Optimization of Handover Algorithms for Wireless Networks

Wireless Networks

SILVIA DE LUCIA

Master’s Degree Project

Stockholm, Sweden 2010

We are witnessing a continuous development of heterogeneous wireless net- works, such as cellular systems (e.g., the LTE, or long term evolution of the third generation wireless system), sensor networks, and satellite networks.

The coexistence of all these networks requires the design of control mech- anisms to allow the seamless communication from one network to another, and even between communication standards. The handover is one such con- trol mechanism. Specifically, the handover is the mechanism of transferring the connection of a mobile node from an access point (which could be a base station or a network using some communication standard) to another AP.

During the handover, a node that is being disconnected from an AP and con- nected to another AP may experience a sudden degradation of the bandwidth or outage of the communication. This leads to frequent and unnecessary han- dovers that may reduce significantly the quality of service perceived by the node. Therefore, in order for the node to achieve high quality of service, an optimization of the handover algorithm is essential. In this thesis we study some optimization algorithms for the handover procedure. We show that the variables available at the node to control the handover are the hysteresis margin, which is used to compare the quality of signals of the APs, and the estimation window length, which is the number of samples to estimate with a desired accuracy these signals. Then, we study two optimization problems in which the parameters that affect the handover are optimized by considering as a cost function a convex combination of the probability of outage and the probability of handover. The first problem is based on a static optimization, whereas the second problem on a dynamic optimization. We show that both problems offer better performance with respect to existing algorithms from the literature. We show that the dynamic optimization gives better results, but at the cost of an increased computational complexity.

I

Abstract . . . . I

1 Introduction to Handover 1

1.1 Handover procedure . . . . 1

1.2 Types of Handover . . . . 2

1.3 Handover initiation . . . . 2

1.4 Handover Decison . . . . 3

1.5 Handover in Microcellular Propagation Environment . . . . . 5

1.6 Handover in Multihop Cellular Networks . . . . 6

1.7 Handover scenarios in MCNs . . . . 6

1.8 Handover in IP Networks . . . . 9

1.9 Mobility in 6LoWPAN . . . 11

1.9.1 Application methods . . . 12

1.9.2 Mobile IPv6 on 6LoWPAN . . . 13

1.9.3 NEMO . . . 13

2 Problem Formulation 15 3 Modeling 18 3.1 System Model . . . 18

3.2 Least Square Algorithm: Performance Measures . . . 19

3.2.1 Handover decision an probabilities . . . 21

3.3 The effect of the Hysteresis Margin . . . 24

3.3.1 Probability of Base Station Connection . . . 24

3.3.2 Probability of Handover and Probability of Outage . . 26

3.4 Approximations . . . 27

3.4.1 First Bound . . . 27

3.4.2 Second Approximation: proposed bound . . . 28

II

4.2 Static Handover and Outage Pareto Optimization . . . 31

4.3 Optimization of the Probability of Handover . . . 32

4.4 Probability of Outage Optimization . . . 33

4.5 Dynamic Handover and Outage Pareto Optimization . . . 33

5 Simulation Results 36 5.1 Probability of Handover and Probability of Outage . . . 36

5.2 Estimate Window Length Pareto Optimization Results . . . . 39

5.3 Approximations results . . . 40

5.3.1 First approximation . . . 40

5.3.2 Second approximation . . . 41

5.4 Performance Evaluation of Hysteresis Optimization . . . 44

5.4.1 Static Optimization Results . . . 44

5.4.2 Dynamic Optimization Results . . . 48

5.4.3 Improving Factors . . . 50

6 Conclusions 52

III

1.1 Soft and Hard handover . . . . 2 1.2 Signal strength and hysteresis between two adjacent BSs for po-

tential handover. . . . . 3 1.3 Handover scenarios in Microcellular Networks (MCN) . . . . 5 1.4 Handover scenarios in MCN; BS is the base station, MS is the

mobile station, RS is the relay station and a Cell is a coverage area of the BS . . . . 6 1.5 Optimized vertical handover framework . . . 11 3.1 The hysteresis of the handover decision. . . . 24 4.1 Trellis diagram for the computation of outage probabilities in the

dynamic optimization. . . . 34 4.2 Trellis diagram for the computation of handover probabilities in

the dynamic optimization. . . . 34 4.3 Optimization scheme . . . 34 5.1 Probability of handover versus distance d ₀ for LS method. The

parameter is the window length n _w . . . . 37 5.2 Probability of outage versus distance d 0 for LS method. The pa-

rameter is the window length n w . . . . 37 5.3 Maximum values of outage probability versus estimate window

length. . . . 37 5.4 Maximum values of handover probability versus estimate window

length. . . . 37 5.5 Handover and outage probability evaluate at n w optimum for the

problem having the objective function in (4.6) . . . 39 5.6 Probability of BSs connection versus distance for the first bound

proposed in Section 3.4 . . . 40

IV

n _w = 10 . . . 41 5.8 Probability of handover versus distance evaluated by using the sec-

ond approximation given by equation (3.19) in Chapter 3, with n _w = 40 . . . 41 5.9 Probability of outage versus distance evaluated by using the second

approximation given by equation (3.19) in Chapter 3, with n w = 10 42 5.10 Probability of outage versus distance evaluated by using the second

approximation given by equation (3.19) in Chapter 3, with n _w = 40 42 5.11 Comparison between handover probability values for different

m . . . 43 5.12 Comparison between outage probability values for different m 43 5.13 Handover probability evaluated at optimum h for the static opti-

mization problem. . . . 44 5.14 Outage probability evaluated at optimum h for the static optimiza-

tion problem. . . . 44 5.15 Objective function evaluated at optimum h for the static optimiza-

tion problem. . . . 45 5.16 Handover and Outage probabilities evaluated by setting a fixed

histeresis margin h=2: no optimization . . . 46 5.17 Objective function evaluated by setting h=2: no optimization . . . 46 5.18 Handover and Outage probability evaluated by setting a fixed

histeresis margin h=8: no optimization . . . 46 5.19 Objective function evaluated by setting h=8: no optimization . . . 47 5.20 Comparison between objective functions evaluate at h opt by apply-

ing dynamic and static optimization techniques with a weigthing cofficient equal to 0.5 . . . 48 5.21 Comparison between objective functions evaluate at h _opt by apply-

ing dynamic and static optimization techniques with a weigthing cofficient equal to 0.8 . . . 48 5.22 Comparison between objective functions evaluate at h _opt by apply-

ing dynamic and static optimization techniques with a weigthing cofficient equal to 0.2 . . . 49 5.23 Improving factor between objective function evaluated by fixing

an hysteresis margin h = 2 and the objective function at h _opt computed by using the static optimization . . . 51

V

computed by using the dynamic optimization . . . 51 5.25 Improving factor between objective function evaluated by fixing

an hysteresis margin h = 8 and the objective function at h _opt computed by using the dynamic optimization . . . 51 5.26 Improving factor between objective function evaluated by fixing

an hysteresis margin h = 8 and the objective function at h opt

computed by using the dynamic optimization . . . 51 5.27 Improving factor between objective function evaluated by using

static and dynamic optimization. The value of the parameter α is 0.2 . . . 51

VI

Introduction to Handover

In a wireless network the need of managing the receivers mobility and ensur- ing continuous service led to the develop of procedures for a good network performance. This is achived by supporting the handover (or handoff ) al- gorithm, which is a procedure that can be applyed to every kind of wireless communication network and it is also possible within the same radio system or between heterogeneous systems wich are standardized by protocols devel- oped by different standardization bodies. In this chapter, we present the main characteristics of the handover algorithm.

