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    Cell to switch assignment in UMTS

    Contains page

    1. Introduction 2

    1.1 Introduction to the wireless cellular network (WCN)

    1.2 Evolution of WCN

    1.3 Introduction to First generation

    1.4 Introduction to Second generation

    1.5 Introduction to third generation

    1.6 Handoff

    2

    4

    4

    6

    8

    11

    2. Literature survey 12

    2.1 Handoff in PCN2.2 Complex handoff

    2.3 Simple handoff

    2.4 Handoff in UMTS

    2.5 Horizontal and vertical handoff in overlay network

    2.6 Complex and Simple handoff in UMTS

    2.7 Cell two Switch Assignment

    2.8 Single homing

    2.9 Dual homing

    2.10 Cost function generation for Single homing cell to switch assignment in

    PCN

    2.11 Cost function generation for Dual homing cell to switch assignment in

    PCN2.12 single homing in UMTS

    2.13 Dual Homing in UMTS

    2.14 Heuristic method

    2.15 Tabu search

    1212

    13

    13

    15

    15

    16

    16

    17

    18

    19

    20

    2323

    23

    3. Future work 24

    4. Reference 25

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    1. IntroductionOver the recent years, telecommunication has been a fast growing industry. Because it draws a

    great deal of attention since it was introduced. Several warless technologies were developed in

    different times. They have different characteristics and different applications.

    Before going into detail of cell to switch assignment for UMTS network, however, it is

    appropriate to review mobile communication in general, as well as first and second generation

    systems

    The first generation mobile system was introduce in 1978 in Chicago as Advanced Mobile Phone

    Services (AMPS), operating in 800-MHz and again it re-launch in 1983 for public use in Chicago.

    Meanwhile many other country introduce their own 1st

    generation mobile system, so, at a gross

    1st

    generation system experience success far greater than anyone had expected. But the main

    problem with the 1

    st

    generation technology is its limited capacity.

    Unlike 1st

    generation system which is analog 2nd

    generation comes with digital system. The

    most successful 2nd

    generation technologies are

    1. Interim Standard 136 (IS-136) TDMA.

    2. IS-95 CDMA

    3. Global System for Mobile communication (GSM)

    In third generation mobile system comes under 3GPP group. UMTS support high speed data as

    well as voice communication that is why it is a popular chooses among people.

    So as the demand increases services provider need to increases their services area. So there is a

    cost for routing the call through their company network and which they have to bear.

    Throughout this article, how could a services provider can minimize the cost associated with

    the call routing through their network by optimally assigning its cells and switches, is disused

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    1.1 Introduction to Wireless Cellular Networks (WCN)The WCN is a telephone system which provides wireless connection to a large number of users

    over a large geographical area with a limited frequency spectrum. In WCN, the whole converge

    area is divided into a number of hexagonal cells so that the same radio channels may be reused

    by another base station located some distance away

    Now, each cell contains a base station,a fixed station in a mobile radio system used for radio

    communication with mobile stations ,Base stations are located at the center or on the edge of a

    coverage region and consist of radio channels and transmitter and receiver antennas mounted

    on a tower [1]. So base stations are the responsible for channel allocation of the users in that

    particular cell. A portion of the total number of available channels is allocated to each BSC and

    ne nearby BSCs is assigned different groups of channels so that there is no interference

    between neighboring BSCs. channels of each BSC are well separated so the simultaneous

    communication is possible and this increase the capacity. Spectrum reuse technique protectsthe services provider from buying more spectrum band which is too costly.

    A group of cell is connected with a switch which is known as MSC or mobile switch center,

    Switching center which coordinates the routing of calls in a large service area. In a cellular

    radio system, the MSC connects the cellular base stations and the mobiles to the PSTN. An MSC

    is also called a mobile telephone switching office (MTSO).[1] So basically MSC routed the call

    trough out the company network. MSC is interconnected with the PSTN via Landline trunks line

    and a switch. The inter communication between MSC is done by some dedicated channel for

    exchanging of information like location update, validation, cell signaling information shown in

    fig.1. In cellular telephone systems, voice traffic carried on the PSTN and the signaling

    information is carried on the SS7 network. MSC is also connected with HLR (home location

    register) i.e. it keeps track of the user of that particular MSC or cells. MSC is also connected

    with VLR (visitor location resistor) and AuC(Authentication center) .by this MSC can keep track

    the new subscriber in that cell and authenticated him or her

    The global cellular network is required to keep track of all mobile users that are register

    throughout the network. So that it is possible to forward incoming calls to rooming users at any

    location. When the user roams the new market with completely new service provider the

    wireless network must register the new user and cancel the previous one for the sleek of the

    routing calls through the new network because it is in different MSC coverage area.

