Inter-arrival Time Distribution for Channel Arrivals in Cellular Telephony

Transcription

1 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin Iner-arrival Time Disribuion for Channel Arrivals in Cellular Telephony José Ignacio Sánchez, Francisco Barceló and Javier Jordán Absrac-- In his paper differen probabiliy densiy funcions are fied o he iner-arrival ime in a channel of a Cellular Mobile Telephony sysem. The approach is enirely experimenal: he daa se o be fied has been obained on an acual sysem in operaion. The Kolmogorov-Smirnov (K-S) goodness-of-fi es is used in order o esablish a ranking of he bes fiing probabiliy densiy funcions. From his sudy i can be concluded ha he arrivals o a channel in a cell are according o a smooh process. Index erms-- Traffic modelling, cellular elephony, mobile elephony, voice modelling. I. INTRODUCTION The design and performance evaluaion of Mobile Telephony Neworks is usually carried ou by using some basic conceps of Queuing Theory as well as assuming cerain saisical disribuions for he iner-arrival ime and he channel occupancy ime processes. A poor knowledge of hese disribuions conribues o an inefficien design of he nework resources because he engineer mus be conservaive o cope wih he possible error margin. On he oher hand, he accurae knowledge of he disribuions allows an accurae design of he resources. This shows up in a beer use of he radio resources used in his kind of neworks and makes i possible he developmen of new services offered hrough cellular neworks such as daa services. Some cellular neworks make use of he excess of capaciy in heir cells in order o ransmi daa or shor daa messages during he periods in which he channels are no being used for voice raffic (voice idle periods). As public cellular neworks mus achieve a very low Grade of Service (GoS) measured as blocking probabiliy, he channel load should be kep low and idle periods are significan. To effecively design his kind of neworks a beer undersanding of he duraion and frequency of hese idle periods is needed. Because of his, voice raffic saisics in hese neworks need o be beer undersood. In he pas, i has been widely used he negaive exponenial E.T.S. de Ingeniería de Telecomunicación de Barcelona (UPC) Mod. C3 Campus Nord, c/ Gran Capián, s/n 834 Barcelona. barcelo@ma.upc.es disribuion o model he call and channel holding ime. However differen sudies showed ha he call duraion is beer fied by oher disribuions in fix elephony [], public cellular mobile [2] and privae mobile radio (PMR) [3]. The channel holding ime has been also showed o fi lognormal disribuions beer han he exponenial [4, 5, 6]. The arrival processes have usually been considered o be Poissonian. Bu he pariculariies found in Cellular Mobile Telephony Sysems make his assumpion very suspicious. This ask has been enered upon in oher sudies from a heoreical poin of view: in [7] i is concluded ha he arrival raffic is Poissonian while in [8] he same raffic appears o be smooh raffic. In [6] a field sudy of he inerarrival ime o he channel is presened showing a good fi wih he exponenial disribuion which agrees wih he Posissonian assumpion. In [9] an experimenal approach o he iner-arrival ime in PAMR sysems proves ha he arrival raffic is smooh, bu he measured sysems are no cellular and have some very specific pariculariies. In our sudy we deal wih he saisical modelling of he iner-arrival ime process in Cellular Mobile Telephony sysems. An experimenal approach is proposed, based on he analysis of real raffic daa. In Secion II a review of some previous resuls is presened along wih he heoreical model of he sysem. Secion III describes he main deails of he empirical procedure followed in our sudy and he saisical procedure seleced o analyse he colleced daa. In Secion IV he fiing disribuions are presened along wih he reasons o selec some disribuions while disregarding ohers. The fiing of he seleced disribuions is presened in Secion V for differen channel loads. In Secion VI oher saisical resuls over he iner-arrival and idle ime processes are presened. The obained resuls and conclusions of Secion VII end o confirm ha he arrival raffic o a channel in cellular elephony is smooh in agreemen wih some resuls reached by analyical models like he one developed in [8]. II. THEORETICAL MODEL AND PREVIOUS RESEARCH A. A simple heoreical model In every cell of a Cellular Mobile Telephony nework he offered raffic is he resul of wo raffic sreams of differen sources: 245

