Modelling Service Time Distribution in Cellular Networks Using Phase-Type Service Distributions

Transcription

1 Modellng Servce Tme Dstrbuton n Cellular Networks Usng Phase-Type Servce Dstrbutons runa Jayasurya, Davd Green, John senstorfer Insttute for Telecommuncaton Research, Cooperatve Research Centre for Satellte Systems, Unversty of South ustrala, Mawson Lakes, S 595, ustrala Teletraffc Research Centre, Unversty of delade, delade S 5, ustrala Tel: Fax: aruna@sprlevelsunsaeduau bstract -- Thrd generaton moble communcaton networks are desgned to provde a varety of hgh data rate servces wth hgher Qualty of Servce (QoS) than second-generaton systems Handover becomes one of the major problems n such a moble envronment as t s of utmost mportance to provde a hgher guarantee that the users are able to contnue ther servce durng the entre length of the transmsson, wthout t beng blocked durng the handover or loss of qualty or data Careful dmensonng of the network and the underlyng teletraffc analyss plays a major role n determnng the varous Grade of Servce (GoS) parameters that servce provders can provde at varous network loads The channel holdng tme of a cell s one of the major parameters that needs to be accurately modelled n the teletraffc analyss Ths paper focuses on usng Phase-type dstrbutons of Generalsed Erlang form to model channel holdng tme n a moble envronment We also present the Quas-Brth- Death process, whch characterses the queung models wth Generalsed Erlang servce and exponental nterarrval dstrbutons Further we nvestgate the use of channels exclusvely reserved for handover users to mprove the handover performance I INTRODUCTION One of the major problems that needs to be addressed n moble communcaton networks s the contnuty of a servce durng a handover wthout any data loss, as the user moves from cell to cell Ths s called seamless handover [] Most of the new servces, such as vdeo-conferencng and e-commerce applcatons requre a hgher guarantee of beng able to contnue the servce across cell boundares, than s provded by current moble servces [] The blockng probablty encountered at handover s an mportant Grade of Servce (GoS) parameter for moble users Due to the fact that hgher GoS s requred from future networks, t s of utmost mportance to carefully dmenson the network to provde the guaranteed GoS levels Durng ths dmensonng phase, the number of channels needed to guarantee acceptable blockng probabltes for users wll be estmated under varous network loads Furthermore the sze of the cells wll be selected to provde enough resources to servce the predcted amount of traffc n a partcular area Secton II brefly descrbes some of the technques that were used to model moble communcaton networks and some of ther drawbacks Ths s followed by a statement of a model for the channel occupancy tme n moble networks Secton III ntroduces the Phase-Type (PH) dstrbutons, whch are the dstrbutons that we used to characterse channel occupancy tme n ths study, followed by the phase-type approxmaton for the channel occupancy tme Secton IV brefly descrbes the Quas-Brth-Death (QBD) processes, whch result from mult-server queue wth phasetype nterarrval and servce dstrbutons We then descrbe how to derve the QBD process for the moble cell server under nvestgaton Secton VI ntroduces the use of channels reserved exclusvely for handover users to mprove the handover performance Ths s followed by the results obtaned n ths study, whch shows that the optmum terrestral system capacty that can be acheved s 8% of the total capacty, wth a sngle channel reserved exclusvely for handover users Secton VIII concludes the paper wth closng remarks and some future research drectons II TELETRFFIC NLYSIS OF MOBILE NETWORKS Channel holdng (occupancy) tme s a very mportant parameter n analysng moble communcaton networks Prevously, analyses used exponental dstrbutons to model channel holdng tme for the sake of tractablty [] But expermental data showed that actual channel occupancy dstrbutons are sgnfcantly dfferent from exponental dstrbutons used n these analyses [3] Varous other methods such as the Sum of Hyperexponental (SOHYP) [4] and general dstrbutons [5] were proposed but the complexty of the analyss has ncreased consderably wth these technques Channel Holdng Tme n Moble Communcaton Networks Users n moble communcaton networks can be grouped nto two major categores accordng to the tme they occupy resources n a partcular cell; namely new and handover users ssumng call duraton s exponentally dstrbuted wth parameter µ Fang and Chlamatac [5] showed that the Laplace transform of the dstrbuton of servce tme for new users, f nh, and the Laplace transform of the dstrbuton of servce tme for handover users, f hh, are gven by ( ) = µ s f nh s + f r ( + µ ) s + µ s + µ s, () where f r ( s) s the Laplace transform of the dstrbuton of tme between the start of a call and the frst handover; and µ s f hh ( s) = + f c ( s + µ ), () s + µ s + µ

2 where f c ( s) s the Laplace transform of the dstrbuton of cell resdue tme ssumng new and handover users have exponental nterarrval tme dstrbutons wth parameters λ n and λ h, then the channel holdng tme, t, n a cell can be expressed as [5] λn λh f t ( s) = f nh( s) + f hh( s) (3) λn + λh λn + λh where f t ( s) s the Laplace transform of the dstrbuton of t Ths result shows that the channel holdng tme n a cellular network can be expressed as a combnaton and convoluton of several exponental dstrbutons In ths research we approxmate the channel holdng tme usng Phase-type dstrbutons Phase-type (PH) dstrbutons represent a set of dstrbutons that are combnatons and convolutons of dfferent exponental dstrbutons III PHSE-TYPE DISTRIBUTIONS Phase-type dstrbutons can be used to approxmate vrtually any renewal process, wth the dmensonalty of the phase-type dstrbuton ncreasng wth the complexty of the partcular process beng modelled [6,7] ny contnuous dstrbuton, X, on [, ), whch can be obtaned as the dstrbuton of tme untl absorpton n a contnuous tme fnte state Markov chan (whch has a sngle absorbng state nto whch absorpton s certan) s sad to be of phase-type The ntal state may be chosen randomly and all states are transent except the absorbng state [6-7] From the above defnton we can defne a Markov process on the states {,,n} wth ntal probablty vector ( τ, ) and nfntesmal generator or rate transton matrx Q =, (4) t T where s a row vector of sze n, T s an n n matrx, and t s a column vector of sze n s Q s a rate transton matrx of a Markov process [6], Te + t =, where e s a column vector of s Further elements of T and t satsfy the followng, T <, t, T for j n j For an ntal probablty vector ( τ, ) ; we also have that τ + H = The above phase-type dstrbuton s denoted as PH(τ,T) Modellng Channel Occupancy Tme Usng Phase-Type Dstrbutons In ths study t was ntended to characterse the channel holdng tme of cellular users usng phase-type dstrbutons moble network envronment was set up n OPNET network modeller to obtan typcal user channel holdng tmes n moble envronments Ths network conssts of four cells wth rad of kms Total of 5 users were randomly dstrbuted wthn the network These users were gven dfferent moblty patterns to smulate the moblty patterns followed by cellular users n urban areas Ths was smulated for a - hour perod Interarrval tmes for handover and new users n dfferent cells, and the tmes new and handover users spend n each cell were recorded The tmes new and handover users spent n each cells were used to create the probablty dstrbuton for channel holdng tme for the above network Fang and Chlamatac show that an Erlang dstrbuton s much better suted than an exponental dstrbuton to characterse the channel holdng tme n cellular networks [5] Therefore n ths study a phase-type dstrbuton of Generalsed Erlang form was used to approxmate the channel holdng tme dstrbuton obtaned from the smulaton data The Expectaton Maxmsaton (EM) method was used to fnd the parameters of the Generalsed Erlang dstrbuton that best fts the expermental data [8] lthough the accuracy of the approxmaton ncreases wth the dmensonalty of the dstrbutons, t also ncreases the complexty of the analyss Therefore a dstrbuton of two phases was selected n ths study as t was observed that the error between the actual and the ftted