    So from the previous discursion we can see that whenever one user moving from one cell to

    other cell a new channel is allocated for that particular user. Before BSC allocated some channel

    MSC must aware about the call and then finally handed over that call from one cell to another

    cell

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    Fig.1. Block diagram of a cellular radio network [1]

    1.2 Evaluation of WCNWCN has gradually improved since it was initiated as most advanced technologies have been

    introduced. The demand of WCN increases with rapid worldwide growth in cellular telephonesubscribers. The wide spread success of WCN has led to the development of advanced WCN

    systems

    1.3 First Generation (1G)This is the one of the most prolific voice communication platforms that has been deployed

    within the last two decades. In this generation some key concept employed in wireless

    communication such as

    1. Frequency reuse

    2. Mobility of the subscriber

    3. Handoff

    There are several 1st

    generations mobile system were develop but among them two most

    popular system was advance mobile phone system (AMPS) & total access communication

    services (TACS).

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    AMPS TACS

    Base Tx MHZ 869-894 935-960

    Base Rx MHZ 824-849 890-915

    Multiple access method FDMA FDMA

    Modulation FM FM

    Radio channel spacing KHZ 30 25

    Number channels 832 1000

    Spectrum allocation MHZ 50 50

    Comparative study between AMPS and TACS

    So as discus previously BSCs are connected with MSCs and two MSCs connected with the land

    line via PSTN & for signaling porous SS7 was there. HLR, VLR and AuC are there for user

    registration and authentication

    Some key features of 1st

    generation architecture is

    Analog traffic channel.

    Frequency modulation (FM) technique is used.

    Operated on FDMA/FDD.

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    It can handle only voice signal

    Use circuit switching

    MSC serves

    Call processing

    Power control

    Hand off control

    Billing

    The main disadvantages are

    Very low in capacity

    High load on MSC

    1.4 Second generation (2G)Second generation design is generalize used to describe the advantage of the digital mobile

    communication for cellular mobile system.

    The issues regarding 2G deployment are as follows:

    Capacity

    Spectrum utilization

    Infrastructure change Subscriber unit upgrades

    Subscriber upgrade penetration rates

    The fundamental binding issues with 2G are the utilization of digital radio technology for

    transporting the information content.

    The digital techniques for cellular communication can be deviled into two major groups, AMPS

    and TACS. For AMPS TDMA or CDMA are use for TACS GSM is preferred. Other then

    AMPS/TACS, IDEN radio access platform is available and in the SMR (specializes mobile redio)

    band, which is neither cellular nor PCS. In the table below represent some of the specificationof different technology used in second generation WCN

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    IS-136 IS-95 GSM

    Base Tx MHZ 869-894 869-894 925-960

    Base Rx MHZ 824-849 869-894 880-915

    Multiple Access

    Method

    TDMA/FDMA CDMA/FDMA TDMA/FDMA

    Modulation Pi/4DPSK QPSK .3 GMSK

    Radio Channel

    Spacing KHZ

    30 1.25 200

    Users /Channel 3 64 8

    Number Channels 832 9(A)

    10(B)

    124

    Spectrum Allocation

    MHZ

    50 50 50

    Comparative study for different technology

    From the fig.3 we can see that one ISDN (Integrated System Digital Network) is connected with

    GMSC (Gateway MSC) for data communication.

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    Some key features are

    Digital network

    Use digital voice coding

    Digital modulation

    Higher Capacity

    Higher security

    Caller identification , short Massaging Service(SMS)

    It is designed to provide voice & other data services.

    Operated on TDMA/FDD or CDMA/FDD.

    Network controlling structure is more distributed

    Hand off process is mobile control (MAHO).

    Introduce Base Station Controller(BSC)

    Higher Mobility

    Lower power radio signal require.