1.1 Handover procedure

The handover is the key to enable function for mobility and service conti- nuity among a variety of wireless access technologies. It is is the process of changing channel (frequency, time slot, spreading code, or combination of them) associated with the current connection while a communication is in progress. In cellular telecommunications the term handover refers to the procedure of transferring a call or a data section while the mobile station is moving away from a coverage area, called cell, to another cell. This process is carried out to avoid the interruption of an in-progress call when the mobile gets outside the range of the cell. In fact a handover initiation is the process by which a handover is started as a consequence of the fact that the cur- rent link is unacceptably degraded and/or another base station can provide a better communication link.

1

BS0 BS1 BS0 BS1

MS MS

a. Soft handover b. Hard handover

Figure 1.1: Soft and Hard handover

1.2 Types of Handover

It is possible to distinguish between Hard Handover and Soft Handover.

In the soft handover the mobile station is simultneously connected to two different base stations (see Fig. 1.1a). On the contrary, in the hard the connection to the serving base station is interrupted while the new base station takes on the connection (Fig. 1.1b). Furthermore hard handover can be divided into intracell handover and intercell handover; the soft handover can also be divided into multiway soft handover and softer handover. Soft handover is highly expensive for the network, but offers potentially a higher performance with respect to the hard handover. Nevertheless, the recent standardizations of new technologies of Long Term Evolution (LTE) and High Speed Packet Access (HSPA) suggest the use of hard handover only.

1.3 Handover initiation

The decision to initiate a handover at a mobile station may be based on

different measurements, e.g. the recived signal strenght from the serving base

station and neighboring base stations (BS’s), the distance from BS’s, and the

bit error rate. Handover is critical in cellular communication systems because

neighboring cells are always using a disjoint subset of frequency bands, so

negotiations must take place between the mobile station (MS), the current

BS, and the next potential BS.

Relative signal strength

This method works by selecting the BS from which the strongest signal is received. The decision is based on the computation of the average of the measurements of the received signal. It has been observed that this method generates many unnecessary handovers, even when the signal of the current BS is still at an acceptable level. Some variations have been proposed, as we see in the following.

Relative signal strength with threshold

This method allows a MS to hand off only if the current signal is sufficiently weak (less than threshold) and the other is the stronger of the two. The effect of the threshold depends on its relative value as compared to the signal strengths of the two BSs at the point at which they are equal.

Signal strength Signal strength

BS0 BS1

T1

T2 T3

A B C D

h

MS

Figure 1.2: Signal strength and hysteresis between two adjacent BSs for potential handover.

If the threshold is higher than this value, say T1 in Fig. 1.2, this scheme

performs exactly as the relative signal strength scheme, so the handoff occurs

at position A. If the threshold is lower than this value, say T2 in Fig. 1.2, the

MS would delay handoff until the current signal level crosses the threshold at

position B. In the case of T3, the delay may be so long that the MS drifts too

far into the new cell. This reduces the quality of the communication link from

BS 1 and may result in a dropped call. Moreover, this results in additional interference to cochannel users. Thus, this scheme may create overlapping cell coverage areas. A threshold is not used alone in actual practice because its effectiveness depends on prior knowledge of the crossover signal strength between the current and candidate BSs.

Relative signal strength with hysteresis

This scheme allows a user to hand off only if the new BS is sufficiently stronger (by a hysteresis margin, h in Fig. 1.2) than the current one. In this case, the handover would occur at point C. This technique prevents the so-called ping-pong effect, the repeated handover between two BSs caused by rapid fluctuations in the received signal strengths from both BSs. The first handover, however, may be unnecessary if the serving BS is sufficiently strong.

1.4 Handover Decison

There are numerous methods to perform handover, at least as many as the type of network entities that maintain the state information. The decision- making process of handover may be centralized or decentralized (i.e. the handover decision may be made at the MS or at in the core network). From the decision process point of view, it is possible to find at least three different kinds of handoff decisions.

Network-Controlled Handover

In a network-controlled handover protocol, the network makes a handover

decision based on the measurements on the MSs at a number of BSs. In

general, the handover process (including data transmission, channel switch-

ing, and network switching) takes 100–200 ms. Information about the signal

quality for all users is available at a single point in the network that facili-

tates appropriate resource allocation. Network-controlled handoff was used

in first-generation analog systems such as AMPS (advanced mobile phone

system), TACS (total access communication system), and NMT (advanced

mobile phone system).

Mobile-Assisted Handover

In a mobile-assisted handover process, the MS makes measurements that are transmitted to the network to let it able to make the decision. In analog systems handover procedure is entirely carried on by the network, in the circuit-switched GSM (global system mobile) is a mobile assisted procedure.

This mainly means that handover algorithm can operate more efficiently ac- cording to the larger number of available parameters. The handover time between handover decision and execution in a circuit-switched GSM is ap- proximately 1 second.

Mobile-Controlled Handover

In mobile-controlled handover, each MS is completely in control of the han- dover process. This type of handover has a short reaction time (on the order of 0.1 second). MS measures the signal strengths from surrounding BSs and interference levels on all channels. A handover can be initiated if the signal strength of the serving BS is lower than that of another BS by a certain threshold.

1.5 Handover in Microcellular Propagation Environment

In a microcellular propagation environment there are two possible handovers:

handover in line-of-sight (LOS) and handover in non line-of-sight (NLOS).

To study the role of the hysteresis, Murase [8] analized the two cases: for LOS handover the MS always mantains a LOS with both the serving and the target BS. This is the case, for example, when a MS traverses along a route from BS ₀ to BS ₂ in Fig. 1.3. NLOS handover, instead, arises when the MS suddenly loses the LOS component with the serving BS while gaining a LOS component with the target BS. This phenomenon is called the corner effect, since it occurs while turning corners in urban microcellular settings as the one shown in Fig. 1.3 where the MS traverses along a route from BS 0 to BS 1 . In this case, the average recived signal strength can drop by 25-30 dB over distance as small as 10 m. Corner effects may also cause link quality imbal- ances on the forward and reverse channels due to the following mechanism.

Quite often the co-channel interference will arrive via a NLOS propagation

BS0

BS1

BS3

BS2

250 m

Figure 1.3: Handover scenarios in Microcellular Networks (MCN)

path. Hence, as a MS rounds a corner, the recived signal strength at the sertving BS suffers a large decrease while the NLOS co-channel interference remains the same, i.e., the corner effect severly degrades the carrier to inter- ference ratio (C/I). Meanwhile, the corner will cause the same attenuation to both the desired and interfering signals that are recived at the MS. There- fore, unless there are other sources of co-channel interference that become predominant as the MS round the corner, the C/I on the forward channel will remain about the same. If the handover requests from rapidly moving MSs in microcellular network are not processed quickly, then excessive dropped calls will occur. Fast temporal based handover algorithms can partially solve this problem; these algorithms make use of temporal averaging windows that are used to detect drops in signal strength. However, the effectiveness of these algorithms depends on the velocity of the MS. Adaptive handover algorithms can overcome these problems in propagation environments that are typical of urban microcellular networks.

