Intelligent Transportation Systems Telecommunications and Transportation.
Real time scheduling in Intelligent Transportation Systems
Transcript of Real time scheduling in Intelligent Transportation Systems
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.
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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
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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
6.2 Configuration 2 results . . . . . . . . . . . . . . . . . . . . . . 64
6.3 Configuration 3 results . . . . . . . . . . . . . . . . . . . . . . 66
6.4 Configuration 4 results . . . . . . . . . . . . . . . . . . . . . . 68
Contents iv
6.5 Configuration 5 results . . . . . . . . . . . . . . . . . . . . . . 70
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
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List of figures vi
5.7 Generalization with three users . . . . . . . . . . . . . . . . . 59
5.8 Generalization with three users . . . . . . . . . . . . . . . . . 60
6.1 Geometric representation of Eq. (5.3) . . . . . . . . . . . . . . 62
6.2 Optimal tau configuration 1 . . . . . . . . . . . . . . . . . . . 63
6.3 Geometric representation of Eq. (5.5) . . . . . . . . . . . . . . 64
6.4 Optimal tau configuration 2 . . . . . . . . . . . . . . . . . . . 65
6.5 Geometric representation of Eq. (5.8) . . . . . . . . . . . . . . 66
6.6 Optimal tau configuration 3 . . . . . . . . . . . . . . . . . . . 67
6.7 Geometric representation of Eq. (5.11) . . . . . . . . . . . . . 68
6.8 Optimal tau configuration 4 . . . . . . . . . . . . . . . . . . . 69
6.9 Geometric representation of Eq. (5.13) . . . . . . . . . . . . . 71
6.10 Optimal tau configuration 5 . . . . . . . . . . . . . . . . . . . 72
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.
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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)]
2.2 Wireless channel model 7
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
2.2 Wireless channel model 8
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
2.2 Wireless channel model 9
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
2.2 Wireless channel model 10
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
2.2 Wireless channel model 11
• 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-
3.3 Communication protocols 18
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
3.3 Communication protocols 19
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
3.3 Communication protocols 20
• 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
3.3 Communication protocols 21
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].
3.3 Communication protocols 22
• 802.11 original sample, 1997;
– datatransfer 1 or 2MBps;
• 802.11a Physical layer extension, 1999;
– datatransfer 54MBps;
• 802.11b Physical layer extension, 1999;
– datatransfer 11MBps;
• 802.11g Physical layer extension, 2003;
– datatransfer 54MBps;
• 802.11n;
– datatransfer 540MBps;
• 802.11p 802.11a extension. It is used in VANET;
– datatransfer 27MBps;
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
3.3 Communication protocols 23
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:
3.4 Space time division multiple access (STDMA) 25
• 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
3.4 Space time division multiple access (STDMA) 26
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)
4.1 Convex optimization 29
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
4.1 Convex optimization 30
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
4.1 Convex optimization 31
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
4.3 Branch and Bound 35
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)
4.3 Branch and Bound 36
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 Branch and Bound 37
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-
4.4 Dynamic programming 39
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.
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
5.1 Proposed scheduling algorithm 45
• 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])).
5.1 Proposed scheduling algorithm 46
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.
5.1 Proposed scheduling algorithm 47
5.1.2 Configuration 2
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]
5.1 Proposed scheduling algorithm 48
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′
RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])
Ak
(5.5a)
subject to2∑
k=1
τk(hk[mTs]) ≤ Ts (5.5b)
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′
RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])
Ak
(5.6a)
subject toK∑
k=1
τk(hk[mTs]) ≤ Ts (5.6b)
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
5.1 Proposed scheduling algorithm 49
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.
5.1.3 Configuration 3
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
5.1 Proposed scheduling algorithm 50
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
τk(hk[mTs])] = A2[mTs] + Q2[(m− 1)Ts] (5.7)
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
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
k=1 τk(hk[mTs])] = Q1[mTs + 2]
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])
5.1 Proposed scheduling algorithm 51
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′
RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])
Ak
(5.8a)
subject to2∑
k=1
τk(hk[mTs]) ≤ Ts (5.8b)
τk(hk[mTs]) ≥ 0 (5.8c)
τ1(h2[mTs])r2(h2[mTs]) ≤Q2[mTs + 1]
Ts
(5.8d)
τ2(h1[mTs])r1(h1[mTs]) ≤Q1[mTs + 2]
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)
Let us analyze now each constraint.