    1.5 Third generation (3G)Analog cellular phones were the first generation while digital marked the second generation.

    3G is loosely defined, but generally includes high data speeds, always-on data access, and

    greater voice capacity. The high data speeds are possibly the most prominent feature and

    currently most hyped. They enable such advanced features as a live streaming video

    3G and all it is meant to be are defined in ITU specification International Mobile

    Telecommunication- 2000 (IMT-2000). IMT-2000 is a radio and network access specification

    defining several method or technology platforms that meet the overall goals of thespecification.

    When the International Telecommunication Union solicited solutions meet the requirement

    laid down for IMT-2000, a number of technology were proposed by various stander groups.

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    The European Telecommunications Stander Institute (ETSI) agreed on a WCDMA solution using

    FDD. In Japan, WCDMA solution is also proposed, with both TDD and FDD option. In Korea two

    different type of CDMA solution were proposed. So a number of groups working on the very

    similar technology and it were fairly obvious that the most effective way forward was pool

    resource. This led to the creation of two group 3GPP (Third generation partnership project) :works on UMTS and other one is 3GPP2 works on CDMA200 .

    UMTSis aimedto be a system capable ofproviding a variety of mobile services (up to 2Mbps)

    offered by (dedicated) paging, cordless, and cellular Communications systems separately. UMTSis tobe operational in domestic, business, vehicular, and (multi-operator) public environments.

    For UMTS architecture two new units are added, those are RNC (Radio Network Controller):

    between BSC and MSC & SGSN (serving GPRS support node): work as an internet gate way

    The major tasks of RNCs:

    management of radio resources

    channelization code allocation

    QoS monitoring

    handover of users between cells on the same site (softer handover)

    handover of users between cells on different sites (soft handover)

    handover of users between different UMTS carriers (hard handover)

    handover of users to GSM networks (hard handover)

    power control management of user and Node B equipment

    network alarm correlation

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    The optimal cellular architecture is determined by traffic and mobility characteristics. In a

    certain cover- age area the traffic and mobility characteristics are varying. Rather than defining

    a conventional cellular architecture to cope with this mixed traffic, a mixed cell layout

    consisting of an over levering of micro and Pico cells by macro (umbrella) cells is being

    considered [2]. The following advantages are foreseen:

    highly mobile traffic may be allocated to macro cells thus reducing the handover

    signaling load

    Macro cells can cover black spotsnot covered by Pico or microcells

    Macro cells introduce some redundancy thus increasing system availability

    Macrocells can accommodate. for microcell handover failures

    The disadvantage of macro cells isreduced spectrum efficiency.

    Fig.5. micro and umbrella cell [1]

    So the some key feature of 3G:

    Generate one world standards.

    CDMA use as air interface.

    Provide better system capacity both in terms of user & bandwidth

    Support multimedia services with high speed.

    o Telephony(voice)

    o Internet

    o Location service

    o Video conferencing

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    o Live streaming

    Better Power Control.

    1.6 Handoff

    From the figure we can see that cell-1, 2 are connected with MSC-1 and cell-4,3 are connected

    with MSC-2. Now consider some subscriber is moving from cell-1 to cell-4 while taking with

    someone over phone (that call is now served by BSC-1). Now, BSC-1 can measure the signal

    strength of that call & inform MSC-1 about it. MSC-1 has also knowledge about the signal

    strength of cell-1 and cell-4. So, when ever MSC-1 can sense that the signal strength of that

    particular call goes down below some pre define threshold and simultaneously the signals

    strength of cell-4 became more prominent, MSC-1 can understand that the vehicle is now

    entering cell-4 and BSC-1 cannot serve that call. So for the sleek of continuing the call MSC-1

    transfers all the data regarding that call to MSC-2. MSC-2 registers the subscriber and finally

    BSC-2 allocated some channel.

    So, in short MSC-1 handed over that call from cell-1 to cell-4 and this is known as handoff

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    2 Literature surveyWe were motivated to work in the field of wireless networks as we have gone through some

    published literatures. Different types of works were done in this field by many researchers in

    different times. Lots of Journals, transaction papers were published and are available in this

    context. We got information required to start our work through some literature survey. We

    explain about our survey in the following sections

    2.1 Handoff in PCNSo in one line the handoff is the process of transferring a mobile station from one channel or

    base station to another.

