Channel Holding Time Distribution in Public Cellular Telephony *

Transcription

1 ITC6 / P. Kelly and D. Smih (Ediors) 999 Elsevier Science B.V. Channel Holding Time Disribuion in Public Cellular Telephony * Francisco Barceló a and Javier Jordán b a Deparameno de Maemáica Aplicada y Telemáica (UPC), c/ Jordi Girona,, Barcelona, Spain, barcelo@ma.upc.es. b Caalana de Telecomunicacions. Sociea Operadora de Xarxes. c/ Ciencies 49, L Hospiale de LLobrega, Barcelona, Spain. This paper examines he channel holding ime of public cellular elephony sysems. This is he ime ha he Mobile Saion (MS) remains in he same cell, a fracion of he call holding ime. The sudy is based on acual daa aken from a working sysem. The probabiliy disribuion ha fis he empirical sample bes when applying he Kolmogorov-Smirnov es is a mixure of lognormals. Combinaions of memory-less sages are also esed in he paper.. INTRODUCTION The duraion of channel holding ime (dwell ime in he cell) in cellular elephony sysems is only a fracion of he oal call duraion. This is due o he fac ha he physical channel is assigned only for he period ha he Mobile Saion (MS) remains wihin he same cell. The average channel holding ime is equal o he average call holding ime divided by he average number of handoffs per call plus one: he number of cells crossed by an average call. Bu nohing can be said a priori abou he relaionship of furher momens of he channel holding ime or abou is whole probabiliy disribuion. Facors such as mobiliy and cell shape and size cause he dwell ime o have a differen probabiliy disribuion funcion o ha of call duraion, his difference being greaer for higher mobiliy and smaller cell sizes. The limi siuaion occurs for a sopped MS or an exremely large cell size; in hese cases he dwell ime is equal o he call duraion. Recenly, empirical approaches have again been used o look ino he probabiliy disribuion of call holding ime in elephony sysems. In his paper in ITC4 [], Boloin menioned up o 8 papers appearing in ITC3 which assumed an exponenially disribued call holding ime. The auhor showed, however, ha mixures of lognormals fi he call holding ime beer han he exponenial * This work was funded by Spanish CICYT Projec TIC

2 08 disribuion when applying he Kolmogorov-Smirnov (K-S) goodness-of-fi es o an empirically obained sample. In ITC5 Chlebus used he Anderson-Darling (A-D) es o show ha call duraion in mobile elephony follows he same paerns as shown by Boloin for fixed elephony, as could be expeced []. In he pas, several auhors researched ino models for channel holding ime in cellular sysems making use of analyical ools and simulaion resuls; all of hem assumed an exponenially disribued call holding ime. In [3] he auhors obained analyical resuls for channel holding ime disribuion, assuming a specific mobiliy paern: uniform speed and direcion which changes a he cell borders. The disribuion obained was complex, and he auhors used he negaive exponenial o approximae i and hus furher invesigae oher performance figures. In [4] Guerin simulaed a large geographical exension wih round cells. When applying he K-S es o he simulaion resuls he exponenial disribuion gave a saisfacory fi. In he same paper oher more complex disribuions were analyically obained for a more resricive mobiliy paern. In [5] Seele and Nofal analyically obained an exponenially disribued channel holding ime for a Manhaan model. In [6] he K-S es was again applied o simulaion resuls and he exponenial disribuion was again he bes fi. In his case he mobiliy paern was as presened by he same auhors in [7]. I is obvious ha channel holding ime disribuion depends on call holding ime, bu only parly. Rappapor s words in [8] reveal he complexiy involved in finding his disribuion: Clearly, dwell ime depends on many facors such as propagaion condiions, he pah a mobile plaform follows, is velociy profile along his pah, and especially he definiion of he communicaion range. Bu even if all of hese were known, he dependency is so complex and burdensome ha one would evenually have o resor o empirical findings in some way. To he auhors knowledge, he only approach based on real measuremens exising in he open lieraure is he paper presened by Jedrzycki and Leung [9], which acceps he lognormal disribuion as being he bes fi. In his paper a field sudy of he channel occupancy of a cellular elephone sysem in Barcelona is performed. The resuls presened in he paper are an exended version of he resuls presened in [0]. The paper is organised as follows. Secion explains in deail how real channel holding ime daa was colleced. Secion 3 briefly describes he saisical ools chosen by he auhors o invesigae he subjec and how hese are used. The equipmen and saisical ools were previously used for he sudy presened in []. The exponenial disribuion is compared wih empirical daa in Secion 4. Secion 5 provides he resuls of applying he K-S es o oher probabiliy disribuions. Oher saisical resuls conneced wih channel holding ime and obained in he same environmen sudied are presened in Secion 6. Secion 7 summarises he main poins and gives some conclusions.