1.6 Handover in Multihop Cellular Networks

In conventional cellular systems handover occurs only when a mobile station

(MS) moves to different cell or different sectors of the same cell. The intro-

duction of next generation wireless systems, such as fourth generation (4G),

Cell 1 ^{Cell 2} MS1

MS2 MS3

MS4

MS5

BS1 BS2

RS1a

RS1b

RS1c RS1d

RS2a

RS2b

RS2c

RS2d

Scenario1

Scenario2 Scenario3

Scenario5 Scenario4

Figure 1.4: Handover scenarios in MCN; BS is the base station, MS is the mobile station, RS is the relay station and a Cell is a coverage area of the BS

Third Generation Partnership Project Long-Term Evolution (3GPP LTE), and IEEE 802.16m includes the development of multihop cellular networks (MCN) to increase the cell radius or combat the shadowing effect, which is mainly caused by large obstacles between transceivers.

As shown in Fig. 1.4 in MCN additional handover occur between the BS and relay stations (RSs) or two different RSs. These additional MCN han- dovers can cause serious ping pong problems and increase signalling overhead.

Although there are works where relay handover problems is explained, the main interest of using the multihop concept is not in cellular networks but in ad-hoc networks; unfortunately doesn’t exist a rich body of literature about applications of handover in multi-hop networks.

1.7 Handover scenarios in MCNs

There are several possible handover scenarios in MCNs regardless of the RS

deployment structure. Handovers in MCNs can occur when an MS moves

between different BSs, between different RSs, or between a BS and an RS as

shown in Fig. 1.4

Intracell handover

Case 1: intracell RS-RS handover

In this case, an MS performs handover between two different RSs in the same cell. The BS easily can control the handover process because both the serving RS and the target RS are under their own control, and inter-BS information or signaling is not required. To prevent packet loss during the handover process, automatic retransmission request (ARQ) status should be consistent between a serving RS and a target RS. In this scenario, transfer of the ARQ status to the target RS is not required during handover if the ARQ function is located in the BS. In addition, the current layer 3 (L3) address can be used after the handover.

Case 2: intracell BS-RS handover

MS changes its communication node from a BS to the RS of the same cell, or vice versa. In this case, the BS easily can control the handover process as in case 1, and the ARQ status transfer and the L3 address renewal are not required.

Intercell handover

Case 3: intercell BS-BS handover

This case is exactly the same as the inter-BS handover in the conventional cellular systems. The ARQ status should be transferred during the han- dover process if the ARQ is controlled by the BS, and the L3 address is also reassigned when the subnet is changed by the handover.

Case 4: intercell RS-RS handover

In case 4, a MS performs handover from an RS to the RS of different cells.

This scenario can cause relatively larger signaling overhead than other sce-

narios because it requires inter-BS signaling and RS-BS signaling in both

cells. In addition, the channel quality of the MS in this handover region can

be seriously attenuated by the intercell interference from the adjacent BSs

and RSs. The ARQ status and the L3 address management are similar to

those of case 3.

Case 5: intercell BS-RS handover

In this case an MS moves from a BS to the RS of different cells, or vice versa.

Inter-BS signaling is also required in this scenario, and the ARQ status and the L3 address management are the same as scenario 3 except for additional signaling between the RS and the BS.

Received-Signal-Strength Measurement and Handover Decision

The received-signal-strength (RSS) measurement process can cause packet- loss or packet-transmission delay because an MS cannot receive data packets from the serving BS during the RSS measurement for the neighbor BSs. In an MCN, the RSS measurement overhead increases because an MS should measure the RSS of the neighbor RSs, as well as the neighbor BSs. There- fore, packet-loss or packet-transmission delay caused by handover can be increased in an MCN compared to a singlehop cellular network (SCN). The handover decision is based on the RSS measurement result. When the serving BS receives the measurement report from the MS or the RS, it determines the handover execution and direction based on the RSS and the resource availability in the target BS or RS.

Handover Procedures and Signaling

Regardless the exchange of specific messages during the handover execution which are different according to the scenario, it is possible to say that the handover procedure consists of several steps such as measurement, handover decision, handover ranging, association, and resource allocation.

Handover Latency and Service-Interruption Time

There are two important performance metrics of handover schemes:

• handover latency

• service-interruption time

Handover latency is defined as the duration between the time when the

measurement procedure starts and the time when the Handover-Complete

message (which indicates the end of handover process) is transmitted to a serving BS. To clarify, handover starts with the mesurement procedure per- formed by the MS and after wich a serving BS determines the handover ex- ecution and direction, whereas an HandoverComplete message comes before the Connection release step wich completes the whole handover procedure.

This parameter can be calculated differently according to the scenario.

The service-interruption time can be defined as the duration from suspend- ing data transmission to resuming data transmission, triggered by the RSS measurement and handover completion, respectively. In intracell handover scenarios, the service interruption occurs only during the measurement pro- cedure if the puncturing or addition finger in the receiving module of an MS is not considered. However, in intercell handover scenarios, the service interruption can occur during the measurement and handover execution pro- cedures.

Considerations on MCNs handover

Several design principles can affect the performance of handover schemes in an MCN. The location and number of RSs in a cell for an MCN have the most direct impact on the performance of handover in terms of throughput and handover latency. With an appropriate handover scheme it is possible, for example, improve the overall cell throughput, simplifie the handover pro- cess, reduce handover signaling overhead and handover latency, and improve significantly the service-interruption time. The choice of the scheme should be a compromise between technical issues and network complexity.

1.8 Handover in IP Networks

Cellular networks are evolving to provide not only the traditional voice ser- vice but also data services. IP appears to be the base technology of future networks, to provide all kind of services and through different access tech- nologies, both fixed and mobile. In Ip networks the management of terminal mobility is achived by using the Mobile IP (MIP) protocol wich makes use of two addresses for the mobile:

• a long term address (home address)

• a short-term local address (care of address, CoA)

The home address belongs to the MT’s home domain, while CoA belongs to the visiting network and it is used by the mobile when it is away from the home network. The home address is used as endpoint identifier for the transport layer, while the CoA is used as location identifier.

