Ethernet passive optical network (EPON) has gained a great amount of interest both in industry and academia as a cost-effective solution for broadband access networks, as illustrated by the formation of several forums and working groups, including the EPON forum and the Ethernet in the First Mile (EFM) alliance. EPON carries data encapsulated in Ethernet frames, which makes it easy to carry IP packets and eases the interoperability with installed Ethernet local area networks (LANs). EPON represents the convergence of low-cost Ethernet equipment [switches, network interface cards (NICs)] and low-cost fiber architectures. Furthermore, given the fact that more than 90% of today's data traffic originates from and terminates in Ethernet LANs, EPON appears to be a natural candidate for future first-mile solutions.

The main standardization body behind EPON is the IEEE 802.3ah task force. This task force developed the so-called multipoint control protocol (MPCP) which arbitrates the channel access among central office (CO) and subscribers. MPCP is used for dynamically assigning the upstream bandwidth (subscriber to service provider), which is the key challenge in the access protocol design for EPON. Note that MPCP does not specify any particular dynamic bandwidth allocation (DBA) algorithm. Instead, it is intended to facilitate the implementation of DBA algorithms.

To understand the importance of dynamic bandwidth allocation in EPON, note that the traffic on the individual links in the access network is quite bursty.

In this and following chapters, we will discuss random processes. After a brief introduction to this subject in Section 12.1, we will give an overview of various random processes in Section 12.2 and then discuss (strictly) stationary and wide-sense stationary random processes and introduce the notion of ergodicity. The last section focuses on complex-valued Gaussian processes, which will be useful in the study of communication systems and other applications.

Random process

There are many situations in which the time dependency of a set of probability functions is important. One example is a noise process that accompanies a signal process and should be suppressed or filtered out so that we can recover the signal reliably and accurately. Another example is the amount of outstanding packets yet to be processed at a network router or switch, which may lead to undesirable packet loss due to buffer overflows if not properly attended to in time.

Such a process can be conveniently characterized probabilistically by extending the notion of a random variable (RV) as follows: we assign to each sample point ω ∈ Ω a real-valued functionX(ω, t), where t is the time parameter or index parameter in some range T, which may be, for instance, T = (-∞, ∞) or T = {0, 1, 2, …} (see Figure 12.1). Imagine that we can observe this set of time functions {X(ω, t); ω ∈ Ω, t ∈ T} at some instant t = t1.

Worldwide interoperability for microwave access (WiMAX) has been envisioned by the WiMAX forum as a single worldwide adopted standard for high-speed wireless metropolitan area networking. The term WiMAX denotes wireless metropolitan area network (WMAN) technology based on IEEE 802.16 specifications. In this chapter, we provide an overview of the salient features and most important specifications of legacy and next-generation WMANs.

Fixed WiMAX

The initial IEEE 802.16 WiMAX standard was established in the frequency band 10–66 GHz, providing up to 75 Mb/s line-of-sight (LOS) connections for both point-to-multipoint and mesh modes. Table 7.1 summarizes the IEEE 802.16 WiMAX standard family.

PHY layer

IEEE 802.16a provides non-LOS connections in the frequency band of 2–11 GHz (licensed and unlicensed). The WiMAX physical (PHY) layer supports the following four different modulation schemes: WirelessMAN-SC (single carrier), WirelessMAN-SCa (single carrier access), WirelessMAN-OFDM (orthogonal frequency division multiplexing), and WirelessMAN-OFDMA (orthogonal frequency division multiple access). WhileWirelessMAN-SC has been designed for the frequency band 10–66 GHz, the other modulation schemes can be used for the frequency band 2–11 GHz. Additionally, the WiMAX PHY layer transfers bidirectional data by means of time division duplex (TDD) or frequency division duplex (FDD).

MAC layer

IEEE 802.16 is a connection-oriented standard, i.e., prior to transmitting data between subscriber stations (SSs) and base station (BS) connections must be established.

The wireless mesh front-end of fiber-wireless (FiWi) access networks provides multiple paths to route traffic coming from and going to the optical backhaul. In this chapter, we review a variety of recently proposed routing algorithms that aim at optimizing the network performance in terms of delay, throughput, packet loss, load balancing, and other important metrics such as path availability and power consumption. The considered routing algorithms cover either only the wireless front-end or both the wireless and optical domains of FiWi access networks. Furthermore, this chapter elaborates on various techniques to provide service differentiation and end-to-end guaranteed quality-of-service (QoS) and enable QoS continuity across the optical–wireless interface of FiWi broadband access networks.