2 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin On one hand we have fresh raffic due o he calls which are originaed inside he limis of he cell (T). T is enirely caused by new calls. The infinie populaion hypohesis could be acceped for he T sream if he number of erminals which are able o sar a new call inside he acked cell is large (much larger han he number of available channels). This happens very ofen in public elephony environmens. In his case he number of call aemps due o T can be assumed o follow a Poisson disribuion On he oher hand, he hand-off raffic is due o he calls handed-off from surrounding cells (T2). I is more difficul o accep ha he raffic creaed by T2 aemps is Poissonian. This is because T2 is a raffic sream ha has already been carried by channels in he neighbouring cells and herefore coming from a populaion of no more han he oal number of channels of all he surrounding cells ogeher. As T2 is originaed by a finie populaion i should be modelled as smooh raffic. T 2 T 22 T 23 T T 24 This can be observed in Figure.a, where a ypical urban scenario scheme of a cellular nework is shown. I can be seen how he cenral cell receives hand-off raffic from jus six neighbouring cells if an hexagonal paern is used. If all cells are assumed o have he same number of channels, he number of raffic sources able o originae hand-off raffic T2 is six imes he number of channels in he acked cell. As he number of sources is no much larger han he number of channels in he Base Saion (BS), his raffic should never be considered o be Poissonian. This is even more obvious in a highway scenario as he one shown in Figure.b. In his paricular case hand-off raffic arrives from only wo neighbouring cells. Therefore, even if he number of channels assigned o he BS is higher due o he higher mobiliy of he mobile erminals in his scenario, he maximum number of incoming hand-off calls will sill be low (wice he number of channels per cell). Theoreically, he handed-off raffic in his laer case should be very smooh, specially if he number of channels per BS is no very high. Following he former discussion, as he overall raffic is he resul of adding T and T2 sreams - his is adding a Poisson raffic and a smooh one - his overall raffic should be beer modelled as smooh raffic. The smoohness obviously depends on facors such as cell shape and size (a larger size reduces he hand-off and he share of T2), speed and mobiliy of erminals (higher speed means more handoff and hen higher T2), raffic inensiy, ec. T = 6 2 T2 i i= T 26 T 25 a) Urban nework scenario T 2 T 22 T T2 = T2 + T22 b) Highway scenario Fig. Differen raffic sreams in a cellular nework B. Analyical sudies In [7] a heoreical model of he cellular nework is proposed in order o model he overall raffic. Traffics, boh fresh and hand-off, are characerised by is firs momen only (average ime beween aemps o size a channel). The Poisson assumpion is acceped because he analyical resuls agree wih hose reached hrough simulaions. Tha is o say, offered raffic is handled like if i was Poisson and as he resuls obained in such a way agree wih hose obained from simulaions he conclusion is ha hand-off raffic can be handled like Poisson raffic. In [8] handed-off raffic is modelled by he firs wo momens and i is shown ha his raffic is in fac a smooh raffic process (wih coefficien of variaion of he inerarrival ime lower han one). The wo momens represenaion of he hand-off raffic is superior o he single momen represenaion. The overall raffic in his case is showed be smooh for he case of hexagonal cell shape. Peakedness as low as.85 are obained, and i can be easily concluded ha much lower peakedness would be obained for he highway scenario. C. Field sudies The main problem found in field sudies is ha while 246

