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1 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. NEW APPROXIMAION FOR ANDOFF RAE AND NUMBER OF ANDOFF PROBABILIY IN CELLULAR SYSEMS UNDER GENERAL DISRIBUIONS OF CALL OLDING IME AND CELL RESIDENCE IME Mohammed Alwaeel Compuer Scence Deparmen, Faculy of Compuers and Informaon echnology, abou Unversy Saud Araba, abou 734, P.O. Box 458 ABSRAC In cellular newors, quanes such as number of handoff probably and handoff rae are very mporan parameers n he cellular sysem performance analyss. In prevous leraure, several echnques were nroduced o evaluae hese parameers; however, here are some lmaons n he nroduced echnques. In hs paper, we approxmae number of handoff probably and handoff rae n cellular sysems n whch he call holdng me and cell resdence me follow arbrary sascal dsrbuons. Specfcally, we derve approxmae expressons o evaluae he probably mass funcon of handoff number and handoff rae. he echnque used does no requre nowledge of he dsrbuon of he cell resdence mes bu only her frs wo momens, whch may be deermned easly from emprcal daa. he analycal resuls are valdaed by compuer smulaon. Keywords: Number of andoff probably, andoff Counng, andoff Rae. INRODUCION Recen advance n wreless communcaons and cellular sysems mae possble for cellular newors o suppor a wde varey of servces o he user on he move. 4G sysems and fuure wreless newors wll enable he user o mae voce, daa, mulmeda calls, or mae an nerne connecon o surf he web, and rereve daa. hese advance servces have movaed he sudy of newor's qualy-of-servce (QoS), n cellular newors he followng QoS measures are he mos mporan measures used o specfy he qualy of he connecons: New call blocng probably ( P ): defned as he probably ha a new call reques be dened for lac of resources. emaure call ermnaon probably (CP) : defned as he probably ha an acceped on gong call s ermnaed due o lac of recourses. Call droppng probably (CDP): defned as he probably ha a call wll experence eher premaure call ermnaon or new call blocng. andoff falure probably ( P ): defned as he probably ha a handoff reques s dened for lac of resources. Some of hese measures may be specfed n he desgn, for example, In second generaon cellular sysems, he premaure call ermnaon probably s lower han 5%, and he handoff falure probably s lower han % for voce calls []. o evaluae hese QoS measures several sysem parameers such as call holdng me, cell resdence me, handoff counng, and handoff rae mus be defned. he handoff number (handoff counng), defned as he number of handoff requesed durng a call connecon, s an mporan parameer n such a sysem snce has a drec mpac on handoff arrval raffc and call admsson conrol polcy desgn [], and he handoff rae s defned as he average number of handoffs underaen durng he acual call connecon. In order o deermne hese wo 34

2 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. parameers, he call holdng me and he cell resdence me need o be defned clearly, where as shown n Fg. : he call holdng me ( ) s defned as he perod from he nsan he acceped call sars o he nsan he call complees. Cell resdence me n he orgnaon cell ( r ) s defned as he me ha he moble user ravels from he pon where he call s orgnaed o he edge of he cell. Cell resdence me n handoff cell (, =, 3, 4... ) s defned as he me ha he moble user ravels hrough a cell (edge o edge) reached afer (, =,3,4,... ) handoff(s). Many prevous leraure nroduced echnques and expressons o evaluae number of handoff probably and handoff rae. In early leraure he followng assumpons are commonly used : he call holdng me and he cell resdence me are assumed o be exponenally dsrbued, and calls arrval s a Posson process [3,4,5,6]. owever, because of echnologcal advances and he growng neres n personal communcaon servces, and because of new mareng servce plans (e.g. fla-rae servce), moble users behavor paern was changed such ha hey use here moble devces for longer perod of me and more frequenly. ence, he exponenal dsrbuon may no longer appropraely models he servce me or he nerarrval me of praccal 3G newors [7,8,9,]. In recen leraure, several echnques were nroduced o deermne number of handoff probably and handoff rae where more general call holdng me and cell resdence me dsrbuons were assumed [,,], however, here are some lmaons n he nroduced echnques. For example, n [,], he auhors nroduced echnques o deermne number of handoff probably and handoff rae for general call holdng me and cell resdence me, however, s assumed ha he Laplace Seljes ransform s exsed for call holdng me, whch may no be sasfed for heavy aled call holdng me dsrbuons such as gamma and Log-Normal dsrbuons []. In [], he auhors used ransform Approxmaon Mehod (AM) o approxmae he Laplace Seljes ransform of he call holdng me, however, hs echnque requres recursve procedure o deermne he approxmaed ransform whch may requres more me, resources, and effor. In hs paper, we nroduced a new approxmaon ha may be used o evaluae he number of handoff probably and handoff rae n sysems wh arbrary call holdng me and cell resdence me dsrbuons. Unle pas researches whch requre ha he probably densy funcon (pdf) of he cell resdence me and/or he pdf of he call holdng me are nown and have nown raonal Laplace ransforms, our echnque only requres ha he frs wo momens of he cell resdence me, and he cumulave dsrbuon funcon (cdf) of he call holdng me are nown. he res of hs paper s organzed as follow: In secon, he analycal formula for handoff number probably mass funcon (pmf) and handoff rae are derved. In secon 3, he developed expressons are appled, and numercal resuls based on he analyss as well as compuer smulaon are presened. Fnally, some concludng remars are made n secon 4.. NUMBER OF ANDOFF PROBABILIY AND ANDOFF RAE Le () be he number of handoffs of a nonbloced call durng he call connecon, hen for nonbloced call, = f he cell resdence me n he orgnaon cell r s longer han he call holdng me ( ), = f he call ermnaed because of he frs handoff falure or he call mae he frs successful handoff and compleed successfully n he new cell, and so on. hen, for a nonbloced call he handoff number probably mass funcon may be found as < r, = = = ( r +++< r.. + )( P ) + ( >+++ r.. )( P ) P whch may be smplfed as ( ) > r, = = = ( > r )( P ). ( > r )( P ) he man ey o evaluae ( ) () () = usng () s o fnd he probably densy funcon of he random varable η m = r m, assumng 35