3 09. EQUIPMENT USED TO DETECT CHANNEL ACTIVITY The hardware used o regiser he necessary daa is exremely simple as shown in Figure. The equipmen is based on a scanner receiver uned o one of he carrier frequencies of a Base Saion (BS). The scanner is conneced o a Personal Compuer (PC) which regisers all he aciviies of a single channel in a PC file. The moniored sysem uses he TACS sandard (very similar o AMPS excep for some minor deails) and FM deecion of he down-link carrier frequency is sufficien o find ou he channel occupancies. The power ransmied in he down-link is higher and more sable han in he up-link. This fac helps o reduce he annoying effecs of noise, fading and inerference. Each record of he PC file includes fields such as he saring and ending ime of every channel occupancy and he carrier srengh a he moniored frequency. A very simple process generaes a new file conaining only he lenghs of he channel occupancies. RS-3C PC Monioring he radio-channel Scanner Record of all aciviies deeced PC Daa ready for saisical analysis: parameer esimaion and goodness-of-fi es Figure. Equipmen used o obain he necessary daa. Before performing he saisical analysis, a cleaner file is obained in which some values of he original sample are eliminaed. Firs, aciviy values under seconds are considered o be caused by noise or inerference and hus suppressed from he daa se. Aciviies separaed by a silence of less han second were considered o be shor cus due o signal fading and were herefore joined. In fac, he TACS sysem is proeced agains hese fading effecs and holds he assigned channel for a longer ime if a handoff is no required. Boh bounds were carefully esablished by aural monioring in an aemp o minimise he number of false daa. Wih hese bounds, suppression of acual aciviies or rue silences smaller han he bounds were reduced o less han 5%, and more han 95% of false aciviies or silences were deeced. 3. STATISTICAL TOOLS Firs of all i mus be decided which candidae or heoreical probabiliy disribuions are o be saisically esed agains he empirically obained daa collecion. In all cases he coefficien of variaion of he empirical holding ime

4 0 was found o be larger han one, so only disribuions which can achieve such coefficiens should be esed. The candidae disribuions are he same as in Secion 5 of [] bu he hyperexponenial disribuion was excluded because he fi proved o be exremely poor. These probabiliy densiy funcions (p.d.f) can be classified as follows: Exponenial and shifed exponenial. Shifed exponenial is he simples sep forward from he exponenial when exremely low values of are no possible o be obained. For d=0 one has he exponenial p.d.f. ( d) f () = e for d () Combinaion of memory-less sages are appreciaed by researchers o be used along wih analyical ools. Erlang-j,k of Equaion () and erlang-k- (also called hyper-erlang-) of Equaion (3) can boh achieve coefficiens of variaion higher han uniy: f () = p j j k k e ( j )! + ( p) e ( k )! for 0 () f () = p k k e ( k )! + ( p) k k e ( k )! for 0 (3) Lognormal as in Equaion (4) and mixures of lognormals as in Equaion (5) for lognormal-3. (log( ) µ ) f () = e σ for > 0 (4) πσ 3 i= p i [ p. d. f. lognormal] i (5) Once a candidae disribuion has been proposed, is parameers mus be esimaed according o he empirical daa. In his paper he Maximum Likelihood Esimaion (MLE) is used []. The only reason o use MLE in his work as opposed o oher mehods is ha in our case MLE gives beer confidence figures han ohers when fiing wih he empirical disribuion. To selec a goodness-of-fi es he auhors considered ha he observed phenomenon is a non-naural one, and is modulaed and disored by many

