Real time scheduling in Intelligent Transportation Systems562645/FULLTEXT01.pdf · To achieve an e cient network utiliza-tion while ensuring acceptable performance, it is instrumental

Degree project in

Real time scheduling in Intelligent Transportation Systems

GIADA MEOGROSSI

Stockholm, Sweden 2011

XR-EE-RT 2012:029

Automatic ControlMaster's thesis

Abstract

In recent years Intelligent Transportation Systems leveraged numerous

applications in vehicular networks. To achieve an efficient network utiliza-

tion while ensuring acceptable performance, it is instrumental to design the

transportation systems and to optimize network resources. In this thesis,

we focus on real time scheduling algorithms for Intelligent Transportation

Systems. The proposed scheduling algorithms consider TDMA based MACs,

and aim at minimizing the average delay. Each algorithm allocates the re-

sources based on the channel conditions: a user with good channel should

transmit for longer time than a user with bad channel condition. The schedul-

ing algorithms are devised by solving a related linear programming problem.

It is shown how the average delay can be minimized by using appropriate

multi-hop configurations.

i

Contents

Abstract i

List of figures vi

1 Introduction 1

1.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Outline of this work . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Wireless channel 4

2.1 The wireless channel . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Wireless channel model . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Delay spread and coherence bandwidth . . . . . . . . . 9

2.2.2 Doppler spread and coherence time . . . . . . . . . . . 10

2.2.3 Wireless channel typologies . . . . . . . . . . . . . . . 10

3 Wireless ad hoc networks 12

3.1 Mobile ad hoc network (MANET) . . . . . . . . . . . . . . . . 12

3.2 Vehicular Ad Hoc Network (VANET) . . . . . . . . . . . . . . 15

3.3 Communication protocols . . . . . . . . . . . . . . . . . . . . 17

3.3.1 Standard IEEE 802.11 . . . . . . . . . . . . . . . . . . 18

3.3.2 Physical layer . . . . . . . . . . . . . . . . . . . . . . . 18

3.3.3 MAC . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3.4 Standard IEEE 802.11 and its extensions . . . . . . . . 21

3.3.5 Standard IEEE 802.11e . . . . . . . . . . . . . . . . . . 22

ii

Contents iii

3.3.6 Standard IEEE 802.11p . . . . . . . . . . . . . . . . . 23

3.4 Space time division multiple access (STDMA) . . . . . . . . . 24

4 Optimization 27

4.1 Convex optimization . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Quadratic problems . . . . . . . . . . . . . . . . . . . . 29

4.1.2 Linear program . . . . . . . . . . . . . . . . . . . . . . 29

4.2 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2.1 Lagrange dual of quadratic problem . . . . . . . . . . . 32

4.2.2 Lagrange dual of linear program . . . . . . . . . . . . . 33

4.3 Branch and Bound . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3.1 Branching rules . . . . . . . . . . . . . . . . . . . . . . 37

4.3.2 Exploration tree rules . . . . . . . . . . . . . . . . . . . 37

4.4 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . 38

5 Scheduling algorithms 41

5.0.1 Scheduling for vehicle-infrastructure communications . 41

5.0.2 Scheduling in real time traffic . . . . . . . . . . . . . . 42

5.0.3 A comparison . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 Proposed scheduling algorithm . . . . . . . . . . . . . . . . . . 44

5.1.1 Configuration 1 . . . . . . . . . . . . . . . . . . . . . . 45

5.1.2 Configuration 2 . . . . . . . . . . . . . . . . . . . . . . 47

5.1.3 Configuration 3 . . . . . . . . . . . . . . . . . . . . . . 49

5.1.4 Configuration 4 . . . . . . . . . . . . . . . . . . . . . . 52

5.1.5 Configuration 5 . . . . . . . . . . . . . . . . . . . . . . 55

5.1.6 Possible generalizations . . . . . . . . . . . . . . . . . . 58

6 Simulations and results 61

6.1 Configuration 1 results . . . . . . . . . . . . . . . . . . . . . . 62




Contents iv


Conclusions and future works 73

Bibliografy 77

List of Figures

2.1 Multipath propagation . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Probability density function of Rayleigh distribution . . . . . 8

2.3 Probability density function of Rice distribution . . . . . . . . 9

3.1 MANET, courtesy of [12] . . . . . . . . . . . . . . . . . . . . . 13

3.2 VANET, courtesy of [8] . . . . . . . . . . . . . . . . . . . . . 17

3.3 MAC architecture . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.4 Frame architecture . . . . . . . . . . . . . . . . . . . . . . . . 20

3.5 MAC frame, courtesy of [5] . . . . . . . . . . . . . . . . . . . . 21

3.6 CSMA/CA vs STDMA courtesy of [22] . . . . . . . . . . . . . 25

4.1 Geometric representation, courtesy to [13] . . . . . . . . . . . 32

4.2 Block diagram for Branch and Bound algorithm . . . . . . . . 35

4.3 Solutions tree . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Search tree of Breadth first . . . . . . . . . . . . . . . . . . . . 38

4.5 Search tree of Depth first . . . . . . . . . . . . . . . . . . . . . 38

4.6 Block diagram for Dynamic program algorithm . . . . . . . . 40

5.1 Possible representation of configuration one . . . . . . . . . . . 45

5.2 Possible representation of configuration two . . . . . . . . . . 47

5.3 Possible representation of configuration three . . . . . . . . . . 49

5.4 Possible representation of configuration four . . . . . . . . . . 52

5.5 Possible representation of configuration five . . . . . . . . . . . 55

5.6 Generalization with three users . . . . . . . . . . . . . . . . . 59

v

List of figures vi



6.1 Geometric representation of Eq. (5.3) . . . . . . . . . . . . . . 62

6.2 Optimal tau configuration 1 . . . . . . . . . . . . . . . . . . . 63





6.7 Geometric representation of Eq. (5.11) . . . . . . . . . . . . . 68


6.9 Geometric representation of Eq. (5.13) . . . . . . . . . . . . . 71


Chapter 1

Introduction

In recent years many technological breakthroughs have occurred, in elec-

tronics, signal processing and communications. A significant development

was achieved by the Intelligent Transportation Systems (ITS). The increas-

ing number of vehicles has greatly increased the possibility of traffic jams,

accidents, as well as CO2 emissions. The Intelligent Transportation Systems

improve the current transportation systems in many aspects: by increasing

the information that reaches the driver, and by reducing the driving loads

and route-enhancing management. One of the main causes of congestion are

car accidents. To reduce accidents we can introduce automation systems in

the car: for example an Intelligent Transportation System module may col-

lect information from adjacent vehicles using appropriate sensors, then sends

a text message to inform the driver. Another application is a speed control.

The adaptive cruise control (ACC) has a function to control the speed of the

host vehicle depending on the inter-vehicle distance and the relative speed

to a preceding one. The inter-vehicle distance and the relative speed are

measured with a lidar or a millimeter wave radar [16]. To achieve this goal

an important role is played by vehicular communications and right resources

allocation.

1

1.1 Problem formulation 2

1.1 Problem formulation

An appropriate allocation of resources is the basis of the implementation

in the aforementioned applications.

The main goal of this thesis is to study scheduling policies. The objective

of scheduling is to allocate the resources optimally, reducing inefficiencies

and waste of resources.

The first objective is to study the vehicle to vehicle communication(V2V)

and vehicle to infrastructure communication(V2I), which are widely described

in Chapter 3. The basic idea is the following: a vehicle communicates with

other vehicle. The vehicle more prone to communication, broadcast to road-

side station, which will manage optimally the resources.

In this thesis a new scheduling algorithm for optimal resources allocation

is proposed. The core of the problem is formulated as an optimization prob-

lem, and is presented in Chapter 5, after brief but thorough introduction to

the optimization tools in Chapter 4.

The idea can described as follows. It is supposed to have a certain number

of users and time slots. Each time slot is divided into appropriate fractions of

time, associated to each user. Each fraction is a function of the transmission

channel, whose general description is presented in Chapter 2. The channel

influences the transmission as follows: a user who has a better channel con-

ditions (meaning higher gain), should be able to transmit for a longer time

than a user with a worst channel.

For the purposes of this thesis we consider different configurations, and

for each of one an appropriate optimization problem has been formulated

and solved.

1.2 Outline of this work

The rest of thesis is organized as follows.

• Chapter 2 is a brief description of wireless channel communication.

1.2 Outline of this work 3

• In Chapter 3, we describe communication protocols of standard IEEE802.11

with particular interest in 802.11e, which emphasizes the QoS, and in

802.11p, used for vehicular communications. In the same chapter a

description of MANETs and VANETs is also presented.

• Chapter 4 focuses on optimization tecniques, with particular attention

to Linear Program problems.

In the same chapter a method to solve Linear Program problems, called

Branch and Bound, is described. In this chapter a brief description of

Dynaminc Programming is also presented.

• Chapter 5 is the core of this thesis. In this chapter after analyzing

the existing scheduling algorithms a new scheduling algorithm with

possible application in the ITS is formulated.

• In Chapter 6 we show the results of implementation in MATLAB of

algorithms, and a results validation.