dstrbuton does not mprove sgnfcantly after the number of phases were ncreased beyond Parameters of the PH dstrbuton selected are gven below = T and = [ ] IV QUEUING MODEL FOR CHNNELS OF CELLULR NETWORKS cell n a moble network havng n channels can be modelled as a PH/PH/n/n queue The rate transton matrx for ths queue has a characterstc block tr-dagonal form s both new and handover arrvals are assumed to be exponental n cellular networks, the collectve channels n moble cells can be modelled as a M/PH/n/n queue wth the rate transton matrx, Q, of the form gven n (5) ssumng an exponental arrval dstrbuton wth parameter λ and phase-type servce dstrbuton wth parameters PH(τ,T), matrces B, B and B are gven n (6) B B Q = B,n,n,n,n,n (5)

3 B = λ, B = λ τ, B = t (6) Where symbol represents the Kronecker product between the two matrces However, the constructon of, and for general phase-type dstrbutons are very complex and are beyond the scope of ths artcle k-,m-k λ k µ k-,m-k+ k,m-k (m-k) µ λ k,m-k+ k+,m-k V M/PH/N/N SERVER WITH GENERLISED ERLNG SERVICE DISTRIBUTION drawback n usng the above model to represent the moble network cells s the fact that sze of the rate transton matrx ncreases exponentally wth the number of channels avalable n a cell [9] Future moble networks ntend to provde a potentally large number of channels per cell to support the hgh data rate servces that wll be avalable n the future Therefore the methods avalable n [9] cannot be used to fnd the blockng probabltes experenced by new and handover users n the moble network envronment smplfcaton can be made to the Q matrx by observng some propertes of the Erlang dstrbuton and the behavour of servers n moble cells In servers wth Erlang dstrbutons, the users always start the servce n the frst phase and move on to the next phase wth probablty once the sojourn n that phase s over Users who fnsh the sojourn n the last phase depart the system When there are m users n the system t s rrelevant whch user of these m users s at whch server and smlarly who fnshes the servce frst Ths leads us to combne all the users n the same servce phase to a sngle server wth the servce rate equvalent to the combned rate of all the servers n that phase Usng ths smplfcaton and a servce dstrbuton of two phases, the state of the system can be represented by (n, n ), where n s the number of users n phase and n s the number of users n phase In order to obtan the characterstc tr-dagonal form of the Q matrx, t s necessary to perform a lnear orderng of the states of ths two-dmensonal contnuous tme Markov chan In ths case t s the smple orderng {, (,), (,), (,), (,), (,),, (n,), (n-,),, (,n-), (, n)} We can defne levels where level m s the combnaton of all states when the number of users n the system are m and there are m+ possble states at level m The events, whch change the state of the system and rates of leavng the current state at those events are shown n table Fg shows the transtons lsted n table The rate transton matrx for the M/PH/n/n QBD process can be constructed by observng the transtons between states gven n fg and usng table to get the transton rates between dfferent states TBLE : TRNSITION RTES BETWEEN DIFFERENT STTES From To Rate Event, λ rrval (n,n ) (n +,n ) λ arrval n (n,n ) (n,n -) n µ departure n (n,n ) (n -,n +) n µ phase change n,,n k,m-k- (m-k) µ (k+) µ k+,m-k- m- users n the system m users n the system m+ users n the system Fg State transtons for M/PH/n/n queue The orderng gven above allows us to generate the Q matrx of the form (5) and the matrces,, are gven below For - µ µ T = µ s a matrx of sze ( + ) ( + ) = s a matrx of sze ( + ) ( + ) ζ µ ζ = ( ) µ ζ where for k,,, = k ( k) µ µ ζ = ζk λ + ( k) µ + kµ s a matrx of sze ( + ) = - (9) (7) (8) () n extra boundary condton s requred at level n, where n s the total number of channels avalable n the system New arrvals are not accepted to the system at ths level thus makng the arrval rate zero Therefore the ζ s for k =,,,n n the matrx n have the followng form ζ = ( kµ () k n k) µ + Statonary probabltes, x, of ths QBD process can be found by solvng (), whch states that effectve probablty flow out of a state s balanced by the probablty flow nto the state k

