1. technology and computing
2. hardware
3. computer networking
4. router

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Contents (page numbers are not final ones)
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CONTENTS
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AUTHOR BIOGRAPHY
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PREFACE TO THE SECOND EDITION
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ACKNOWLEDGMENTS
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PART I: TELECOMMUNICATION NETWORKS
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1. Introduction to Telecommunication Networks
1.1 Milestones in the evolution of telecommunications
1.2 Standardization bodies in telecommunications
1.3 Telecommunication networks: general concepts
1.3.1 Transmissions in telecommunication networks
1.3.2 Switching techniques in telecommunication networks
1.3.3 The ISO/OSI reference model
1.3.4 Traffic engineering: general concepts
1.3.5 Queuing theory in telecommunications
1.4 Transmission media
1.4.1 Copper medium: the twisted pair
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3. IP-based Networks and Future Trends
3.1 Introduction
3.2 The Internet
3.2.1 Introduction to the Internet protocol suite
3.2.2 TCP/IP protocol architecture
3.3 IP (version 4) addressing
3.3.1 IPv4 datagram format
3.3.2 IP subnetting
3.3.3 Public and private IP addresses
3.3.4 Static and dynamic IP addresses
3.3.5 An example of local area network architecture
3.3.6 IP version 6
3.4 Domain structure and IP routing
3.4.1 Routing algorithms
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2. Legacy Digital Networks
2.1 Introduction to digital networks
2.1.1 X.25-based networks
2.1.2 ISDN
2.1.3 Frame relay-based networks
2.2 B-ISDN and ATM technology
2.2.1 ATM protocol stack
2.2.2 Cell format
2.2.3 ATM protocol stack
2.2.4 Traffic classes and ALL layer protocols
2.2.5 ATM switches
2.2.6 ATM switch architectures
2.2.7 Management of traffic
2.2.8 ATM physical later
2.2.9 Internet access through ATM over ADSL
2.3 Bibliographic references
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1.4.2 Copper medium: the coaxial cable
1.4.3 Wireless medium
1.4.4 Optical fibers
1.5 Multiplexing hierarchy
1.5.1 FDM
1.5.2 TDM
1.5.3 The E1 bearer structure
1.6 The classical telephone network
1.6.1 Digital transmissions through POTS
1.6.2 Switching elements in PSTN
1.7 Bibliographic references
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PART II: QUEUING THEORY AND APPLICATIONS TO
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4. Survey on Probability Theory
4.1 The notion of probability and basic properties
4.2 Random variables: basic definitions and properties
4.2.1 Sum of independent random variables
4.2.2 Minimum and maximum of random variables
4.2.3 Comparisons of random variables
4.2.4 Moments of random variables
4.2.5 Random variables in the field of telecommunications
4.3 Transforms of random variables
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3.4.2 Routing implementation issues
3.5 QoS provision in IP networks
3.5.1 IntServ
3.5.2 DiffServ
3.6 IP traffic over ATM networks
3.6.1 The LIS method
3.6.2 The Next Hop Routing Protocol
3.6.3 The integrated approach for IP over ATM
3.7 Multi-Protocol Label Switching technology
3.7.1 Comparison between IP routing and label switching
3.7.2 Operations on labels
3.7.3 MPLS header
3.7.4 MPLS nested domains
3.7.5 MPLS forwarding tables
3.7.6 Protocols for the creation of an LSP
3.7.7 IP/MPLS over ATM
3.7.8 MPLS traffic management
3.7.9 GMPLS technology
3.8 Transport layer
3.8.1 TCP
3.8.2 UDP
3.8.3 Port numbers and sockets
3.9 Next-Generation Networks
3.9.1 NGN architecture
3.9.2 Geographical core/transport networks
3.9.3 Current and future satellite networks
3.10 Future Internet concepts
3.11 Bibliographic references
3.12 Exercises on Part I of the book
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6. M/G/1 Queuing Theory and Applications
6.1 The M/G/1 queuing theory
6.1.1 The M/D/1 case
6.1.2 The M[comp]/G[b]/1 queue with bulk arrivals or bulk service
6.2 M/G/1 system delay distribution in the FIFO case
6.3 Numerical inversion method of the Laplace transform
6.4 Impact of the service time distribution on M/G/1 queue
6.5 M/G/1 theory with state-dependent arrival process
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5. Markov Chains and Queuing Theory
5.1 Queues and stochastic processes
5.1.1 Compound arrival processes and implications
5.2 Poisson arrival process
5.2.1 Sum of independent Poisson processes
5.2.2 Random splitting of a Poisson process
5.2.3 Compound Poisson processes
5.3 Birth-death Markov chains
5.4 Notations for queuing systems
5.5 Little theorem and insensitivity property
5.5.1 Proof of the Little theorem
5.6 M/M/1 queue analysis
5.7 M/M/1/K queue analysis
5.7.1 PASTA property
5.8 M/M/S queue analysis
5.9 M/M/S/S queue analysis
5.10 The M/M/∞ queue analysis
5.11 Distribution of the queuing delays in the FIFO case
5.11.1 M/M/1 case
5.11.2 M/M/S case
5.12 Erlang-B generalization for non-Poisson arrivals
5.12.1 The traffic types in the M/M/S/S queue
5.12.2 Blocking probability for non-Poisson arrivals
5.13 Exercises
5.14 Bibliographic references
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4.3.1 The probability generating function
4.3.2 The characteristic function of a pdf
4.3.3 The Laplace transform of a pdf
4.4 Methods for the generation of random variables
4.4.1 Method of the inverse of the distribution function
4.4.2 Method of the transform
4.5 Exercises
4.6 Bibliographic references
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7. Local Area Networks and Analysisis
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7.1 Introduction
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7.1.1 Standards for local area networks
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7.2 Contention-based MAC protocols
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7.2.1 Aloha protocol
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7.2.2 Slotted-Aloha protocol
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7.2.3 The Aloha protocol with ideal capture effect
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7.2.4 Alternative analytical approaches for Aloha protocols
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7.3.5 Packet Reservation Multiple Access (PRMA) protocol
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7.3.6 Efficiency comparison: CSMA/CD vs. token protocols
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7.4 Fixed assignment protocols
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7.4.1 Frequency Division Multiple Access (FDMA)
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7.4.2 Time Division Multiple Access (TDMA)
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7.4.3 Code Division Multiple Access (CDMA)
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7.4.4 Orthogonal Frequency Division Multiple Access (OFDMA)
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7.4.5 Resource reuse in cellular systems
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6.6 Applications of the M/G/1 analysis to ATM
6.7 A survey of advanced M/G/1 cases
6.8 Different imbedding options for the M/G/1 theory
6.8.1 Imbedding at slot end of the output line
6.8.2 Imbedding at transmission end of low-priority cells
6.8.3 Imbedding at transmission end of low-priority messages
6.9 Continuous-time M/G/1 queue with ‘geometric’ messages
6.9.1 Imbedding at packet transmission completion
6.9.2 Imbedding at message transmission completion
6.10 M/G/1 theory with differentiated service times