Enhanced Mobile Internet Protocols

The basic MIP scheme is not efficient in certain circumstances, this due to an handover delay introduction wich is often unaccetable for real time applications. Wether to make faster some procedures like movement detec- tion, new CoA configuration and location update, and to improve mobile communications, enhancements to the Mobile IP technique, such as MIPv6, Faster MIPv6 (FMIPv6) and Hierarchical MIPv6 (HMIPv6), are being de- veloped. All these protocols are host-based in the sense that the MTs needs to implement mobility-related functionality to perform handover and loca- tion management signaling when they move between network subnets. Re- cently the Internet Engineering Task Force IETF specified a network-based localized mobility management protocol that allows MTs to move between subnets within the same access network without requiring changes in their IPv6 protocol stack. This protocol is called Proxy MIPv6. Although the current trend in network-layer mobility management protocol design is to enhance the level of coordination between distinct network entities, efficient handover is required also for close interaction of functions inside the entities themselves. This interaction is essential to reduce the delays associated with handover operation.

IEEE 802.21 based vertical handover framework

The IEEE 802.21 is the standard that provides services facilitating handovers

between heterogeneous networks and an optimized handover framework that

leverages genric link-layer intelligence independent of the specifics of mo-

bile nodes or radio access networks. In this regard, the mobility manage-

ment protocol stack of the network elements engaged in handover signaling

is readdressed, and a logical entity is introduced between the link and up-

per layers. This entity, called MIH function (MIHF), provides tree kind of

services: event, command and information services to its users. To provide

these services a group of primitives included in a media-independent service

Upper Layers

MIH_SAP MIHF MIH_LINK_SAP

802.11 802.16 3GPP DVB

Mobile terminal

Figure 1.5: Optimized vertical handover framework

access point (SAP) MIH-SAP are used; on the other hand, to communicate with link layers the MIHF uses primitives that are defined in the media- dependant MIH-LINK-SAP and mapped to technology-specific primitives.

The architecture is shown in Fig. 1.5. MIHF facilitates handover initiation (network discovery, network selection, handover negotiation) and handover preparation (link 2 and link 3 connectivity, resource reservation).

1.9 Mobility in 6LoWPAN

Today’s increasing use of embedded devices, also called smart objects, that are universally becoming IP enabled and an integral part of the internet (Internet-connected), led to the definition of the so-called Internet of Things.

6LoWPAN is a set of standards defined by the Internet Engineering Task

Force (IETF) enabling the efficient use of IPv6 over low-pass, low-rate wire-

less networks on simple embedded devices through an adaptation layer and

the optimization of related protocols [3]. There is a wide range of applications

where 6LoWPAN technology may be used, for example home and building

automation, real-time environmental monitoring and forecasting, industrial

automation, personal health and fitness. In these scenarios is useful to con- sider the mobility because in some cases, such as with body area networks, the network itself may even be mobile. Mobility is often associated with topology change so there are several reasons why it happens:

• Physical Movement: the devices change their point of attchment.

• Radio Channel: change in radio propagation may be caused by en- vironmental change and this often require different topology.

• Network performance: in a wireless network poor signal strength, node congestion and collisions cause packet loss and delay that reduce network performance and induce a node to change its attachment point.

• Sleep schedules: in a wireless embedded network, to save battery power, node use sleep schedules. If a node is attached to a sleeping router it will move to a better point of attachment.

• Node failure: the failure of a router causes a topology change for nodes using it as their default router.

In IP networks, mobility is the change of the attachment point of the node.

Mobility may be distinguish in two kinds [3]:

• Roaming: node moves to another network typically without packets stream.

• Handover: node changes its attachment point during the packets trans- mission to and from the network. In order for the mobile node to com- municate again, this process may include operations at specific link layers as well as at the IP layer.

Further, it is possible to distinguish between micro-mobility and macro-

mobility; in 6LoWPAN micro-mobility refers to the mobility of a node within

a LoWPAN where the IPv6 prefix does not change thus such a mobility re-

quires only the handover, macro-mobility, instead, refers to mobility between

LoWPANs in which the IPv6 changes, and it requires both routing and han-

dover. Another class of mobility is the network mobility that concerns the

changing of the attachment point of an entire network.

Solutions for mobility

As well known each node manage the mobility involving different layers of the protocol stack. Depending on the type of mobility that occurred and the handover and roaming prcedures that are in use there are more or less layers involved. If the node movement is contained within the same LoWPAN network the link layer may be able to deal with the mobility without any noticeable change to the network layer. Common link layers that deal with the mobility include cellular systems such as GPRS or UMTS which perform handovers maintaining the same IP address; low-power wireless link-layer technologies such as IEEE 802.15.4 tend to leave mobility to be dealt with by the network layer. Networks such as IEEE 802.15.4 work in a manner that all topology changes are node-controlled rather than network-controlled (such as in cellular systems). If the node moves between different LoWPANs it needs to change the IPv6 address; if a node acts as a client, a simple way to deal with such an address change is for the applications to restart when detecting a change in IP address. A real challenge is to manage the macro- mobility if a node acts as a server: a way to deal with this in 6LoWPAN applications is making use at application layer of session initiation protocol (SIP), uniform resource identifiers (URIs) or a domain name server (DNS).

1.9.1 Application methods

If the LoWPAN network layer does not use mechanisms for the mobility, the handover causes effects at the application layer. Dealing with mobility at this layer may even provide better optimization for some applications. For an application server communicating with mobile 6LoWPAN Nodes with changing IP address to function, it needs to use some sort of unique and stable identifier for each node. Examples include the EUI-64 of the node’s interface, a URI, a universally unique identifier (UUID) or for example a domanin name.

1.9.2 Mobile IPv6 on 6LoWPAN

As explained in the previous section, the mobility of nodes on the Internet

is managed by using the MIP protocol. As 6LoWPAN is an adaption for

IPv6, it would make sense to apply MIPv6 for dealing with the mobility in

6LoWPAN Nodes while maintaining a costant IPv6 address. The limit to

the use of the MIPv6 for 6LoWPAN is due to the node complexity, power consumption and limited bandwidth of wireless links that make it difficult to apply on this field. In order to be applied to 6LoWPAN Node mobility, the MIPv6 would be to be implemented on LoWPAN Nodes but this problem is still under investigation. In this regards there are some proposals:

• To full MIPv6 messages are compressed and decompressed by edge routers; this technique would need a standardization;

• The edge router or some other entity on the visited network would proxy MIPv6 functions on behalf of the LoWPAN Nodes (proxy method).

1.9.3 NEMO

IP, IPv4, IPv6 were not designed to take into account mobility of users and

terminals. But some IP-layer protocols, such as MIPv6, that the IETF has

defined to enable terminal mobility in IP networks exists. Nevertheless, these

protocols do not support the movement of a complete network that moves

as a whole changing its point of attachment to the fixed infrastructure, that

is, network mobility. A solution to this problem is provided by a working

group called NEMO or network mobility [4]. The NEMO protocol works

by introducing a new logical entity called the mobile router that is able to

deal with MIPv6 and is responsible for handling its function for the entire

mobile network. The network consists of a number of nodes called mobile

network nodes (MNNs) and each node does not need to run Mobile IP be-

cause the router are attached to runs Mobile IP. When the entire LoWPAN,

including edge router and MNNs associated to it, move togheter to a new

point of attachment, applying NEMO means having benefits because inside

the LoWPAN no change can be noticed due to network mobility. The edge

router, in fact, acts as a NEMO mobile router: using MIPv6 binds its new

care-of address in the visited network, and in addition the home LoWPAN

prefix. This means that the LoWPAN continues to use the same prefix as in

the home network.

