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Chapter 1: Problem 9

Consider the discussion in Section \(1.3\) of packet switching versus circuit switching in which an example is provided with a \(1 \mathrm{Mbps}\) link. Users are generating data at a rate of \(100 \mathrm{kbps}\) when busy, but are busy generating data only with probability \(p=0.1\). Suppose that the \(1 \mathrm{Mbps}\) link is replaced by a 1 Gbps link. a. What is \(N\), the maximum number of users that can be supported simultaneously under circuit switching? b. Now consider packet switching and a user population of \(M\) users. Give a formula (in terms of \(p, M, N\) ) for the probability that more than \(N\) users are sending data.

Short Answer

Expert verified
a. 10,000 users. b. Probability: \( 1 - \sum_{k=0}^{N} \binom{M}{k} p^k (1-p)^{M-k} \).

Step by step solution

01

Understand Circuit Switching

In circuit switching, a fixed portion of the link is allocated to each user for the duration of the connection. Given a 1 Gbps link and each user requires 100 kbps, we can calculate the number of users supported by dividing the total link bandwidth by the bandwidth required per user.
02

Calculate Maximum Users for Circuit Switching

Compute the maximum number of users, \( N \), by dividing the total link capacity (1 Gbps) by the bandwidth requirement per user (100 kbps): \[ N = \frac{1,000,000 \, \text{kbps}}{100 \, \text{kbps}} = 10,000 \]Thus, the maximum number of users that can be supported simultaneously is 10,000.
03

Understand Packet Switching

In packet switching, users share the link capacity. Here, each user sends data with probability \( p \). We need to compute the probability that more than \( N \) users are actively sending data at the same time.
04

Express Probability for More Than N Users in Packet Switching

For packet switching, each of the \( M \) users independently sends data with probability \( p \). If more than \( N \) users are sending data, the event count follows a binomial distribution. The probability that more than \( N \) users are simultaneously sending data can be expressed as:\[ P(X > N) = 1 - \sum_{k=0}^{N} \binom{M}{k} p^k (1-p)^{M-k} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Circuit Switching
Circuit switching is a type of network switching where a dedicated communication path is established between two parties for the entire duration of their interaction. This method ensures that the communication channel is reserved exclusively for the two parties, much like having a private phone line.
This means that once the path is set up, it remains connected until the communication is complete.
  • Each user is allocated a fixed portion of the bandwidth.
  • It offers a constant data rate without interruptions.
  • Commonly used for voice communication systems.
The biggest advantage of circuit switching is its reliability and consistent performance.
However, it can be inefficient because resources are tied up regardless of whether data is being actively transmitted, and only a fraction of the paths' potential may be utilized.
Packet Switching
Packet switching is a more flexible and efficient method than circuit switching.
In this approach, data is divided into packets before being sent out on the network. Each packet can take any path available and may arrive out of order, but they are reassembled at the destination.
  • Bandwidth is shared dynamically among users.
  • More efficient use of network resources.
  • Each user sends data in bursts, not requiring a constant connection.
Packet switching is the backbone of most modern digital communications, including the Internet.
Because it allows networks to handle many operations simultaneously, it accommodates a wide range of data and user types.
Bandwidth Allocation
Bandwidth allocation plays a crucial role in both circuit and packet switching.
It determines how network resources are divided among users. In circuit switching, bandwidth is allocated on a per-user basis, providing each user with a fixed rate of data transfer.
  • In circuit switching: Allocation is static, predetermined, and exclusive.
  • In packet switching: Allocation is dynamic and depends on the active users at any time.
  • Packet switching supports more users but may result in variable performance due to shared bandwidth.
By understanding how bandwidth is allocated, network engineers can optimize service quality, ensuring efficient use of available resources.
Probability Distribution
Probability distribution is a statistical function that describes all the possible values and likelihoods that a random variable can take within a given range.
In the context of packet switching, probability distribution helps in predicting how many users are accessing the network at any time.
  • A critical aspect is the binomial distribution used to model the number of users sending data.
  • It considers users' actions as independent events with a probability, \( p \), of being busy with data sending.
  • Useful in calculating the probability of scenarios like more than a certain number of users transmitting data simultaneously.
These distributions enable network engineers to design systems that can handle variances in user behavior, maintaining service quality and efficiency.

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Most popular questions from this chapter

Consider a packet of length \(L\) which begins at end system A and travels over three links to a destination end system. These three links are connected by two packet switches. Let \(d_{\vec{r}} s_{\vec{i}}\), and \(R_{i}\) denote the length, propagation speed, and the transmission rate of link \(i\), for \(i=1,2,3\). The packet switch delays each packet by \(d_{p \text { moc }}\). Assuming no queuing delays, in terms of \(d_{i} s_{i} R_{i}\), ( \(i=1,2,3\) ), and \(L\), what is the total end-to-end delay for the packet? Suppose now the packet is 1,500 bytes, the propagation speed on all three links is \(2.5\). \(10^{8} \mathrm{~m} / \mathrm{s}\), the transmission rates of all three links are \(2 \mathrm{Mbps}\), the packet switch processing delay is \(3 \mathrm{msec}\), the length of the first link is \(5,000 \mathrm{~km}\), the length of the second link is \(4,000 \mathrm{~km}\), and the length of the last link is \(1,000 \mathrm{~km}\). For these values, what is the end-to-end delay?

Consider an application that transmits data at a steady rate (for example, the sender generates an \(N\)-bit unit of data every \(k\) time units, where \(k\) is small and fixed). Also, when such an application starts, it will continue running for a relatively long period of time. Answer the following questions, briefly justifying your answer: a. Would a packet-switched network or a circuit-switched network be more appropriate for this application? Why? b. Suppose that a packet-switched network is used and the only traffic in this network comes from such applications as described above. Furthermore, assume that the sum of the application data rates is less than the capacities of each and every link. Is some form of congestion control needed? Why?

What is the difference between a host and an end system? List several different types of end systems. Is a Web server an end system?

(a) Visit the site www.traceroute.org and perform traceroutes from two different cities in France to the same destination host in the United States. How many links are the same in the two traceroutes? Is the transatlantic link the same? (b) Repeat (a) but this time choose one city in France and another city in Germany. (c) Pick a city in the United States, and perform traceroutes to two hosts, each in a different city in China. How many links are common in the two traceroutes? Do the two traceroutes diverge before reaching China?

Design and describe an application-level protocol to be used between an automatic teller machine and a bank's centralized computer. Your protocol should allow a user's card and password to be verified, the account balance (which is maintained at the centralized computer) to be queried, and anprotocol entities should be able to handle the all-too-common case in which there is not enough money in the account to cover the withdrawal. Specify your protocol by listing the messages exchanged and the action taken by the automatic teller machine or the bank's centralized computer on transmission and receipt of messages. Sketch the operation of your protocol for the case of a simple withdrawal with no errors, using a diagram similar to that in Figure \(1.2\). Explicitly state the assumptions made by your protocol about the underlying end-to-end transport service.
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