6.10.1 The differentiated theory applied to compound arrivals
6.11 M/D[b]/1 theory with batched service
6.12 Exercises
6.13 Bibliographic references
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8. Networks of Queues
8.1 Introduction
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8.1.1 Traffic rate equations
8.1.2 The Little theorem applied to the whole network
8.2 Tandem queues and the Burke theorem
8.3 The Jackson theorem
8.3.1 Analysis of a queue with feedback
8.4 Traffic matrices
8.5 Network planning issues
8.6 Exercises
8.7 Bibliographic references
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EXTRACT FROM PART I
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IP-BASED NETWORKS AND FUTURE TRENDS
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3. IP-BASED NETWORKS AND FUTURE TRENDS
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3.1 Introduction
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3.2 The Internet
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A growing number of people are using the Internet, the network of the
networks; this is also evident from the different bandwidth-intensive
applications supported by Internet and by the considerable number of
Internet books, video, etc. that have become available during these years.
The widespread diffusion of social networks (Facebook, YouTube, etc.),
peer-to-peer traffic, and cloud applications have further contributed to the
impressive growth in the Internet use. IP traffic has globally grown eight
times in the period 2008-2012 (five years) and is expected to increase
threefold in the next three years. The annual global IP traffic will surpass the
Zettabyte (i.e., 1021 bytes) threshold by the end of 2016 [1]. This Chapter
focuses on the protocols and the network technologies to support Internet
traffic.
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J. C. R. Licklider of the Massachusetts Institute of Technology (MIT)
proposed a global network of computers in 1962 and moved to the Defense
Advanced Research Projects Agency (DARPA) to lead a project to
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interconnect Department of Defense (DoD) sites in the United States of
America. L. Kleinrock of MIT (and, later, University of California, Los
Angeles, UCLA) developed the theory of packet-switching, which is at the
basis of Internet traffic. In 1965, L. Roberts of MIT connected a
Massachusetts computer with a California computer by means of a dial-up
telephone line. He showed the feasibility of wide area networking, but also
that the telephone circuit-switching was inadequate for this traffic, thus
confirming the importance of the Kleinrock packet-switching theory. These
pioneers (as well as other people) are the actual founders of the Internet. The
Internet, then known as ARPANET, was brought online in 1969, initially
connecting four major sites (computers), under a contract held by the
renamed Advanced Research Projects Agency (ARPA).
Once the initial sites were installed, representatives from each site met
together to solve the technical problems concerning the interconnection of
hosts by means of protocols. A working group, called Network Working
Group (NWG), was in charge of defining the first ‘rules’ (i.e., protocols) of
the network. The open approach adopted by the first NWG meeting
continued in a more formalized way by using meeting notes, called Request
For Comments (RFC). These documents are intended to keep members
updated on the status of several things concerning Internet protocol. They
were also used to receive responses from researchers.
The Internet was designed to provide a communication network able to
work even if some sites are destroyed. The early Internet was used by
computer experts, engineers, scientists, and librarians. There were no
personal computers and no massive use in those days. Different ‘initial’
applications and protocols were conceived to exploit ARPANET. E-mail
was adopted for ARPANET in 1972. The telnet protocol, allowing us to log
on a remote computer, was defined in 1972 [2]. The FTP protocol, enabling
file transfers between Internet sites, was published as RFC 354 in 1972
[3],[4] and from then further RFCs were made available to update the
characteristic of the FTP protocol. RFCs are today the method used to
standardize every aspect of the Internet; they are freely accessible in the
ASCII format through the Internet Engineering Task Force (IETF) Web site
[5]. RFCs are approved after a very strong review process. IETF is an open,
all-volunteer organization (started its activities in 1983), with no formal
membership nor membership requirements. It is divided into a large number
of working groups, each dealing with a specific Internet issue.
In 1974, a new suite of protocols was proposed and implemented in the
ARPANET, based on the Transmission Control Protocol (TCP) for end-toend communications. In 1978, a new Internet design approach was
conceived with the division of tasks between two protocols:
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 The new Internet Protocol (IP) for routing packets and device-to-device
communications (i.e., host-to-gateway or gateway-to-gateway);
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Since TCP and IP were originally conceived as working in tandem, this
protocol suite is commonly denoted as TCP/IP. The original versions of both
TCP and IP were written in 1981 [6],[7].
As long as the number of Internet sites was small, it was easy to keep
track of the resources of interest that were available. But as more and more
universities and organizations connected, the Internet became harder to
track. There was the need for tools to index the available resources. Starting
from 1989, significant efforts were pursued in this direction. In particular, T.
Berners-Lee and others at the European Laboratory for Particle Physics (i.e.,
CERN) laid the basis to share documents using browsers in a multi-platform
environment. In particular, three new technologies were incorporated into
his proposal: (i) the HyperText Markup Language (HTML) used to write
documents (also named ‘pages’) for the Internet; (ii) the HyperText Transfer
Protocol (HTTP), an application layer protocol to transmit documents in
HTML format; (iii) a browser client software program to receive and
interpret HTML documents and to display the results. His proposal was
based on hypertext, i.e., a system of embedding links, that is Internet
addresses, in the text to refer to other documents Internet documents.
In 1991, the World Wide Web was born because the first really-friendly
interface to the Internet was developed at the University of Minnesota; it was
named ‘gopher’, after the University of Minnesota mascot, the golden
gopher. In 1993, the development of the graphical browser, called Mosaic,
by M. Andreessen and his team at the National Center For Supercomputing
Applications (NCSA), a research institute at the University of Illinois, gave a
strong boost to the Web. Starting from this browser, new ones rapidly spread
and made the Web a worldwide success. Further developments to the Web
were represented by the Web search engines, as already discussed in Chapter
1 (Section 1.1).
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 The TCP protocol for reliable, end-to-end communications.
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3.2.1 Introduction to the Internet protocol suite
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The goal of TCP/IP was to interconnect different physical networks to
form what appears to the user as a universal network. Such a set of
interconnected networks is called an Internet [8]-[11]. Communication