Wireless routing algorithms

In this section, we describe various recently proposed routing algorithms for the wireless front-end of FiWi access networks. All of the discussed wireless routing algorithms aim at finding the optimal path through a wireless mesh front-end by meeting one or more objectives.

DARA

A pro-active routing algorithm, referred to as the delay-aware routing algorithm (DARA), which minimizes the average packet delay between a router and any wireless mesh gateway was presented in (Sarkar et al. [2007b, 2008]). DARA is a link-state routing algorithm, where each wireless mesh router and gateway periodically advertises their link conditions (i.e., traffic intensity and link capacity with time stamp) in link state advertisement (LSA) messages. Upon reception of an LSA message, each router/gateway predicts the traffic intensity, which is used to predict the state of the corresponding link until the next LSA message arrives.

Fiber access networks have in general one of the following three architectures: (i) point-to-point architecture, (ii) active star architecture, or (iii) passive star architecture. In the point-to-point architecture, each home or building is connected to the central office (CO) via one or two dedicated fibers. This type of architecture provides improved privacy and ease of service upgrade for individual subscribers, but requires a large number of fibers and transceivers since network equipment is not shared among subscribers. As a consequence, footprint and power consumption may become serious problems at the CO. This shortcoming is avoided in star architectures, where a single feeder fiber runs from the CO to a remote node, from which individual distribution fibers branch out to connect the subscribers. The feeder fiber carries all the traffic of the attached subscribers and its cost can be shared among them. In doing so, the number of required fibers and transceivers at the CO can be reduced significantly. Depending on the nature of the remote node, the star architecture may be either active or passive. In the active star architecture, the remote node is an active device, e.g., Ethernet switch, and needs powering and maintenance. Conversely, in the passive star architecture, the active node is replaced with a passive optical splitter/combiner. Using a completely passive splitter/combiner at the remote node avoids the need for powering and maintenance and thereby helps reduce the capital expenditures (CAPEX) and in particular operational expenditures (OPEX) of fiber access networks (Koonen [2006]).

GPON and EPON, described at length above in Chapter 3 and Chapter 4, respectively, represent the two most important Gigabit-class passive optical networks (PONs) that are widely deployed in the United States, Europe, and Asia Pacific region. Given the ever increasing bandwidth demand from consumer and business applications, current PONs are expected to evolve into next-generation PONs (NG-PONs) over the next couple of years. GPON and EPON are expected to coexist for the foreseeable future as they evolve into NG-PONs. Clearly, one way to realize NG-PONs is to increase the line rate of current Gigabit-class PONs to 10 Gb/s. A good example of this approach is the IEEE 802.3av 10G-EPON standard, which was approved in September 2009 (see Section 4.4). NG-PONs are mainly envisioned to (i) achieve higher performance parameters, e.g., higher bandwidth per subscriber, increased splitting ratio, and extended maximum reach, than current GPON/EPON architectures, and (ii) broaden GPON/EPON functionalities to include, among others, the consolidation of optical access, metro, and backhaul networks, and the support of topologies other than conventional tree structures. Network operators are seeking NG-PON solutions that can transparently coexist with legacy PONs on the existing fiber infrastructure and enable gradual upgrades in order to avoid costly and time-consuming network modifications and stay flexible for further evolution paths.

In Sections 4.2.4 and 4.3.1 we defined the normal (or Gaussian) distributions for both single and multiple variables and discussed their properties. The normal distribution plays a central role in the mathematical theory of statistics for at least two reasons. First, the normal distribution often describes a variety of physical quantities observed in the real world. In a communication system, for example, a received waveform is often a superposition of a desired signal waveform and (unwanted) noise process, and the amplitude of the noise is often normally distributed, because the source of such noise is usually what is known as thermal noise at the receiver front. The normality of thermal noise is a good example of manifestation in the real world of the CLT, which says that the sum of a large number of independent RVs, properly scaled, tends to be normally distributed. In Chapter 3 we saw that the binomial distribution and the Poisson distribution also tend to a normal distribution in the limit. We also discussed the CLT and asymptotic normality.

The second reason for the frequent use of the normal distribution is its mathematical tractability. For instance, sums of independent normal RVs are themselves normally distributed. Such reproductivity of the distribution is enjoyed only by a limited class of distributions (that is, binomial, gamma, Poisson). Many important results in the theory of statistics are founded on the assumption of a normal distribution.