Problem Formulation

The handover is the procedure that allows user’s mobility in the context of heterogeneous -or not- wireless networks by ensuring continuity of service; in this scenario outage events, that are defined as periods in which the recived signal level is not sufficiently high to prevent the session interruption, must be considered to achive a good quality of service. In a wireless network, it is necessary to minimize outage events, because they reduce the quality of the communicaiton, and optimize the number of handovers, because they are very expensive to manage. This thesis focus on the probability of handover and the probability of outage optimization with respect to important parameters in the handover initiation: the estimate window length and hysteresis margin.

In effect, a variety of parameters such as bit error rate (BER), carrier-to- interference ratio (C/I), distance, traffic load, signal strength, and various combination of them have been suggested in literature for evaluating the link quality and deciding when a handover should be performed, as we explained in the previous chapter. Signal strength measurement is one of the most common criterion, and it is considered in this work. To improve performance, optimization with respect to some key parameters of the handover algorithm are studied in this thesis. We consider the LS algorithm proposed in [1], and we will first evaluate the probabilities of handover (P H ) and outage (P O ) for a mobile station (MS) moving between a certain number of base stations (BSs); then we will study an optimization of the probabilities with respect to the value of the estimate window length n w , which is the number of samples to use to make an estimate of the power of the received signals from the base stations. This parameter plays a foundamental role in terms of performance since it affects P H and P O . As will be shown in Chapter 5, an increase of

16

n w causes a decrease of P H and an increase of P O , thus a suitable balance between higher values of P O and lower values of P H has to be found. The optimal n _w is obtained by solving a static optimization problem, which makes use of the following objective function:

F obj (n, n w ) = α · P H (n, n w ) + (1 − α) · P O (n, n w ),

where P H (n) is the handover probability at time n, P O (n) is the outage probability at time n, α is a weighting coefficient. We study how to chose n _w so that the objective function is minimized at each time n. Moreover, we study the case when an hysteresis margin h is introduced (see the previous chapter). We then study how to optimize the objective function for such an hysteresis margin. We will consider two optimizations: a static one and a dynamic one. Static optimization means that the minimization looks only at present time and does not take into account what happens in future moments, that is what is done in the dynamic one. By using the hysteresis margin, the cost function is defined as follows:

J(n, h(n)) = α · P ^H (h(n), n) + (1 − α)P ^O (h(n), n). (2.1) The static optimization problem is expressed by:

min h(n) J(n, h(n)),

Notice that the left hand side of (2.1) depend on h(n). The obtained results will be discussed in Chapter 5. The dynamic optimization problem uses the following cost function:

J(b(n), h(n)) =

l+K

X

n=l

α · P ^H (h(l), l) + (1 − α)P ^O (h(l), l),

where the objective function is defined over the finite time interval [l, l + K]

and it also depends on the histeresys margin and base station b(n) at wich the mobile is connected to in the future evolutions. Hence the constrained optimization problem can be written as:

min h(n) J(b(n), h(n))

s.t b(l + 1) = f (b(n), h(n)).

In Chapter 4, how to solve this problem will be studied. The evaluation of

the handover and outage probability after the introduction of the histeresis

margin, implies the computation of joint probabilities composed by a large

number of elements, and this could be temporally and computationally pro-

hibitive; for this reason some approximations of both handover and outage

probability will be proposed in Chapter 4.

Modeling

To provide high quality of service in mobile communication systems is ness- esary to develop and optimize algorithms that manage critical procedures in wireless networks such as handover procedure. In this chapter, we study the analytical model of the expressions of the probability of outage and of the probability of handover. These expressions will be then used to optimize the handover.

3.1 System Model

A system model, consisting of two BSs and a mobile terminal moving from one base station to the other is assumed. A typical lognormal fading envi- ronment is considered, where the attenuation of the power of the transmitted signals follow a lognormal distribution.In this scenario the probability of out- age Po and the probability of handover Ph are evaluated along the trip from the current to the target BS at regular time intervals. The number of han- dovers and outages should be minimized but the decision to make a handover should not be delayed too long should be correctly taken, since the quality of the communication link can degrade. Using the LS algorithm proposed in [1]

which is based on the least squares (LS) estimate of path-loss parameters for each MS-BS link, it is necessary to know only the distance of the MS from the surrounding BS’s.

19

3.2 Least Square Algorithm: Performance Mea- sures

The handover problem is formulated here for the case of two BS’s, BS ₀ and BS 1 , separated by D meters, and with an MS moving from BS 0 to BS 1

along a straight line at a constant speed v. The mobile measures the signal strength from each BS at constant time intervals T (frame period). The value of the power of the received signal level is the sum of two terms—one due to path loss and the other due to lognormal (shadow) fading. Rayleigh fading, which has a much shorter correlation distance with respect to shadow fading, is neglected here because it gets averaged out at the time scale under consideration. Hence, the signal levels [in decibels, p 0 (n) and p 1 (n)] the mobile terminal receives from BS 0 and BS 1 at time nT, are given by

p s (n) = α s − β ^s log[d s (n)] + u s [d s (n)], s = 0, 1 (3.1) where d 0 and d 1 are the distances of the MS from BS 0 and BS 1 , respectively.

Notice that d ₀ (n) = vT n and d ₁ (n) = D − d 0 (n), n=1,2. . . N with N =

⌊D/(vT )⌋ (⌊z⌋ stands for the highest integer less than or equal to the real z). In (3.1), α s − β ^s log[d s (n)] is due to the path loss and α s and β s are the parameters of the mean signal strength for the MS-BS link. u 0 and u 1

model the shadow fading processes. They are zero-mean stationary Gaussian processes, independent of each other. According to [1] for the fading process is assumed an exponential autocorrelation function

˜

r u

(∆) = E[u s (d)u s (d − ∆)] = σ u ²

e ^{−|∆|/ ¯ d} s = 0, 1

where ¯ d is the decorrelation distance. This model is appropriate for cellular environments, wich are widely used in system performance evaluation, but as shown in [1] it can be extended to microcell and picocell system models.

Let l s (n), s=0,1 be a linear estimate of the signal strength received from the BS _s , obtained from the sequence {p s (i) }, i = 1, 2 . . . n

l s (n) =

n

X

i=n

p s (i) · G ^s (n, i) + G s,∞ (n)

where n b = max {1, n − n w + 1 } and n w is the estimation window length.