Chapter 2

Wireless channel

Essential for vehicle to vehicle communications and vehicle to infrastruc-

ture communications is the wireless channel. If the propagation happens in

free space, there are not edges between transmitter and receiver, and the

atmosphere is considered as a uniform and not absorbing medium:

PR = L (d)PT

L(d)= GRGT( λ4πd

)2

Where PR is the received power, PT is the trasmitted power, L is the path

loss, λ the wave lenght, GT is transmitting antenna gain and GR receiving

antenna gain. This easy model is not adequate to describe a wireless channel.

The motivation of this affirmation will be shown in the following section.

2.1 The wireless channel

A wireless channel has two features: it is time varying, and is hard to

individuate a exact mathematical model to describe it. The wireless channel

can be modelled only statistically, describing an average behavior of the

system. The radio propagation is characterized by two phenomena:

• multipath propagation shown in Figure 2.1

4

2.1 The wireless channel 5

• time varying conditions: the main reason for the time variance is the

user’s motion respect to the base station.

Figure 2.1: Multipath propagation

The wireless channel variation can be divided in two types:

• large scale: usually is frequency indipendent. The signal power de-

creases proportional to 1dk

, where k=2 in open space, and k=4 in harsh

environments.

• small scale fading: is frequency dipendent. The small scale fading

describes the intesity variation of signal, due to constructive and de-

structive interferences.

The signal propagation in a wireless channel happens for multipaths. In

multipath propagation the received signal is a combination of the line of

sight(LOS) and a non line of sight(NLOS) paths. LOS refers to the possible

direct path. NLOS is due to signal interaction with the surrounding envi-

ronment. The electromagnetic waves can interact with the environment in

three manners. The interaction depends on λ, and it is possible distinguish

three phenomena:

2.2 Wireless channel model 6

• Reflection: the signal encounters objects bigger than λ. The model

used for this phenomenon is the geometrical optics.

• Diffraction: the signal encounters objects with sharp edges. The wave-

front is altered and acts as a secondary source of spherical waves.

• Scattering: the signal encounters objects that have the same dimension

of λ.

2.2 Wireless channel model

Considering the multipath propagation it is possible to formulate a math-

ematical model for wireless channel. Consider a signal s(t) with carrier fre-

quency fc. The signal s(t) can be expressed using the complex envelope s′(t):

s(t) = Re[s′(t)ej2πfct]

The received signal r(t) is linear and it can be written as the sum of routes:

r(t) =∑

n(αn(t)s[t− τn(t)])

With some mathematical manipulations and considering the complex enve-

lope r′(t), it is possible obtained the following expression [14]:

r′(t) =∑

(αn(t)ejφn(t)s′[t− τn(t)]) with φn = 2πfcτn(t)

αn is the amplitude of received rays, φn is the phase delay and τn propagation

delay. The propagation delay can assume the following values:

• 1 µs urban environment

• 10 µs extraurban environment

This relation can be expressed in the following form:

r′(t) =∫ +∞−∞ c′(τ ; t)s′(t− τ)dτ

c′(τ ; t) =∑

n αn(t)ejφn(t)δ[τ − τn(t)]


This expression is the LTV in-out relation that can be expressed in frequency

domain:

C ′(f ; t) =∫ +∞−∞ c′(τ ; t)e−j2πfτdτ

The expression of LTV can be written also as follows:

c′(τ ; t) = c′LOS(t; τ) + c′NLOS(τ ; t)

where

c′LOS(t; τ) = α0(t)ejφ0(t)δ[t− τ0(t)]

c′NLOS(τ ; t) =∑

n 6=0 αn(t)eφ0(t)δ[τ − τn(t)]

The NLOS component is stochastic while the LOS is deterministic. The

system defined in the expression of LTV is an aleatory LTV with gaussian

impulse response. A guassian process is characterized by its mean value and

autocorrelation function. Hence, the wireless channel can be described by

mean value and autocorrelation function of c′(t, τ). Assuming that αn(t) and

φn(t) are indipendent the average can be written as follows:

E[c′NLOS(τ ; t)] =∑

n E[αn(t)]E[ejφn(t)] = 0

while

E[c′(τ, t)] = c′LOS(τ, t)

In NLOS propagation, the signal attenuation follows a Rayleigh distribution

with d.d.p. shown in Figure 2.2:

f(ρ) = ρσ2 e− ρ2

2σ2

The phase φ is described by a random variable uniformly distributed in

[−π, π]. The d.d.p. is:

f(φ) = 12πrect φ

2π


In LOS propagation the signal attenuation follows a Rice distribution with

d.d.p. shown in Figure 2.3:

f(ρ) = ρσ2 e−kI0(

√2k ρ

σ)

I0 is a Bessel function of first kind and order zero, k is Rice factor:

k =ρ21/2

σ2

k is the ratio between average power of the direct component and average

power of the widespread component. k=0 is a NLOS propagation. The

phase φ does not follow a uniform distribution. Its polaritazion varies as a

function of E[·]. The model described above, allows to describe statistically

the wireless channel. The followig parameters characterize a wireless channel:

• delay spread

• coherence band

• doppler spread

• coherence time

Figure 2.2: Probability density function of Rayleigh distribution


Figure 2.3: Probability density function of Rice distribution

2.2.1 Delay spread and coherence bandwidth

Suppose to transmit an ideal impulse s′(t) = δ(t):

r′(t) =∑

n αn(t)ejφnδ[t− τn(t)]

r′(t) is a sequence of impulses centred in τn(t). The channel is dispersive

in time becuase s′(t) is dispersed on larger time interval. It is possible to

measure the time dispersion as the difference between the longest and shortest

path:

Tm = τmax − τmin

This difference is called delay spread. The delay spread depends on the fre-

quency coherence, which shows how quickly the channel varies in frequency.

The coherence bandwidth is defined as follow:

Bc = 1Tm

The coherence bandwidth is the maximum time interval in which the fre-

quency response is constant.

It is possible to write Bc �W, where W is the signal bandwidth. This rela-

tion highlights that the channel is not temporally dispersive if its coherence


bandwidth is larger than the signal bandwidth. The input-output relation

for a not temporally dispersive channel is [14, 18, 17]:

r′(t) ≈∑

n αn(t)ejφn(t)]s′(t)

2.2.2 Doppler spread and coherence time

The frequency dispersion depends on the transmitter and receiver mobil-

ity. There are two phenomena: a spectrum shift relative to LOS component,

and a dispersion relative to NLOS component. This phenomena are typical

of Doppler effect. The Doppler spread Bd can be compute as follow:

Bd = fcvc

Where v is the velocity and c is the light velocity. The reciprocal of Doppler

spread is the coherence time:

Tc = 1Bd

The coherence time is a temporal interval in which the channel is constant.

Consider a signal s(t) with bandwidth W if:

W� Bd

the Doppler spread is negligible and the channel can be considered as LTI

system.

2.2.3 Wireless channel typologies

On the basis of the previous consideration, it is possible to characterize

the wireless channel as follows:

• W� Bc −→ flat fading

• W� Bc −→ frequency selective fading

• Tm � Tc −→ unspread


• Tc � τ−→ slow fading

• Tc � τ −→ fast fading

In conclusion it is possible to say that the wireless channel is very impor-

tant for radio communication and its characterization is not trivial. The only

characterization of wireless channel is not sufficient for a correct communica-

tion between the users. It is necessary define an appropriate communication

protocol and the best network configuration possible. In the next chapter

some wireless tecnologies including the ad hoc networks will be studied .

Chapter 3

Wireless ad hoc networks

An Ad Hoc Network is a network with nodes that configure themselves

without reference to a structure. The operation are managed by distributed

algorithms because there is no infrastructure. To improve coverage, is possi-

ble to use multi-hop, where nodes act as relay nodes. The possibility to use

relay nodes can significantly reduce power consumption, but there can be

problems related to delays. An Ad Hoc Network can be easily reconfigured.

Thanks to its distributed nature the Ad Hoc Network is particularly robust.

3.1 Mobile ad hoc network (MANET)

A MANET is defined as a system of mobile nodes connected by wireless

links. The nodes are free to move in the space, so the network configuration

can change rapidly and unpredictably. A MANET can constitute an infras-

tructure, but this is not strictly necessary. The IP protocol is not convenient

for the routing. The IP address and subnetting are not suitable for MANET

because the phisical address not corrispond to geografic address. In Figure

3.1 is represented a MANET.

12

3.1 Mobile ad hoc network (MANET) 13

Figure 3.1: MANET, courtesy of [12]

Routing protocols for mobile ad hoc network

The routing in Ad Hoc Network results be really complex. The routing is

complex because the network is not static but dynamic, therefore the routing

needs to be dynamically reconfigured. The main MANET routing algorithm

are divided in:

• Flooding

– The packet is broadcasted to every node in the communication

range. When a node receives a packet, it broadcasts it to its

neighboard. The exchange continues until the packet arrives to

destination. This technic is good for networks with few packets

and high mobility, but presents an high power consumption.

• Proactive

– Centralized proactive algorithms

∗ A central station is informed on channel and network condi-

tion from nodes. When the information has been received, the

station processes the routing tables for all nodes and trans-

mits it back to them. The tables processing requires time.