3 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin analyical research can deal wih he aemps of sizing a channel, field sudies can only see he acual seizures. Aemps which happen during blocking periods are los in he field research, so arrivals and aemps are more differen he higher is he blocking probabiliy. Noe ha public cellular sysems should achieve a very low blocking probabiliy o give accepable GoS. In [6] he arrivals o he se of resources in a cell are sudied, and he negaive exponenial disribuion is fied o he iner-arrival ime disribuion wih a significance beer han 5%. This agrees wih he Poisson arrival hypohesis. Figure 3); he busy hour was aken from he high rae busy period around en hours which has he higher ineres for raffic sudies. Noe ha his firs spike is more naural. Spikes a 6 and 2 hours are coinciden wih he beginning of he lower rae period and are in fac delayed raffic (raffic ha wais o ge economical advanage bu which would have appeared before if he rae was kep). % 8% 6% III. DATA ACQUISITION, PRE-PROCESSING AND STATISTICAL TOOLS A. Daa Acquisiion The real raffic daa used in his sudy were obained hrough a scanning receiver conrolled by a Personal Compuer (PC). The program ha conrols he scanner generaes.log repor of aciviy files. Each line of hese files describes he ime of he beginning of he aciviy, modulaion and srengh of he signal and duraion of he aciviy. A scheme of he working saion is shown in Figure 2. The sysem moniored is a public cellular sysem in Barcelona based on he TACS sandard (very similar o AMPS). Frequency Modulaion (FM) is used, so he deecion of he carrier in he down-link is sufficien for he knowledge of he channel occupancy. The advanage of deecing he down-link is he higher and more sable power level which helps o reduce he annoying effecs of noise and inerference. PC Scanner Colleced Daa Fig. 2 Working Saion Tweny hree frequencies belonging o differen cells were scanned during a period of one monh. Because of he cochannel and adjacen-channel inerference presen in cellular neworks, we had o selec he appropriae frequencies o be scanned. This was done by doing an aural survey of he channels in order o selec frequencies free of inerference ha could disor he colleced daa. In a firs sage he frequencies were scanned during whole day periods in order o deermine he busy hours. As he invesigaed sysem provides hree differen rae periods, he daily occupancy shows hree spikes along he day (see 4% 2% % Fig. 3 Load along he ime of he day (relaive o he maximum load). Among all he colleced daa hree samples were seleced for our sudy. We will refer o hese samples as Heavy, Medium and Ligh Load. The Heavy Load sample corresponds o calls made in channels of average channel load around.6 Erlangs. In he same way, he Medium and Ligh Load samples correspond o calls in channels of average channel load around.5 Erlangs and.4 Erlangs respecively. To build up he hree samples we have considered daa in he. Erlang inervals [.35,.44], [.45,.54] and [.55,.64]. All hese hree samples where buil up by he aggregaion of he daa colleced a differen scanned frequencies. This could be done so because all he scanned channels belong o he same cell and herefore presen he same saisical properies. We haven considered heavier loads in order o avoid masking arrivals ha may occur during acive periods of he scanned channel. Anyway, loads heavier han.65 Erlangs represened less han a % of all he amoun of daa (4 ou of 45 hours ), so we hink ha he resuls obained from he seleced daa accuraely represen he naure of he iner-arrival process. Noe ha in order o keep a low blocking probabiliy, loads can never be very heavy in a sysem for public voice service. B. Pre-processing Alhough he scanned channels are suppose o be free of inerference, some shor inerference and cus may sill occur and disor he samples. Therefore he samples were 247

4 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin filered in order o eliminae inerference, shor ransmission cus and oher undesired effecs. This was basically done in wo ways by a simple program wrien in C language. Shor cus ha ofen occur due o fading were eliminaed by merging regisered aciviies separaed by idle periods of less han 2 seconds. And inerference was sripped by eliminaing aciviies of less han second. These bounds of 2 and second respecively were checked by aural monioring o give very good resuls for he desired purpose: more han 95% of he rejeced daa were acual cus or inerference and more han 95% of cus and inerference were rejeced. The