services are provided by Internet protocols, which operate between the link
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layer and the application one. The architecture of the physical networks is
hidden to the users.
To be able to interconnect two networks, we need a ‘computer’ that is
attached to both networks and that can forward packets from one network to
another and vice versa; this device, called router, has two important
characteristics:
 From the network standpoint, a router is a normal host.
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 From the user standpoint, routers are invisible; the user sees only a larger
internetwork.
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Each host has an address assigned, the IP address, to identify it in the
Internet. When a host has multiple network adapters, each adapter has a
separate IP address.
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Although there is no universal agreement on how to describe TCP/IP
with a layered model, it is generally regarded as being composed of fewer
layers than the seven layers of the classical OSI model. Most descriptions of
TCP/IP define three to five functional levels in the protocol architecture
[12]; a four-layer TCP/IP model is shown in Figure 3.1.
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Figure 3-1. Simplified Internet protocol suite. The acronyms in this figure will be described
along this Chapter; this figure will be taken as a reference.
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As in the OSI model, data are passed down through the stack when they
are sent to the network, and passed up through the stack when they are
received from the network. Each layer treats the information it receives from
the layer above as data and adds its own header in front of that information
Thunderbird
Silverlight
Fire Fox
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HTTP
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Media players
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Waist of the Internet protocol
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to ensure the proper management of these data. The operation to add the
header (containing control information) is called encapsulation.
The network layer is the lowest layer of the TCP/IP protocol hierarchy.
The protocols of this layer provide the means to route data to other network
devices. Unlike higher-level protocols, network layer protocols must know
the details of the underlying network (its packet structure, addressing, etc.)
to correctly format the data being transmitted to comply with local network
constraints.
The Internet protocol stack has a layered architecture resembling an
hourglass (see Figure 3.2): the reason for this denomination of the Internet
protocol model is that there are many PHY and MAC layer protocols and
there are many application and transport layer protocols, while on the waist
of the hourglass at the network layer there are very few protocols, basically
the IP protocol. The hourglass model expresses the concept that the IP
protocol is the glue, the basic building block of the Internet. The protocols of
the waist are those to which we are referring mainly when talking about the
Internet ‘ossification’; this is seen today mostly as a limit to the flexibility
and security, because all information is forced through a small set of midlayer protocols.
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IEEE 802.11
PHY layer
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IEEE 802.3
Coaxial cable
Optical fiber
Wireless link
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Figure 3-2. The Internet protocol stack and the hourglass model (note that not all the
protocols have been shown at the different layers, but just some of them).
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The Internet Protocol (IP) originally defined in RFC 791 [6] is the heart
of the Internet protocol suite and the most important protocol of the network
layer. IP provides the basic packet delivery service for the networks. All the
higher-layer protocols (and the related data flows) use IP to deliver data. Its
functions include:
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 Defining the IP packet (i.e., a datagram, the basic transmission unit in the
Internet),
 Defining the Internet addressing scheme,
 Moving data between network and transport layers,
 Routing datagrams to remote hosts,
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IP is an unreliable protocol, because it does not perform error detection
and recovery for transmitted data. This does not mean that we cannot rely on
this protocol. In fact, IP can be relied upon to deliver data accurately to the
destination, but it does not check whether data are received correctly or not.
Higher-layer protocols of the Internet protocol stack are in charge of
providing error detection and recovery, if required.
The protocol layer just above the network one is the host-to-host
transport layer. This name is commonly shortened to transport layer. The
two most important protocols at the transport layer are Transmission Control
Protocol (TCP) and User Datagram Protocol (UDP). TCP provides a
reliable, connection-oriented, byte-stream data delivery service; error
detection and error recovery (through retransmissions) are end-to-end
performed. UDP provides a low-overhead, unreliable, connectionless
datagram delivery service. Both protocols exchange data between
application and network layers. Applications programmers can choose the
service that is most appropriate for their specific needs.
UDP gives application programs direct access to a datagram delivery
service, like the delivery service provided by IP. This allows applications to
exchange messages over the network with a minimum protocol overhead.
Applications requiring the transport protocol to provide reliable data
delivery use TCP, since it verifies that data are accurately delivered across
the network and in the right sequence.
The application layer is at the top level of the TCP/IP protocol
architecture. This layer includes all processes that use transport protocols to
deliver data. There are many application layer protocols. Most of them
provide user services; new services are constantly being added at this layer.
The most popular and implemented application layer protocols are:
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 Performing fragmentation and re-assembly of datagrams.
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 Telnet: The network terminal protocol, which allows us to remotely log
on hosts spread in network.
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 FTP: The File Transfer Protocol used for file transfer.
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 SMTP: The Simple Mail Transfer Protocol, which delivers electronic
mail.
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 HTTP: The Hypertext Transfer Protocol, delivering Web pages over the
network.
 Domain Name System (DNS): This is a service to map IP (numeric)
addresses to the names assigned to network devices.
 Network File System (NFS): This protocol permits to share files among
various hosts in the network.
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 Finally, the Open Shortest Path First (OSPF), which is a layer 3 routing
protocol, includes a transfer protocol for the exchange of routing
information among routers and as such (even with some debate) can also
be considered as an application layer protocol.
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IP addresses are used to route datagrams in the network and to allow their