3 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. ha (, 3,.., m ) are ndependen and dencally dsrbued random varables. In leraure, several echnques were used o do so, such as usng Laplace Seljes ransform or usng ransform approxmaon mehod, as we menoned above. In hs paper, we wll sar by approxmang he pdf of he cell resdence mes r,,..., m and hen use he approxmaed resul(s) o derve he pdf of he random varable η m. I was shown ha wo processes are (approxmaely) equvalen f hey have he same frs few momens [], hence, a momen machng echnque may be used o approxmae he pdf of he random varables r,,..., m. One of he mos flexble sandard dsrbuons s he gamma dsrbuon, by changng s parameers can ae many shapes ha may be used o approxmae oher dsrbuons as shown n Fg., n addon, gamma dsrbuon s he general case of several dsrbuons, such as Exponenal dsrbuon and Erlang dsrbuon, whch are used n many leraure o model he call holdng me and he cell resdence me. Based on ha, o derve an approxmaed pdf of he random varable η m, he resdence me n he frs cell r s approxmaed by a gamma dsrbued random varable ha has he same mean µ r, and he resdence me n all subsequen cells are approxmaed by a gamma dsrbued random varable ha have he same mean µ and varance and f r σ (), such ha e, Γ( ) (3) he varance of he random varable found from (4), respecvely, as may be µ, = σ =. (5) Solvng (5) for and we have µ, = µ =. (6) σ Smlarly, for he resdence me n he frs cell we have µ r =, and leng = we have µ r µ = µ r =. (7) σ Based on he above, he random varable η m s a summaon of ( m ) gamma dsrbued random varables ha have he same shape parameer, hen η m s also a gamma dsrbued random varable wh parameers ( m ) m = +, and = =, (8) hence, he pdf of η m may be found as fη m () m m e =. Γ ( m ) (9) f () e, Γ( ) (4) where f x () s he pdf of he random varable x, x and ( x ) Γ = e d. ence, he mean and From () we have ( ) hen > r, = = = ( > η )( P ), ( > η )( P ) () 36

4 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. = = ( ) fη () F () d, = ( ) ( ) P f F d P f F d ( ) η () () ( ) η () (), () where F () s he cumulave dsrbuon funcon of he call holdng me. ence, he mean of may be found as E[ ] = ( = ). () = 3. Illusraon and Dscusson o Illusrae he use of he proposed mehod, we evaluae he probably mass funcon of he number of handoff and he mean of handoff n a cellular newor, n whch he cell resdence mes are assumed o be a generalzed gamma random varables, wh parameers a, b, and c, for he orgnaon cell, and a, b, and c for all oher subsequen cells, meanwhle, he call holdng me s assumed o be exponenally dsrbued random varable wh mean. ence, we have [3] µ and F = e, (3) µ () Γ a + Γ a + c c r, µ = µ =, Γ( ab ) Γ( a ) b (4) and σ = Γ a + Γ( a ) Γ a + Γ( a ) b c c Subsung (4) n (6), (7), (8), and (3) n () we have, = + µ = = ( P ) -( P ),. + µ + µ ( ) (5) From () and (5), he handoff rae may be found as [ ] ( E P ) -( P = ) = + µ + µ = ( P ) ( ) P + µ + µ = + µ hen ( P ) + µ E [ ] = ( P ) + µ + µ ( P ) + µ (6). (7) Fg. 3 shows he probably mass funcon of he number of handoff as a funcon of he number of handoff, as evaluaed usng (), and compuer smulaon. From he fgure, we can see ha he analycal resuls derved usng he nroduced echnque and he compuer smulaon resuls are n very good agreemen for dfferen sysem parameers. he handoff rae derved usng compuer smulaon and analyss, for dfferen values of he call holdng me and cell resdence me means s shown n able, he resuls show ha he approxmaon ha was used n he nroduced analycal mehod leads o a hghly accepable accuracy. ABLE I. ANDOFF RAE FOR DIFFEREN VALUES OF CALL OLDING IME AND CELL RESIDENCE IME MEANS Call holdng me mean Orgnaon cell resdence me mean andoff cell resdence me mean Smulaon handoff rae Analyss handoff rae CONCLUSION 37

