The Trend Of Wsns Applications Computer Science Essay

Published Date: 02 Nov 2017

A wireless sensor network, abbreviated as WSN, consists of a possibly large number of wireless devices able to take environmental measurements such as temperature, light, sound and humidity. With the development of the technology, the wireless devices even can sense the video and audio and transmit it. These sensor readings are transmitted over a wireless channel to a running application that makes decisions based on these sensor readings. It is described in the section that some examples of proposed wireless sensor applications and consider the following questions to motivate and an application-based viewpoint:

\item What is the trend of WSN's applications like?

The one recognized issue is the limited power available to each wireless sensor node, but there are other challenges such as the limited processing and the data backbone problems.

\item What are the requirements of the applications to the wireless sensor networks?

There are already some proposed publications of the traditional wireless sensor network's applications which will be introduced in the following parts. But with the development of the technologies and the higher requirements of the human beings, the more and more applications with new type, new services and new requirements come out recently. As the result, the new requirements are analyzed.

\item Is there any solution to solve the problems of the new requirements?

Based on the new requirements that given by the current or future applications to the wireless sensor networks, it is necessary to find out a solution.

\item Is there a method can measure or evaluate the applications' performance with the new solution?

A sensor network is composed of a large number of sensor nodes, which are densely deployed either inside the phenomenon or very close to it\cite{WSNintr1}\cite{WSNintr2}. Generally, the position of each sensor node in the WSN need not be engineered or predetermined, which means it allows WSN has random deployment in inaccessible terrains or disaster relief operations. Also as the result of it, the WSN's protocols and algorithms must possess self-organizing capabilities. Another feature of WSN is the cooperative effort of sensor nodes. The sensor nodes are fitted with an onboard processor and they use their processing abilities to locally carry out computations and transmit only the required and partially processed data. Furthermore, WSN has some other features which may important:

\begin{itemize}

\item The sensor nodes are limited in power, computation capacities and memory;

\item The sensor nodes are densely deployed;

\item The sensor nodes are prone to failures;

\item The topology of a WSN changes very frequently;

\item The sensor nodes may not have global identification because of the large amount of overhead and large number of sensors.

\end{itemize}

Generally the sensor nodes' architecture in WSN is as Fig.\ref{f:node} shows \cite{WSNintr4app}.

A sensor nodes may be composed of several basic components which including a sensing unit, a processing unit, a communication subsystem, a coordination subsystem, a storage unit and an optional mobility/ actuation unit:

\begin{itemize}

\item The sensing units are usually composed with two subunits: sensors and analog-to digital-converters (ADCs). The analog signals produced by the sensors based on the observed phenomenon are converted into digital signals by the ADC and then fed into the processing unit. The traditional sensors are usually those ones which sense the temperature, humidity, acceleration and some other scalar data. Currently, the with the human beings' higher requirements to the WSN applications, the sensors begin to contain the camera and audio \textit{etc}.

\item The processing unit executes the system software in charge of coordinating sensing and communication tasks and it is interfaced with a storage unit.

\item The communication subsystem interfaces the device to the network and is composed of a transceiver unit and of communication software which usually includes a communication protocol stack and system software such as middleware, operating systems and virtual machines.

\item The coordination subsystem takes the charge of coordinating the operation of different network devices by performing operations such as network synchronization and location management.

\item The optional mobility/ actuation unit can enable movement or manipulation of objects.

\item The power unit is the most important part that powered the whole system and may supported by an energy scavenging unit such as solar cells or others.

\end{itemize}

\subsubsection{Network Layers}

As introduced in Ref.\cite{WSNintr1}, the protocol stack used by WSN is shown in Fig.\ref{f:layers}. This one combines power and routing awareness, integrates data with networking protocol, communicates power effciently through the wireless medium, and promotes cooperative efforts of WSN \cite{WSNintr1}. The layers of WSN are composed of physical layer, data link layer, network layer, transport layer and application layer and have power management plane, mobility management plane and task management plane.

\begin{itemize}

\item Physical layer: it addressed the needs of simple but robust modulation, transmission and receiving techniques.

\item Data link layer: in it a medium access control (MAC) protocol is needed and because of the noisy environment and the mobility, possible feature of the sensor nodes, it must be power-aware and able to minimize collision with neighbour's broadcasts.

\item Network layer: it focuses on the routing the data supplied by the transport layer.

\item Transport layer: it helps to maintain the flow of data if the application of WSN requires it.

\item Application layer: different types of applications software can be built and used on this layer, depending on the sensing tasks.

\item Power management plane: it takes the charge of managing how a sensor nodes uses its power, such as the sensor node may turn off its receiver or even sensor power after receiving a message from a neighbour node or just sensed a data from the environment. This can avoid getting duplicated messages which have less utility.

\item Mobility management plane: it detects and registers the movement of sensor nodes, so a route back to the user is always maintained and the sensor nodes can keep track of who are the neighbours of them. And by knowing it, they can balance their power and task usage. Note that not all the WSN applications allow the nodes to move. But with the higher requirements of the applications, the WSN applications with movable nodes comes out more and more currently.

\item Task management plane: it takes the responsibilities of banlancing and scheduling the sensing tasks. Not all the sensor nodes in WSN are required to perform the sensing task at the same time. As the result of it, soe sensor nodes perform the task more than the others depending on their power level or some other factors. This plane is needed so that the sensor nodes in WSN can work together in a more power-saving way and share resources between the sensor nodes.

\end{itemize}

For its convenience and inexpensive cost of collecting data, recently we have seen tremendous growth in the research and the application of various wireless sensor network (WSN) (e.g. wireless sensor network for military, for industry even underwater wireless sensor network, video wireless sensor network etc.). The growing applications give more bandwidth-intensive services and demand for larger bandwidth supplication from the backbone of the access network. Also more application requires the data to transmit through the network with a longer distance to realize the remote surveillance which is not realistic to achieve only by wireless transmission (one of the important reason is the huge energy consumption\cite{WSNintr3_LEACH}). The proposed applications and the trend of future applications is introduced in the following part and also the requirements are analyzed.

\subsection{Applications}

\subsubsection{Applications}

The applications of wireless sensor network have been applied in many areas for its great potential and they can be classified generally into the following categories.\cite{WSNintr2}\cite{WSNintr4app}

\begin{itemize}

\item Military. The WSN is developed from the military research and application. It is very potential in the applications like tactical surveillance\cite{WSNapp1} (land, sea), tracking troop movement (both sides), making ubiquitous and undetected smart mines, battlefield communication\cite{WSNapp2}, detection of hazardous agents\cite{WSNapp3}\cite{WSNapp4} (explosive, nuclear, biological, poisonous, radioactive) and environmental awareness (terrain mapping) etc.