    Handoff is broadly divided into two categories- hard handoff and soft handoff. In hard handoff

    current resources are released before the new resources are used, that is when two MSCs

    involve in handoff. In case of soft handoff the current and new resources are used

    simultaneously while handoff takes place, i.e. when handoff occurs between two different cellbut both are connected with same MSC. Handoff can be further divided into intra-cell handoff

    and Inter-cell handoff. The soft handoff can also be divided into multi way soft handoff and

    softer handoff.

    2.2 Complex hand off

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    As the vehicle move from cell-1 to cell-4, two MSCs involve in the process. So this type of handoff needs

    transferring data from one MSC to other MSC so it is respectively costly. This type o f hand off is known

    as complex handoff

    2.3 Simple handoff

    In case of simple handoff, consider the case when vehicle moving from cell-1 to cell-2. In this case as the

    both cell are connected with the same MSC, there is no need of data transferring so this type of handoff

    is known as simple handoff

    In [3] they calculated handoff for PCN like this way .The rate of calls originated in each cell was

    generated from a Gamma distribution with mean one and coefficient of variation 0.25. The

    holding time of a call within each cell was taken to be exponentially distributed with mean one.

    If a cell had ICneighbors, then we divided the range (0, 1) into K+ 1 interval by selecting K

    random numbers from a uniform distribution between 0 and 1. The length of the ith intervalrepresented the probability of handoff from the cell to its ith neighbor for i= 1, 2. . .Katthe

    end of the holding time. With probability equal to the length of the K+ 1st interval, the call ends

    2.4 Handoff in UMTSUMTS is a 3 layer architected system. In UMTS, there are 3 different types of handover, as follow.

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    Hard handoff: involving two different MSCs

    Soft handoff : involving two different RNCs and both are connected with same MSC

    Softer handoff: handoff between two cells connected with same RNC

    Fig.11. hard handoff

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    2.5 Horizontal and Vertical Handoff in Overlay Networks

    Fig.12. horizontal and vertical handoff

    Horizontal handoff is a system which allows an active user to move between different = access

    points (AP) of same network technology without any interruption on the ongoing call. Its

    shown in fig.12 Vertical handoff is a system which allows active user to roam between different

    wireless networks in wireless overlay network.

    Vertical handoff happens between the BSC of different wireless network technologies. there

    are two type of vertical handoff describe in [13] ,

    upward vertical handoff

    downward Vertical handoff

    When vertical handoff occurs from network with lower cell size and higher bandwidth, it is

    called upward vertical handoff. And when it happens to a network of lower cell size and higher

    bandwidth from higher cell size and low bandwidth network, then it is called downward Vertical

    handoff.

    Vertical handoff may happen between two immediately higher and lower networks or hopping

    of an overlay may possible. The downward vertical handoff is less time-critical because it could

    always be connected to an upper layer while handoff takes place.

    2.6 Complex and simple handoff in UMTSSome Literatures [9] describe this type of handoff in UMTS network for making an analogy with

    PCN simple and complex handoff.

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    Simple handoff: in this type of hand off both soft and softer handoff included. In this

    type of handoff two RNCs or two BSC may involve but both are connected with same

    MSC & SGSN. So this is basically horizontal handoff

    Complex handoff: in this type of handoff hard hand off is included. so this is basically it is

    vertical type of handoff

    2.7 Cell to switch assignmentIn [5], Merchant and Sengupta define the problem as We are given a group of cells and a

    group of switches (whose locations are known). The problem is to assign the cells to switches in

    an optimum manner. We would like to do the assignment in an attempt to minimize a cost

    criterion. The cost has two components. One is the cost of handoffs which involve two switches

    and the other is the cost of cabling (or trucking).

    2.8 Single homing

    Fig.13. single home cell to switch assignment

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    If we have only one user behavior pattern trough out the day then assigning each cell to a

    switch would make sense. That Is in this type of arrangement each cell can be assign only one

    switch

    2.9 Dual homing

    Fig.14. dual homing

    Consider a situation, say at the evening many subscriber are moving form cell-7 to cell-2 by cell-

    8 and cell-8 is connected with MSC-2 and cell-2 & & are connected with MSC-1 (as in the fig.14).

    So it would make sense if we connect cell-8 with MSC-1 for that time period. Because it will

    reduce the handoff cost i.e. connecting with MSC-1 the handoff between cell-2 and cell-8 and

    cell-8 and cell-7 is became simple handoff. So for the evening time cell-8 connects with MSC-1

    and rest of the day it connected with MSC-2, i.e. cell-8 connects with two different MSCs but

    only one is active in a particular time. So this is known as dual homing.