4 xq = () By takng the transpose Q, t Q and rearrangng the equaton, Q t x =, (3) the statonary probabltes of ths system are gven by the nullspace of Q t, whch was found usng Sngular Value Decomposton (SVD) methods Once the statonary probabltes, x, of the system have been obtaned where xe =, where e s a column vector of s, we can fnd the probablty of havng users n the system p() as follows, S p ( ) = x(), + + p( ) = x ( j) for n (4) S + where S = j = ( + ) / If the system does not dstngush between new and handover users no prorty wll be gven to one class over the other In such a system the blockng probablty for any user, P block wll be gven by, P block = p(n) (5) and the average load of the system at ths blockng probablty s gven by L = n = p( ) (6) VI IMPROVING HNDOVER PERFORMNCE s contnung an exstng call s more mportant than acceptng a new user nto the network, handover users should have prorty over new users durng resource allocaton In ths study, prorty for handover users was acheved by reservng a certan number of channels exclusvely for handover users [] In ths scenaro only handover users are accepted to the system when the total load exceeds a certan threshold Ths s modelled by varyng the nterarrval parameter wth the number of users n the system and blockng probabltes are calculated by addng the probabltes of the states where partcular types of users are not allowed to vst When there are C HO channels exclusvely reserved for handover users, n equatons (7), (8) and (9), the value for λ has to be modfed as follows For =,,, n ; If n C Otherwse, λ = λ HO, h λ = λ n + λ Where n s the total number of channels avalable n the system and, λ n and λ h are the parameters of exponental nterarrval rates for new and handover users respectvely Blockng probabltes for the handover and new users, P block _ ho and P block _ new can be calculated as follows h Pblock _ ho = p( n), P block _ new = p( j) (7) n CHO VII RESULTS The results obtaned through the queung models abovedescrbed are presented n ths secton In partcular analyss has been performed to determne the blockng probabltes that handover and new users experence wth channels exclusvely allocated for handover users In these analyses t was assumed that a typcal terrestral cell has 5 channels It was also assumed that the GoS blockng probabltes for new and handover users must be less than 5% and % respectvely [] The operatng pont of the system s taken to be the maxmum capacty that leads to the blockng probabltes less than or equal to the above stated values Further t was observed that wth an average call length of 3 mnutes users make 8 handovers per call Therefore we selected, λ = 8 h λ n Blockng Probabltes Wth Channels Reserved for Handover Users Fg shows the performance of the system wth varous numbers of channels reserved exclusvely for handover users In the case where no channels are reserved for handover users, to meet the GoS for handover users (% blockng) 79 % (or 39 channels) of the total system capacty s found to be the operatng pont of the system In ths case new users enjoy the same blockng probablty of % whch s 3% less than ther guaranteed GoS blockng probablty Therefore mechansms have to be used to dstngush between the two types of users so that the operatng capacty can be maxmsed whle guaranteeng the GoS Ths can be acheved through reservng a certan porton of the channels exclusvely for handover users Fg shows the results obtaned wth ths strategy It shows that the operatng capacty can be ncreased to 8% when channel s reserved for handover users Subsequent allocaton of more handover only channels leads to a drop n the operatng capacty Ths s due to the fact that wth hgher number of handover only channels, new users have only a smaller number of channels to contend, resultng that the GoS blockng probabltes can only be met at lghter network loads Fg3 shows the behavour of operatng capacty wth the number of channels exclusvely reserved for handover users Ths shows that the operatng capacty drops by 7% when the number of handover-only channels are ncreased from to 4 In terms of the system capacty, ncrease n handover-only channels from to 4 corresponds to reservng 6% more of the total capacty exclusvely to handover users Therefore the optmum operatng condton s to reserve a sngle channel for handover users, whch results n the n