The gigabit passive optical network (GPON) is an outcome of the full service access network (FSAN) alliance and is specified in the ITU-T G.984.x series of recommendations, which were finalized in February 2004. GPON extends the capabilities of its two predecessors, asynchronous transfer mode (ATM) PON, also known as APON, and broadband PON (BPON). Compared with its predecessors, GPON provides larger splitting ratios, higher up- and downstream data rates, longer reach, improved privacy and security through the use of the Advanced Encryption Standard (AES) algorithm, and a new GPON encapsulation method (GEM) to carry synchronous voice services and data services such as Ethernet in a bandwidth-efficient manner (Shumate [2008]). These extended capabilities of GPON are explained in greater detail in the following.

Architecture

Figure 3.1 shows the architecture of a GPON network (Effenberger et al. [2007]). GPON deploys two different wavelength channels for upstream and downstream communication. The upstream and downstream wavelength channels operate at 1310 nm and 1490 nm, respectively. Several upstream and downstream data rates are specified for GPON, with a maximum data rate of 1.244 Gb/s in the upstream direction and 2.488 Gb/s in the downstream direction. The reach of a GPON network can be as high as 60 km, whereby the differential reach between optical network units (ONUs) must not exceed 20 km. The ITU-T recommendations for GPON allow for a splitting ratio of up to 128.

To maximize the performance of FiWi networks and minimize their deployment costs, network planning and reconfiguration play a key role in achieving these design objectives. In this chapter, we describe a number of algorithms that help solve important FiWi network planning problems related to the optimal placement of optical network units (ONUs), mitigation of the detrimental impact of wireless interferences for peer-to-peer communications between wireless end-users, and architectural modifications for the support of direct inter-ONU communications. Furthermore, we discuss previously proposed reconfigurable FiWi network architectures that are able to respond to varying traffic loads.

ONU placement

The optimal placement of ONUs is an important design objective of FiWi networks due to the fact that the cost of laying optical fiber is significantly higher than that of devices attached to either end of the optical fiber, e.g., optical line terminal (OLT).

Several heuristics to solve the problem of optimally placing ONUs in a FiWi access network consisting of a passive optical network (PON) in tandem with a WiFi- or WiMAX-based wireless mesh network (WMN) were studied in (Sarkar et al. [2008]). The first proposed heuristic is a greedy algorithm that aims at finding a suitable placement of multiple ONUs to minimize the average Euclidean distance between wireless end-users and their closest ONU, i.e., this heuristic targets only the wireless front-end and does not take the fiber layout of the optical backhaul into account.

Why study probability, random processes, and statistical analysis?

Many problems we face in daily life involve some degree of uncertainty and we need to use probabilistic reasoning in order to make a sound decision, be it an investment, medical, or social problem. In many cases “probability” may represent our personal judgment about how likely a particular event (e.g., price movement of a stock we own; positive or negative effect of some medicine we may choose to take when we are ill) is to occur. Probability attached to a given event is generally not based on any precise computation but is often a reasonable assessment based on our knowledge or experience. Such probability may be aptly called subjective or qualitative probability, and may not be scientifically estimated, unless the same event happens repeatedly.

The other type of probability we deal with is what we may call objective or quantitative probability, which can be estimated objectively based on empirical evidence from observable events. This philosophical question concerning subjective versus objective probability has been pondered by many probabilists and statisticians, as we will briefly discuss in Section 1.2. Philosophical discussion on subjective probability still continues today as a fascinating topic, as is found in arguments between two schools of statistics, i.e., frequentist statistics and Bayesian statistics, which will be discussed at the end of this chapter.

In this book we will discuss probability based on the modern probability theory established by Kolmogorov in the early twentieth century.

Introduction

In this chapter we shall discuss hidden Markov models (HMMs), which have been widely applied to a broad range of science and engineering problems, including speech recognition, decoding and channel modeling in digital communications, computational biology (e.g., DNA and protein sequencing), and modeling of communication networks.

In an ordinary Markov model, transitions between the states characterize the dynamics of a system in question, and we implicitly assume that a sequence of states can be directly observed, and the observer may even know the structure and parameters of the Markov model. In some fields, such as speech recognition and network traffic modeling, it is useful to remove these restrictive assumptions and construct a model in which the observable output is a probabilistic function of the underlying Markov state. Such a model is referred to as an HMM.

We shall address the important problems of state and parameter estimation associated with an HMM: What is the likelihood that an observed data is generated from this model? How can we infer the most likely state or sequence of states, given a particular observed output? Given observed data, how can we estimate the most likely value of the model parameters, i.e., their MLEs? We will present in a cohesive manner a series of computational algorithms for state and parameter estimation, including the forward and backward recursion algorithms, the Viterbi algorithm, the BCJR algorithm, and the Baum–Welch algorithm, which is a special case of the EM algorithm discussed in Section 19.2.