pre-processing is he same used in [4, 9] where furher informaion can be found. C. Saisical Tools For our saisical analysis we have o follow hese hree seps: -choose he probabiliy densiy funcion o be fied. -fi he chosen funcion o he sample daa. -es he goodness of he fi. In Secion IV we describe he probabiliy densiy funcions chosen and we briefly explain he reasons for our choice. The parameers of he differen probabiliy densiy funcions fied are calculaed by making use of he Maximum Likelihood Esimaion (MLE) []. We have chosen his mehod insead of Minimum Error or Momens mehod. This was moivaed by he beer resuls obained when we compared he fied disribuion wih he sample daa used for he fi (goodness-of-fi es). Moreover we have seen ha he MLE follows he shape of he empirical hisogram beer han he Momens Esimaion (ME). The reason for his is ha he ME uses he momens of he sample bu no he sample iself o calculae he parameers of he fied disribuion, as he MLE does. Once we have he parameers, we es he suiabiliy of he fiing disribuions by making use of he Kolmogorov- Smirnov (K-S) goodness-of-fi es. This is achieved by using Malab programs ha selec, for every disribuion esed, he parameers of he bes fi from he K-S goodness of fi es poin of view. The K-S is very simple o apply and has been widely used o fi elecommunicaions raffic []. The K-S es avoids he problems relaed wih he bin size presen in he chi-squared [6] and he Anderson- Darling [2]. This laer is more powerful han he K-S and chi-squared bu he naure of he measured daa makes he exra complexiy no worhy. The K-S es provides wo figures of saisical ineres, he modified K-S disance D and he level of significance α. The lower (higher) D (α ) is, he beer he fi will be. This is wha will allow us o esablish a ranking among hose fiing disribuions ha pass he K-S es wih a cerain level of significance (5% in our sudy). The level of significance α can be easily obained from D and he size of he sample n wih hese wo formulas []: D Dn =. n n α = 2 (log 2 2 ) () nd n (2) We have seen ha he resuls of he K-S es depend on he size of he sample. For sizes of less han he bes fi was no clearly disinguishable from he oher fied disribuions. As he size of he sample was closer o he bes fi was easily disinguished. If he size of he sample was over 2, he fiing began geing worse for all he disribuions esed. The reason for his is ha he sample daa is no really originaed from he fied disribuion, and as he sample grows i becomes more and more random. This is why he hree samples used in our sudy have sizes around. I is imporan o sand ou ha we use he K-S es jus o esablish a ranking among he differen heoreical disribuions, according o he significance resuling from he K-S es. We don ry o find ou he acual disribuion behind he iner-arrival process. Once we have his ranking, he choice of he disribuion used in any applicaion will be a compromise beween he level of significance and paricular feaures ha may be of our ineres. Some of hese feaures could be, for example, he simpliciy of implemenaion in a compuer program, he shape of he p.d.f. or ha i akes ino accoun a paricular aspec of he sysem. The las one is he reason why we ried he Erlang- -k disribuion, as we will explain in secion IV. IV. FITTING DISTRIBUTIONS In his Secion he se of p.d. funcions seleced as candidaes o fi he empirical disribuion is inroduced. No every possible p.d.f. is an accepable candidae o fi he underlying daa. Two basic conceps have been aken ino accoun o choose he candidaes, obaining he wo following classes of disribuions. A. Markovian sages Saisical disribuions based on Markovian sages allow o idenify some memory-less properies of he source which generaes he sample. Some p.d.f. of his ype which are more complex han he exponenial allow also o idenify differen sreams or populaions generaing arrivals, and his is he case inroduced in Secion II. Exponenial: Is he mos common disribuion used for modelling he iner-arrival ime process in Mobile Telephony Sysems due o is memory-less properies. 248