3.1 Mobile ad hoc network (MANET) 14

This algorithm results good for networks with few nodes and

slowly variable.

– Source driven algorithms

∗ Each node knows the network condition. Knowing the net-

work state the node can decide the best route to send its

packets. The central unit is periodically informed on network

conditions.

– Proactive distributed algorithms

∗ This algorithms supposes that the informations are exchange

among neighboring nodes. The distributed algorithm fits well

to dynamics of network and presents low overhead in com-

munication. The routing tables are obtained according to

specific criteria. There are two mail types of algorithms for

IP routing: Distance Vector Routing and Link State Rout-

ing. Basically, Distance Vector protocols determine best path

on how far the destination is, while Link State protocols are

capable of using more sophisticated methods taking into con-

sideration link variables, such as bandwidth, delay, reliability

and load.

• Reactive

– The routing is done only if the nodes requests the transmission.

The route selection is done only when the node decides to trasmit,

the information are preserved until the node stops the packets

transmission. The reactive algorithms present a longer delay in

packets delivery, because the node does not know precisely the

path when it starts the transmission. The most popular reac-

tive algorithms are: Ad hoc on demand distance vector routing

(AODV), and Dynamic source routing (DSR).

∗ Dynamic source routing (DSR): when a node decides to trasmit

it sends a route request to neighboring nodes. When a neigh-

3.2 Vehicular Ad Hoc Network (VANET) 15

boring nodes receive a request two events can happen: if the

node does not know the route to reach the destination adds

its address to packets and propagates the request to neigh-

bor. When the packet arrives to destination, the delivery is

acknowledge through a route reply.

– Ad hoc on demand distance vector routing (AODV)

∗ It is able to adapt to rapidily links changes. It ensures there

are no routes with the use of cyclic sequence numbers, and

avoiding the problem of counting to infinity, it ensures a fast

convergence when the network topology changes. Each node

keeps in memory a routing table [21].

It is possible to create a hybrid protocol. An hybrid protocol is a pro-

tocol that presents proactive and reactive characteristics. The most popular

MANET applications are:

• data networks

• home networks

• sensor networks

• distributed control systems.

A typical MANET example is VANET.

3.2 Vehicular Ad Hoc Network (VANET)

VANET is a type of MANET, that uses cars as nodes to form the net-

work. In a VANET vehicle to vehicle communication (V2V) and vehicle to

infrastructure communication (V2I) are possible. The possibility of vehicle

to communicate with other vehicles or with infrastructure is important to re-

alize safer streets. A vehicle can communicate with other vehicles to inform

3.2 Vehicular Ad Hoc Network (VANET) 16

them on street conditions, so it is possible to avoid traffic jams and accidents.

It is possible to identify four main applications of VANET:

• active security

– alerts to dangerous street condition: limited visibility,work in

progress, etc...

– alerts to possible collision: lane change, slow forward vehicle, etc..

– accidents: alert and automatic SOS.

• pubblic service security

– support for traffic control

• driving aid

– traffic management

– easy guide

• business entertainment

A typical example of VANET is shown in Figure 3.2

The VANETs communication modes are the following:

• beaconing: each node periodically transmits informations to the neigh-

booring nodes.

• geobroadcast: a node transmits informations to the neighbooring nodes,

which retransmit the informations only in a designed area.

• routing unicast: the network is used to transmit the message point

to point. It is not possible to use classic routing algorithms because

the network is dynamic. In VANET there is a Location Service. If

the Location Service is reactive, when reaches the destination has also

found the routing. If the Location Service is proactive the routing

discovery is done subsequently.

3.3 Communication protocols 17

Figure 3.2: VANET, courtesy of [8]

• information dissemination: the message is memorized for long time and

subsequently retransmitted.

• information aggregation: the received messages are elaborated and ag-

gregated, to enrich the information of the receiver vehicle.

The application of VANETs can be classified in:

• emergency: in this case the messages are not elaborated.

• information and driving aid: in this case the messages are elaborated

before retransmission [3, 10, 7]

3.3 Communication protocols

A communications protocol is a formal description of digital message for-

mats and the rules for exchanging those messages in or between computing

systems and in telecommunications. A protocol defines the syntax, seman-


tics, and synchronization of communication, and the specified behaviour is

typically independent of how it is to be implemented.

3.3.1 Standard IEEE 802.11

The IEEE 802.11 standard defines a standard for ad hoc nets developed by

the group 11 of IEEE 802. The 802.11 family has three dedicated protocols for

the transmission of information (a, b, g). Other protocols are improvements

of the above. Important for ad hoc nets is the Basic Service Set (BSS).

It characterizes the set of basic services. The BSS consists of a series of

stations that use the same protocol defined MAC. The MAC protocol may

be centralized or distributed. The set of services that are extended, the

Extended Service Set (ESS), consists of two or more BSS interconnected. In

the IEEE 802.11 standards, there are three types of stations whose difference

lies in mobility:

• no transition: fixed station

• BSS transition: the stations can move between BSS internal to the

same ESS

• ESS transition: the station can move between BSS belong to different

ESS

3.3.2 Physical layer

The 802.11 physical layer (PHY) is the interface between the MAC and

the wireless media, where frames are transmitted and received. The PHY

provides three functions. First, the PHY provides an interface to exchange

frames with the upper MAC layer for transmission and reception of data.

Secondly, the PHY uses signal carrier and spread spectrum modulation to

transmit data frames over the media. Thirdly, the PHY provides a carrier

sense indication back to the MAC to verify activity on the media. 802.11


provides the following different PHY definitions: Frequency Hopping Spread

Spectrum (FHSS) and Direct Sequence Spread Spectrum (DSSS) [19].

• Direct Sequence Spread Spectrum operates in the ISM band at 2.4 GHz.

The adopted spectrum is divided into 14 channels of 22MHz each. If

the data is transmitted at 1Mbps or 2Mbps is spreading with the long

sequences 11 chip.

• Frequency hopping spread spectrum operating in the ISM band at 2.4

GHz with data rates of 1 Mbps or 2 Mbps.

3.3.3 MAC

The Figure 3.3 represents a MAC architecture. The MAC layer defines

two different access methods: distributed (DCF) or centralized (PCF). The

data transfer without time constraints is obtained using DCF (Distribution

Coordination Function) while in system with time constraints is obtained

using PCF (Point Coordination Function). The DCF is the lowest layer

and operates with a contention algorithm. The PCF is a higher level than

the DCF, on which it operates, and provides a service without contention.

Figure 3.3: MAC architecture

The information basic unit exchanged between different MAC entities is the

frame. There are three types of frames:

• data frame: used for data transmission


• control frame: used for the medium access

• management frame...

Each frame is divided into subtypes. The Figure 3.4 represents the frame

architecture:

The preamble is dependent on the physical level and includes Synch and

Figure 3.4: Frame architecture

SFD. Synch is a sequence of 80 bits, with 0 and 1, used to physical level to

select the optimal antenna, while SFD is a binary sequence of 16 bits, used

to determine the beginning of the frame.

PLCP header is always transmitted at 1 Mbps and contains logical informa-

tion used by the physical layer to decode the frame.

The Data MAC presents the structure shown in Figure 3.5

CRC is a field of 32 bits, containing a 32 bits Cyclic Redundancy Check .

The system time unit is the time slot, whose duration depends on the physical

layer. The time intervals between transmissions are called IFS (Interframe

Spaces). There are four IFS types:

• SIFS: separates the same dialogue transmission;

• PIFS: offers priority to PCF;

• DIFS: used in the stations attending free channel;

• EIFS: used in the stations whose phisical level notifies the MAC layer

that a transmission has not been understood.

The DCF uses a carrier sense multiple access (CSMA) algorithm: when a

station wants to transmit it must verify that the channel is free. There are


Figure 3.5: MAC frame, courtesy of [5]

two DCF types: base DCF and DCF with handshaking. The base DCF

operates as following: when a transmitter wants to send a packet it listens

to the channel for a DIFS time: if the channel is free the frame can be sent,

and the AP sends an ack upon correct reception within SIFS time. When

the channel is busy, the transmitter must wait until the channel is free.

The transmissions may fail due to collisions: a collision avoidance scheme is

implemented based on ARQ stop and wait with a back off procedure [19].

The DCF with handshaking allows channel reservation to avoid collisions. In

PFS mode, the central unit performs the queries handled by PIFS. The PIFS

controls the channel and blocking traffic during the asynchronous queries and

waits for response. It is possible to avoid an excessive traffic stop by using a

time interval called superframe.

3.3.4 Standard IEEE 802.11 and its extensions

The 802.11 standard for WLANs is a family of communication proto-

cols. Currently the family consists of 11 samples: 802.11, 802.11a, 802.11b,

802.11c, 802.11d, 802.11e, 802.11f, 802.11g, 802.11h, 802.11i, 802.11j [19].


• 802.11 original sample, 1997;

– datatransfer 1 or 2MBps;

• 802.11a Physical layer extension, 1999;

– datatransfer 54MBps;

• 802.11b Physical layer extension, 1999;


• 802.11g Physical layer extension, 2003;


• 802.11n;


• 802.11p 802.11a extension. It is used in VANET;


There are other 802.11 extensions already mentioned.