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    So, if we have two different users behavior patterns throughout the day then we could assign

    each cell to one or two MSCs but only one MSC is active for a particular time is known as dual

    homing problem

    2.10Cost function for single homing cell to switch assignment in PCNIn [5], Merchant and Sengupta well define about the cost function .some assumptions are as follow

    There are n {1,2,n} cells to be assigned to m{1,2,m} switch

    hij is the handoff cost between cells i & j

    cik be the amortized cost of cabling per unit time for ith cell with kth switch

    i denote the number of calls that cell i handles per unit time

    Mk denote the call handling capacity of switch k

    We are interested min. cabal costing as well as cost due to complex handoff. we shall not

    consider the simple handoff cause cost due to it respectively chip

    Now consider we arrange our cells and switch in some way and it is look like fig.13.

    Now define

    (1)

    Constrain on the call handling capacity

    For k=1 m (2)

    This prove is obvious say consider k=2 then equation (2) shows

    (3)

    I.e. the left hand side of the inequality (3) gives the sum of call handle capacity per unit time of

    those cells which are connected with MSC-2 and the RHS shows that the call handling capacity

    per unit time for MSC-2 . so it is obvious that for successfully handling a call inequality (2) must

    hold true

    For I, j=1 n (4)

    The cabling cost for ith cell is

    1 ' ' ' '

    0ik

    x if cell i is assigned to switch k

    otherwise

    1

    n

    i ik k

    i

    x M

    53 4 6 2M

    1 ' ' & ' '

    0

    ijy if i j connected with a common switch

    otherwise

    1

    m

    ik ik

    k

    C c x

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    (5)

    This is also very obvious like (2)

    Handoff cost from ith cell can be express as

    (6)

    For proving it 1st notice that

    ( ) If i and j are connected with same MSC

    ( ) Otherwise

    I.e. we eliminate the simple handoff from cost function.

    So

    (

    ) if there is simple handoff and

    (

    )

    if there is complex hand

    off

    So the RHS of (6) shows the total cost due to complex handoff from cell i to other cell in the

    network

    So the total cost is (7)

    2.11Cost function for dual cell to switch assignment in PCNAccording to the principal of double homing, a cell could be connected to one or two switch,but in each pattern only one switch is active at a time. We must then find two assignment

    patterns for each part of the day, by indicating whether it is cost-effective to have a cell

    connected to two switches or to have a cell connected with one switch

    Some assumptions are

    For some time in a day the parameters are

    For some other time it is

    are constant

    So we have two single home problem in the cost function but we can/t simply write them side

    by side because we may over calculated the cabling cost . So we define ne variable as

    (8)

    1

    (1 )n

    ij ij

    j

    H h y

    1 1 1 1

    (1 )n m n n

    ik ik ij ij

    i k i j

    c x h y

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    Here is the or operation

    So the cost function is

    ' '

    1 1 1 1 1 1

    (1 ) (1 )ij ij

    n m n n n n

    ik ik ij ij

    i k i j i j

    c w h y h y

    (9)

    2.12Single homing in UMTSThe basic problem is that UMTS is three layer, so we cannot simply formulated its cost function

    Here complex handoff is hard handoff and soft and softer are called simple handoff as discuss

    before

    Fig.15. typical UMTS architecture

    The assumptions are

    There are I{1,2,n} cells , J{1,2,r} RNC, K{1,2,m} MSCs and L{1,2,s} SGSN

    12

    ijc cabling cost between ith BSC and jth RNC

    23jkc cabling cost between jth RNC and Kth MSC

    24

    jlc cabling cost between jth RNC and lth SGSN

    2

    vjw circuit switching cap. For of jth RNC

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    2djw packet switching cap. For of jth RNC

    3

    vkw capacity of MSC k

    4dl

    w capacity of SGSN l

    12vif amount of voice traffic of node B

    12dif amount of data traffic of node B

    23vj

    f amount of voice traffic of RNC j

    24dj

    f amount of data traffic of RNC j

    '

    3

    iih complex handoff involving MSCs

    '