correct delivery to destination. An IP version 4 (IPv4) address is formed of
32 bits, written by dividing the bits in groups of 8 and taking the
corresponding decimal number. Each of these numbers is written separated
by a dot (i.e., dotted-decimal notation) and can range from 0 to 255. For
example, 1.160.10.240 could be an IP address. The specification of IP
addresses is contained in RFC 1166 [13]. An IP address can be divided in a
pair of numbers (the length of these fields depend on the IP address class):
M
ed
3.3 IP (version 4) addressing
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IP address = <network identifier> + <host identifier> .
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There are five classes of IP addresses, as described in Figure 3.3. Classes
are introduced to divide the space of IP addresses in groups of a limited
number of addresses (i.e., that can support a limited number of hosts). This
is carried out for an efficient use of IP addresses and takes the name of
‘classful’ IPv4 addressing.
166
QUEUING THEORY AND TELECOMMUNICATIONS
0
8
16
24
31
0
Class A
network
host
10
Class B
network
host
110
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Class C
host
ew
network
1110
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Class D
M
Class E
ed
multicast
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1111
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reserved
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Figure 3-3. IPv4 address classes.
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For classes A, B, and C, the address of a network has all the host bits
equal to ‘0’. Whereas, the broadcast address of a network is characterized by
all the host bits equal to ‘1’. The number of hosts addressable in a network is
therefore related to the number of available combinations for the bits of the
host field minus two addresses for network and multicast purposes.
om
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Class A:
 First bit set to ‘0’ plus 7 network bits and 24 host bits
 Initial byte ranging from 0 to 127
 Totally, 128 (= 27) Class A network addresses are available (0 and 127
network addresses are reserved)
 16,777,214 (= 2242) hosts can be addressed in each Class A network
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Class B:
 First two bits set to ‘10’, plus 14 network bits and 16 host bits
 Initial byte ranging from 128 to 191
 Totally, 16,384 (= 214) Class B network addresses
 65,534 (= 2162) hosts can be addressed in each Class B network
Chapter 3
167
Class C:
 First three bits set to ‘110’ plus 21 network bits and 8 host bits
 Initial byte ranging from 192 to 223
 Totally, 2,097,152 (= 221) Class C network addresses
 254 (= 282) hosts can be addressed in each Class C network
N
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Yo
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Class D:
 First four bits set to ‘1110’ plus 28 multicast address bits
 Initial byte ranging from 224 to 247
 Class D addresses are used for multicast flows
us
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A router receiving an IP packet extracts its IP destination address, which
is classified by examining its first bits. Once the IP address class has been
determined, the IP address can be broken down into network and host bits.
Intermediate routers ignore host bits and only need to match network bits
within their routing table to route the IP packet along the correct path in the
network. Once a packet reaches its target network, its host field is examined
for the final local delivery.
IPv4 addressing space is limited: this is a significant problem because of
the continued spread of the Internet. In order to address this issue, possible
approaches are: IP subnetting (see Section 3.3.2), the use of private IP
addresses (see Section 3.3.3), and the new IP version 6 (see Section 3.3.6).
ne
ss
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Class E:
 First four bits set to ‘1111’ plus 28 reserved address bits
 Initial byte ranging from 248 to 255
 This address class is reserved for experimental use.
er
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3.3.1 IPv4 datagram format
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Data transmitted over the Internet using IP addresses are organized in
variable-length packets, called IP datagrams. Let us consider here the IPv4
datagram format, defined in RFC 791 [6]. An IPv4 datagram is divided into
two parts: the header and the payload. The header contains addressing and
control fields, while the payload carries the actual data to be sent. Even
though IP is a relatively-simple, connectionless, “unreliable” protocol, the
IPv4 header carries some control information that makes it quite long. It is
minimum 20 byte long and can be even longer with the options. The IP
datagram format is shown in Figure 3.4, where each row corresponds to four
168
QUEUING THEORY AND TELECOMMUNICATIONS
bytes (i.e., a word of 32 bits). The meaning of the different header fields is
explained below.
32 bits
4 bits
Type of Service
Time To Live
Flags
Fragment Offset
Protocol
rk
20 bytes
Identification
Total Length
Header Checksum
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Version H.length
ew
Source Address
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Destination Address
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Options
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M
Padding
lity
MF
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ghput
DF
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Reserved
Delay Throu- Reliabi- Reserved
Precedence
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Data
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Figure 3-4. IPv4 datagram format.
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 Version (4 bits): Identifies the IP version of the datagram. For IPv4,
obviously this field contains the number 4. The purpose of this field is to
ensure compatibility among different devices, which may be running
different IP versions. In general, a device running an older IP version will
reject datagrams created by newer implementations.
or
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 IHL, Internet Header Length (4 bits): Specifies the length of the IP
header in 32-bit words. This length includes any optional field and
padding. The normal value of this field when no options are used is 5
(i.e., 5 words of 32 bits, corresponding to 20 bytes).
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 ToS, Type of Service (8 bits): A field carrying information to support
quality of service features, such as prioritized delivery of IP datagrams.
The ToS byte is divided into four sub-fields, as shown in Figure 3.4:
- The first three bits are used for the precedence field (value of 0 for a
normal priority, up to a value of 7 for control messages);
Chapter 3
-
169
The delay bit specifies whether a low delay is required for the
datagram transfer (D = 1) or if the delay is not critical (D = 0);
The throughput bit T = 1 when a high throughput is needed, instead
T = 0 if the throughput is not a critical issue;
The reliability bit R = 1 when a high reliability is required, instead R
= 0 if reliability is not needed.
The last two bits are unused.
-
+B
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 TL, Total Length (16 bits): This field specifies the total length of the IP
datagram in bytes. Since this field is 16 bits wide, the maximum length of
an IP datagram is 65,535 bytes (typically, they are much smaller to avoid
fragmentation due to MAC layer constraints). The most common IP