The constraints in Eq. (5.8b) and in Eq. (5.8c) corresponds to constraints
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
5.1 Proposed scheduling algorithm 52
fraction. The constraint can be written also as:
τ3(h2[mTs])r2(h2[mTs]) ≤Q2[mTs + 1]
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.
5.1.4 Configuration 4
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
5.1 Proposed scheduling algorithm 53
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
τk(hk[mTs])] = A1[mTs] + Q1[(m− 1)Ts] (5.10)
where
Q1[mTs +∑K
k=1 τk(hk[mTs])] = Q1[mTs + 1]
This expression indicates the queue lenght at the time slot mTs plus one
fraction of time slot. In Eq. (5.10) appears the queue lenght in the previous
slot, that can be explicit as follows:
Q1[(m− 1)Ts] = A1[(m− 1)Ts]− τ1(h1[(m− 1)Ts])r1(h1[(m− 1)Ts]
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.
5.1 Proposed scheduling algorithm 54
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′
RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])
Ak
(5.11a)
subject toK∑
k=1
τk(hk[mTs]) ≤ Ts (5.11b)
τk(hk[mTs]) ≥ 0 (5.11c)
τ1(h1[mTs])r1(h1[mTs]) ≤Q1[mTs + 1]
Ts
(5.11d)
τ2(h2[mTs])r2(h2[mTs]) ≤Q1[mTs + 2]
Ts
(5.11e)
τ3(h3[mTs])r3(h3[mTs])− τ2(h2[mTs])r2(h2[mTs])
≤ Q2[mTs + 3]
Ts
(5.11f)
Q2[mTs + 3]
Ts
+ τ2(h2[mTs])r2(h2[mTs]) ≤ Qmax (5.11g)
Let us analyze now each constraint.
The constraints in Eq. (5.11b) and in Eq. (5.11c) corresponds to constraints
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
5.1 Proposed scheduling algorithm 55
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.
5.1.5 Configuration 5
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
5.1 Proposed scheduling algorithm 56
We describe now the expression to update the queues lenght.
Q1[mTs +K∑k=1
τk(hk[mTs])] = A1[mTs] + Q1[(m− 1)Ts] (5.12)
where
Q1[mTs +∑K
k=1 τk(hk[mTs])] = Q1[mTs + 1]
In Eq. (5.12) appears the queue lenght in the previous slot, which can be
explicit as follows:
Q1[(m− 1)Ts] = A1[(m− 1)Ts]− τ1(h1[(m− 1)Ts])r1(h1[(m− 1)Ts]
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:
5.1 Proposed scheduling algorithm 57
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′
RT,k(dk[mTs])τk(hk[mTs])rk(hk[mTs])
Ak
(5.13a)
subject to2∑
k=1
τk(hk[mTs]) ≤ Ts (5.13b)
τk(hk[mTs]) ≥ 0 (5.13c)
τ1(h1[mTs])r1(h1[mTs]) ≤Q1[mTs + 1]
Ts
(5.13d)
τ2(h2[mTs])r2(h2[mTs]) ≤Q1[mTs + 2]
Ts
(5.13e)
τ3(h3[mTs])r3(h3[mTs])− τ2(h2[mTs])r2(h2[mTs])
≤ Q2[mTs + 3]
Ts
(5.13f)
τ4(h3[mTs])r3(h3[mTs])− τ2(h2[mTs])r2(h2[mTs])
+ 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)
Let us analyze now each constraint.
The constraints in Eq. (5.13b) and in Eq. (5.13c) corresponds to constraints
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
5.1 Proposed scheduling algorithm 58
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:
τ3(h3[mTs])r3(h3[mTs]) ≤ Q2[mTs+3]Ts
+ 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:
τ4(h3[mTs])r3(h3[mTs]) ≤ Q2[mTs+3]Ts
+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.
5.1 Proposed scheduling algorithm 59
Figure 5.6: Generalization with three users
Figure 5.7: 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;
• channel gain of user 2 h2=0.4;
• time slot fraction user 1 τ1=0.1 ms;
• time slot fraction user 2 τ2=0.2 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:
6.1 Configuration 1 results 63
• 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.
