B}\}$ denote the probability that a user "switches" from Bing to Bing (i.e., staying with Bing), $P\{\mbox{B to G}\}$ denote the probability that a user switches from Bing to Google, $P\{\mbox{G to B}\}$ denote the probability that a user switches from Google to Bing, and $P\{\mbox{G to G}\}$ denote the probability that a user stays with Google. You need to calculate $B_2$ and $G_2$ (i.e., the shares on day 2) from $B_1$, $G_1$ and these transition probabilities?
2. What search engine "will" people use the day after tomorrow? (Show your work.)
Suppose your new company's WWW server receives requests at a rate of 20 per minute and can process those requests at a rate of 50 per minute. What is the expected (steady state) time in system for a request?
Suppose your new company's WWW server now receives requests at a rate of 60 per minute and can process those requests at a rate of 50 per minute. What is the expected (steady state) time in system for a request? (Hint: Be careful and thoughtful.)
Suppose your new company has purchased two additional WWW servers (for a total of three) that provide the exact same services. From a performance perspective, should you have three different URLs and let users divide up the load (i.e., send their requests to one of the three servers), or should you have one URL (and an appropriate device) and send requests to the next available server? (Explain carefully!)

		Next Search
		Bing	Google
	Bing	0.40	0.60
Previous Search
	Google	0.20	0.80