packet length is 1500 bytes to be compatible with the maximum Ethernet
payload size.
rk
The ToS byte has never been used as originally defined. A great deal of
experimental, research and deployment work has focused on how to use
these 8 bits (ToS field), which have been redefined by IETF for use by
Differentiated Services (DiffServ) and by Explicit Congestion
Notification (ECN); see also the following Section 3.5 and sub-Sections
3.7.8.2, 3.7.8.3.
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 Identification (16 bits): This field contains a 16-bit value, which is
common to each fragment belonging to the same message. It is filled in
for originally-unfragmented datagrams, in case they have to be
fragmented at an intermediate router along the path. Such field is used by
the recipient to reassemble messages in order to avoid an accidental
mixing of fragments coming from different messages, since the IP
datagrams can be received out of order.
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 Flags (3 bits): It contains three control flags, but only two of them are
used: Do not Fragment (DF) flag and More Fragments (MF) flag. If DF =
1, the datagram should not be fragmented. MF = 0 denotes the last
fragment of a datagram.
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 Fragment Offset (13 bits): When a message is fragmented, this field
specifies the position of the current data fragment in the overall message.
It is specified in units of 8 bytes (64 bits). The first fragment has an offset
of 0.
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 TTL, Time To Live (8 bits): This field specifies how long a datagram is
allowed to “live” in the network in terms of router hops. Each router
decrements the TTL value of one, before transmitting the related
datagram. If TTL becomes zero, the datagram is not forwarded, but
is
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EXTRACT FROM PART II
Chapter 6
ss
M
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M/G/1 QUEUING THEORY AND APPLICATIONS
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6. M/G/1 QUEUING THEORY AND APPLICATIONS
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6.1 The M/G/1 queuing theory
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The M/G/1 theory is a powerful tool, generalizing the solution of
Markovian queues to the case of general service time distributions. There are
many applications of the M/G/1 theory in the field of telecommunications;
for instance, it can be used to study the queuing of fixed-size packets to be
transmitted on a given link (i.e., M/D/1 case). Moreover, this theory yields
results, which are compatible with the M/M/1 theory, based on birth-death
Markov chains.
In the M/G/1 theory, the arrival process is Poisson with mean arrival rate
, but, in general, the service time is not exponentially distributed. Hence,
the service process has a certain memory: if there is a request in service at a
given instant, its residual service time has a distribution depending on the
time elapsed since the beginning of its service. Let us refer to a generic
instant t. The system is described by a two-dimensional state S(t),
characterized as follows:
tf
 Number of requests in the system at instant t, n(t);
Th
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sa
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no
 Elapsed time from the beginning of the service of the currently-served
request, (t). Note that in the Markovian M/M/1 case, the pdf of the
residual service time does not depend on  (t) because of the memoryless
property of the exponential distribution.
454
QUEUING THEORY AND TELECOMMUNICATIONS
Hence, S(t) = {n(t),  (t)}. In order to characterize these queues, we study
their behaviors at specific time instants i where we obtain a monodimensional simplification of state S(i). The M/G/1 queue is studied at
specific imbedding instants, where we obtain again a Markovian system; this
is a so-called imbedded Markov chain [1],[2]. Different alternatives are
available to select instants i. It is not requested that instants i be equally
spaced in time. Typical choices for i instants are:
1. Service completion instants;
Yo
rk
2. Arrival instants {as done for G/M/1 queues to study the waiting time
[3]};
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ed
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It makes a difference how we select the imbedding points: different
imbedding options in general do not allow to achieve the same results. In
this study, let us refer to the first type of imbedding points: let i denote the
service completion instant of the i-th request arrived at the queue. We have
that  (i)  0 i, since at instant i a request has completed its service and
no new request has yet started its service. Hence, at these instants i the state
becomes mono-dimensional: S(i)  n(i) = ni, where ni denotes the number
of requests in the queue soon after the service completion of the i-th request.
Let ai denote the number of requests arrived at the queue during the service
time of the i-th request (ending at instant i). Note that ni and ai random
variables are also used with different imbedding points, but the distributions
of both ni and ai depend on the imbedding instants selected.
ia
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3. Regularly-spaced instants for cases with service based on time slots.
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New arrival
e.
nj = 0
al
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ai+ 1
i+2
ci
i+1
er
i
©
Depa rture instants
ni+1
ni  0
j+1
time
Case (b)
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Case (a)
j
aj+ 1
tf
Figure 6-1. Time diagram of service completion events and new arrivals.
Th
is
sa
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Let us refer to the situation depicted in Figure 6.1. If ni  0 [i.e., case (a)
in Figure 6.1], the following balance is valid at the next service completion
instant: ni+1 = ni – 1 + ai+1. Instead, if ni = 0 [i.e., case (b) in Figure 6.1], we
455
Chapter 6
have to wait for the next arrival, which is served immediately, so that at the
next completion instant i+1 the system contains only the arrivals occurred
during the service time of the last request; this number is still represented by
variable ai+1. Hence, we have: ni+1 = ai+1.
Let us recall that the indicator (Heaviside) function is defined as: I(x) = 1
for x > 0; I(x) = 0 for x ≤ 0. By means of function I(x), we can represent ni+1
with an expression, which is valid for both ni  0 and ni = 0, as shown below
where we have also provided alternative notations adopted in the literature:
ni 1  ni  I ni   ai 1  maxni  1, 0  ai 1  ni  1  ai 1 (6.1)
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The difference equation (6.1) describes the behavior of the M/G/1 queue
at imbedding instants. Since the variables at the instant i+1 depend only on
the variables at instant i , equation (6.1) characterizes the M/G/1 system by
means of a discrete-time Markov chain (or, more correctly, an imbedded
Markov chain). Note that the method of imbedding instants is quite general
and has also been applied to study G/M/1 queues (general iid interarrival
times; exponentially-distributed service times; one server). In this case, the
chain is imbedded at the arrival instants of the input process [3].
Let G(t) denote the PDF of the service time, X: G(t) = Prob{X  t}. Let
g(t) denote the pdf of the service time: g(t) = dG(t)/dt. The mean service
time is indicated as E[X].
Let us assume that the M/G/1 queue admits a steady state with Pn
denoting the probability (at regime) to have n requests in the queue at
imbedding instants:
rk