5 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. In hs paper we nroduced a new approxmaon echnque ha may be used o evaluae he probably mass funcon of handoff number and handoff rae n cellular sysem. he advanages of hs echnque are, frsly, s flexbly such ha may be used wh any assumpons for call holdng me and cell resdence me dsrbuons. Secondly, unle oher echnques, whch requre he dsrbuon of he cell resdence me whch may no be avalable n praccal applcaons, hs echnque requres only he frs momen of he orgnaon cell resdence me, and he frs wo momens of subsequen cells resdence me o evaluae he handoff probably mass funcon and he handoff rae. he resuls obaned from compuer smulaon and he analyss show ha he nroduced approxmaon echnque produces a hghly accurae resuls. 5. REFERENCES [] Y. Fang, "Modelng and performance analyss for wreless moble newors: A new analycal approach", IEEE/ACM rans. on Neworng, Vol. 3, No. 5, Oc. 5, pp [] Z. Yan and S. Boon-ee, "andoff counng n herarchcal cellular sysem wh overflow scheme", Compuer Newors journal, Vol. 46, 4, pp [3] R. A. Guern, "Channel occupancy me dsrbuon n a cellular rado sysem", IEEE rans. Veh. echnol., vol. 35, no. 3, Mar. 987, pp [4] D. E. Ever, "raffc engneerng of he rado nerface for cellular moble newors", oc. IEEE, vol. 8, no. 9, Sep. 994, pp [5].-S. P. Yum and K. L. Yeung, "Blocng and handoff performance analyss of dreced rery n cellular moble sysems", IEEE rans. Veh. echnol., vol. 44, no. 3, Jun. 995, pp [6] D. ong and S. S. Rappapor, "raffc model and performance analyss for cellular moble rado elephone sysems wh prorzed and nonprorzed handoff procedures", IEEE rans. Veh. echnol., vol. V-34, no. 3, 986, pp [7] M. Rajaranam and F. aawra, "Nonclasscal raffc modelng and performance analyss of cellular moble newors wh and whou channel reservaon", IEEE rans. Veh. echnol., vol. 49, no. 3, May, pp [8] C. Jedrzyc and V. C. M. Leung, "obably dsrbuons of channel holdng me n cellular elephony sysems", oc. IEEE Vehcular echnology Conf., Alana, GA, May 996, pp [9] J. Jordan and F. Barcelo, "Sascal modelng of channel occupancy n runed PAMR sysems", oc. 5h In. eleraffc Conf., 997, pp [] V. A. Bolon, "Modelng call holdng me dsrbuons for CCS newor desgn and performance analyss", IEEE J. Sel. Areas Commun., vol., no. 3, Mar. 994, pp [] R. Rodrguez-Dagnno and. aag, "Counng handovers n a cellular moble communcaon newor: equlbrum renewal process approach", Performance Evaluaon Journal, No. 5, Aprl 3, pp [] S. Subramanam, A. Soman, M. Azzoglu, and R. Barry, "he benefs of wavelengh converson n WDW newors wh non-posson raffc", IEEE Comm. Leers, vol. 3, March 999, pp [3] I. Gradsheyn, M. Ryzh, and A. Jeffrey, able of Inegral, Seres, and oducs. New Yor: Academc, 994. Call oldng me Channel oldng me Channel oldng me Channel oldng me Cell # (Orgnaon cell) Cell Resdence me r Cell # Cell #3 Cell Resdence me Cell Resdence me 3 Fgure. mng Dagram for Cell Resdence me and Call oldng me. 38

6 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. Gamma PDF for dfferen ses of parameers.5 Gamma PDF me () sec Fgure. Gamma obably Densy Funcon.3 Smulaon Analycal o. of andoff Average call holdng me =4 mnues Average orgnaon cell resdence me = mnue Average handoff call holdng me =.5 mnues andoff falure probably = andoff Number Fgure 3.a andoff obably Mass Funcon. 39

7 Journal of heorecal and Appled Informaon echnology 5-9 JAI. All rghs reserved. Smulaon Analycal o. of andoff Average call holdng me =8 mnues Average orgnaon cell resdence me = mnue Average handoff call holdng me =.5 mnues andoff falure probably = andoff Number Fgure 3.b andoff obably Mass Funcon. ABOU AUOR: Dr. Mohammed M. Alwaeel receved he B.S. degree n Compuer Engneerng and he M.S. degree n Elecrcal Engneerng n 993 and 998, respecvely, boh from Kng Saud Unversy, and he Ph.D. degree n Elecrcal Engneerng n 5 from Florda Alanc Unversy. From 994 o 998, he was employed as Communcaons Newor Manager a he Naonal Informaon Cener n Saud Araba. From 999 o, he was employed by Kng Adulazz Unversy as a lecurer and as vce dean of abu Communy College. e s now he dean of Compuers and Informaon echnology college a Unversy of abu. s curren research neress nclude eleraffc analyss, moble saelle communcaons, and cellular sysems. 4