\item Surveillance. Video and audio sensors are more and more used to enhance and complement existing surveillance systems against crime and terrorist attacks. In Ref.\cite{WSNapp5}, problems related to the transport of an audio signal through a wireless channel and sensor nodes are analyzed and a project for an audio surveillance system is presented. In most all the other applications introduced in this part also use the multimedia sensor as the monitor to 'feel' the world: locating missing persons, criminals or terrorists, inferring and recording potentially relevant activities such as thefts, car accidents, traffic violations, and other smart home surveillance\cite{WSNapp6}. Furthermore an application was developed in Ref.\cite{WSNapp7} which used the sensor network (that consists of 21 mobile nodes, 29 reference nodes, and one gateway) to track 21 medical assets so that the nurses can use more times to take care of patients and the patients can receive better medical services. Ref.\cite{WSNapp22} studied how to use WSN to satisfied the requirements of protection of critical infrastructures. And an innovative solution for ship intrusion detection was proposed in Ref.\cite{WSNapp23}.

\item Traffic Monitoring. It will be possible to monitor car traffic in big cities or on highways and deploy services that offer traffic routing advice to avoid congestion or identify violations\cite{WSNapp24}. In Ref.\cite{WSNapp8} the researchers proposed a smart parking advice systems based on WMSNs to detect available parking spaces and provide drivers with automated parking advice. Some other researchers considered realize traffic surveillance using video camera sensors\cite{WSNapp9}. Ref.\cite{WSNapp10} focuses on congestion control but while previous works considered scalar sensor nodes which only report events in the size of a few bytes, they addressed congestion control for information-intensive flows such as video flows for surveillance applications in pervasive wireless multimedia sensor networks.

\item Advanced Health Care. Wireless sensor network can be used to monitor and study the behaviour of elderly people as a means to identify the causes of illnesses that affect them like the research in Ref.\cite{WSNapp11} to monitor the patients suffering from dementia. Ref.\cite{WSNapp12} developed a prototype for remote health monitoring, and a four-levels hierarchical wireless body sensor network (WBSN) system is designed for biometrics and healthcare applications in Ref.\cite{WSNapp13}. Ref.\cite{WSNapp14}\cite{WSNapp20}\cite{WSNapp21} gives the description and experimental analysis of the application which is based on the main idea of using simple distributed sensor nodes in a home environment, to provide home assistants, nurses, healthcare centres and relatives with a degree of "understanding" and information about the actual persons health and activity status in order to fast determine what kind of help is needed. And Ref.\cite{WSNapp25} tried to avoid the congestions between the sensor nodes.

\item Gaming. Networked gaming is emerging as a popular recreational activity. WMSN can help to find applications in future prototypes that enhance the effect of the game environment on the game player\cite{WSNapp15}.

\item Environmental and Industrial. Wireless sensor network, especially wireless multimedia sensor networks can be used for time-critical, industrial, process control. Some projects and testbeds have be developed: Arrays of video sensors are used by oceanographers to determine the evolution of sandbars via image processing techniques\cite{WSNapp16}; Algorithm is designed for reliable video transmission within mine tunnel to enhance the reliability of the end-to-end monitoring of mine tunnel\cite{WSNapp17}; some researches focused on the environmental and structural monitoring such as structural health monitoring of bridges or buildings\cite{WSNapp18}; and some ones aiming at tracking the pollutions using underwater wireless sensor network\cite{WSNapp19} in the harbours. Ref.\cite{WSNapp26} proposed a solution to the remote mine tunnel monitoring challenge of its multipath and diffraction effect due to unreliable channel and limited capacity.

\end{itemize}

\subsubsection{The trend and requirements}

As introduced above, in recent years, the growing interest in the wireless sensor network has resulted in thousands of new applications \cite{WSNintr2}\cite{WSNintr4app}\cite{WSNintr10} that not only measured the scalar data from physical phenomena such as temperature, pressure, humidity or location of objects, but more aimed at revisiting the real world with the description of multimedia content, for example, audio and video streams and still images. Because of the new methods of collecting data, there are also many new challenges to the wireless sensor network.

With the trends of increasing number of various applications which come out with the developing of the wireless sensor network, the challenges become more obvious when designing the wireless sensor network and guaranteeing it applications quality of service.

\begin{itemize}

\item Larger deployed area vs. network's lifetime

The trends of application need the network to be deployed in lager areas to monitor the physical world. But it means the sensor nodes have to send their data through a long distance to the sink and the increasing consumption is unavoidable even it sends the data by multi-hops. With the increasing energy consumption level, the lifetime becomes shorter and shorter definitely. When the lifetime decreases to some extent, the whole network is useless and the application is failed.

\item Remote surveillance application vs. network's lifetime, data security and QoS

In some other conditions, the application only requires to surveillance an appropriate area for wireless sensor network but it need to transmit the data to some other place which is very far away from the surveillance area (such as monitoring the nuclear pollution, the volcanoes, forests fire etc. ). It is also impossible for sensor nodes to send the service data through the distance like this even it can send by the multi-hops. Especially for this type of applications it is hard or unexpected to recharge the nodes battery and it is usually expected to deploy just once. Therefore the saving energy and prolong the lifetime is the most significant to guarantee the application is working. In the other hands, the long distance wireless transmission always has company with high packet's loss rate and lower security level.

\item Delay sensitive application vs. one-sink structure

As the sensor density increases to infinity and more sensors send data especially multimedia data to the central controller. Most of the application chooses the sensor nodes to transmit using the CSMA/CD. But with the increasing amount of sensor nodes and the larger packet size of multimedia service, the occupancy rate of the wireless change is also increasing, which means the services' data have to wait for a longer time to access into the channel and the transmission delay will have a huge increase. For some delay sensitive applications (such as the ones for military, the ones with audio services etc.), it is unacceptable.

\item Bandwidth requirements vs. sink???s constant capacity

With more multimedia sensors are used in the applications, the bandwidth requirements to the sink in the wireless sensor network also became a big challenge. The applications of wireless sensor network trends to using the event driven algorithms to save more energy and make the network more effective which makes the requirement to the sink in the network is different. For the sink in the wireless sensor network, if it has a large capacity then it is a kind of waste when the event does not happen; on the contrary, if it has a small capacity it cannot fulfil the requirements of the bandwidth to transmit enough service packets.