i 
©
i 
Sp
r
lim Pni 1  lim Pni  Pn
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e.
Hence, we have:
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i 
er
i 
al
lim E ni 1   lim E ni   En, where E n denotes the regime value.
om
m
By taking the expected values of both sides of (6.1), we have:
tf
or
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E ni 1   E ni   E I ni   E ai 1 
(6.2)
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Hence, if we take the limit of both sides for i  , we obtain regime
values as:
456
QUEUING THEORY AND TELECOMMUNICATIONS
E n   E n  E I n   E a   E a   E I n 
We can evaluate E[I(n)] by means of the state probability distribution as:


n 0
n 1
E I n    I n Pn   Pn  1  P0
def
(6.3)
Yo
P0  1  E a 
rk
By using (6.3) and the expression at regime corresponding to (6.2), we
can obtain probability P0 as:

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M
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Let us recall that on the basis of the PASTA property P0 (or 1P0) is the
probability that a new arrival finds an empty (or a non-empty) M/G/1 queue.
The mean number of arrivals during the service time of a request, E[a],
can be obtained as the mean number of Poisson arrivals conditioned on a
given service time X = t, E[a | X = t] = t, and, then, by removing the
conditioning by means of the pdf g(t) of X:
ia
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(6.4)

(6.5)
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0
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E a    E a | X  t g t dt    tg t dt  E X 
ci
er
i
m
z ni 1  z ni  I ni  ai 1 ,
al
us
e.
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Sp
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From (6.5) we note that E[a] corresponds to the traffic intensity
expressed in Erlangs, . The M/G/1 queue is stable if P0 > 0 or, equivalently
on the basis of (6.4) and (6.5), if  < 1 Erlang.
We focus here on the solution of the difference equation (6.1) in the z
domain by means of PGFs. First of all, we consider the equality obtained by
taking the exponentiation with base z on both sides of (6.1) for any index i
value:
Th
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tf
or
c
om
Then, we multiply both sides by the joint distribution Prob{ni+1=h, ni=k,
ai+1=j} and we sum over h, k, j. The summations on k and j can be removed
on the left side; moreover, the summation on h can be removed on the right
side. Details are as follows:
457
Chapter 6
left side :
 z
h
k
ni 1
Pni1 , ni , ai 1   z ni1  Pni1 , ni , ai1   z ni 1 Pni1
h
j
k
j
h
right side :
 z
h
k
ni  I  ni  a i 1
Pni1 , ni , ai1   z ni  I ni  ai 1  Pni1 , ni , ai1 
k
j
j
h
  z ni  I ni  ai1 Pni , ai1
rk
j
Yo
k
In order to solve the imbedded Markov chain we make the following
assumptions:
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2. Arrival process independent of the number of requests in the queue: ni
and ai+1 are independent variables (2).
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e.
j
(6.7)
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k
om
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al
h
Pni 1   z ni  I ni  Pni  z ai 1 Pai 1
Sp
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The above assumptions are quite general and can be met by many
systems. In particular, they are verified in the special case of Poisson arrivals
and general service time, which are both independent of the queue state.
Under the previous assumption #2, Prob{ni=k, ai+1=j} = Prob{ni=k} 
Prob{ai+1=j}. Therefore, the left side in (6.6) can be rewritten as:
ni 1
1
is
sa
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no
tf
or
c
In the case of continuous-time processes, we have to consider Poisson (or compound
Poisson) processes. Instead, in the case of discrete-time processes, we have to consider
Bernoulli or Binomial arrival processes on a slot basis (in this respect, symbol M used to
denote the arrival process at the queue has to be considered in a wider sense and as such it
will be substituted by ‘M’).
2
Note that it is also possible to solve (6.6) by removing such assumption: we obtain a
recursive formula to determine the state probabilities Pn at imbedding instants. More details
are provided in the following Section 6.5.
Th
N
us
i
1. Memoryless arrival process (1);
z
ia
(6.6)
j
ed
k
M
h
Pni 1   z ni  I ni  ai 1 Pni ,ai 1
ss
ni 1
ne
z
ew
By equating the two expressions above, we obtain:
458
QUEUING THEORY AND TELECOMMUNICATIONS
Let P(z) denote the PGF at regime of the state probability distribution at
the imbedding instants. Let A(z) denote the PGF at regime of the number of
arrivals during the service time of a request. Moreover, note that:
n i  I  ni 
k 0