    4iih complex handoff involving two SGSN

    So as previously define

    12 1ijx If I ith BSC is connected with jth MSC

    12 0ijx Otherwise

    231

    0

    jkx if jthRNCconnectwithKthMSC

    otherwise

    24 1

    0

    jlx if jthRNCconnect withlthSGSN

    otherwise

    Also define

    ' '

    12 1 &

    0

    iiy if both i i conectedwithsameRNC

    otherwise

    ' '

    23 1 &

    0

    jjy if both j j conected withsameMSC

    otherwise

    ''

    241 &

    0

    jjy if both j j conected withsameSGSN

    otherwise

    Considering two classes of traffic, the traffic generated by all Nodes Bs connected with RNC j

    cannot exceed the RNC capacities in term of voice and data traffic which is expressed as follow

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    12 12 2

    12 12 2

    vi ij vj

    i I

    di ij dj

    i I

    f x w j J

    f x w j J

    (10)

    Similarly, voice traffic from the RNCs to MSC k cannot exceed the MSC capacity, which is

    express as follows:

    23 23 3

    vj jk vk

    j Jf x w k K

    (11)

    Also, data traffic from the RNC to SGSN l cannot exceed the SGSN capacity, which is express as

    follow

    24 24 4

    dj jl vl

    j Jf x w l L

    (12)

    Like the previous solution we can express our cabling cost in similar way but the only problem is

    we shod consider three case BSC RNC, RNCMSC and RNCSGSN

    So our cabling cost is

    12 12 23 23 24 24

    ij ij jk jk jl jl

    i I j J j J k K j J l L

    C c x c x c x

    (13)

    Now in case of handoff cost again we three layer architecture now considering the liner

    programming we can express the BSCRNCMSC handoff is as

    '

    ' '

    1 3 12 23(1 )(1 )i iii ii jj

    i I j J i I j J

    f h y y

    (14)

    Because 1st

    consider the term'

    3 12(1 )i iii ii

    i I i I

    h y

    this is just like PCN network handoff cost

    formulation. The only difference is that instead of BSC, MSC connection it is BSC, RNC

    connection. Now'

    23(1 )jjy is 1 if both the RNC doesnt control by the same RNC.

    So the total term is express as like (14)

    Similarly for the BSCRNCSGSN, the handoff cost is as

    '

    ' '

    2 4 12 24(1 )(1 )i iii ii jj

    i I j J i I j J

    f h y y

    (15)

    So the total cost function is express as

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    ' '

    ' ' ' '

    1 2 12 12 23 23 24 24 3 12 23 4 12 24(1 )(1 ) (1 )(1 )i i i i

    ij ij jk jk jl jl ii ii jj ii ii jj

    i I j J j J k K j J l L i I j J i I j J i I j J i I j J

    C f f c x c x c x h y y h y y

    (16)

    2.13

    Dual homing in UMTSIn case of dual homing in we have two options

    We may connect cells with one or two RNC but only one RNC is in action at a time

    We may connect RNCs with one or two MSCs and SGSN but only one MSC and SGSN is in

    action at a time

    In [10] authors are formulated cost function for dual homing in RNC in UMTS network

    2.14Heuristic MethodAssigning cell to switch is like Bin-picking problem in mathematics, so this is NP-hard problem

    and can be solve by heuristic method. In [8] authors solve cell to switch assignment problem for

    PCN network for both the single & dual home by greedy method & their own describe heuristic.

    In [9],[14] authors solve the same problem by Tabu search method. In case of UMTS network

    [9],[14] authors solve it by tabu and the use three level Tabu search for getting the better

    solution. In [10] they compeer different Meta heuristic technique for giving us better

    understanding

    2.15Tabu searchTabu search is an adaptive technique used in combinatorial optimization problem the besic

    Tabu search sis as follow

    INITIALIZATION: choose an initial solution S in X

    : = S ( )

    nbiter: = 0 ( )

    bestiter : = 0

    ( )

    T := null (T is Tabu list)

    Continue: = true

    While Continue do

    If (nbiter-bestiter>kmax) or (V-T=null)

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    Then continue: = false

    Else nbiter : = nbiter+1

    Generate (N(S) is the neighborhoods of S)

    Find the best solution S in

    S: = S

    Update T

    If then

    bestiter := nbiter

    End if

    end while

    3 Future workSolving the cell to switch problem for UMTS network by some Meta heuristic method and also

    involve kth step random walk model

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    References

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    April 2010.

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