The analyzed handover algorithm makes use of the LS estimate of the path-

loss parameters, assuming that the distances are known. In a nonuniform

propagation environment the estimate of α s and β s should be carried for each link; the estimate of the coefficients, here denoted by ˆ α s (n) and ˆ β s (n), minimize the function

n

X

i=n

(p s (i) − ˆα s (n) + ˆ β s (n) log(d s (i))) ² (3.2)

By setting to zero the first-order partial derivatives of (3.2) with respect to ˆ

α s and ˆ β s , the following system of equations turns out:

n

X

i=n

(p s (i) − ˆα ^s (n) + ˆ β s (n) log d s (i)) = 0

n

X

i=n

(p s (i) − ˆα ^s (n) + ˆ β s (n) log d s (i)) log d s (i) = 0

whose solutions are ˆ

α _s (n) = 1

D s (n) − C s (n) ² · [P s (n)D _s (n) − Q s (n)C _s (n)]

β ˆ s (n) = 1

D s (n) − C ^s (n) ² · [P s (n)C s (n) − Q s (n)]

where

P s (n) = ^△ 1 n − n b + 1

n

X

i=n

p s (i)

Q s (n) = ^△ 1 n − n b + 1

n

X

i=n

p s (i) log d s (i)

C s (n) = ^△ 1 n − n b + 1

n

X

i=n

log d s (i)

D s (n) , 1 n − n b + 1

n

X

i=n

(log d s (i)) ² .

Let

A s (n, i) = ^△ 1

n − n b + 1 · D s (n) − C ^s (n) log d s (i) D s (n) − C s (n) ² B s (n, i) = ^△ 1

n − n b + 1 · C s (n) − log d s (i)

D s (n) − C s (n) ²

it follows

l s (n) =

n

X

i=n

p s (i)A s (n, i) − p s (i)B s (n, i) log d s (n) and

G s (n) = A s (n, i) − B s (n, i) log d s (n) G ∞ (n) = 0.

Considering that the decision process is expressed by y(n) = l ₀ (n) − l 1 (n),

it follows that for the LS algorithm the decision variable to start the handover has the following expression:

y(n) =

n

X

i=n

p ₀ (i) · G 0 (n, i) −

n

X

i=n

p ₁ (i) · G 1 (n, i) (3.3)

3.2.1 Handover decision an probabilities

The handover decision is taken on the base of the process y(mK) defined as y(mK) , l 0 (mK) − l 1 (mK), m = 1, 2, · · · N/K − 1

every KT seconds (considering KT as the period of signalling multiframe with K integer).

Let define the following connection rule based on y(mK)

b(mK) =

( 0, if y(mK) > 0 1, if y(mK) < 0

where b(mK) is the BS the mobile is connected to at time samples n = mK + 1, · · · (m + 1)K. A handover occurs every time b(mK) 6= b(m + 1)K.

To evaluate performance of the algorithm, the probability of outage P O and the probability of handover P H are defined as

P _O (mK + k) = P [p _b

(mK + k) < β]

P H (mK) = P [b(mK) 6= b(mK − K)] .

Since in the described scenario is reasonable to assume that the mobile is initially assigned to BS 0 , the following expressions hold:

P O (mK + k) =



 



 



P [p 0 (k) < β] , if m = 0

P [p 0 (mK + k) < β, y(mK) > 0] +

P [p 1 (mK + k) < β, y(mK) < 0] , if m > 1

P H (mK) =



 



 



P [y(k) < β] , if m = 1

P [y(mK) > 0, y(mK − K) < 0] +

P [y(mK) < 0, y(mK − K) > 0] , if m > 2

P O (n + 1) =



 



 



P [p ₀ (n) < β] , if n = 0 P [p 0 (n + 1) < β, y(n) > 0] +

P [p 1 (n + 1) < β, y(n) < 0] , if n > 1

P _H (n) =



 



 



P [y(n) < β] , if n = 1 P [y(n) > 0, y(n − 1) < 0] +

P [y(n) < 0, y(n − 1) > 0] , if n > 2

The above probabilities can be evaluated numerically by integrating a bivari- ate Gaussian probability density function. To compute them is necessary to know means, variances and correlation coefficients of every variable. Con- sider that σ ² _y (n) is the variance of the decision process y(n), σ ² _p

(n) is the variance of the signal level p ₀ and σ _p ²

(n) is the variance of the signal level p 1 , then

σ _y ² (n) =

n

X

i=n

n

X

j=n

r _u

(i − j) · G 0 (n, i) · G 0 (n, j) + r _u

(i − j) · G 1 (n, i) · G 1 (n, j) (3.4) σ _p ²

(n) = σ _u ²

σ _p ²

(n) = σ _u ²

where r u

(l) , ˜ r u

(lvT )=σ ² _u

exp( − |l| /( ¯ d/vT )) is the autocorrelation func- tion of u s (n).

And for the expectations it results m y (n) =

n

X

i=n

(α 0 − β 0 log(d 0 (i))) · G 0 (n, i) − (α 1 − β 1 log(d 1 (i))) · G 1 (n, i)

m p

(n) = α 0 − β 0 log(d 0 (n)) m p

(n) = α 1 − β ¹ log(d 1 (n))

By defining z(n) = y(n) − m y (n) for n = mK the normalized correlation coefficients of the handover decision variable y are given by

ρ y (mK) = E [z(mK) · z(mK − K)]

σ _z(mK) σ _z (mK − K) ρ p

,y (mK + k) = E [u 0 (mK + k) · z(mK)]

σ _u

σ _z (mK)

ρ p

,y (mK + k) = E [u 1 (mK + k) · z(mK)]

σ u

σ z (mK) and from (3.1) and (3.3)

E [z(n) · z(n − K)] =

n

X

i=n

n−K

X

j=n

r u

(i − j) · G 0 (n, i) · G 0 (n − K, j) +r u

(i − j) · G 1 (n, i) · G 1 (n − K, j) where n ^′ _b = max {1, n − K − n w + 1 }

E [u 0 (n + k) · z(n)] =

n

X

i=n

r u

(n + k − i) · G ⁰ (n, i)

E [u ₁ (n + k) · z(n)] = −

n

X

i=n

r _u

(n + k − i) · G 1 (n, i)

Given the expressions above, we can evaluate the performances of the LS algorithm in terms of probability of outage and probability of handover;

moreover the expected number of outages and handovers along the trip are simply given by

O = ¯

N −1

X

n=1

P O (n)

O = ¯

N/K−1

X

m=1

P H (mK)

3.3 The effect of the Hysteresis Margin

Up to now LS algorithm was analyzed in the absence of any hysteresis mar- gin but in handover decision criteria some hysteresis margin is allowed to avoid the ping-pong effect that may occur in the edge of cells and hence fre- quent handover requests from single user. The value of the hysteresis margin plays a key role in handover performance and affects the coverage of cellular system, therefore it must be chosen to optimize handover performance: if this margin is too small numerous unnecessary handovers may be processed, on the contrary if it is too high the long handover delay may result in a dropped-call or low QoS. In the proposed algorithm handover decisions are based on the comparison of y(n) with an hysteresis margin h(n) as will be explained in next sections. For the same scenario of a MT moving between only two BSs, BS ₀ and BS ₁ , let define by ε(n) the event of the MT being connected to BS 1 at time n, namely ε(n)= {b(n) = 1}, and anlogously let

¯

ε(n)= {b(n) = 0}, where the event ε(n) is defined as follows:

ε(n) = {y(n) < −h(n)} ∪ {y(n) < h(n), ε(n − 1)} (3.5) and its complementary is

¯

ε(n) = {y(n) > h(n)} ∪ {y(n) > −h(n), ¯ε(n − 1)} (3.6)

b(n)

y(n) -h(n) h(n)

1

Figure 3.1: The hysteresis of the handover decision.