3.3.5 Standard IEEE 802.11e

The 802.11e is an 802.11 extension to support Quality of Service (QoS).

The primary purpose of QoS is to protect high priority data from low pri-

ority data. The IEEE 802.11e allows two modes to support the application

with QoS. Since DCF and PCF do not differentiate between traffic types or

source, the IEEE developed enhancements in 802.11e to both coordianation

modes to facilitate QoS. The enhancement to DCF (EDCF) introduces the

concept of access categories. By EDCF the station with high priority traf-

fic waits less time than a station with a low priority traffic. The priority

differentation is possible altering the time during wich the station listens

the channel, and altering the Contention Window (CW) lenght [19]. EDCA


provides contention-free access to the channel for a period called Transmit

Opportunity (TXOP). TXOP is the time interval during which a station can

send as many frames as possible. Using EDCF, the station try to send data

when the channel is free, and after a time, called Arbitration Interframe

Space (AIFS), defined by the traffic category. Another 802.11e feature is the

Hybrid Coordination Function (HCF). The HCF controlled channel access

(HCCA) works a lot like PCF but there are differences. In PCF, the interval

between two beacon frames is divided into two periods of CPS and CP, the

HCCA allows for CFPs being initiated at almost anytime during a CP. This

CFP is called CAP. During a CAP the Hybrid coordinator (HC) controls the

access to the medium. In the HCCA are defined the Traffic Class (TC) and

the Traffic Stream (TS). The station can give information about the queues

lenght for each Traffic Class. The HC uses this information to give priority.

There is another difference to PCF, in HCCA the stations are given a TXOP.

Using HCCA is possible to configure with carefully the QoS [19].

3.3.6 Standard IEEE 802.11p

The IEEE 802.11p is an 802.11 extension that adds WLAN access in the

vehicular networks. The MAC protocol in IEEE 802.11p uses the EDCF,

described in the previous section. There are four priority classes that ensure

the different QoS levels: background traffic (BK), best effort traffic (BE),

voice traffic (VO) and video traffic (VI). For each data traffic class is possi-

ble to choose different values of AIFS and CW. It is necessary to ensure that

packets with high priority get access to the channel first. In the same class

of data traffic packets collisions are possible. After a packet collision has

occurred, a backoff time is randomly chosen from an interval. The window

size depends on the priority level. In a Vehicular Network the high mobility

leads to sacrifice the identification and authentication procedures that are

usually part of the IEEE 802.11. In the 802.11p, a WAVE Basic Service

Set (WBSS) is realized around an RSU. The WBSS existence is announced

through the WAVE Service Annuncement (WSA). The WSA is a generated

3.4 Space time division multiple access (STDMA) 24

beacon from the WBSS leader. When a vehicle hears this beacon configures

itself according to the informations contained in the beacon frame and is

immediately ready to communicate with RSU. No authentication or associ-

ation is required. Furthermore to privacy reasons, a mobile node (MN) in

vehicular network changes its MAC address regulary. The MAC address is

determinated randomly [24, 20].

3.4 Space time division multiple access (STDMA)

In the IEEE 802.11p standard considered for VANETs, a CSMA protocol

is used. This protocol has an unboundend delay and are not collision free.

The CSMA protocol is not appropriate for real time communications. To

solve this problem it is possible to use an extension of TDMA, called STDMA.

The STDMA is found in a standard for the shipping industry, automatic

identification system (AIS). In STDMA the space is divided into virtual

geographic cells called space slots that are grouped in space frames, in order

to facilitate spatial reuse, and time slots are assigned to space slots. Every

space slot is assigned its own time slot, and each node simply inherits the

time slot assigned to its current space slot. The STMA is a 2-tier hierarchical

TDMA protocol: in the first TDMA tier, time is looping over space slots; in

the second TDMA tier, time is looping over the node IDs located within the

same space slot. In STDMA two simultaneus transmitters will never have

any receiver in common [1]. The algorithm can be summarized as follows:

• nodes that entered in the network exchange local information with its

neighbors;

• the node with highest priority in its local surrondings assigns itself a

time slot;

• the local schedule is the update and a new node has highest priority.

The nodes act in three manners:


• Active: the node has the highest priority in its local neighborhood and

it will subsequently assign itself a time slot.

• Waiting: a node wants to assign itself a time slot, but another link has

higher priority.

• Asleep: there are no available slot for the node [9].

STDMA is a decentralized, predictable MAC method with a finite channel

access delay, making it suitable for real time ad hoc vehicular networks. The

STDMA algorithm grants packets channel access since slots are reused if all

slots are currently occupied within the selection interval of a node. When a

node is forced to reuse a slot, it will choose the slot that is used by a node lo-

cated further away. Hence, there will be no packet drops, the channel access

delay is always buonded and relatively small [11]. In Figure 3.6 a comparison

between CSMA/CA and STDMA is presented.

Figure 3.6: CSMA/CA vs STDMA courtesy of [22]

After describing the main types of communication protocols and Ad Hoc

Networks, it is necessary to understand how to formalize in a rigorous manner


the typical communication networks problems. A mathematical tool widely

used, and really important to formalize the problem presented in this thesis,

is the Optimization. The main optimization techniques will be described in

the following chapter.

Chapter 4

Optimization

The Optimization is a matematical theory that studies the techniques to

find the maximum or minimum of a fuction. To indicate an optimization

problem we use the following notation:

min f0(x)

subject to fi ≤ 0 i = 1, . . . , p

hi(x) = 0 i = 1, . . . , p (4.1)

where x ∈ Rn is the optimization variable, f0 : Rn → R is the objective func-

tion, fi : Rn → R, are the inequality functions fi(x) ≤ 0 are the inequality

constraints and hi : Rn → R are the equality functions hi(x) = 0 are the

equality constraints. If there are no constraints the problem Eq. (4.1) is

unconstrained. The optimal value p of problem Eq. (4.1) is defined as [15]:

p = inf{f0(x)|fi(x)→ 0, i = 1, ..,m, hi(x) = 0, i = 1, . . . , p}

4.1 Convex optimization

Convex optimization is a class of optimization problems where:

• the objective function must be convex

27

4.1 Convex optimization 28

• the inequality constraint functions must be convex

• the equality constraint functions must be affine.

Definition 4.1. A function f : Rn → Rm is affine if it is a sum of a linear

function and a constant, i.e., if it has the form f(x)=Ax+b where A ∈ Rm×nand b ∈ Rm.

Definition 4.2. A function f : Rn → R is convex if domf is a convex set

and if for all x, y ∈ f results [15]:

f((1− β)y+) ≤ (1− β)f(y) + βf(z), β ∈ [0, 1]

The function f is said strictly convex if for y, z ∈ domf, y 6= z, results:

f((1− β)y+) < (1− β)f(y) + βf(z), β ∈ (0, 1)

Definition 4.3. A function f : Rn → R is concave on a convex set, if for all

y, z ∈ domf ,results [15]:

f((1− β)y+) ≥ (1− β)f(y) + βf(z), β ∈ [0, 1]

The function f is strictly concave if for all y, z ∈ domf, y 6= z, results:

f((1− β)y+) > (1− β)f(y) + βf(z), β ∈ (0, 1)

Theorem 4.1.1 (Necessary and sufficient convexity conditions [15]). f is

differentiable its gradient 5f exists at each point in domf ,which is open.

Then f is convex if and only if domf is convex and

f(y) ≥ f(x) +5f(x)T (y − x)

holds for all x, y ∈ domf .

f is strictly convex on C if and only if, for all pairs of point x, y ∈ C with

y 6= x has:

f(y) > f(x) +5f(x)T (y − x)


Theorem 4.1.2 (Necessary and sufficient convexity conditions [15]). Assume

that f is twice differentiable, that is, its Hessian or second derivative 52f

exists at each point in dom f, which is open. Then f is convex if only if dom

f is convex and its Hessian is positive semidefinite: for all x ∈ domf

52f ≥ 0

A convex optimization problem can be written as follows:

min f0(x)

subject to fi ≤ 0 i = 1, . . . ,m

aTi x = bi i = 1, . . . , p (4.2)

An important property is that a convex optimization problem has only global

solution. This property is called ” Absence of local optima”.

4.1.1 Quadratic problems

Quadratic functions are more interesting for optimization problems. An

optimization problem becomes a quadratic optimization problem when the

objective function is quadratic and the constraint functions are affine. It can

be written as follows:

min (1/2)xTPx+ cTx+ r

subject to Gx ≤ h

Ax = b (4.3)

If the constraint functions are quadratic the problem is quadratically con-

strained quadratic program [15].

4.1.2 Linear program

A particular case of quadratic optimization are the Linear programs. We

can obtain it putting P=0 in Eq. (4.3). A Linear program (PL) is an


optimization problem with the following properties:

• the objective function f(x) is linear

• the admissible set is defined by linear constraints.