5 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin For his reason we use i as a reference, alhough we show ha i doesn fi well he daa. f ( ) = e (3) where (he mean ime) is a scale parameer. Erlang-k : This disribuion has a simple inerpreaion as he sum of k independen exponenial random variables, i.e., i is he resul of a succession of memory-less exponenial sages. Is coefficien of variaion (cv )is always lower han one, so i will be a good candidae for our smooh samples. f ( ) = k k e ( k )! where is a scale parameer and k a shape parameer. Erlang-n,k: Displays a grea versailiy, fiing random variables of coefficien of variaion lower or greaer han one. Noe ha we expec smooh raffic and hence coefficiens lower han one as saed in Secion II. (4) raffic sream and p as he share of he T2 raffic sream. B. Lognormal based disribuions The skew shown by he empirical disribuion does no allow he normal disribuion as a candidae. In such a case he lognormal and combinaions represen a good choice. In addiion, hese p.d. funcions give he bes fi in previous research on he channel holding ime [,5]. Lognormal: Displays a long ail wih an iniial spike, and suis very well wih he shape of he filered sample hisogram. f ( ) = 2 ( log( ) µ ) 2 e 2σ 2 2πσ where σ is a shape parameer and µ a scale parameer. Lognormal-k: Provides wih a grea power o fi difficul empirical disribuions, wih big iniial spikes ha can be followed by oher disribuions. (7) f ( ) = p n n k k e ( n )! + ( p) e ( k )! (5) k f p p. d. f.log normal, ( ) = ( σ µ ) i= [ ] i i i (8) where n and k are shape parameers, and is a scale parameer. p and p are he probabiliy parameers of he Erlang-n and Erlang-k componens respecively. HyperErlang-k-2: A combinaion of wo Erlang-k. Is an alernaive o he Lognormal-2 (see nex paragraph) wih he advanage ha i can be represened as a combinaion of memory-less sages. f ( ) = p k k k 2 k e + ( p) 2 e ( k )! ( k )! where he parameers have he same inerpreaion ha for he Erlang-n,k. Erlang--k : This is a paricular case of he Erlang-n-k wih n=. We have chosen his disribuion because of i s heoreical ineres. If his disribuion would be he bes fi of he sample or a leas be a reasonably good fi, hen he incoming raffic of a cell could be described as a resul of wo flows, one Poissonian (he one corresponding o n=) and he oher Erlang-k. The Poissonian flow could be idenified wih he fresh raffic originaed inside he limis of he cell, and he Erlang-k would be idenified wih he non-poissonian (smooh) hand-off raffic. Here, he probabiliy parameer will be inerpreed as he share of he T (6) where σ i and µ i have he same inerpreaion ha for he Lognormal disribuion and he p i are probabiliy parameers of he differen Lognormal componens. V. STUDY OF THE INTER-ARRIVAL TIME In his Secion he numerical resuls of our sudy are shown. We classify hese resuls according o he nominal load of he sample used in every case as menioned in Secion III: heavy (.6), medium (.5) and ligh (.4) load. We call nominal load ρ n o he cener of he considered inerval. I is imporan o sand ou ha he nominal load ρ n is no necessarily equal o he average load ρ. Noe ha he laer is he average load of he whole se of sudied channels along he observaion inerval. In all hree cases, we have found ha he coefficien of variaion of he iner-arrival ime process is lower han one. This fac ends o corroborae ha he arrival process is smooh as conjecured in Secion II. A. Heavy Load Sample ( ρ n =. 6 Erlangs) The Heavy Load Sample consiss of 223 iner-arrival imes colleced among differen scanned channels 249

6 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin belonging o he same cell. All he scanned channels have a load in he range [.55,.64] wih average load (acual load) of.58 Erlangs. ρ() Fig. 4 Insananeous load over a channel In Figure 4 i can be seen ha he iner-arrival ime is he sum of he channel holding ime and he idle ime. Therefore, as he average load of he channel is he average amoun of ime ha he channel is being used in relaion o he oal ime, i can be found by calculaing he raio beween he average channel holding ime and he average iner-arrival ime: Tch ρ = T + T ch id T = T Tch ch in Tin Tid where T ch is he average channel holding ime, T id he average idle ime and T in he average iner-arrival ime. Obviously Equaion (9) can also be direcly applied o he whole se of sudied channels by adding all he colleced daa for he channel holding ime and iner-arrival or idle ime from all he considered channels. In Table I i can be seen ha he lowes value of he saisical disance D (and hen he bes significance and fi) is obained for he Erlang-3,8 disribuion. I is followed by a Lognormal-2, an Hyper-Erlang-3-2 and a Lognormal. I can also be seen ha he Exponenial disribuion is far from fiing he sample daa. Table I: Momens and fiing for iner-arrival ime. Fiing Momens:m = , cv 2 =. 