\item Other challenges

There are challenges also face the wireless sensor networks when it is used in more application.

\end{itemize}

Therefore our research aims at solving or relieving the challenges above by proposing a new convergence of multi-sink wireless sensor network and passive optical network structure and study the queueing model under a given structure. The other related published work is introduced in the following sections.

\subsection{Backbone Problems}

The applications and the possible applications in future also result in the backbone problems of WSN and it has already be focused recently. Some researchers already studied the method to solve the backbone problems, such as: A novel sleep-scheduling technique called Virtual Backbone Scheduling (VBS) is designed for WSNs which has redundant sensor nodes\cite{backbone1}. In VBS, traffic is only forwarded by backbone sensor nodes, and the rest of the sensor nodes turn off their radios to save energy. Ref.\cite{backbone2} propose BEES, a lightweight bio-inspired backbone construction protocol, that can help mitigate many of the typical challenges in sensor networks by allowing the development of simpler network protocols. Ref.\cite{backbone3} tried to solve the backbone problem for museum monitoring and surveillance. A distributed routing protocol \cite{backbone4} that aims at constructing an energy efficient backbone being convergecast efficient at the same time was proposed and the assistant methods for the backbone were also proposed\cite{backbone5}\cite{backbone6}.

Although all of the above introduced work has tried to solve the backbone problems of WSN and many of them have the good results, they all considered it only in the WSN but not further area. Actually, all the applications are not only inside the WSN but more areas such as access network or core network, because the surveillance and monitoring applications always require the sensed data be transmitted to the human beings who is possible far away the area or parameters which are interested. And the data need to go through all the access and core network to arrive at the monitoring terminal. Consequently, only consider the backbone problem in WSN area is not enough to solve the QoS guarantee requirements and it is also the reason that we tried to consider it in WSN-PON converged structure to solve it.

\section{Passive Optical Networks}

In the following part, the PON and the DBA (Dynamic Bandwidth Allocation) algorithms are introduce because PON is a significant part of the architecture of the converged network and a DBA algorithm based on the modelling work is proposed. Hence, a briefly basic knowledge of them is necessary.

\subsection{PON introduction}

Passive optical network (PON) has been considered as a solution for the subscriber access network for quite some time, even before the Internet spurred bandwidth demand \cite{PONbook1}. It offers low cost and high bandwidth solutions in the last mile service of the Internet access\cite{PONintr3}. Fiber to the Home/ Curb/ Building (FTTx) solutions of PONs can meet the requirements of the services such Internet Protocol (IP) telephony, IP television (IPTV), video on demand and http\cite{PONintr4}. Therefore, deploying a PON between service providers and customer premises can provide a cost efficient and flexible infrastructure that will provide the required bandwidth to customers\cite{PONintr2}.

Generally, PONs are a network in which a shared fiber medium is created using a passive optical splitter/ combinner in the physical plant. Sharing the fiber medium means reduced cost in the physical fiber deployment, and using passive components in the physical plant means reduced recurring costs by not maintaining remote facilities with power. These reduced cost make PONs an attractive choice for access networks, which are inherently cost sensitive\cite{PONintr5}.

All the transmission in a PON are performed between an optical line terminal (OLT) and optical network units (ONUs). The OLT connects the optical access network to the metropolitan-area network or wide-area network (WAN), also known s the backbone or lang-haul network. The ONU is located either at the end-user location (as FTTx).

At a top level, PONs can be classified into several subset: By providing the advantages of low maintenance cost and adaptability to higher bit rates, Ethernet PON (EPON) seems a promising PON technology which has been standardized in IEEE 802.3ah\cite{802.3ah}; Gigabit-capable PON (GPON) is another attractive technology which is standardized in ITU-T G.984 \cite{G.984}. To avoid collision in the PON, time division multiplexing (TDM) or wavelength division multiplexing (WDM)can be used\cite{PONintr3}. Consequently, PONs can also classified as TDM-PON and WDM-PON.

The PON used in the proposed WSN-PON convergence architecture is the TDM-EPON with a physical tree topology, and the mentioned PON in the following part of this thesis is used to represent it. In a PON, the OLT is connected to the ONUs with a feeder fiber that is subsequently split using a 1:N optical splitter/ combiner to enable the ONUs to share the optical fiber. And the transmission direction from OLT to ONU is reffered to as donwstream and operates as a broadcast medium. The transission direction form the ONUs to the OLT is reffered to as upstream, or called uplink.

\begin{itemize}

\item Downstream: In the downstream direction, packets trasmitted by the OLT pass through the 1:N splitter and reach each ONU. The value of N is typically between 4 and 64. This behaviour is similar to a shared-medium network: Packets are broadcast by the OLT and selectively extracted by their destination ONU.

\item Upstream: In the upstream direction, due to the directional properties of a passive optical combiner, data packets from any ONU will reach only the OLT and not other ONUs. In this sense, in the upstream direction, the behaviour of EPON is similar to that of a point-to-point architecture. However, unlike a true point-to-point network, in EPON, all ONUs belong to a single collision domain. Namely, data packets from different ONUs transmitted simultaneously still my collide.

\end{itemize}

As the results of introduced collision problem in upstream above, in the upstream direction, PON needs to employ some arbitration mechanism to avoid data collisions and fairly share the channel capacity among ONUs. That is the DBA algorithm which is introduced in the following section.

\subsection{PON's DBA Algorithms}

The dynamic bandwidth allocation (DBA) algorithms \cite{PONintr2} are defined as the process of providing statistical multiplexing among ONUs and they have been researched a lot. Its major branches of the taxonomy are grant sizing, grant scheduling and queue scheduling.

The DBA algorithms will affect the performance of the convergence network in a large amount of aspects, but might not be the key contribution of our research work, so the algorithms are only introduced generally.

\begin{itemize}

\item \textbf{Grant sizing}

Grant sizing can be divided into four major categories: gated, limited, limited with excess distribution and exhaustive using queue size prediction. Ref.\cite{DBAintr4} and \cite{DBAintr5} gave the analysis of the fixed grant-sizing scheme and Ref.\cite{DBAintr6} and \cite{DBAintr8} analyzed deeply about the delay in the gated ones. The limited grant-sizing technique is studied in Ref.\cite{DBAintr4}\cite{DBAintr9} and the research results illustrate that there is no average packet delay difference between gated and limited grant sizing. Also, some other researchers studied the limited with excess distribution and the result can be found at Ref.\cite{DBAintr10}\cite{DBAintr11}. Queue size prediction is concerned with estimating the traffic that is generated during the period between the REPORT message transmission by the ONU and the beginning of the gated transmission window. Some research used control theory to drive the gap between predicted and actual queued traffic to zero\cite{DBAintr12}, and a higher order liner predictor for predicting traffic during the waiting period at an ONU is proposed in Ref.\cite{DBAintr13}.