A Linear Program can be written as follows:

min cTx+ d

subject to Gx ≤ h

Ax = b (4.4)

Where A is a real matrix m×n, b ∈ Rm and x ∈ Rn. d is usually omitted

because it does not influence the optimal set.We can have linear programs in

standard form [15]:

min cTx

subject to Ax = b

x ≥ 0 (4.5)

and inequality Linear Program,where there isn’t equality constraints. Usu-

ally the inequality Linear Programs are written as [15]:

min cTx

subject to Ax ≤ b (4.6)

Geometric interpretation

When a Linear Program has only few variables, it is possible to represent

it on the Cartesian plane and compute the solution by geometric considera-

tions. Suppose to have a Linear Program in two variables, so the objective


function is an expression: c1x1 + c2x2. This expression can be maximize or

minimize. To represent the previous expression we consider a family of par-

allel lines c1x1 + c2x2 = C. If we want to minimize the objective function, we

must find the lower value for C that verifies the condition on x1 and x2. If

we want to maximize the objective function, we must find the higher value

for C that verifies the condition on x1 and x2. In a maximization problem we

consider the traslations to the increasing direction of the objective function,

vice versa for a minimization problem. Each constraint of the problem is

represented as a line that identifies a semiplane. The intersection of those

semiplanes is the admissible region. The admissible region is a convex set.

Each point in this region is a possible problem solution. We can have two

possible situations: the problem admits an optimal solution, or the problem

does not admit optimal solution. If the solutions set is not empty the opti-

mal solution will be a vertex of the polygon. If the solutions set is empty or

not limited, the problem does not admit solution [4, 6]. We can consider the

following example:

max 2.5x1 + 2.02x2

subject to x1 + 2x2 ≤ 8

3x1 + 2x2 ≤ 9

x1, x2 ≥ 0

The geometric interpretation is represented in Figure 4.1, where the grey

area is the admissible set.

4.2 Duality 32

Figure 4.1: Geometric representation, courtesy to [13]

4.2 Duality

The basic idea of duality is to define a new function L : Rn×Rm×Rp ∈ R,

associated with the Eq. (4.1). The function L is called Lagrangian, with this

we associate to Eq. (5.1) constraints a Lagrange multiplier [15]:

L(x, λ, ν) = f0(x) +∑m

1 (λifi(x)) +∑p

1(νihi(x))

λi is a Lagrange multiplier associated with fi(x) ≤ 0 while νi is a Lagrange

multiplier associated with hi(x) = 0. For each fixed value of λ and ν we can

define the following function, called Lagrange dual function:

g(λ, ν) = infx∈D) L(x, λ, ν)

The function g(·) assigns the inf to each pairs (λ, ν). The dual problem

becomes:

max g(λ, ν)

subject to λ ≥ 0

4.2.1 Lagrange dual of quadratic problem

Consider the following quadratic problem:

4.2 Duality 33

min 12xTQx+ cTx

subject to Ax ≥ b

The Lagrangian function is:

L(x, λ) = 12xTQx+ cTx+ λ(b− Ax) = bTλ+ (c− ATλ)Tx+ 1

2xTQx

g(λ) = infx∈Rn L(x, λ)

We have

g(λ) = infx∈Rn L(x, λ)

The Lagrangian function has only one minimum. The minimum is obtained

by putting the Lagrangian gradient equal to zero.

5xL = Qx + (c − ATλ) the solution is x = −Q−1(c − ATλ). Replacing this

value in the Lagrangian function we obtain the dual problem [15]:

g(λ) = −12(c− ATλ)TQ−1(c− ATλ) + bTλ

The dual problem is:

max − (1/2)(c− ATλ)TQ−1(c− ATλ) + bTλ

subject to λ ≥ 0

4.2.2 Lagrange dual of linear program

Consider the following linear program [15]:

min cTx

subject to Ax ≥ b

The Lagrange function is:

L(x, λ) = cTx+ λT (b− Ax) = λT b+ (c− ATλ)Tx

We have:

g(λ) = infx∈Rn [cTx+ λT (b− Ax)] =

{−∞ ifATλ 6= c

λT b ifATλ = c

4.3 Branch and Bound 34

The dual problem is:

max bTλ

subject to ATλ = c

λ ≥ 0

4.3 Branch and Bound

The Branch and Bound (B&B) is a method to solve an optimization prob-

lem. The used approach is top down: it divides the problem into subprob-

lems. This method permits to enumerate explicitly or implicitly all problem

solutions. The steps to follow are:

1. Build the feasible solutions tree. This operation is called ”branch”

2. Find a good feasible solution

3. Exstimate the objective function for each solution found. This opera-

tion is called ”bound”.

Is possible summarize the Branch and Buond algorithm in Figure 4.2 defining:

• G0 feasible solution set,

• Ui optimal value of relaxed problem,

• Yi optimal solution obtained by relaxation,

• Z∗ lower bound

• X∗ optimal solution,

• W activeset


Figure 4.2: Block diagram for Branch and Bound algorithm

The branch and bound method is applicable to problems of combinatorial

optimization. A combinatorial optimization problem has a finite space of

feasible solutions, to solve it is possible to use the following steps:

1. generate all possible solutions

2. verify the solutions

3. estimate f(x)


4. choose the optimal solutions

To generate all possible solutions it is possible divide X in subproblems.

This division is an iterative process, that permits to represent the problem

trough a solutions tree as represented in Figure 4.3 Q0 is a combinatorial

Figure 4.3: Solutions tree

optimization problem and G0 = X is the solutions set of the problem, exactly

G0 is the root while Gi is the solutions set on node i. Dividing the node Gi

is possible to obtain the son nodes. Each solutions present in a father node

must be present in at least a son node. The creation of son nodes is called ”

branch”. Is not possible explore entirely the solutions tree, because usually

there are a lot of leaves. The best solution would be to explore only the good

areas of feasible region. To do this is necessary fix a bound. The bound is a

value estimate of objective function in all solutions present in the same node.

If the considered problem is a minimization problem the bound is a lower

bound, and it represents a value under which is not possible go, otherwise if

is a maximization problem the bound is an upper bound, and it represents

the value over wich is not possible go. Having a bound is possible define

”implicitly explored” the nodes that not have the optimal solution, that will

be pruned. [2]


4.3.1 Branching rules

Consider a problem Q with feasible solutions set G. Applying the branch

and bound method is possible obtain some subproblems Qi with solutions

Gi. The subproblems solutions must be satisfied the following property:⋃i Ei = E. The above property ensures that the optimal solution is at least

in one of son nodes.

The subproblems Gi are always smaller, and each node takes the parents

features.

4.3.2 Exploration tree rules

After to have built the tree is necessary explore it. There are a lot of

exploration methods to decide which node to visit:

• Depth first: the best node is the deeper. The tree is developed in depth,

in Figure 4.5.

• Best bound first or best node: the node with best bound is choosen.

• Breadth first: the tree is developed in width. First consider the nodes

on the same level and then those of the underlying layer in Figure 4.4.

• Mixed rules: is possible to use the above criteria as needed.

The Branch and Bound method stops when all nodes are fathomed. [2]

4.4 Dynamic programming 38

Figure 4.4: Search tree of Breadth first

Figure 4.5: Search tree of Depth first

4.4 Dynamic programming

The Dynamic Programming is applicable to decomposable optimization

problems. This method solves the problem by putting together the solutions

of subproblems. Usually the subproblems are not independent. It is possible

to avoide to solve several times the same problem using a bottom up ap-

proach: first solve a smaller problem then the bigger one. The solutions are

memorized in tables and are always available. To solve a Dymanic Program-


ming it is necessary to find a collection of subproblems of the orginal problem.

This collection must not be too complex, because the original problem so-

lution can be found only after the subproblems solutions have been found.

The problems that can be resolved with Dynamic Programming must satisfy

the following properties:

• the optimal solution contains the optimal solutions of subproblems

• there are common subproblems that are solved only one time

It is possible to summarize a generic dynamic programming in Figure 4.6.

With the exposition of Optimization problems and B &B method, the

background of this thesis is completed. In the next chapters, the original part

of this thesis: ”Scheduling Algorithms” will be illustrated and accurately

described, having now to disposition all the necessary tools for quick and

correct understanding of what will be proposed.


Figure 4.6: Block diagram for Dynamic program algorithm

Chapter 5

Scheduling algorithms

In this chapter we focus our attention on the core of this thesis: the

scheduling algorithms. A scheduling algorithm is an algorithm for the al-

location of resources among multiple users. Our analysis is based on the

works in [10] and [23], with particular attention for [23]. In the following, a

description of the two scheduling algorithms is presented, then we introduce

our proposed algorithm.

5.0.1 Scheduling for vehicle-infrastructure communi-

cations

Consider a central station that decides how to allocate the resources

among the vehicle under coverage. The aim is to deliver as much packets

as possible during the period in which the vehicle is under coverage, and to

empty the queues of vehicles leaving the coverage area. This problem is a

minimization problem that is equivalent to throughput maximization. The

scheduling algorithm is formulated as an optimal control problem [10]. The

decisions of the scheduler are reflected in the evolution of the queues lenght.