49 ρ n =. 6 Exponenial D : 6.2 :64.77 Erlang-3,8 D :.59 :7.84 p:.875 Lognormal D :.3 µ : 3.94 σ :.69 Lognormal-2 D :.6 µ : 4.2 σ :.63 p :.9 µ 2 : 3.28 σ 2 :.8 H-Erlang-3-2 D :.67 : : 33 p :.7 Erlang-2 D :.3 : 32.4 Erlang-,3 D :.94 :2.27 p:.99 (9) In Figure 5 we can see he hisogram of he filered sample and he probabiliy densiy funcion (p.d.f.) of he bes fi. Alhough i is no a good fi, in Figure 6 we show he hisogram of he filered sample along wih is exponenial fi. The value of he modified disance (D ) for he bes fi is.59. For his value of D, he level of significance (α ) is 4.6%. This is a very good resul, considering ha in elecommunicaions i is normal o work wih levels of significance from 5 [] o 5% [6]. I is ineresing o see ha for his case we find he smalles iner-arrival ime average and bigges coefficien of variaion among he hree sudied samples. The smalles average of he iner-arrival ime can be easily explained by he heavier load over he scanned channel. Noe ha he average channel holding ime is always he same under all loads as he samples are all aken a he same ime of he day and cells are very similar. densiy iner-arrival ime, sec Fig. 5 Hisogram of he Heavy Load sample along wih he bes fi pdf. The explanaion for he high coefficien of variaion is no so simple. The iner-arrival ime can be spli in wo componens, he channel holding ime and he idle ime (see Figure 4): As he load increases, he idle periods are shorer and he iner-arrival akes saisical properies from he channel holding ime. In [4] i is shown ha he channel holding ime in cellular sysems has a coefficien of variaion bigger han one. So as he load grows, he iner-arrival ime becomes more similar o he channel holding ime and herefore is coefficien of variaion is bigger. When he load decreases he correlaion beween he iner-arrival and idle imes is higher. The coefficien of variaion of he idle ime measured in his work is lower for ligher loads as shown in Secion VI. Then boh effecs end o increase he coefficien of variaion when he load grows and decrease i for ligher loads. 25

7 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin densiy iner-arrival ime, sec Fig. 6 Hisogram of he Heavy Load sample and is exponenial fi. densiy iner-arrival ime, sec Fig. 7 Hisogram of he Medium Load sample along wih he bes fi p.d.f. B. Medium Load Sample ( ρ n =. 5 Erlangs) This sample consiss of 85 iner-arrival imes colleced among differen scanned channels of he same cell. All he scanned channels have a load in he range [.45,.54] wih average (acual) load of.48 Erlangs obained according o Equaion (9). Table II: Momens and fiing for iner-arrival ime. Fiing Momens:m = , cv 2 =. 38 ρ n =. 5 Exponenial D : 6.52 :84.25 Erlang-3,6 D :.6 :2.97 p:.722 Lognormal D :.25 µ : 4.24 σ :.64 Lognormal-2 D :.64 µ : 4.6 σ :.462 p :.5 µ 2 : 3.88 σ 2 :.586 H-Erlang-4-2 D :.57 : : 26.6 p :.4 Erlang-3 D :.4 : 2 Erlang-,3 D :.7 :29.52 p:.927 In Table II we see how ha he bes fi is an Hyper-Erlang- 4-2 disribuion wih a level of significance of 45.6%. I is followed by an Erlang-3,6 wih a level of significance of 37%. I can also be seen ha funcions such as he Lognormal-2 or he Erlang-,3 fi he iner-arrival ime sample quie well. Here we can see how he average is bigger and he coefficien of variaion smaller han in he previous case. The explanaion is he same given for he heavy load case. In Figure 7 he empirical hisogram of he Medium Load sample along wih he p.d.f. of he bes fi - he Hyper- Erlang are represened. C. Ligh Load Sample ( ρ n =. 