5 parameers ha depend on very differen maers and even on sysem and nework seings. In his siuaion, simpliciy was preferred o exreme accuracy. As in [, ] he K-S goodness-of-fi es is used in is all parameers known version in his paper. The K-S es was also applied o analyically model he holding ime of mobile elecommunicaion sysems [4, 6] because of is power and simpliciy. The K-S es works on he c.d.f. insead of he p.d.f. and avoids he dependency of he significance figures on he seleced bin-widh found in oher ess such as he chi-squared used in [9] or he A-D used in []. The modified K-S disance D and he significance level α can be compued as: 0. D = ε( n ) n ε α = ( ) i in e (6) i= where ε represens he maximum difference beween he heoreical and empirical c.d. funcions and n is he number of daa []. I is common pracice o esablish he allowable level of significance before carrying ou a saisical analysis. The proposed heoreical disribuion is no rejeced if and only if he significance α is higher han he desired level (α=5% in [, 6] or α=5% in [9]). One of he goals of his paper is o esablish he ranking in which reasonable candidae disribuions fi he empirical daa: a simpler disribuion may fi well enough for a paricular purpose. The significance level is hus no fixed beforehand, bu all he candidae p.d. funcions are compared according o heir significance α or modified K-S disance D. 4. THE DATA SET AND THE EXPONENTIAL DISTRIBUTION Many daa samples were obained hroughou he busy hour and all of hem feaure very similar saisical properies. The sample used o illusrae his paper was obained in June 996. The sample size is n=,445, he average holding ime m =40.6 seconds, and he squared coefficien of variaion of he sample is.7. The spikes observed in he empirical hisogram of Figure make parameer esimaion and fi more difficul. These spikes are due o he hyseresis ime ha he sysem requires before rerying he handoff, as was observed in [9]. This ime sep is necessary o avoid insabiliy and coninuous handoffs when he MS is on he border beween wo cells. I is possible o remove he spikes by esimaing he percenage of channel holding imes due o immediae handoff, as done in [9]. In his work we prefer o keep he sample unalered as long as fiing resuls are saisfacory. When he negaive exponenial disribuion is fied wih he empirical daa he probabiliy of very shor occupancies is overesimaed, while he area wih he highes probabiliy in he empirical hisogram is underesimaed. This behaviour can be observed in Table and in Figure.

6 When he occupancy remaining ime is examined as a funcion of he occupancy elapsed ime, he average remaining ime is far from being independen of he elapsed ime, as should occur if he empirical daa follows an exponenial disribuion paern because of he memory-less propery of he laer. Figure 3 shows how he average remaining ime is greaer he longer he elapsed ime. This behaviour is similar o wha was observed in [] for he call duraion in convenional elephony and in [] for he ransmission duraion in PMR sysems. The disconinuous shape for long elapsed imes is due o he fac ha fewer values remain o be averaged for longer elapsed imes, leading o fewer and more dispersed values. Table Percenage of accumulaed probabiliy: empirical vs. exponenial. Time (seconds) Empirical Exponenial Number of aciviies Average remaining ime (s.) Channel holding ime (s.) Elapsed ime (s.) Figure. Empirical vs. exponenial channel holding ime disribuion. Figure 3. Average remaining vs. elapsed channel holding ime 5. NUMERICAL RESULTS In Table he K-S disance D and he significance α are abulaed for he candidae probabiliy disribuions along wih he MLE parameers. The bes fi is aained by lognormal-3 disribuion, wih a significance of almos 0%. Noe

7 3 ha he K-S es is argeed on coninuous funcions while in his case here are large spikes in he empirical hisogram. Combinaions of memory-less sages fi wih negligible levels of significance, bu he exponenial fis much worse han ohers. When he researcher is o simulae a cellular sysem, he single lognormal disribuion represens a good rade-off beween simpliciy and accuracy. The lognormal- disribuion no included in Table gives a significance beween ha of he lognormal and lognormal-3. Table Momens and fiing of channel holding ime: Sample size=,445. Momens of channel holding ime: m : s. cv :.70 Fiing Exponenial D: 5.54 α: : Shifed Exp. D:.05 α: : 3.37 d: 4.63 Erlang-jk D:.7 α: : 5.38 j: k: 7 p: 0.95 Erlang-k- D:.59 α: : 5.07 k : : 64.4 k : 4 p: 0.89 Lognormal D:.55 α: 0.06 µ: 3.9 σ: 0.89 Lognormal-3 D:.3 α: µ : 3.33 σ :.04 p : 0.5 µ : 3.55 σ : 0.50 p : 0.33 µ 3 :.44 σ 3 : Densiy F() F n + F n Channel holding ime (s.) Channel holding ime (s.) Figure 4. Empirical hisogram and bes fi (lognormal-3). Figure 5. Conours of 5% significance. In Figure 4 he empirical hisogram is ploed agains he bes fi. The lognormal-3 disribuion follows he whole shape of he empirical daa much beer han he exponenial. In Figure 5 he proposed c.d.f. is ploed along wih he conours of 5% significance: empirical increased and decreased by he ε