Denote with xi(k) the queues lenght in bits, while with xk = (x1(k), ..., xN(k))

the state of system. The state changes after each MAC frame acording to the

TD (TXOP duration). The control vector Vk = (TD1(k), ...,TDN(k)) is the

scheduling decision at each MAC frame. Denote with PERi(k) the packet

41

42

error rate. The state at frame k+1 is given by:

xi(k + 1) = xi(k)− PERi(k)× R× TDi(k)

It is possible to formulate the expression above as follows:

xk+1 = Akxk + BkVk

The matrix Ak is the system matrix and corresponds to identity matrix, while

Bk is a diagonal matrix. The queue lenght can be multiplied by a weighting

factor αi that assigns different weights to vehicles. A vehicle that spends

more time under coverage has a greater weight of a vehicle that spends less

time under coverage. Also the control vector can be multiplied by a weighting

factor βi that reflects the link quality. Therefore, the minimization problem

can be formulated as a quadratic problem in the form:

xTk+1Qk+1xk+1 + VTk RkVk

The optimal control vector can be obtained by [10]:

Vk = Lkxk

where Lk is the gain matrix:

Lk = −(Rk + BTk Qk+1Bk)

−1BTk Qk+1Ak

The contraint for the objective function, is given by the MAC frame limited

capacity, called CAPlimit:∑TDi +

∑OHi ≤ CAPlimit

5.0.2 Scheduling in real time traffic

Consider K users that want transmit. Scheduling algorithms for real time

traffic can be formulated with either absolute or average delay requirements.

We focus now on average delay requirements. Given time allocation τ(·)and arrival rate α = [α1, ...., αn]T , denote with w(τ) = [w1(τ), .., wk(τ)]T the

average queue delay [23]. The objective function that we want maximize is:

43

∑Kk=1 URT,k(wk)

The utility function URT,k must be choosen concave and monotonically de-

creasing: the utility of user k should decrease as wk increases. Assuming that

each user has a large input queue, denote with qk[n] the queue size at the

beginning of time slot n. The queue lenght can be update as follows:

qk[n+ 1] = qk[n]−min{τk(hk[n])rk(hk[n])Ts, qk[n]}+ αk[n]

where τk(hk[n])rk(hk[n]) is the departure rate, Ts is the time slot duration,

and hk[n] is the channel gain. The average delay can be estimate as follows:

wk[n+ 1] = (1− β)wk[n] +β

αk

(qk][n] + αk[n]−min{τk(hk[n])rk(hk[n])Ts, qk[n]})− wk[n]

After a first order approximation of its Taylor’s expansion, and denoting the

Lagrange multiplier it is possible to obtain the following problem:

maxτ(h[n])

K∑k=1

(−U′

RT,k(wk[n]) + λk[n])τk(hk[n])rk(hk[n]) (5.1a)

subject toK∑

k=1

τk(hk[n]) ≤ 1 (5.1b)

τk(hk[n])rk(hk[n]) ≤ qk[n]

Ts

for each k (5.1c)

5.0.3 A comparison

In this section the differences between two previous scheduling policies

are highlighted. In [10], the scheduling objective is to transmit more data

are possible during the period in which the vehicle is under coverage. The

decisions of scheduler are reflected in the evolution of the queue lenghts,

which must be minimized. In [10] each vehicle under coverage can transmit

regardless of channel conditions. In [23], the scheduling objective is minimize

the average delay. The decisions of scheduler are reflected in the time slot

5.1 Proposed scheduling algorithm 44

fractions. Each time slot fraction is a channel function so in this article, all

users don’t trasmit in the same time. The scheduler assigns the entire slot

to user 1 with maximum weighted rate. If only part of slot is required to

serve all data in 1’s queue, the remaining fraction will be assigned to user 2.

This allocation continues until the entire slot is assigned to users or data in

all user queues are empty [23].

5.1 Proposed scheduling algorithm

Here we extend the scheduling algorithm in [23]. In our algorithm we

propose multi-hop configurations, to improve the performance. The proto-

col considered is 802.11p with STDMA, which is a decentralized, predictable

MAC method with a finite channel access delay, making it suitable for real

time ad hoc vehicular networks. Different configurations have been consid-

ered, for each of which an optimization problem has been formulated and

solved. For simplicity only two users have been considered, but a possible

development could be the generalization of the algorithms to more users.

Suppose to indicate with ”k” the number of users, and with Ts the time

slot duration. The time slot is divided in fractions ”τ(·)”. Each time slot

fraction is a function of channel gain ′′h(·)′′. The scheduling politicy is the

following. The user transmits for time slot fractions and the duration of its

transmission depends by the channel gain. A user with a higher channel gain

should transmit more time than a user with a lower channel gain. The first

configuration considered is really similar to the problem shown in [23], while

the other ones are an extentions. In the following sections the individual

configurations and for each of them the corresponding optimization problem

will be presented. Let us define:

• the queue lenght Q

• the arrival rate A

• the average queue delay d


• the utility function U′

RT,k, must be choosen concave and monotonically

decreasing: the utility of user k should decrease as dk increases.

5.1.1 Configuration 1

We consider the situation shown in Figure 5.1

Figure 5.1: Possible representation of configuration one

The queue lenght Qk(mTs) at time mTs, time slot index, plus the fractions

of time slot τk(hk[mTs]), can be written as follows:

Qk[mTs +∑K

k=1 τk(hk[mTs])] =

Qk[mTs]− τk(hk[mTs])rk(hk[mTs])Ts + Ak[mTs]

Where τk(hk[mTs])rk(hk[mTs]) is the departure rate, while rk(hk[mTs]) is the

transmission rate, obtained from Shannon’s formula:

rk(hk[mTs]) = log2(1 + (hk[mTs])).


To simplify the notations, we write:

Qk[mTs +∑K

k=1 τk(hk[mTs])] = Qk[mTs + 1].

This expression indicates the queue lenght at the time slot mTs plus one time

slot fraction. The average queue delay can be update as follows:

dk[mTs +K∑

k=1

τk(hk[mTs])] = (1− β)dk[mTs]β

Ak

(Qk[mTs]

+ Ak[n]− τk(hk[mTs])rk(hk[mTs])Ts)−Qk[mTs] (5.2)

It is possible now to formulate the optimization problem:

minτ(h[mTs])

K∑k=1

U′

RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])

Ak

(5.3a)

subject toK∑

k=1

τk(hk[mTs]) ≤ Ts (5.3b)

τk(hk[mTs]) ≥ 0 (5.3c)

τk(hk[mTs])rk(hk[mTs]) ≤Qk[mTs]

Ts

(5.3d)

Qk[mTs]

Ts

≤ Qmax (5.3e)

Let us analyze now the meaning of each constraint.

Eq. (5.3b) imposes that the sum of all time slot fraction must be less than

or equal to time slot duration.

Eq. (5.3c) limits each fraction of time slot to be positive or zero.

Eq. (5.3d) tells that the user cannot transmit more than the available data.

In Eq. (5.3e) we set that the queue lenght cannot exceed a maximum queue

capacity.



We consider a situation shown in Figure 5.2

Figure 5.2: Possible representation of configuration two

In Figure 5.2 the user 1 does not transmit directly to the roadside station. It

sends its data to the user 2, so the queue expression and the constraints will

be different respect to Eq. (5.3e). The queue of user 2 now at time slot mTs,

where m is the time slot index, plus the fractions of time slot τk(hk[mTs]),

can be written as follows:

Q2[mTs +K∑

k=1

τk(hk[mTs])] = A2[mTs] + Q2[(m− 1)Ts] (5.4)

where

Q2[mTs +∑K

k=1 τk(hk[mTs])] = Q2[mTs + 1]


This expression indicates the queue lenght at the time slot mTs plus one

time slot fraction. In Eq. (5.4) the queue lenght in the previous slot can be

explicit as follows:

Q2[(m− 1)Ts] = A2[(m− 1)Ts]− τ2(h2[(m− 1)Ts])r2(h2[(m− 1)Ts]

The average queue delay can be update as Eq. (5.2). The optimization

problem can be expressed as follows:

minτ(h[mTs])

2∑k=1

U′


Ak

(5.5a)

subject to2∑

k=1


τk(hk[mTs]) ≥ 0 (5.5c)

τ2(h2[mTs])r2(h2[mTs])− τ1(h1[mTs])r1(h1[mTs])

≤ Q2[mTs + 1]

Ts

(5.5d)

Q2[mTs + 1]

Ts

+ τ1(h1[mTs])r1(h1[mTs]) ≤ Qmax (5.5e)

It is possible to generalize the problem Eq. (5.5):

minτ(h[mTs])

K∑k=1

U′


Ak

(5.6a)

subject toK∑

k=1


τk(hk[mTs]) ≥ 0 (5.6c)

τk(hk[mTs])rk(hk[mTs])− τk−1(hk−1[mTs])rk−1(hk−1[mTs])

≤ Qk[mTs + 1]

Ts

(5.6d)

Qk[mTs + 1]

Ts

+ τk−1(hk−1[mTs])rk−1(hk−1[mTs]) ≤ Qmax

(5.6e)

Let us analyze now each constraint.