5 maxmum capacty whle satsfyng GoS for both new and handover users Fg shows that the handover users experence very low blockng probabltes (3 wth 4 handover only channels) at the GoS operatng ponts wth hgher number of handover only channels In thrd generaton networks these operatng condtons may be of nterest due to the need of very low handover blockng probabltes for servces such as vdeo conference and e-commerce applcatons VIII CONCLUSIONS ND FUTURE DIRECTIONS In ths study we modelled the channel holdng tme of a cellular network as a generalsed Erlang dstrbuton We also generated the block tr-dagonal rate transton matrx for the M/PH/n/n queue, whch we used to model the collectve channels of a sngle cell n a cellular network lso we nvestgated the performance of the system when handover user Operatng capacty Blockng Probabltes for new and handover users channel reserved for handover = 3 handover users new users System capacty or utlsaton (%) Fg Blockng probabltes all users wth channels exclusvely reserved for handover users Number of handover only channels ( total number of channels = 5) Fg 3 Operatng capacty wth number of channels allocated for handover users 4 were gven prorty n the system through reservng channels exclusvely for handover users These results show that the system can operate at a maxmum capacty of 8 % whle guaranteeng that the new users wll suffer blockng probablty of less than 5 whle handover users wll experence blockng probabltes less than Ths operatng condton s acheved when only terrestral channel s reserved exclusvely for handover users In thrd generaton networks some servces may requre very small blockng probabltes, lower than that guaranteed n current systems These results show that ths can be acheved by lowerng the operatng pont consderably untl the requred GoS s met nother possble strategy s to make the handover decson makng strategy more ntellgent n order to support GoS requred by dfferent types of users One of the technques that can be utlsed for system mprovement s the use of moblty predcton to estmate the probablty of a user requrng a handover to a partcular cell and dynamcally reservng a certan porton of the resources from these cells The authors are currently nvestgatng the use of ntellgent systems to mprove the handover procedures through movement predctons and early reservatons [] CKNOWLEGMENT Ths work was supported by the ustralan Government s Cooperatve Research Centre Program REFERENCES [] UMTS Task Force Report, tech rep, Mar 996 [] D Hong and S S Rappaport, Traffc Model and Performance nalyss for Cellular Moble Rado Telephone systems wth Prortzed and Non-Prortzed Handoff Producers, IEEE Trans Veh Tech, pp 77-9, ug 986 [3] C Jedrzyck and V Leung, Probablty Dstrbuton of Channel Holdng Tme n Cellular Telephony Systems, Proc IEEE Veh Technol Conf, pp 47-5, May 996 [4] P Orlk and S Rappaport, Model for Teletraffc Performance and Channel Holdng Tme Characterzaton n Wreless Cellular Communcaton wth General Sesson and Dwell Tme Dstrbutons, n IEEE J Sele reas of Commun, vol 6, pp , June 998 [5] Y Fang and I Chlamtac, Teletraffc nalyss and Moblty Modellng of PCS Networks,'' n IEEE Trans Commun, vol 47, pp 6-7, July 999 [6] M Neuts, Matrx-Geometrc Solutons n Stochastc Models, John Hopkns Publshers, pp 4-9, [7] G Latouche, Introducton to Matrx nalytc Methods n Stochastc Modellng, S-SIM publshers, pp 3-5, 996 [8] S smussen, O Nerman, and M Olsson, Fttng Phase-type Dstrbutons va the EM algorthm, Scandnavan Journal of Statstcs, vol 3, pp 49-44, 996 [9] V Ramaswam and D Lucanton, lgorthms for mult-server queue wth phase-type servce, Comm Statstc Stochastc Models, vol, no 3, pp , 985 [] Jayasurya et al, Protocol for Internetwork Handover between Terrestral UMTS and Satellte UMTS networks, Usng Intellgent Systems, Proc IEEE Int workshop on Intellgent sgnal Process and Commun Systems, pp 47-4, Nov 998