\item \textbf{Grant scheduling}

Since grant scheduling works at the inter-ONU level and is coupled with the process of grant sizing, it is performed at the OLT. Typically, to change the scheduling order from round robin, the OLT must wait for all REPORTed queue sizes from the ONUs and then determine the grant order. This requires the use of interleaved polling with stop or the offline DBA framework. There are many published work focused on it: ONU transmissions ordered longest queue first (LQF)\cite{DBAintr9}\cite{DBAintr14} or earliest packet first\cite{DBAintr9} (EPF) have been examined; Simulation results\cite{DBAintr14}\cite{DBAintr15} using Poisson traffic show that both LQF and EPF provide lower average delay at medium loads compared to a round robin scheduler.

\item \textbf{Queue scheduling}

Intra-ONU scheduling is concerned with scheduling the multiple queues of Ethernet frames at an ONU, for transmission within the ONU???s granted transmission window. If the number of queues in an ONU is relatively small, this intra-ONU scheduling can be performed at the OLT. However, as the number of queue increases, scheduling is typically make hierarchical\cite{DBAintr16} with the inter-ONU scheduling at the root of the hierarchy in the OLT and one level of branches. The ideal scheduler should allow statistical multiplexing, but guarantee a minimal portion of the available bandwidth to each priority queue. The generalized processor sharing\cite{DBAintr17} (GPS) achieves this goal for the fluid traffic model, where packets are infinitesimally small. Unfortunately, in practical systems with finite-size packets, the ideal GPS link sharing is not directly applicable because a packet must monopolize the server while in service. The improved versions of GPS are proposed soon, such as WFQ, SFQ and M-SFQ etc.

\end{itemize}

The DBA algorithm proposed in this thesis belongs to the Grant sizing one and some classic DBA algorithms which are compared with the proposed one in Chapter 5 are introduced as the following:

\begin{itemize}

\item Fixed: Each ONU is allocated the same grant size $G_i^{max}$ for every cycle, namely the maximum grant size for that ONU.

Analysis and simulation results \cite{DBAintr4}\cite{DBAintr5} demonstrate markedly inferior performance to the three dynamic techniques below.

\item Balanced: The OLT calculates each ONU's grant size by using the queue length $R_i$ reported by each ONU as a weight:

$G_i=t^{max}_{cycle}R_i/\sum R_i$.

\item Gated: The grant size for an ONU is simply the queue length reported by that ONU: $G_i=R_i$. The average delay is low, but there is insufficient control to ensure fair resource allocation between ONUs. The delay introduced by this scheme has been analysed \cite{DBAintr6}\cite{DBAintr7}.

\item Limited: The grant size is set to the reported queue size up to a maximum grant size for that ONU: $G_i=\min(R_i,G_i^{max})$. Although there is no difference in average packet delay between gated and limited grant sizing \cite{DBAintr4}, limited grant sizing can assist in providing fair access between ONUs.

\end{itemize}

\section{Models of Queueing Theory in WSN and PON}

There are some great contributions of previous researchers who focused on the Modelling work of WSN and PON respectively. But it is hard to find out the related work of modelling work of converged network of WSN and PON. In the following part, the existed models are introduced. Beforehand, some basic knowledge of queueing theory which is used in the modelling work need to be introduced.

\subsection{Used knowledge of Queueing Theory}

\subsubsection{General Introduction of Queueing System}

A queueing system is a place where customers arrive according to an ''arrival process'' to receive service form a service facility\cite{book1}. The service facility may contain more than one server and it is assumed that a server can serve one customer at one time. If an arriving customer finds the service facility occupied (all the servers in this facility are busy or in vacation mode), it joins the waiting queue. This customer will receive its service later in time, either when he reaches the head of the waiting queue or according to some service disciplines (such as higher priority served first), and then leave the system upon completion of his service.

Also, in the paragraph above the words ''customer'' and ''server'' have the different meanings differ from the various application. For example, in the case of computing network (wired or wireless one), the ''customers'' are more often to be used to present the packets (or messages, or cells) that arrive at a switching node (or relay node, or router, or transmission channels \textit{etc}.) which is presented as the ''servers''.

Basically, a queueing system can be broken down into three major components\cite{book1}:

\begin{itemize}

\item the input process;

\item the system structure;

\item the output proces;

\end{itemize}

In the following parts we will discuss the model in this order. Beforehand, the basic structure of the queueing model and the random variables is given.

\begin{figure}

\centering

\includegraphics[width=\textwidth]{Ch2/Modle.pdf}

\caption{Basic queuing system}

\label{f:Basic queuing system}

\end{figure}

\subsubsection{Input process}

We concern ourselves with the following three aspects of the arrival process:

\begin{itemize}

\item[A.] The size of the arriving population;

\item[B.] Arriving patterns;

\item[C.] Behaviour of the arriving customer;

\end{itemize}

In the following part, these three parts are introduced respectively.

\textbf{A. The size of the arriving population}

Generally, the size of the arriving customer population may be infinite in the sense that the number of potential customers form external sources is very large as compared to those in the system, or it may be finite in the sense that the arrival rate is affected by the size.

The size of arriving population gives a significant impact on the queueing results. An infinite population tends to render the queueing analysis more tractable. On the other hand, the analysis of a queueing system with finite customer population size is more involved because the arrival process is affected by the number of customers already in the system \cite{book1}.

Based on it, we will discuss that whether the customer population is infinite or finite in the environment of WSN-PON convergence network, especially for the QoS guarantee of remote surveillance application. Also, the basic random variables of a queueing system are given in Table \ref{t:basicqueuepara} for understanding the analysis of the queueing model.