The constraints in Eq. (5.6b) and in Eq. (5.6c) corresponds to constraints


in Eq. (5.3b) and in Eq. (5.3c)

In Eq.(5.6c) the constraint is different from in Eq. (5.3d), because in Qk

there are also the Qk−1 data. The constraint can be written also as:

τk(hk[mTs])rk(hk[mTs]) ≤ Qk[mTs+1]Ts

+ τk−1(hk−1[mTs])rk−1(hk−1[mTs])

The departure rate to k step must be less or equal to sum of queue lenghts

to k step and one fraction of time slot, plus the departure rate to k-1 step.

In Eq. (5.6d) the queue lenght is influenced by the data in the queue of users

previous hop. This quantity cannot exceed a maximum capacity.


Consider the situation shown in Figure 5.3

Figure 5.3: Possible representation of configuration three

The situation shown in Figure 5.3 is similar to the Figure 5.2. The data of


user 2 include also the data of user 1 like in the previous case. The difference

now is that the time slot is divided in three fraction and not in two, and the

scheduling. Now the user 2 is the first to trasmit, then the user 1 that sends

its data to user 2 , and then again the user 2 that transmits its data plus data

of user 1 to roadside station. The scheduler will define the lenght of time

slot fraction to assign to each user. Let us formalize mathematically which

said above. The queue of user 2 at time slot mTs, where m is the time slot

index, plus the fractions of time slot τk(hk[mTs]), can be written as follows:

Q2[mTs +K∑

k=1


where

Q2[mTs +∑K



fraction of time slot. In Eq. (5.7) the queue lenght in the previous slot can

be explicit as follows:

Q2[(m− 1)Ts] = A2[(m− 1)Ts]− τ1(h2[(m− 1)Ts])r2(h2[(m− 1)Ts]−τ3(h2[(m− 1)Ts])r2(h2[(m− 1)Ts]

The queue lenght of user 1 can be written as follows:

Q1[mTs +∑K

k=1 τk(hk[mTs])] = A1[mTs] + Q1[(m− 1)Ts]

where

Q1[mTs +∑K


The queue of user 1 has been written as sum of arrival rate at actual slot

plus the queue lenght in the previous slot plus a fraction, because now user

1 is not the first to transmit but the second one.

Q1[(m− 1)Ts + 1] =

A1[(m− 1)Ts + 1]− τ2(h1[(m− 1)Ts + 1])r1(h1([(m− 1)Ts + 1])


The average queue delay can be update as Eq. (5.2).

The optimization problem can be expressed as follows:

minτ(h[mTs])

2∑k=1

U′


Ak

(5.8a)

subject to2∑

k=1


τk(hk[mTs]) ≥ 0 (5.8c)

τ1(h2[mTs])r2(h2[mTs]) ≤Q2[mTs + 1]

Ts

(5.8d)


Ts

(5.8e)

τ3(h2[mTs])r2(h2[mTs])− τ2(h1[mTs])r1(h1[mTs]) (5.8f)

+ τ1(h2[mTs])r2(h2[mTs]) ≤Q2[mTs + 1]

Ts

(5.8g)

Q2[mTs + 1]

Ts

− τ1(h2[mTs])r2(h2[mTs])

+ τ2(h1[mTs])r1(h1[mTs]) ≤ Qmax (5.8h)



in Eq. (5.3b) and in Eq. (5.3c)

In Eq. (5.8d) the departure rate of user 2 in first time slot fraction must be

less or equal to sum of queue lenght. It is not possible to transmit more than

the available data.

In Eq. (5.8e) the departure rate of user 1 in the second time slot fraction

must be less or equal to sum of queue lenghts. It is not possible to transmit

more than the available data.

In Eq. (5.8g) the queue of user 2 in the third time fraction is influenced by

the departure rate of user 1, which increments the quantity of data in user

2 queue, and by the departure rate of user 2 in the first and third time slot


fraction. The constraint can be written also as:


Ts

+ τ2(h1[mTs])r1(h1[mTs])− τ1(h2[mTs])r2(h2[mTs])

(5.9)

In Eq. (5.8h) the queue lenght is influenced by data transmitted by user 1,

minus the data transmitted by user 2 in the first time slot fraction. This

quantity cannot exceed a maximum queue capacity.


We consider the situation shown in Figure 5.4

Figure 5.4: Possible representation of configuration four

In the configuration shown in Figure 5.4 the time slot is divided in three

fractions. The data of user 1 are transmitted during two fractions of time

slot. It is possible that during the first fraction all data of user 1 will be


transmitted. The quantity of data transmitted depends by channel condi-

tions. If the channel from user 1 to roadside station is optimal all data could

be transmitted, while if that channel is not really good, one part of data

will be transmitted directly to roadside station during the first fraction of

time slot, while the other part will be send to user 2 during the second time

slot fraction, and in the end transmitted to roadside station during the third

time slot fraction. The data arriving to roadside station from user 2 are the

sum of data in the queue of user 1 and the data in the queue of user 2. The

queues lenght of user 1 and 2 can be written as follows:

Q1[mTs +K∑

k=1


where

Q1[mTs +∑K



fraction of time slot. In Eq. (5.10) appears the queue lenght in the previous

slot, that can be explicit as follows:


The data presents in the queue of user 1 are transmitted in two fraction of

time slot, those transmitted during the first fraction in Eq. (5.10), and the

data transmitted during the second fraction, where the queue lenght has the

following expression:

Q1[mTs + 2] = A1[mTs] + Q1[(m− 1)Ts + 1]

This expression indicates the queue lenght at the time slot mTs plus two

fractions of time slot where:

Q1[(m−1)Ts+1] = Q1[(m−1)Ts]−τ2(h2[(m−1)Ts+1])r2(h2[(m−1)Ts+1])

Once described the expressions of queue lenght of user 1, it is necessary to

show the expression of queue lenght of user 2. The user 2 transmits during

the third time slot fraction. The data of user 2 are the sum of quantity of

data presents in queue of user 1 and its data.


Q2[mTs + 3] = A2[mTs] + Q2[(m− 1)Ts]

This expression indicates the queue lenght at the time slot mTs plus three

fractions of time slot where:

Q2[(m− 1)Ts] = A2[(m− 1)Ts]− τ3(h3[(m− 1)Ts])r3(h3[(m− 1)Ts])

The average queue delay can be updated as in Eq. (5.2).

The optimization problem can be expressed as follows:

minτ(h[mTs])

K∑k=1

U′


Ak

(5.11a)

subject toK∑

k=1


τk(hk[mTs]) ≥ 0 (5.11c)


Ts

(5.11d)


Ts

(5.11e)


≤ Q2[mTs + 3]

Ts

(5.11f)

Q2[mTs + 3]

Ts

+ τ2(h2[mTs])r2(h2[mTs]) ≤ Qmax (5.11g)



in Eq. (5.3b) and in Eq. (5.3c).

In Eq. (5.11d) the departure rate of user 1 at first time slot fraction must be

less or equal to queue lenght. It is not possible to transmit more than the

available data.

In Eq. (5.11e) the departure rate of user 2 at second time slot fraction must

be less or equal to queue lenght. It is not possible to transmit more than the

available data.

In Eq. (5.11f) the queue of user 2 in the third time fraction is influenced by


the departure rate of user 1 in the second time slot fraction, that increments

the quantity of data in user 2 queue, and by the departure rate of user 2 in

the third time slot fraction. The constraint can be written also as:

τ3(h3[mTs])r3(h3[mTs]) ≤ Q2[mTs+3]Ts

+ τ2(h2[mTs])r2(h2[mTs])

In Eq. (5.11g) the queue lenght is influenced by data transmitted by user 1

in the second time slot fraction. This quantity cannot exceed a maximum

capacity.


We consider a situation shown in Figure 5.5.

The configuration shown in Figure 5.5 is a particular case of the configuration

in Figure 5.4. In this case the time slot is divided in four fractions, where

the user 2 transmits for two consecutive time slot fractions. The behavior of

the system is in principle the same as above.

Figure 5.5: Possible representation of configuration five


We describe now the expression to update the queues lenght.

Q1[mTs +K∑k=1


where

Q1[mTs +∑K


In Eq. (5.12) appears the queue lenght in the previous slot, which can be

explicit as follows:


The data in the queue of user 1 are transmitted in two fractions of time

slot, those transmitted during the first fraction where the queue lenght has

the expression written previous, and the data transmitted during the second

fraction, where the queue lenght has the following expression:

Q1[mTs + 2] = A1[mTs] + Q1[(m− 1)Ts + 1]

where

Q1[(m−1)Ts+1] = Q1[(m−1)Ts]−τ2(h2[(m−1)Ts+1])r2(h2[(m−1)Ts+1])

Once described the expressions of queue lenght of user 1, it is necessary to

show the expression of queue lenght of user 2. The user 2 transmits during

the third time slot fraction. The data of user 2 are the sum of quantity of

data presents in queue of user 1 and its data. The transmission happens in

two consecutive fractions of time slot. The queue lenght at third time slot is

given from:

Q2[mTs + 3] = A2[mTs] + Q2[(m− 1)Ts]

where

Q2[(m− 1)Ts] = A2[(m− 1)Ts]− τ3[(m− 1)Ts]r3[(m− 1)Ts]

The queue lenght at fourth time slot is given from:


Q2[mTs + 4] = Q2[mTs + 3]− τ4(h3[(m− 1)Ts])r3(h3[(m− 1)Ts])

where

Q2[(m− 1)Ts] = A2[(m− 1)Ts]− τ3[(m− 1)Ts]r3[(m− 1)Ts]

The average queue delay can be updated as Eq. (5.2). The optimization

problem can be expressed as follows:

minτ(h[mTs])

2∑k=1

U′


Ak

(5.13a)

subject to2∑

k=1


τk(hk[mTs]) ≥ 0 (5.13c)


Ts

(5.13d)


Ts

(5.13e)


≤ Q2[mTs + 3]

Ts

(5.13f)


+ τ3(h3[mTs])r3(h3[mTs]) ≤Q2[mTs + 3]

Ts

(5.13g)

Q2[mTs + 3]

Ts

− τ2(h2[mTs])r2(h2[mTs])

− τ3(h3[mTs])r3(h3[mTs]) ≤ Qmax (5.13h)



in Eq. (5.3b) and in Eq. (5.3c).