Pni  P0i   z ni 1Pni  P0i  z 1  z ni Pni 
k 1
k 1
(6.8)


 P0i  z  z ni Pni  P0i 

k 0
rk
1

Yo

z
ss
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us
i
z  1A( z )
+B
(6.10)
z  A( z )
nc
e
P z z  A( z )  P0  z  1A( z )  P z   P0
sa
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The PGF of the state probability distribution in (6.10) represents a quite
general formula, which can be applied to all the imbedded Markov chains
fulfilling (6.1) and the previous assumptions #1 and #2. In particular, the
PGF in (6.10) is valid for any service policy, provided that the conditions of
the insensitivity property are fulfilled (see Section 5.5).
Since P0 is determined from (6.4), the PGF of the state probability
distribution depends only on the PGF A(z), which, in turn, depends on both
the arrival process and the imbedding instants. The state probability
distribution can be obtained by inverting (6.10). This is not an easy task,
since there may not be a closed form solution: the PGF in (6.10) typically
does not correspond to a classical distribution. A possible approach to invert
P(z) is to adopt the method of the Taylor series expansion centered at z = 0,
as show in Section 4.3.1: the coefficients of the expansion represent the state
probability distribution. This approach requires a numerical method based on
the Matlab® symbolic toolbox. Another method to invert (6.10) is described
in Section 6.5.
By means of (6.4), the stability condition P0 > 0, can be expressed as
follows, noticing that E[a] = A'(z=1):
is
N
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(6.9)
Finally, we can solve P(z) in (6.9):
Th
ed

P z   P0  z 1 P z   P0  A( z )
M

ew
By considering the situation at regime (i.e., for i  ), we can eliminate
subscript i in equations (6.7) and (6.8). Then, we substitute (6.8) in equation
(6.7) where we use the PGFs P(z) and A(z):
459
Chapter 6
P0  1  A' (1)  0 
A' (1)  1 Erlang
Under the assumption of Poisson arrivals and imbedding at the service
completion instants, A(z) can be derived considering the PGF of the number
of arrivals in a given interval X = t, A(z | X = t) = et(z1) and then removing
the conditioning by means of the general pdf of the service time X, g(t):

A( z )   et  z 1 g t dt  s    z  1
rk
(6.11)
Yo
0
  '    z  1 z 1    ' 0   E  X 
M
(6.12)
ss
z 1
 
2
+B
(6.13)
ie
2
Sc
  ' ' 0    E X
2
nc
e
d 2 A( z )
d
  '    z  1  2 ' '   z  1 z 1
2
dz z 1 dz
z 1
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dA( z )
dz
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where (s) denotes the Laplace transform of the pdf g(t). On the basis of
the expression of A(z) in (6.11) we can evaluate A'(1) and A''(1) as follows:
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Note that (6.12) is equivalent to (6.5).
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The PGF in (6.10) has a singularity at z = 1 (a removable singularity
according to the Abel theorem), which causes some problems for both the
normalization condition and the derivation of the moments of the
distribution. Of course, we can use the Hôpital theorem to prove that P(z =
1) = 1 (normalization). Moreover, the moments of the state probability
distribution can be obtained by taking subsequent derivatives on both sides
of the leftmost expression in (6.10). With the first derivative, we have:
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P '  z z  A( z )  P z 1  A' ( z )  P0 A( z )  P0  z  1A' ( z )
(6.14)
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If we evaluate (6.14) at z = 1, we obtain: P0 = 1 – A'(1); this is the same
expression as in (6.4).
If we derive again (6.14) on both sides with respect to z we obtain:
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QUEUING THEORY AND TELECOMMUNICATIONS
P ' '  z z  A( z )  2 P'  z 1  A' ( z )  P z  A' ' ( z ) 
 2 P0 A' ( z )  P0  z  1A' ' ( z )
(6.15)
If we evaluate (6.15) at z = 1 and we use (6.4) for P0, we have:
2 P' 11  A' (1)  A' ' ( z )  2 P0 A' (1)
(6.16)
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The mean number of requests in the queue at imbedding instants, N,
depends on the first two derivatives of A(z) computed at z = 1. Let us recall
that the stability condition is met if 1 – A'(1) > 0, i.e., traffic intensity is
lower than 1 Erlang. Note that (6.16) is a general expression, which could
also be applied to memoryless arrival processes different from the Poisson
one provided that the imbedded system is characterized by (6.1). If we refer
to Poisson arrivals (i.e., the classical M/G/1 queue) and imbedding points at
service completion instants, we can substitute (6.12) and (6.13) in (6.16),
thus yielding:
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A' ' ( z )
21  A' (1)
ne
 N  P ' 1  A' (1) 
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2 E X 2 
N  E  X  
21  E  X 
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(6.17)
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E X 2 
21  E  X 
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
 E X  
(6.18)
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We can derive the mean delay to cross the queuing system, T, by
applying the Little theorem to (6.16) for the more general case or to (6.17)
for the Poisson arrival case. In particular, referring to (6.17), we obtain the
well-known Pollaczek-Khinchin formula for the mean queuing delay
[1],[2],[4]:
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Note that in (6.18) the first contribution to the mean delay is E[X], i.e.,
the
mean
service
time;
instead,
the
second
contribution
E[X2]/{2[1E[X]]} represents the mean waiting time. The mean queuing
delay is related to the second moment of the service time distribution. In
particular, the mean waiting time increases with the variance of the service
time, considering a certain fixed mean service time. If the traffic intensity of
the input arrival process, E[X], tends to 1 Erlang (stability limit), the mean
delay tends to infinity.
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In the case of exponentially-distributed service times (mean rate ), the
above formulas (6.17) and (6.18) yield the same expressions of the M/M/1
queue as shown in Chapter 5. In this case, we have (s) = /(+s), E[X] =
1/ and E[X2] = 2/As shown in [1],[2], this result permits to conjecture
that the state probability distribution obtained for an M/G/1 system at the
imbedding instants is also valid in general for the continuous-time chain.
These considerations can be supported more formally introducing the
Kleinrock principle [1]: for queuing systems where the state changes at most
by +1 or −1 (we refer here to actual changes in the number of requests in the
queue and not to what happens only at imbedding instants), the state
distribution as seen by an arriving customer is the same as that seen by a
departing customer. Hence, the state probability distribution at departure
instants is equal to the state probability distribution at arrival instants.
Moreover, by applying the PASTA property (in the Poisson arrival case), the
state probability distribution at arrival instants is also valid at generic
instants (random observer). Hence, by means of both the Kleinrock principle
and the PASTA property, we can conclude that the state probability
distribution at service completion instants coincides with the distribution of
the continuous-time system (random observer). As for discrete-time
(Markov) systems, the equivalent BASTA property can be adopted to
determine the probability that an arrival finds the queue in a certain state by
means of the corresponding state probability.
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Chapter 6
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6.1.1 The M/D/1 case
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In this system the requests have a fixed, constant service time, x. This is
for instance the case of the transmission of packets of a given size on a link
with constant capacity. Therefore, the pdf of the service time becomes g(t) =
(tx), where (.) denotes the Dirac Delta function. The corresponding
Laplace transform is (s) = exs. By using (6.11), we have: A(z) =
(s)|s=(z1) = ex(z1). Note that x is the intensity of the input traffic in
Erlangs. By substituting this expression of A(z) in (6.10), we obtain P(z)
with imbedding points at the service completion instants as:
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
z  1e x  z 1
P z   1  x 
x  z 1
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ze
(6.19)
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Note that the PGF of an M/D/1 queue in (6.19) cannot be antitransformed in closed form, so that numerical methods (as those discussed in
Section 4.3.1) are needed to obtain the state probability distribution.
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QUEUING THEORY AND TELECOMMUNICATIONS
Finally, the mean number of requests in the queue N can be expressed
according to (6.17) as:
2 x 2
x
2 x 2
N  x 