In the converged network of WSN and PON, as long as the sensor nodes have the energy, the data will be sent to the ONU-Sink. Therefore, from the view of ONU-Sink or the relay nodes in the WSN, it will receive the applications' data continually until the WSN's energy is consumed out. Because the amount of the services' data is much more larger than the one in the system, and for making the system's analysis simple, we suppose that the population of the arriving data is infinite. When the WSN is dead (without energy), it is meaningless to solve the QoS guarantee problem and consider the arriving population. Therefore, the arriving population of the data in the queueing model is considered as infinite.

\textbf{B. Arriving patterns}

Customers may arrive at a queueing system either in some regular patterns or totally random fashions. However, it is significant to fit a statistical distribution to the arriving pattern in order to render the queueing analysis mathematically feasible.

There are probability distributions that are commonly used to describe the arrival process, as shown in Table \ref{t:distributions}.

%\begin{itemize}

%\item $M$:& Memoryless, imply the Poisson process;

%\item $D$:& Deterministic, fixed inter-arrival times;

%\item $E_k$:& erlang distribution of order k;

%\item $G$:& General probability distribution;

%\item $GI$:& General and independent (inter-arrival time) distribution;

%\end{itemize}

In the case of QoS guarantee for remote surveillance application in the convergence network, there are two types of data need to send: one is the scalar data, the other one is the media data. We will discuss it separately.

The scalar data, which describes the temperature, humidity, light intensity or other physical parameters that human beings are interested in, or the information of the sensor nodes such as current energy, routing information and so on, is supposed to be sent to the ONU-Sink when the interested events happen or with regular internal. So it is a process with various stationary probabilities. From the view of ONU-Sink, as it receives many nodes' data as the sum of many point processes, the input process tends to Poisson process\cite{theorembook2}.

Another one is the media data. It has some differences between the scalar and media data:

\begin{itemize}

\item[i.] One of the characters of the media data is its huge amount of the packages and the highly requirement of the time-delay. For remote surveillance always requires to be monitored by voice or video, the amount of data is much larger than the scalar data. Additionally, for the requirements of QoS of media services, the media data requires low time-delay level during the transmission.

\item[ii.] Transmitting the media data need to consume a large amount of energy and need more bandwidth resource. As it is introduced before the media services always have a large amount of data need to send. So it may consume a lot of energy in the sensor node which generates the media services. Meanwhile, the service will be sent to the ONU-Sink by multi-hops. Therefore it will also consume abundant energy of the nodes on its path to the ONU-Sink. On the other hand, because of the highly time-delay requirement, more bandwidth resource is essential in the ONU-Sink so that it can upload more packages in once and decrease the services' waiting time in the queue.

\item[iii.] Last but not least, the media services' data is batch arrived. For the media services are only generated when the special events are happening, the media data is not always received by the ONU-Sink. But once the event happens, the media data will be generated continually for some time since the events are always likely to last for some time. So if the ONU-Sink received a media service data, it is likely to receive another one in the next receiving period. If we consider the each batch as one package, then it follows the Poisson process as we discussed before in scalar data part. Therefore, the media service data follows the batch arrivals according to a Poisson process.

\end{itemize}

\textbf{C. Behaviour of the arriving customer}

Customers arriving at a queueing system may behave differently when the system is full (due to a finite waiting queue) or when all servers are busy. If an arriving customer leaves and is considered lost without entering the system when the system is full, that queueing system is referred to as a blocking system. The analysis of blocking system is more involved.

In our case, the behaviour of the arriving customer depends on the capacity of the queue in each ONU-Sink and the amount of the services' data. Though the amount of the data of the services from the sensor nodes is huge, the capacity of the buffer of the queue in the ONU-Sink is much larger and ONU-Sink always sends the data to OLT very quickly and effectively. So the behaviour of the arriving customer in our special application will be considered as always can get into the queue and wait, namely the capacity of the queue's buffer is infinite.

\subsubsection{System structure}

basic, single-multi multi single-single, open close jackson ...

Fig.\ref{f:Basic queuing system} shows the basic structure of the queueing system (single queue $\&$ single server). Besides this, the queueing system has many different structures. From the view of queue system itself, it has:

\begin{itemize}

\item single queue $\&$ single server in the service facility;

\item single queue $\&$ multi servers in the service facility;

\item multi queues $\&$ single server in the service facility;

\item multi queues $\&$ multi servers in the service facility;

\item hybrid queues $\&$ multi servers in the service facility;

\end{itemize}

The structures listed above can be found in the Fig.\ref{f:queuestr1}.

From the topology point of view queueing networks can be categorized into several classes:

\begin{itemize}

\item open queueing network

\item closed queueing network

\item Jackson queueing network

\end{itemize}

And the description of them is as following:

\begin{itemize}

\item[i.] \textbf{Open queueing networks}: In an open queueing network, customers arrive from external sources outside the domain of interest, go through several queues or even revisit a particular queue more than once, and finally leave the system as depicted in Fig.\ref{f:opennetwork}. Furthermore, the total sum of arrival rates is equal to the total departure rate under steady state conditions.

The open queueing networks are good models for analyzing circuit-switching and packet switching data networks. In a data network, packets or messages usually travel from node to node across several links before they reach their destination. Superficially, Fig.\ref{f:pctnetwork} gives us an impression that the transmission of data packets across several links can in a way be modelled as a cascade of queues in series. In reality, certain simplifying assumptions can be made. If we focus on a single virtual circuit then it generally has no feedback loop and can be approximately modelled as a series of queues in tandem as shown in Fig.\ref{f:tandemwork}.

\item[ii.] \textbf{Closed queueing network}: It has no external arrivals nor departures, as Fig. shows. This type of networks is often termed as single customer class networks as contrast to those multi-class networks in which customers belong to different classes and each class has its own service demand distribution.

\item[iii.] \textbf{Jackson queueing network}: A Jackson queueing network \cite{Jackson} is a network of M single-server state-independent queueing systems (hereafter referred to as a queueing node or simply node) as shown in Fig.\ref{f:jacksonnetwork} with the following features:

\begin{itemize}

\item[a.] Customers from the exterior arrive at each node $i$ according to a Poisson process with rate $\gamma_i\geqslant 0$.

\item[b.] All customers belong to the same class; meaning that their service times at node $i$ are all exponentially distributed with mean $\mu_i$. The service times are independent from that at other nodes and are also independent of the arrival process.

\item[c.] A customer upon receiving his service at node $i$ will proceed to node $j$ with a probability $p_{ij}$ or leave the network at node $i$ with probability:

\begin{equation}\label{e:jackson}

p_{i0}=1-\sum_{j=1}^Mp_{ij}

\end{equation}

\item[d.] The queue capacity at each node is infinite so there is no blocking.