4 Erlangs) This sample consiss of 27 iner-arrival imes colleced among differen scanned channels of he same cell. All he scanned channels have a load in he range [.45,.44] wih average.397 Erlangs obained from Equaion (9). In Table III we see how he Lognormal-2 has become he bes fi, wih a level of significance of 28%. In Figure 8 we can see he hisogram of he Ligh Load sample depiced along wih he pdf of he bes fi. In his case he square coefficien of variaion is lower han in he firs wo cases. I is.33 and boh he Erlang-3,5 and he Erlang-,3 are good fis. In fac, if we look a he fiing of he Erlang-n,k along he hree cases we see ha n is always a 3. And when we force n= (Erlang-,k) hen k for he bes fiing happens o be 3. Moreover he Erlang-3 pars of his disribuions are much beer weighed han he oher so i can be said ha hey are almos Erlang-3 funcions ha have cv 2 =. 33. Table III: Momens and fiing for iner-arrival ime. Fiing Momens:m = 9. 59, cv 2 =. 33 ρ n =. 4 Exponenial D : 6.24 :9.59 Erlang-3,5 D :.68 :27.99 p:.882 Lognormal D :.86 µ : 4.32 σ :.63 Lognormal-2 D :.65 µ : 4.6 σ :.44 p :.5 µ 2 : 3.94 σ 2 :.63 H-Erlang-3-2 D :.69 : : 4 p :.9 Erlang-3 D :.8 :3.9 Erlang-,3 D :.79 :3.75 p:.973 As we explained in Secion III i could be ineresing o wach he sample daa as he resul of he addiion of wo 25

8 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin raffic sreams. A Poissonian fresh raffic flow and a non- Poissonian smooh hand-off raffic flow. densiy iner-arrival ime, sec Fig. 8 Hisogram of he Ligh Load sample along wih he bes fi p.d.f. In Figure 9 we show how he Erlang-,3 funcion follows quie well he shape of he sample hisogram. Alhough is level of significance of.6% is no he bes one, i is quie good and worh o be considered due o he heoreical aspecs involved. Even hough, we leave his resul penden for furher sudies where he share of he raffic flows T2 and T migh be sudied. The reason for his is ha, as we have explained in Secion IV.A, he share of he fresh raffic T would be equal o p (Table III) ha in his case is much smaller (.27) han he share of he handed-off raffic T2 (.973). This resul doesn seem o be realisic. A. Iner-arrival ime saisics In Figure he bes fis for he differen average channel loads are described, according o he level of significance resuling from he K-S es. We include he Erlang-,k because of is heoreical ineres explained in Secion IV. Significance (%) Erlang-3,k Lognormal 2 H-Erlang-2-k Erlang-, ,3,4,5,6 Erlang Fig. Ranking of he differen fied disribuion depending on he average channel load. I can be seen ha he Erlang-3,k is he mos sable fi for he hree loads sudied in his paper. Moreover i is a very good fi wih levels of significance among 25 and 4%. Finally, in Figure we show he average remaining ime agains he elapsed ime. Wih his we ry o probe in anoher way, ha he iner-arrival ime is no exponenial. densiy average remaining ime, sec Heavy Load -- Medium Load.. Ligh Load iner-arrival ime, sec elapsed ime, sec Fig. 9 Hisogram of he Ligh Load sample along wih he Erlang-,3 fi pdf. VI. OTHER STATISTICAL RESULTS In his Secion some addiional saisical resuls are presened in order o reinforce he hesis of he nonexponenial naure of he iner-arrival ime. We will classify hese resuls in wo groups: Iner-arrival ime saisics and Idle ime saisics. Fig. Average remaining ime vs elapsed ime We can see ha he average remaining ime clearly depends on he elapsed ime. I has a decreasing dependence on i as opposie as i occurs for he call holding ime [] and he channel holding ime [4, 5]. This means ha as he ime wihou new calls increases he average remaining ime for a call o happen decreases. If he iner-arrival ime would be exponenial, hen his average remaining ime would be consan and equal o is 252

9 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin parameer due o he memory-less propery of his disribuion. As we show ha his is far from being rue we have one more reason, apar from he ones based on saisical fiing, o hink ha iner-arrival ime in Mobile Telephony Sysems is no exponenially disribued. This curve was obained as i is done in [] for fixed Telephony circui holding imes. Each poin of he curve is calculaed by subracing he elapsed ime o all he values of he sample and hen doing he average of he nonnegaive resuling values. This curves resul much less scaered han in [4] for he channel holding ime, he reason for his is ha as i is shown in he menioned paper, he channel holding ime in cellular sysems is hyperexponenial and herefore much more random ha he hypoexponenial iner-arrival ime. B. Idle ime saisics As we have seen in Secion V.A, he iner-arrival ime is he resul of he sum of he channel holding ime and he idle