21  x  1  x 21  x 
(6.20)
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Hence, N has [rightmost term in (6.20)] a contribution corresponding to
that of an M/M/1 queue (with the same mean arrival rate and the same mean
service time) minus a positive term. Hence, the congestion of an M/D/1
queue is lower than that of the corresponding M/M/1 queue. The same
relation holds for the mean system delay given by (6.18). This is consistent
with the fact that for the same mean service time the exponential distribution
has a mean square value two times larger than that of a deterministic
distribution.
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6.1.2 The M[comp]/G[b]/1 queue with bulk arrivals or bulk service
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The queue with bulk (compound) arrivals (as defined in Section 5.1.1)
and imbedding instants at the end of the service of each object entails a
modification in (6.1) when ni = 0: when the service is completed for the first
object of a group arrived at an empty system, the remaining objects in the
queue are not only those arrived during the service time of this object, but
also the remaining objects belonging to the same group arrived at an empty
system. Let m denote the random variable of the length of a group in objects.
Then, the difference equation in (6.1) for ni = 0 becomes: ni+1 = ai+1*, where
ai+1* = m1+ ai+1; this model corresponds to the differentiated service time
case detailed in Section 6.10. According to the previous notations, this
queuing system can be denoted as M[comp]/G/1.
In the bulk service case, b arrivals (= objects) can be serviced together in
the imbedding interval. This is for instance the case of TDM(A)
transmissions with a frame-based allocation of packets having fixed service
time: the imbedding points are here at the end of each frame. With bulk
service, the difference equation (6.1) has to be modified when ni  0, thus
obtaining the following expression: ni+1 = max(ni  b, 0) + ai+1. According to
the previous notations, this queuing system can be denoted as M/G[b]/1; in
the TDM(A) case we have actually an M/D[b]/1 queue.
More details on the solution of these cases (including the consideration of
different imbedding options) will be provided in Section 6.7.
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Chapter 6
6.2 M/G/1 system delay distribution in the FIFO case
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Arrival instant of the black
packet at the queue
finding other packets
inside
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This Section provides an extension of the study made in Section 5.11.1 to
the case of general service times. As long as possible, we keep the same
notations as those used in Section 5.11.1. Let us refer to a queue with FIFO
discipline, Poisson arrivals, general service time, and system imbedded at
service completion instants. The n requests left in the system at the service
completion instant are those arrived during the system delay TD experienced
by the request just served; see Figure 6.2.
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Completion instant of the black
packet leaving n packets
in the queue arrived in the
meanwhile. These are also
the packets in the queue at
at the imbedding instants, as
modeled by PGF P(z) in (6.10)
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After time TD
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Figure 6-2. Relation between random variable TD of the queuing delay and the PGF P(z) of
the number of requests n in the queue at imbedding instants.
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The probability distribution for the n requests in the system at the service
completion instants coincides with the state probability distribution with
PGF P(z) in (6.10). This PGF of random variable n can also be obtained
referring to the fact that these n requests are the arrivals at the system during
the system delay TD, with corresponding pdf fD(t) [note that fD(t) is the
unknown distribution that we need to characterize]. Let us first condition our
study on a given system delay TD = t, so that the PGF of the number of
Poisson arrivals in this interval is: P(z | TD = t) = et(z1). Then, we remove
the conditioning by means of the pdf fD(t) as:
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
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P  z    et  z 1 f D t dt  TD s    z  1
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where TD(s) is the Laplace transform of the pdf fD(t).
(6.21)