3.3.1 Probability of Base Station Connection

Let L(h(n)) = {y(n) ≤ −h(n)}, M(h(n)) = {−h(n) < y(n) ≤ h(n)}, N (h(n)) =

{y(n) ≥ h(n)}; from (3.5) and (3.6), by using the previous definitions and

considering that L(h(n)) ⊆ ¯ N (h(n)) it follows that ǫ(n) = L(h(n)) + ¯ N (h(n))ǫ(n − 1)

= L(h(n)) + M(h(n))ǫ(n − 1)

= L(h(n)) + M(h(n))L(h(n − 1)) + M(h(n))M(h(n − 1))ǫ(n − 2) The proof is obtained considering that

ǫ(n) = L(h(n)) + ¯ N (h(n))ǫ(n − 1)

= L(h(n)) + {L(h(n)) + M(h(n))} ǫ(n − 1)

= L(h(n)) + {L(h(n))ǫ(n − 1)} + {M(h(n))ǫ(n − 1)}

= L(h(n)) {Ω + ǫ(n − 1)} + {M(h(n))ǫ(n − 1)}

= L(h(n)) + ¯ M(h(n))ǫ(n − 2),

where Ω is the probability space. Since it results that

ǫ(n − 1) = L(h(n − 1)) + ¯ N (h(n − 1))ǫ(n − 2)

= L(h(n − 1)) + {M(h(n))} ǫ(n − 2), it follows

ǫ(n) = L(h(n)) + M(h(n)) {L(h(n − 1)) + M(h(n − 1)ǫ(n − 2))}

= L(h(n)) + M(h(n))L(h(n − 1)) + M(h(n − 1))M(h(n − 1)ǫ(n − 2)).

By iterating the procedure until time 0 ≤ m ≤ n it results ǫ(n) =

n

X

j=m+1

L(h(j))

n

Y

k=j+1

M(h(k)) +

n

Y

k=m+1

M(h(k))ǫ(m) (3.7)

¯ǫ(n) =

n

X

j=m+1

N (h(j))

n

Y

k=j+1

M(h(k)) +

n

Y

k=m+1

M(h(k))¯ǫ(m), (3.8) And in term of probabilities

P r [ǫ(n)] =

n

X

j=1

P r

(

L(h(j))

n

Y

k=j+1

M(h(k)) )

+P _r ( _n

Y

k=1

M(h(k))ǫ(0)

) (3.9)

P r [¯ǫ(n)] =

n

X

j=1

P r

(

N (h(j))

n

Y

k=j+1

M(h(k)) )

+P r

( _n Y

k=1

M(h(k))¯ǫ(0)

) (3.10)

The proof follows from (3.7) and (3.8) by setting m = 0 and observing that ǫ(n) and ¯ǫ(n) are given by the sum of events mutually exclusive. The computation of the probabilities (3.9) and (3.10) is given by a multivari- ate Gaussian distribution, since the events L(h(j)), M(h(k)), N (h(j)) are defined over Gaussian (cross correlated) random variables.

3.3.2 Probability of Handover and Probability of Out- age

Based on the events L(h(n)), M(h(n)), N (h(n)) the handover probability at time n is given by

P _H (n) = P _H01 (n) + P _H10 (n) (3.11) where

P _H01 (n) = P _r [ N (n)ǫ(n − 1)] · P r [ǫ(n − 1)] (3.12)

P H10 (n) = P r [ L(n)¯ǫ(n − 1)] · P ^r [¯ǫ(n − 1)] (3.13) Considering that P 0 (n) = {p 0 (n) < β } and P 1 (n) = {p 1 (n) ≤ β} the expression of the probability of outage is given by

P _O (n) = P _O0 (n) + P _O1 (n) (3.14) where

P O0 (n) = P r [ P ⁰ (n) |¯ǫ(n)] = P r [ P ⁰ (n) |¯ǫ(n)]

P r [¯ǫ(n)] , (3.15) P O1 (n) = P r [ P ¹ (n) |ǫ(n)] = P r [ P ¹ (n) |ǫ(n)]

P _r [ǫ(n)] . (3.16)

By combining (3.12),(3.13),(3.15),(3.16) with (3.9) and (3.10) come out

once again multivariate Gaussian distributions; since the use of these distri-

butions may be computationally prohibitive, in the following some bounds

to approximate the probabilities are provided.

3.4 Approximations

To obtain efficient performance, in this chapter we propose some bounds to approximate the probabilities and their optimizations with respect to the estimate window lenght and hysteresis margin that, as seen in previous chap- ters, play a crucial role in terms of performance.

3.4.1 First Bound

To make calculation of Gaussian distribution faster, in this subsection we study a simple bound to the computation of the probability. Considers a Gaussian vector y ∈ R ⁿ having average µ and covariance matrix Σ given by











σ ² _y

ρ y

y

σ y

σ y

· · · ρ y

y

σ y

σ y

ρ _y

_y

σ _y

σ _y

σ ² _y

· · · ρ y

y

σ _y

σ _y

... ... . .. ...

ρ y

y

σ y

σ y

ρ y

y

σ y

σ y

· · · σ _y ²











which is a n-by-n symmetric positive definite matrix.

Let λ max and λ min be the maximum and minimum eigenvalue of Σ, re- spectively. Consider the sets Y l = y l ∈ y l , ¯ y l  for l = 1...n. Then

P r {Y ¹ Y ² . . . Y ⁿ } ≤ λ

n

max

√ detΣ

n

Y

l=1

P r

 y l ∈

 σ l y l

√ λ max

, σ l y ¯ l

√ λ max



, (3.17)

P r {Y ¹ Y ² . . . Y ⁿ } ≥ λ _min

√ detΣ

n

Y

l=1

P r

 y l ∈

 σ l y l

√ λ min

, σ l y ¯ l

√ λ min



. (3.18) The bounds (3.17), (3.18) are obtained by considering that for every x ∈ R ⁿ it holds

kxk ²

λ max ≤ x ^T Σ ⁻¹ x ≤ kxk ² λ min

and

P r {Y ¹ Y ² . . . Y ⁿ } = Z y ¯

y

Z y ¯

y

· · · Z y ¯

y

e ⁻

^(y−µ)

^Σ

^(y−µ)

pdetΣ(2π) ² dy 1 . . . dy n

≤ Z y ¯

y

Z y ¯

y

· · · Z y ¯

y

e ⁻

pdetΣ(2π) ² dy 1 . . . dy n .

We will show in Chapter 5 that this approximation works only for low values of the hysteresis margin. Therefore, we need an alternative approximation, as we discuss next. The graphic that shows this trend will be reported in Chapter 5.