\end{itemize}

\end{itemize}

For each node $i$ in the Jackson network,

\begin{equation}\label{e:jackson2}

\lambda_i=\gamma_i+\sum_{j=1}^M\lambda_jp_{ji} \hspace{4em} j=1,2,\ldots,M-1, M

\end{equation}

And for the network as whole

\begin{equation}\label{e:jackson3}

\sum_{i=1}^M\gamma_i=\sum_{i=1}^M\lambda_ip_{i0}=\sum_{i=1}^M\lambda_i\left(1-\sum_{j=1}^Mp_{ij}\right)

\end{equation}

where $\lambda_i$ is the effective arrival rate to node $i$.

Jackson queueing network is more useful to analyze and model the packet delivering network. Coupled with Little's theorem, it provides us with a simple means of evaluating some of the performance parameters of an queueing network.

As shown previous, the queue system of the special application has many servers and each server has a queue for it. The customer is the services' data and the server is the ONU-Sink. It is apparently that once the data is generated, the sensor node has many choices to send it. Meanwhile the servers (the ONU-Sink) may have many difference parameters, for instance the bandwidth it allocated, and the amount of services in the server's queue etc. Consequently the different choices will make different time-delay and different QoS. What's more, once the data has sent into a queue, it cannot change to another one's queue.

The system capacity refers to the maximum number of customers that a queueing system can accommodate, inclusive of those customers at the service facility. It has been discussed in the arriving customers' behaviour part, which will be considered as having infinite capacity.

More analysis is given in Chapter 4.

\subsubsection{Output process}

Some aspects of the service behaviour influence the departure process greatly. Some important aspects are introduced in the following.

\textbf{A. Queueing discipline}

Queueing discipline refers to the way in which customers in the waiting queue are selected and served by the servers. In general, we have\cite{book3} :

\begin{itemize}

\item First-come-first-served (FCFS);

\item Last-come-first-served (LCFS);

\item Higher-Priority-first-served (HPFS);

\item Processor sharing;

\item Random;

\end{itemize}

The first-come-first-served queueing discipline does not assign priorities and serves customers in the order of their arrival. Apparently this is the most frequently encountered discipline at an ordered queue and therefore it will be the default queueing discipline for all the subsequent systems discussed, unless otherwise stated.

The last-come-first-served discipline is just the reverse of the FCFS one. Customers who come last will be served first. This type of queueing discipline is commonly found in stack operations where items are stacked and operations occur only at the top of the stack.

In the priority queueing discipline, customers are divided into several priority classes according to their assigned priorities. Those who have a higher priority than others are served first and the rest have to wait.

In the processor sharing, capacity is equally divided among all customers in the queue; i.e. when there are $k$ customers, the server devotes $1/k$ of his capacity to each. Equivalently, each customer obtains service at $1/k$ of rate and leaves the system once he completes his service.

In our application, the scalar data and media data have the different requirements of the time-delay and the latter ones requires to be sent more urgently. So we set priority based on the urgent level of the service data: the media data has higher one and the scalar data has lower one. As the same reason, we choose the high-priority-first served (HPFS) as the queueing discipline and the FCFS when between the customers with the same priority.

\textbf{B. Service distribution}

As the arrival patterns, in any queueing system, it needs a probability distribution to describe the service pattern. If all customers require the same amount of service time then the service pattern can be easily described by a single number. But generally, different customers require different amounts of service time, hence, we again have to use a probability distribution to describe the service pattern.

The most commonly assumed service time distribution is the negative exponential distribution. It is also what we supposed in the system. That means the service time of the ONU-Sink follows the negative exponential distribution. (This will be proved or classified in future work.)

A single letter is used to indicate the type of service distributions as Table\ref{t:servedis} shows.

Little's theorem is a very simple and yet powerful result that governs its steady-state performance measures, also called Little's Law or result.

This Law existed as an empirical rule for many yeears and was first proved in a formal way by J.D.C Little in 1961\cite{Little}. Because it has already been widely used in the queueing theory, the following paragraph only introduce it simply.

This theorem relates the average number of customers ($N$) in a steady-state queueing system to the product of the average arrival rate ($\lambda$) of customers entering the system and the average time ($T$) a customer spent in that system as follows:

\begin{equation}

N=\lambda T

\end{equation}

This result was derived under very general conditions. The beauty of it lies in the fact that it does not assume any specific distribution for the arrival as well as the service process, nor does it assume any queueing discipline or depend upon the number of parallel servers in the system. With proper interpretation of $N$, $\lambda$ and $T$, it can be applied to all types of queueing systems including priority queueing and multi-server systems.

Because it is only one of the tools in our research, we do not offer the proof of the Little's theorem here.

\subsection{WSN's Models}

Although WSN has been focused and researched for years, but the work of evaluation with mathematical models is still the minority of research work, especially the ones who use the queueing theory to analyze and model the WSN's performances such as packet delay, lifetime, throughput or some other parameters related with the performances' QoS. Some of the modelling work focused on the link choosing method or its creation, and some others on the sleep-awaken mechanism. In this Section, the existed modelling work of WSN is introduced and the differences between these existed work and the proposed ones in this thesis are explained.

Although there some modelling work of WSN, most of them focused in the energy subarea. Energy saving is an unchangeable topic of WSN and many researchers also proposed some models using the queueing theory to study this problem, but most of them only focused the modelling inside one sensor node but not the whole WSN: A design strategy for optimizing power consumption of sensor node using the N-policy M/G/1 queuing theory was proposed in Ref.\cite{WSNmdl1}. With little or no extra management cost, the proposed queue-based power-saving technique can be expanded and applied to alleviate energy hole problem (EHP) economically and effectively. A mathematical analysis on the optimal control parameters were also made. This one was trying to prolong the WSN's lifetime by saving the energy of sensor nodes which more closed to the sink. But besides that, the evaluation of services QoS is not studied, such as the packets' delay in the WSN. Ref.\cite{WSNmdl2} aimed at developing a framework for modelling the synchronous wakeup patterns of networked sensors so as to provide a basis for studying such design tradeoff. The proposed framework employed the vacation model from queuing theory to model the synchronous wakeup patterns as an M/G/1 queue with server vacations. Based on the model, the optimal operation scenario for the synchronous wakeup operation is first deduced. It also has no contributions to evaluate the whole WSN's performance but only the lifetime. Differ from the ones above, Ref.\cite{WSNmdl3} derived the probability distribution of the number of active nodes and blocking probability of node activation, also presented the mean packet delay, average down period of a node as well as the network throughput by modelling of a wireless sensor network with a TDMA media access protocol with slot reuse. It gave an expression of the average packet delay but it only considered the effects from the nodes situation such as sleep or awake. And the effects give by the other parameters such as the packet size, packet generation rate, the priorities of the packets' applications were not considered. Similar to the previous one, a finite queuing model for the sensor nodes was proposed and the network performance of contention-based sensor networks with the synchronous wakeup patterns was studied\cite{WSNmdl4}\cite{WSNmdl4.1}.