In Eq. (5.13d) the departure rate of user 1 in first time slot fraction must be

less or equal to queue lenght. It is not possible to transmit more than tha

available data.

In Eq. (5.13e) the departure rate of user 2 in second time slot fraction must


be less or equal to queue lenght. It is not possible to transmit more than the

available data. In Eq. (5.13f) the queue of user 2 in the third time fraction

is influenced by the departure rate of user 1, which increments the quantity

of data in user 2 queue, and by the departure rate of user 2 in the third time

slot fraction. The constraint can be written also as:


+ τ2(h2[mTs])r2(h2[mTs])

In Eq. (5.13g) the queue of user 2 in the fourth time fraction is influenced by

the departure rate of user 1, which increments the quantity of data in user 2

queue and by the departure rate of user 2 in the third and fourth time slot

fraction. The constraint can be written also as:


+τ2(h2[mTs])r2(h2[mTs])−τ3(h3[mTs])r3(h3[mTs])

In Eq. (5.13h) the queue lenght is influenced by data transmitted by user

1 in the second time slot fraction, and by data transmitted by user 2 in the

third time slot fraction. This quantity cannot exceed a maximum capacity.

5.1.6 Possible generalizations

Below the possible generalizations of the previous configurations in which

a third user is inserted are presented. The optimization problems are similar

to those seen before. The mathematical formulation is not here reported.

Since the process is iterative it can be generalized to n users taking into ac-

count the considerations made for the case with three users.


Figure 5.6: Generalization with three users




Chapter 6

Simulations and results

In this chapter the numerical results of the configurations presented in

the previous chapter are pointed out. The algorithms solutions is obtained

using the cvx MATLAB toolbox. For each configuration numerical results

and a graphic representation of solution are shown. Before presenting the

numerical results, the geometric interpretation of a possible solution is pre-

sented. This interpratation is important because it can be used to identify

the feasible solutions set. For the algorithms implementation the following

initial numerical values are used:

• arrival rate user 1 A1=0.5 Mbps;

• arrival rate user 2 A2=1.2 Mbps;

• queue lenght user 1 Q1=5 Mb;

• queue lenght user 2 Q2=10 Mb;

• average queue dalay user 1 d1= 5 ms;

• average queue dalay user 2 d2= 10 ms;

• β=0.05;

• time slot duration Ts=1 ms;

• maximum queue capacity Qmax= 70 Mb.

61

6.1 Configuration 1 results 62

6.1 Configuration 1 results

To solve the problem in Eq. (5.3) associated with the configuration in

Figure 5.1, the following values of channel gain and time slot fractions are

used:

• channel gain of user 1 h1=0.1;


• time slot fraction user 1 τ1=0.1 ms;


The feasible solutions set, if it exists, is shown in Figure 6.1, where it is rep-

resented by colored area. The feasible set is obtained by using the geometric

interpretation presented in Chapter 4, Section 4.1.2.

Figure 6.1: Geometric representation of Eq. (5.3)

We obtain the following results. The values of time slot fractions are the

following and are represented in Figure 6.2:


• τ1=0.0647 ms;

• τ2=0.8889 ms;

These values reflect the adopted scheduling policy, a user with higher channel

gain should transmit for more time than a user with less channel gain, and

respect the constraints in Eq. (5.3b) and in Eq. (5.3c). The departure rate,

r(·)τ(·), for each user are the following: r1τ1=0.009 Mbps, r2τ2=0.43 Mbps.

Figure 6.2: Optimal tau configuration 1

The optimal objective function value according to optimal τ is +1.43892e-12.




Figure 5.2, the following values of channel gain and time slot fraction are

used:





The feasible solutions set, if it exists, is shown in Figure 6.3, where it is rep-

resented by colored area. The feasible set is obtained by using the geometric

interpretation presented in Chapter 4, Section 4.1.2.





• τ1=0.3649 ms;

• τ2=0.5640 ms.




r(·)τ(·), for each user has the following values: r1τ1=0.05 Mbps, r2τ2=0.27

Mbps. The user 2 departure rate is higher than that of user 1 because in the

user 2 queue there are the user 1 data.







used:

• channel gain of user 1 h1 = 0.1;

• channel gain of user 2 at first time slot fraction h2=0.4;

• channel gain of user 2 at third time slot fraction h2=0.4;




The feasible solutions set, if it exists, is shown in Figure 6.5. The feasible

solution set has a triangular shape. The feasible set is obtained by using the

geometric interpretation presented in Chapter 4, Section 4.1.2.





• τ1=0.3549 ms;

• τ2=0.2987 ms;

• τ3=0.1699 ms.





Mbps, r2τ3=0.08 Mbps. The user 2 departure rate in third time slot fraction

is higher than that user 1 in second fraction, because it forwards its data and

user 1 data.







used:














• τ1=0.3200 ms;

• τ2=0.0530 ms;

• τ3=0.5782 ms.





Mbps, r2τ3=0.33 Mbps.







used:




• channel gain of user 2 at fourth time slot fraction h2=0.5;








We have obtained the following results. The values of time slot fractions are

the following and are represented in Figure 6.10:

• τ1=0.0615 ms;

• τ2=0.2332 ms;

• τ3=0.3817 ms;

• τ3=0.2504 ms.



These values reflect the adopted scheduling policy, a user with higher chan-

nel gain should transmit for more time than a user with less channel gain,

and respect the constraints in Eq. (5.13b) and in Eq. (5.13c). In this case

the third fraction apparently does not respect the scheduling politicy. This

fraction has an higher channel gain respect to the second one, but transmits

for less time because much of its data are transmitted in the first time slot

fraction. The departure rate, r(·)τ(·), for each user has the following values:

r1τ1=0.009 Mbps, r1τ2=0.11 Mbps, r2τ3=0.22 Mbps,r2τ4=0.14 Mbps.


From each algorithm solution it is possible to note that the best configuration

is the number 2 represented in Figure 5.2 and by Eq. (5.5). This configu-

ration presents the best optimal value +9.12526e-13. The optimal values of

configuration 3 in Figure 5.3 and configuration 1 in Figure 5.1 are very simi-

lar: +1.43892e-12, for configuration 1, and +1.45859e-12, for configuration 3.

Even if the values are very similar, the configurations are very different, and

it is not possible to estabilish a priori which is better. In configuration 1 the

users transmit directly to the station while in the configuration 3 is present



the multi hop transmission. The worst optimal value is obtained by solution

of the problem in Eq. (5.11) associated with configuration 4 in Figure 5.4.

This problem has the highest optimal value +2.55107e-12.

Conclusions and future works

In this work new real time scheduling algorithms with possible application

in Intelligent Trasportation Systems (ITS) have been presented. Scheduling

algorithms for real time traffic are formulated with either absolute or average

delay requirements. The aim of our problem was the average queue delay

minimization. The protocol considered is the 802.11p with STDMA policy,

because the classical CSMA has an unboundend delay and are not collision

free. The transmission happens for time slot fractions and, in the consider

cases, parallel transmissions are not possible. The proposed scheduling algo-

rithms can be used to send message with high priority according to STDMA

policy. An optimization problem has been associated to scheduling algo-

rithms. From results analysis it is possible to highlight that a transmission

scheme single communication does not necessarily guarantee lower average

delay than a configuration with multi-hop. In among various configurations,

a scheme where an user transmits its data to an other user and not directly

to roadside station, has the lowest delay according to the particular setup.

Eventually, it is possible to note that the scheduling order is fundamental to

reach good perfomance, and by changing it, also the resulting delay changes.

Many future developments are perspected. One possible development is the

following. In this work the scheduling algorithms are dependent on the par-

ticular configuration used over the optimal time allocation. An extension to

the algorithm could be to define a new optimization problem to find also

the best configuration (configuration with lower delay). This problem could

be formulated as a combinatorial optimization problem using a Branch and

73


Bound method. Another development could be to formulate an optimization

problem that permits parallel transmissions. Eventually, it would be inter-

esting to implement these algorithms in Intelligent Transportation Systems

(ITS) to evaluate the real performance of what was theoretically formulated.

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Documents

Real time scheduling in Intelligent Transportation Systems562645/FULLTEXT01.pdf · To achieve an e cient network utiliza-tion while ensuring acceptable performance, it is instrumental