3.4.2 Second Approximation: proposed bound

In this section we focus on evaluating the probabilities of handover and outage by using the following approximation: given a Gaussian vector y ∈ R ⁿ having average µ and covariance matrix Σ, the probability of y is expressed by:

P r {Y 1 Y 2 . . . Y n } = P r {Y n Y n−1 . . . Y n−m } · P r {Y n−m−1 } · · · P r {Y 1 } (3.19) Where m is such that m ≤ n and Σ ^n−m is the matrix obtained by taking the first n − m rows and n − m columns of Σ. Make use of this bounds means to consider the first n − m components of the Gaussian vector related and the rest of them as if they were independent. In other words, m indicates the number of components belonging to the joint probability which are effec- tively considered to be joint. This make sense considering that at a generic instant time nT there will be a strong correlation with more recent instants time (n-1)T, (n-2)T, (n-3)T and a weaker correlation with those passed. By starting from the expression of the probability given in (3.14), here we study some expressions that we need to use in the probability of handover for the simulator. By combining (3.12) with (3.9) after simple algebra and for a generic time nT it results that the first term of the product in (3.12) is given by:

P r [ N (n)ǫ(n − 1)] =

= P {N (h(n))L(h(n))}

= P {N (h(n))L(h(n − 1))M(h(n))}

= P {N (h(n))L(h(n − 2))M(h(n − 1))M(h(n))}

= P {N (h(n))L(h(n − 3))M(h(n − 2))M(h(n − 1))M(h(n))}

...

= P {N (h(n))L(h(n − 2))M(h(n − 2)) · · · M(h(n − i)) · · · M(h(n))} ,

therefore by applying the equation (3.19) to these joint probability and by

considering the events correlated up till the time (n − 4) (according to the

approximation), for example for the last expression, we obtain:

P {N (h(n))L(h(n − 2))M(h(n − 2)) · · · M(h(n − i)) · · · M(h(n))} = P {N (h(n)))M(h(n))M(h(n − 1))M(h(n − 2))M(h(n − 3)} ·

· M(h(n − 4)) · · · M(h(n − i))L(h(n − 2)) (3.20) Analogously, from (3.13) and (3.10) we have:

P r [ L(n)¯ǫ(n − 1)] =

= P {L(h(n))N (h(n))}

= P {L(h(n))N (h(n − 1))M(h(n))}

= P {L(h(n))N (h(n − 2))M(h(n − 1))M(h(n))}

= P {L(h(n))N (h(n − 3))M(h(n − 2))M(h(n − 1))M(h(n))}

...

= P {L(h(n))N (h(n − 2))M(h(n − 2)) · · · M(h(n − i)) · · · M(h(n))} , hence again it is possible to apply the approximation to have:

P {L(h(n))N (h(n − 2))M(h(n − 2)) · · · M(h(n − i)) · · · M(h(n))} = P {L(h(n)))M(h(n))M(h(n − 1))M(h(n − 2))M(h(n − 3)} ·

· M(h(n − 4)) · · · M(h(n − i))N (h(n − 2)) (3.21) In this way the approximation has been applied to the other joint proba- bilities involved, as the probabability of outage and base station connection probabilities.

For comparison purposes in Chapter 5 we illustrate the results obtained by

using a different value of m which indicates the number of the joint compo-

nents in the provided approximation. Will be shown how chosing a smaller

value of m can degrade noticeably the performance.

Optimization

We formulate a static optimization problem with respect the estimate window length and both static and dynamic optimizazion problems with respect to hysteresis margin.

4.1 Pareto Optimization of the Estimate Win- dow Length

One of the most important parameters of LS algorithm is the estimate win- dow lenght (n w ). The variation of this parameter causes opposite effects on the handover and outage probability; we will show that an increase of n w de- termines a decrease of handover probability (P H ) and an increase of outage probability (P _O ) hence is necessary to find a tradeoff between lower values of P H and higher values of P O . An approach to solve the problem is to use an objective function defined in terms of outage and handover probabilities, is given by:

F obj (n w ) = α · P H (n w ) + (1 − α) · P O (n w ), (4.1) where P _H (n) is the handover probability at time n, P _O (n) is the outage prob- ability at time n, α is a weighting coefficient to tradeoff the performance in terms of outages or handovers. The objective function is therefore a weighted sum of handover and outage probability. The (4.1) is used in the following minimization problem

min n

F obj (n w ).

31

The optimization problem expressed above is an unconstrained static op- timization problem: the optimum value of n w can be evaluated for each instant time nT not taking into account optimum values chosen at previous and next time instants. The choice of the optimum value is carried out by considering that given a value of α included between the interval [0.01,0.99]

it is possible to determine the minimum value of the objective function with respect to the parameter n w for each instant value nT. As in D/2 falls the worst case for values of handover and outage probability, for the parameter n such a value is assumed in the formulated minimization problem. Ones the optimum values of n w are evaluated it results that n wopt ⇒ n ^wopt (a), in D/2; by plotting handover and outage probability evaluated in n wopt (a) P H (n wopt (a)), P O (n wopt (a)) a fair value of the weighting coefficient can be choosen graphically. The results will be shown in Section 5.2.

4.2 Static Handover and Outage Pareto Op- timization

LS algorithm uses an hysteresis for the handover decision. It is easy to see that performance of LS can be improved by adapting the hysteresis. To select the hysteresis level, we could use a simplified method similar to the one proposed for the estimate window length. The optimization problem is formulated by considering the following cost function:

f obj (n, h(n)) = α · P ^H (n, h(n)) + (1 − α) · P ^O (n, h(n)), (4.2) which has to be minimized with respect to the values that h assumes in each n, thus we have to find

min h(n) f obj (n, h(n))

without any constrains. As it is a static optimization problem, to solve it we need to find the optimum value of h that, for every n, minimizes the objective function, without considering what happens in the past and in the future. Consider the following values:

• α ∈ [0, 01; 0, 99]

• h ∈ [2; 10]

the optimization algorithm can be described as follows:

1. for each value of α we evaluate P H and P O for every value of n and every value of h. The values are collected in a matrix of dimensions [n × h];

2. as probabilities values are known it is possible to compute numerically the cost function and find its minimum value corresponding to a specific value of h. The minimum-find is processed for each n and so optimum values of h are known.

3. finally, by iterating step 2 for each value of α, it is possible to see the variation of the cost function evaluated in h optimum with respect to n,a.

The results obtained by applying the above algorithm will be presented in Chapter 5.

4.3 Optimization of the Probability of Han- dover

In this section we propose the optimization of the handover probability un- der outage constraints. More specifically, here we investigate the following dynamic optimization problem:

min h(n) n+m

X

l=n

P Hb(l) (l)

s.t P Ob(l) (l) ≤ P out , l = n, · · · , n + m

b(l + 1) = f (b(l), h(l)), l = n, · · · , n + m.

In such a problem, the decision variables are the hysteresis thresholds h(l) for l = n, . . . , n+m, which we collect in the vector h(n) = [h(n)...h(n+m)] ^T . The problem is posed such that at each time instant n, the mobile station tries to minimize the probability of handover from the current moment up until a future instant that is m sampling times from n. The handover probability is minimized taking into account outage events, which motivates the outage probability constraint for ensuring an adequate quality of the communication.

In other words, we impose that at each time instant l, l = n, . . . , n + m, the

outage probability must be below a maximum value Pout. The last constraint

of the optimization problem gives the base station b(l+1) at which the mobile