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TRANSITION DIAGRAMS A transition diagram is a tree or graph (see 15) whose internal nodes represent different states or situations that may occur during a multistage process. As in a decision tree, each leaf represents a different outcome from the process. Each branch is labeled with the conditional probability that the resulting child event will occur, given that the parent event has occurred. EXAMPLE 10.4 The Game of Craps

In addition to reliable transmission, the data link layer, unlike the physical layer, provides structure to the messages that are being transmitted The physical layer is concerned only with the delivery of the individual bits that make up the messages, and does not care about the structure of those messages The data link layer, on the other hand, often divides messages into frames and adds additional information to the frames to provide synchronization and flow control By synchronization we mean that the sender and receiver must agree on the start and end of each frame, and by flow control we mean that the receiver must have a mechanism for pacing the sender, so that the sender does not flood the receiver with data at a pace faster than the receiver can accept it.

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The game of craps is a dice game played by two players, X and Y. First X tosses the pair of dice. If the sum of the dice is 7 or 11, X wins the game. If the sum is 2, 3, or 12, Y wins. Otherwise, the sum is designated as the point, to be matched by another toss. So if neither player has won on the first toss, then the dice are tossed repeatedly until either the point comes up or a 7 comes up. If a 7 comes up first, Y wins. Otherwise, X wins when the point comes up. The transition diagram shown in Figure 10.6 models the game of craps.

When a pair of dice is tossed, there are 36 possible outcomes (6 outcomes on the first die, and 6 outcomes on the second for each outcome on the first die) Of those 36 outcomes, 1 will produce a sum of 2 (1 + 1), 2 will produce a sum of 3 (1 + 2 or 2 + 1), and 1 will produce a sum of 12 (6 + 6) So there are a total of 4 chances out of 36 of the event 2, 3, or 12 happening That s a probability of 4/36 = 1/9 Similarly, there are 6 ways that a sum of 7 will occur and 2 ways that a sum of 11 will occur, so the probability of the event 7 or 11 is 8/36 = 2/9 The other probabilities on the first level of the tree are computed similarly.

The network layer is concerned with end-to-end communication between computers that are not necessarily directly connected one to another When machines that wish to communicate are not directly connected, they must rely on other machines that are directly connected to relay their messages The network layer is the lowest layer of networking protocol with a focus on exchanging messages by means of relaying communications between intervening computers For example, you have probably passed notes in class If the person to whom you want to pass the note is on the other side of the classroom, you have to rely on the people sitting between you to pass the message It may be the case that you know that your message will be safely exchanged between yourself and the person sitting next to you (ie.

To see how the probabilities are computed for the second level of the tree, consider the case where the point is 4 If the next toss comes up 4, X wins If it comes up 7, Y wins Otherwise, that step is repeated The transition diagram shown in Figure 107 summarizes those three possibilities The probabilities 1/12, 1/6, and 3/4 are computed as shown in the transition diagram in Figure 107: Figure 107 The game of craps P(4) = 3/36 = 1/12 P(7) = 6/36 = 1/3 P(2,3,5,6,8,9,10,11, or 12) = 27/36 = 3/4 So once the point 4 has been established on the first toss, X has a probability of 1/12 of winning on the second toss and a probability of 3/4 of getting to the third toss.

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