ime. Therefore, if we show ha his idle ime is no exponenial hen he iner-arrival ime could never be considered exponenial independenly of he naure of he channel holding ime. We will ry o show his in hree differen ways: by sudying he second order saisics of he idle ime of he sudied samples, by sudying he bes fiings of he idle ime and finally by sudying he average remaining ime as a funcion of he elapsed ime. 2) Average remaining ime As i was done above for he iner-arrival ime, we ry o show ha he idle ime is no exponenial by showing ha he average remaining ime as a funcion of he elapsed ime is no consan. In Figure 3 we can see ha he average remaining ime decreases as he elapsed ime increases. average remaining ime, sec Fig. 3 Average remaining ime vs. Elapsed ime for he idle ime process. 3) Second order saisics - Heavy Load -- Medium Load.. Ligh Load elapsed ime, sec For he hree sudied samples he coefficien of variaion is lower han one. Acually, he values of his parameer are.8,.63 and.58 for he heavy, medium and ligh load samples respecively. I can be seen ha he idle ime is non-exponenial. ) Bes fiings Among all he candidae disribuions, he exponenial is far from being he bes fi. In Figure 2 we show a ranking of he bes fiings along he hree differen channel loads. Significance (%) Erlang-n,k Lognormal 2 H-Erlang-2-k ,3,4,5,6 Erlang Fig. 2 Ranking of he bes idle ime fiings as a funcion of he channel load. VII. CONCLUSION The coefficien of variaion is cv<, i.e., he offered raffic is smooh. This ends o confirm of he resuls shown in [8] from an analyical poin of view. Alhough he above menioned sudy deals only wih he hand-off raffic (T2), he fac ha his raffic is smooh implies ha he oal offered raffic (T2+T) canno be, in any case, exponenial. On he measured daa we conclude ha he iner-arrival ime process is no exponenial. Alhough his would imply ha he design and performance evaluaion of hese neworks based on he assumpion of an exponenial iner-arrival ime process overesimaes he blocking probabiliy, furher sudies over greaer samples would be needed o confirm his in an experimenal way. The resuls presened in his paper differ from ha presened in [6] where he exponenial disribuion agrees wih he measured daa. The difference mus no cause surprise o he reader, because many facors influence on he iner-arrival ime. Among hem we can menion he mobiliy paern, cell shape and size, mobile speed and ohers. The main conclusion of his paper is ha all hese 253

10 Proceedings of 5 h Inl. Workshop on Mobile Mulimedia Communicaion MoMuc 98, Ocober , Berlin maers mus be aken ino accoun and measures are needed o effecively design he number of channels and save radio specrum. ACKNOWLEDGEMENT This work was funded by Spanish CICYT Projec TIC REFERENCES [] V. Boloin, Telephone circui holding ime disribuions, Proc. 4 h Inernaional Teleraffic Congress, 994, Anibes, pp [2] Edward Chlebus, Empirical validaion of call holding ime disribuion in cellular communicaions sysems, In Tellerafic Conribuions for he Informaion Age, pp , Elsevier, 997. [3] Javier Jordán, Francisco Barceló, Saisical of Channel Occupancy in Trunked PAMR Sysems, In Tellerafic Conribuions for he Informaion Age (ITC 5), pp , Elsevier, June 997. [4] Francisco Barceló, Javier Jordán, Channel holding ime disribuion in cellular elephony, Elecronics Leers, Vol.34 No.2, pp , 998. [5] Francisco Barceló, Javier Jordán, Channel holding ime disribuion in cellular elephony, Proc. 9 h Inernaional Conference On Wireless Communicaions, 997, Calgari, pp [6] C. Jedrzycki, V. C. M. Leung, Probabiliy Disribuion of Channel Holding Time in Cellular Telephone Sysems, Proc of he IEEE Vehicular Technology Conf. VTC 96, pp , 996. [7] Edward Chlebus, Wieslaw Ludwin, Is Handoff Traffic Really Poissonian?, Proc of he IEEE ICUPC 95, pp , Nov [8] M. Rajaranam, F. Takawira, Hand-off Traffic Modelling In Cellular Neworks, Proc of he IEEE GLOBECOM 97, pp. 3-37, Nov [9] Francisco Barceló, Sergio Bueno, Idle and Iner-arrival Time Saisics in Public Access Mobile Radio (PAMR) Sysems, Proc of he IEEE GLOBECOM 97, pp. 26-3, Nov. 98. [] A.O. Allen, Probabiliy, Saisics, and Queuing Theory wih Compuer Science Applicaions, Academic Press 99. [] A.M. Law, W.D. Kelon, Simulaion Modelling and Analysis, McGraw-Hill