There are some other research work which is based on the queueing theory and related to the energy in the WSN: Ref.\cite{WSNmdl5} modelled the solar radiation process by a stochastic process (i.e., a Markov chain) and a linear battery model with relaxation effect was used to model the battery capacity recovery process. Developed based on a multidimensional discrete-time Markov chain, the presented model is used to analyze the performances of different sleep and wakeup strategies in a sensor/mesh node.

Some others focused on the relay link choose and packet lost: Ref.\cite{WSNmdl6} presented a novel queueing analytical framework to study the tradeoff between the energy saving and the QoS at a relay node. Specifically, by modelling the bursty traffic arrival process as a MAP (Markovian arrival process) and the packet service process as having a phase-type (PH) distribution, the authors modelled each node as a MAP/PH/1 nonpreemptive priority queue. It focused more on one node and more on looking for the tradeoff between the sleep mechanism which is energy saving and the packet's lost but not the average packet delay which is the WSN's applications performance measurement.

Some others focused on the WSN's performance: To investigate the end-to-end delay distribution, Ref.\cite{WSNmdl7} a comprehensive cross-layer analysis framework, which employs a stochastic queueing model in realistic channel environments, was developed. Although it was a great work with a lot of analysis and experimental work, it studied the delay distribution my modelled the CSMA/CA MAC protocol only. Another model\cite{WSNmdl8} which studied the MAC protocols is also proposed and it focused on the queueing analysis of polling based sleep-wake cycles.

The rest of the exist work deployed in all the other subareas of WSN, such as: Ref.\cite{WSNmdl10} investigated the use of proactive multipath routing to achieve energy-efficient operation of ad hoc wireless networks while Ref. \cite{WSNmdl11} considerd an on-demand data collection scenario, in which sensor nodes broadcast service requests when their buffer is about to be full. On receiving such requests, the ME moves toward the sensor nodes to collect data, and uploads the data to the sink when possible. An M/G/1 queue-based analytical model is presented, and analytical results on several important system performance metrics are derived. Although it considered the multi-hops, it did not provide a clear analysis of the effect given by the parameters of WSN.

Ref.\cite{WSNmdl12} developed queuing and Markov chain models for clusters that are interconnected via the slave-slave bridge under carrier-sense multiple access with collision avoidance and guaranteed time slots bridge access modes, respectively. It also addressed acknowledged and non acknowledged data transfers under both bridge access modes. It only gave the effect analysis given by the parameters, but all the others are not. A novel power-saving scheme was proposed in Ref.\cite{WSNmdl13} to alleviate the EHP based on the D-policy M/G/1 queuing model. With little management cost, the proposed queue-based powersaving technique can be applied to prolong the lifetime of sensor network economically and effectively. As most of the modelling work, it only analyzed the energy consumed in WSN.

Furthermore, in wireless sensor networks (WSNs), nodes are often scheduled to alternate between a working mode and a sleeping mode from the energy efficiency point of view. When delay is tolerable, it is not necessary to preserve network connectivity during activity (working or sleeping) scheduling, in order to enable more sensors to be switched to sleeping mode and thus more energy savings. In Ref.\cite{WSNmdl14}, authors modelled and analyzed nodal behavior in such delay-tolerant WSNs (DT-WSNs) with MG1K model. Ref. \cite{WSNmdl15} focused on how to arrange and schedule the movement of mobile elements throughout the sensing field is of ultimate importance. In that paper, the online scenario where data collection requests arrive progressively is investigated, and the data collection process is modeled as an M/G/1/c-NJN queuing system, where NJN stands for nearest-job-next, a simple and intuitive service discipline. Ref. \cite{WSNmdl16} developed a stochastic model for data in a wireless sensor network using random field theory. The model captures the space-time behaviour of the underlying phenomenon being observed by the network. The authors of Ref.\cite{WSNmdl16} studied the size and spatial distribution of the regions of the network that sense statistically extreme values using the theory of extreme excursion regions. Analytical expressions are found for the average size of the data load in a variety of scenarios.

The whole lifetime of WSN is deteriorated because of such an uneven node power consumption patterns, leading to what is known as an energy hole problem (EHP). The EHP is an embedded risk and would compromise the lifetime security of WSN. Most research works have focused on how to optimally increase the probability of sleeping states using various wake-up strategies to prolong the lifetime of WSN while Ref.\cite{WSNmdl17} proposed a novel power-saving scheme to alleviate the EHP based on the N-policy queuing theory. With little or no extra management cost, the proposed queue-based power-saving technique can be applied to prolong the lifetime of the WSN economically and effectively. Like most of the work introduced above, it only focused on energy consumption and no WSN performance analysis.

Ref.\cite{WSNmdl18} was trying to find a solution to the problem that how can we optimize the number of deployed nodes (sensor node and sink) with a QoS guarantee (minimal end-to-end delay and packet drop). It proposed a deployment optimization model for non-beacon-mode 802.15.4 sensor networks by using the M/M/1 queuing model. Different from the others, it considered the effect given by the sensor nodes and the sinks in WSN to analyze effect to the average packet delay. And in Ref.\cite{WSNmdl19}, the authors outlined an approach to improve the lifespan of a wireless sensor network by introducing a variant to standard sleep synchronization protocols. A multi-layered architecture was used. To ensure even higher scalability and lower message size in any particular layer, number of layers is limited to four and each layer is broken into grids.

In summary, many researchers have done great contributions in the modelling work with queueing theory. But the studies on this point is still on its first steps. Most of them studied the energy problem and some of them studied the packet delay. Only a few of them studied the throughput of the WSN and seldom of them researched the multi-hop issues. The researches which focused on the multi-priorities packets transmission and multi-sink in the WSN are nearly none, neither the effect analysis of the parameters give to the WSN's performance. Therefore, the one of the contributions of this thesis is giving a first attempt to model the WSN in the multi-hop, multi-priorities and multi-sink conditions with packet delay and throughput analysis.