8 4 corresponding o 5% in Equaion (). The proposed bes fi c.d.f. always remains wihin he bounds. The same saisics were obained for differen imes of day, loads and cells, and he ranking in Table is always mainained, alhough he average and coefficien of variaion may vary, as explained in Secion 6. This gives our work more general scope, as he conclusion ha lognormal disribuions fi beer han he memory-less ype relies on saisical analysis for samples from many differen siuaions. 6. OTHER STATISTICAL RESULTS Alhough he main goal of his paper is o invesigae he probabiliy disribuion of dwell ime in cellular sysems, oher saisical resuls were obained which can be useful when applying he resuls of previous secions o he design of cellular sysems. In Table 3 he mean and squared coefficien of variaion of he dwell ime are shown for wo differen charging periods. As he duraion is measured in he same cell he larger average corresponds o a lower mobiliy and/or o a longer call duraion. The mobiliy is acually lower a nigh. The average unencumbered call duraion is also greaer due o he higher proporion of non-professional calls and he lower charging rae. Table 3 Momens for differen charging periods. Time Day: 7 o Nigh: o 7 Average (seconds) cv.70.9 In Table 4 he channel holding imes of he same daa sample are classified according o he occupancy ype: he average and squared coefficien of variaion of he channel holding ime are abulaed for each ype. Sar-handoff means, for insance, ha he call sars in he observed cell and coninues in anoher one. The ype wih handoff includes he firs hree, while whole call means ha he call begins and finishes wihin he cell considered. No available indicaes ha he called pary is no conneced and ohers includes calls which can no be assigned, such as erroneous calls, calls blocked by nework overload, ec. From Table 4 i can be concluded ha 89%=76/(76+9.6) of he occupancies belonging o answered calls have a leas one handoff; his can be considered o be a very high mobiliy value. If we accep he 3 seconds found in [] as he average call duraion each mobile visis 3/40.6=.78 cells on average. In he sysem examined he auhors found a slighly longer average call holding ime bu canno guaranee ha i corresponds o he calls of he same sample. Mos of he erminals ha generae an occupancy classified as a whole call are probably sopped, his being he reason for he longer average holding ime.

9 5 7. CONCLUSSION Alhough he use of analyical ools can lead o he conclusion ha channel occupancy in cellular elephony conforms o a negaive exponenial disribuion paern, field sudies show ha he exponenial disribuion is far from maching empirical daa. A mixure of lognormal disribuions which fis call duraion in convenional elephony very well also gives he bes resul for dwell ime in mobile elephony. There are oher simpler p.d.f. which fi much beer han he exponenial disribuion: single lognormal, shifed exponenial and erlang-j,k. The laer can be represened as a combinaion of memory-less sages, his being an advanage when analyical research is performed. Table 4 Classificaion of occupancies: average and squared coefficien of variaion. m (s) % cv Sar-Handoff % 0.55 Handoff-End %.3 Handoff-Handoff %.54 Wih handoff %.4 Whole call %.8 Busy % 0.36 No answer % 0.03 No available % 0.64 Ohers % 0.70 Our sudy proves ha dwell ime follows he same disribuion paern ha he unencumbered call duraion. The average channel holding ime is shorer in our sudy han in [9], leading o he conclusion ha we presumably work wih smaller cells or higher MS speed or boh. This difference can easily be explained as being caused by he difference beween he life-syles of Canada [9] and Europe (ours). The high handoff rae (almos 90% of he occupancies undergo a leas one handoff) shows ha he agreemen beween dwell ime and he unencumbered call duraion disribuions does no rely on a low handoff rae. REFERENCES. V. Boloin, "Telephone Circui Holding Time Disribuions", Proc. 4h Inernaional Teleraffic Congress, pp. 5-34, Elsevier Science B.V., E. Chlebus, Empirical validaion of call holding ime disribuion in cellular communicaions sysems Proc. 5 h Inernaional Teleraffic Congress, pp , Elsevier Science B.V., 997.

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