6.2 Configuration 2 results 64
6.2 Configuration 2 results
To solve the problem in Eq. (5.5) associated with the configuration in
Figure 5.2, the following values of channel gain and time slot fraction are
used:
• channel gain of user 1 h1=0.1;
• channel gain of user 2 h2=0.4;
• time slot fraction user 1 τ1=0.1 ms;
• time slot fraction user 2 τ2=0.2 ms;
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.
Figure 6.3: Geometric representation of Eq. (5.5)
6.2 Configuration 2 results 65
We obtain the following results. The values of time slot fractions are the
following and are represented in Figure 6.2:
• τ1=0.3649 ms;
• τ2=0.5640 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. (5.6b) and in Eq. (5.6c). The departure rate,
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.
Figure 6.4: Optimal tau configuration 2
The optimal objective function value according to optimal τ is +9.12526e-13.
6.3 Configuration 3 results 66
6.3 Configuration 3 results
To solve the problem in Eq. (5.8) associated with the configuration in
Figure 5.3, the following values of channel gain and time slot fraction are
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;
• time slot fraction user 1 τ2=0.1 ms;
• time slot fraction user 2 τ1=0.2 ms;
• time slot fraction user 2 τ3=0.5 ms;
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.
Figure 6.5: Geometric representation of Eq. (5.8)
We obtain the following results. The values of time slot fractions are the
following and are represented in Figure 6.6:
6.3 Configuration 3 results 67
• τ1=0.3549 ms;
• τ2=0.2987 ms;
• τ3=0.1699 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. (5.8b) and in Eq. (5.8c). The departure rate,
r(·)τ(·), for each user has the following values: r2τ1=0.17 Mbps, r1τ2=0.04
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.
Figure 6.6: Optimal tau configuration 3
The optimal objective function value according to optimal τ is +1.45859e-12.
6.4 Configuration 4 results 68
6.4 Configuration 4 results
To solve the problem in Eq. (5.11) associated with the configuration in
Figure 5.4, the following values of channel gain and time slot fraction are
used:
• channel gain of user 1 h1=0.1;
• channel gain of user 1 at first time slot fraction h1=0.4;
• channel gain of user 2 at third time slot fraction h2=0.5;
• time slot fraction user 1 τ1=0.1 ms;
• time slot fraction user 1 τ2=0.2 ms;
• time slot fraction user 2 τ3=0.5 ms;
The feasible solutions set, if it exists, is shown in Figure 6.7. 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.
Figure 6.7: Geometric representation of Eq. (5.11)
6.4 Configuration 4 results 69
We obtain the following results. The values of time slot fractions are the
following and are represented in Figure 6.8:
• τ1=0.3200 ms;
• τ2=0.0530 ms;
• τ3=0.5782 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. (5.11b) and in Eq. (5.11c). The departure rate,
r(·)τ(·), for each user has the following values: r1τ1=0.04 Mbps, r2τ3=0.03
Mbps, r2τ3=0.33 Mbps.
Figure 6.8: Optimal tau configuration 4
The optimal objective function value according to optimal τ is +2.55107e-12.
6.5 Configuration 5 results 70
6.5 Configuration 5 results
To solve the problem in Eq. (5.13) associated with the configuration in
Figure 5.5, the following values of channel gain and time slot fraction are
used:
• channel gain of user 1 h1=0.1;
• channel gain of user 1 at first time slot fraction h1=0.4;
• channel gain of user 2 at third time slot fraction h2=0.5;
• channel gain of user 2 at fourth time slot fraction h2=0.5;
• time slot fraction user 1 τ1=0.1 ms;
• time slot fraction user 1 τ2=0.2 ms;
• time slot fraction user 2 τ3=0.5 ms;
• time slot fraction user 2 τ4=0.3 ms;
The feasible solutions set, if it exists, is shown in Figure 6.9. 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.
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.
6.5 Configuration 5 results 71
Figure 6.9: Geometric representation of Eq. (5.13)
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.
The optimal objective function value according to optimal τ is +1.77844e-12.
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
6.5 Configuration 5 results 72
Figure 6.10: Optimal tau configuration 5
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
Conclusions and future works 74
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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