The existed researches' features can also be found in Table \ref{t:WSN models} and all the abbreviations can be found in the List of Abbreviations or the footnote under the table.

Compared with the modelling work of WSN, the ones of PON mainly focused on the average packet delay analysis. The following part introduces the existed modelling work of PON and also a Table is given to summarize these work.

Ref.\cite{PONmdl1} considered and analysed a generic multi-priority dynamic bandwidth allocation (DBA) algorithm for TDM PONs serving multimedia traffic in an upstream link. PON traffic was served strictly according to its priority. It considered this DBA algorithm using two approaches: (i) the algorithm assigns a fixed service quantum to each priority service and (ii) different service quanta are assigned to different priority services. The mean message delay was evaluated using a multi-queue processor sharing (MPS) model and an MPS with Heterogeneous Traffic (MPS-HT) model for the two approaches respectively. The MPS model is a classical processor sharing model limited by the critical assumption that there is egalitarian service sharing among all users, which is inefficient for multimedia applications in PONs. Also, the authors extended the MPS model to a general MPS-HT model that enables the analysis of message delay performance in the case where the service quanta may be different for different services. It made great work on the analysis, especially the multi-priority services, but it only considered the fixed grant sizing in PON which is not effective one and not useful for most of the PON used in the industry.

In an access node to a multi-service network (e.g., a base station in an integrated services cellular wireless network or the optical line terminal (OLT) in a broad-band passive optical network (PON)), the output link bandwidth is adaptively assigned to different users and dynamically shared between isochronous (guaranteed bandwidth) and asynchronous traffic types. The bandwidth allocation used in Ref.\cite{PONmdl2} was effected by an admission controller, whose goal was to minimize the refusal rate of connection requests as well as the loss probability of cells queued in a finite buffer. The optimal admission control strategies were approximated by means of back propagation feed forward neural networks, acting on the embedded Markov chain of the connection dynamics and the neural networks operate in conjunction with a higher level bandwidth allocation controller which performs a stochastic optimization algorithm.

In PON, the largest part of the energy consumption is due to the equipments at the customer premises. Ref.\cite{PONmdl3} proposed a method based on queueing theory to compute the ONU sleep time for cyclic sleep with service-based variable sleep period in energy efficient PONs. Such method computes the maximum allowed sleep time based on the services subscribed by the ONU and their frame delay constraints. Both average frame transfer delay and frame delay variation were considered.

Ref.\cite{PON} provided derivations of closed-form expressions of the mean packet delay for the gated service and the limited service of dynamic bandwidth allocation in Ethernet passive optical networks (EPONs). Based on the M/G/1 queueing analysis framework of a multi-user cyclic polling system, the authors derived the mean packet delay expressions by modifying the expressions for the reservation time component of the total delay. Also they extended the analysis to demonstrate how the limited service can protect packets transmitted by a light-load user from having excessive delays due to high traffic loads from other users in the same EPON. It also proved that in selecting the maximum length of a scheduling cycle for the limited service, there is a tradeoff between the mean packet delay under uniform traffic and the guaranteed upper bound on the mean packet delay under non-uniform traffic.

Ref.\cite{PONmdl4} investigated the mean queue length and the mean packet delay of a dynamic bandwidth allocation (DBA) scheme in an Ethernet passive optical network (EPON). It focused on the interleaved polling system with a gated service discipline with the assumption that input packets arrive at an optical network unit (ONU) according to Poisson process. It used a continuous time queueing model in order to find the queue length distribution of the gated interleaved polling system with the first stage input queue and the second stage transmission queue. Similar with Ref.\cite{PONmdl4}, Ref.\cite{PONmdl5} proposed an optical line terminal (OLT) centric dynamic bandwidth allocation (DBA) scheme based on individual requests from service queues in optical network units (ONUs) for a quality-of-service (QoS) aware Ethernet passive optical network (EPON). The goal was to provide fairness for allocating bandwidth among different service classes based on their service level agreements (SLAs). The proposed DBA scheme makes use of the excess bandwidth of lightly loaded queues to meet the bandwidth demand of heavily loaded queues. But both these two work focused the gated grant sizing in PON only.

Ref.\cite{PONmdl6} investigated the performance of a simple PON configuration, in respect of the end-to-end packet delay. The network supports multiple service-classes with priorities, in terms of the number of packets that can be transmitted in each transmission period. The authors calculated the mean queueing delay of a packet, by taking into account the fact that packets are serviced in batches. The queueing delay was defined through the formulation of two queuing models: an M/D/1 model for the batches, and an M/D/m model for the individual packets. They also studied the effect of the packet arrival rate of different service-classes, and the effect of the distribution of the packets into a frame, on the end-to-end packet delay.

Interleaved Polling with Adaptive Cycle Time (IPACT) is one of the earliest proposed schemes for bandwidth distribution in Ethernet PON (EPON) and has been extensively used as a benchmark by many subsequent allocation schemes. In Ref.\cite{PONmdl7}, the authors proposed an analytical model which yields approximate values for mean queue length and mean packet delay in an EPON with the Gated IPACT scheme under the assumption of Poisson arrivals. But no special queueing model is clarified. Ref.\cite{PONmdl8} proposed a mechanism to increase the energy efficiency of the upstream channel in TDM-base PONs. Essentially, the ONUs are encouraged to accumulate traffic and transmit data bursts just by increasing the cycle time values artificially. The guard time is enlarged to avoid the case where ONUs are queried by the OLT and have none or few packets to transmit, thus allowing more time to sleep until the next cycle time. This strategy has however the downside effect of a substantial increase in the queueing delay experienced by packets. A basic analysis to maximise the power savings for a given average delay target experienced by the packets are also provided in it.

Furthermore, the authors of Ref.\cite{PONmdl9} attempted to construct a mathematical model of the IPACT scheme under the gated service discipline: For $N=1$ ONU, they derived closed-form expression for the steady state grant size; For $N>1$ ONUs, they considered separately a small and a large load-distance ratio. For the former case, the $N=1$ ONU model holded even for $N>1$. For the latter case, the authors found a closed form expression for the grant size. Their model shows a reasonable match with the values obtained from simulation for the steady state queue size and hence the throughput and delay.

More detailed comparison of the modelling work of PON is shown in Table \ref{t:PON models} as follo