‘Telephone Traffic Time Averages By JOH RIORDAN. (atoms Sond Ato 25,281 sateen i args, rans ee 1. Ttmaoverion ‘OH mathematical slits of tletone tric, He those of ell Jos or ‘leloy which ase woe in tanking enginering, Ube La i considered a6 lo of proline. Inthe pio nf ment etance, tho busy heat, tis fw is ysly epee ae stationary that isto ay, the prode- Nit oF a given numberof busy turks, oe the pry of dle ofan incoming all for any oer puobebity of the sysen whic cones in ques- ‘Hon is ake skype of he partculae momest in the busy hour at ‘which the system ie examined. Tresyer ead oe ts statistical et > ‘For auch theoretical stu the statetin! quate. whieh determine these probate, like the rate a which ells appear, are uf couse taken {stiven, but in the apa ey must be determined by observetions, hc as those being lake Cae eucent reer program of We meas. lrements, Hecr a uly apgeane, To alge the estezsive anoact of heervtingsl maria, either measurements are made of trafic verges Grete small compa tote busy houe but aot smal ov Un he eyes} ofthe mereorements of sntiauous recorders ae avert! I Fad. Ttmey be noticed here lal for penton ofthe rr ives below the stair averages ialand By miaetrerents mse thane of a com incocsdevee wh rconeal afc charges! nl, se eating tlvises, those oolained from mmbsr a “looks 1 piste within he verging interval ul to use Eve measicemenis i leteraing the tine perimeters by standard sansping tory, a corespnsing Viorel suc of te average i necaty ‘Ses sty, nit mie to be eral pesca, i iver bee, No asiempt is mae ta describe the stmpling studs pete from the results reathal, These scom to he of many Kinds, not neceesery 10 lei, bal for 140 re wn sere raeueat yore, crease 1954 conertones it my he rental tht he moe nye, the ome, scnss co be hat of setting comidercelimis forthe average taf. "The toes staportans ofthe limite to this study ere taoe imped by the sesumptons of ettiatie! egibrium ith feed average, and infinite ‘number af trunks. The omer lnits splicaion to psiods in whic roughly speaking, ayeage tates acter ising 32 faling; ta acter jac hed. ‘nly by the extome rantemstisal dies pron iy assuming eter ‘vse The tai variable is the mantber of busy tems na peri fate ‘al ecibsium, For pure chance cal ape, the als dng time shaeacter ino is let astray thrown he develo it rm intrest ie the ene eases of rant oi ne and exonential bold Lime, which are cxumine i datil® For cal oom limited wumsber af rte, rls a la! vl for enfetia ‘Mure pele, 17 0h be theca variable Zar th ur trunks atime Ue variable sted, the avenge scores nt intereal of length Ze a baay oy LP ty 0 woh fap 0 ‘The cat ix; Weal ane the stalin! properton of MCT}? ‘The reclls gve are the st fo cml (ominvaranss) of (7, which on tu ave Ihe simyiol expen, For the convict of Che Fender il mye vod That the fst cunt Ue ey Ue od the tient moment shout the mean which isthe vasanet, Gxt thd the hed roent alt che wean, ae the fourth the Tauei moment about the wea fea tee Lines the agua af te arin, Tall eae the mean of (Pe the mean oh (0) a or fae cree xl inpat ised teenage nother of alsin wet verge holdin ime, "ae other eamulines for pate evarce ell inp Ay, eave the gonecal apres oe ee wih wo from cB Sabe tassel pu oe oye cay aig ‘Peipinoe arcane tcc desta oy Rage Wikies ‘reverse eam rian xorg ust etn ohio ae Toe pete yr ans Bg Cie a eomestl Teg tn. n ta 4 Te sent The pete snng th low, Th sate de ccm ota ao th ny st ter tre dered nate of expen Tete Set noe en dena is marr, 4 [oie = mon 0 wre nwa or te spt ota Sy tested monet mureni= jf [Rwowoem ral st on foe higher saomente Coration ests appear in (8) in FLY <a sd ave ele in She eveogaacnt Ly ‘eration of tas tion probable, shat i thew probable decening she tre lbw fn tte, The Lansiton probally Ps is denned ns the probability of Teavition in ton f valle in prgeae Pay rua! tn cf progres find ies enor see of eat praia illevent tne epochs (Orly for bine ves ef sa thee lili ees eres the a as Wy celeenine tree ple taasiton probabil. yen thee of slouble std ple “resi, shes “he expected values of Iie rides sin quasples 0 nies uf oy tans, and ly he 2 Tassos Promssiaares For exponential olin tn, and iainitesoueces, infinite tru, these watts save ale sen determines! y Coney Pala [S]. Pabn's Mosk as been smal th Ep Feller [J] a%@ Hy Toren (3, and dee Uethes the nhole pony sot sacely the elie coalition, For Ue fauilbriom condition, a eigerent_provalar wile to St used BY event [for anther parpase lle the suri af ante general oling tine characterise 1182 ra ants siete earveat jounnst, deroRRR 195 Wo inte sources, and eae arriving individually and collestivaly at cansort with average ey 4, te wllsnown Jona forthe probity ‘ha exactly als arrive in ime interval the Psu mania “ ‘hen, if Pell; #1 Poe eat praity of transition fam § eo j when balls arsive in five Consider Pf 0), tha isthe (ceditona tenscon probebiities when no calle arsive Let the probably thet «cell asts at leet fe fy 0 tat the averege holding ft i ven by = [Moree au = ivy aw io Th e's ntl in proooe ee inlspeaent of eae. oe: Select on of then ad euppose ee ting foam ie sval ite age) ff Then se pb billy thas st il leo exit ¢ unite later iz che conditional probe- billy Feet. Seen cqubrium coodiion al mersnts of trval hve acal psctabie, che saesponding probability foram eabiery al = of Lane ats = (eno ® oa C0 td its generating fusion ie Plt 250) — DP uth Oe — Wt te te ® rs (8) and (9), far Devils, de argue a g ome [Now suppeee ob cal serves in interval f The moment of aviv ix uflorsy distibuted neha i, fm see moment of arzval Pel <u) Sab da) = aa td the probability that cll nivng alan abitary moment wil bein Me oo - [we -a# 4 [ade-fa-eo on rtaruose zRArMC TaMe aven4cns nat he sarsequoing yenesating function ie 1G = wh 1 ie no fevly since valle ating ene independent, the generating faastion Jor (oll aig in Ih te vor ond Paar Bo + fe Ugh + te — DOr a Hens, fray by (. Pua = E Putts? ikem neon ce ner eet =U Ge Daler tr sae HEA item te Dek = 8 ap he lat sep us 0 Tics the gener‘ingfuretion fr tne singles trascnn probabil, sau Inguite fife Vals esl; idol, fr expose sng rae p= f 2 fhe sabes henaevee ae abt! exe Le gal= ing fest in orca of ok By estan far in Pale wou Iles they tse used hee: the generating finn e mets oe dee tring Un evatagee of rte, 9 wile Felore quia om tothe ober transition pda, ti ntereting to sete veut essa 12) Tn tit cl eer te Ii Tas Peis desi (sn Le) tha isthe presen novation PAC = OR sere — of Thi of connareadependeat of cme Then, i410} has hie Mlenstys 20 shold Na dled far 40 ane the crete pred ins teva (1, This wie’ by Erdaiettin sep oe meng kw ee ee sete mh 18 wn sweat tree Joos, seman LIST Abu, 9) = Teel oot = Wer Pans = Me ie B= © 18 stun td fv vn iso aed a eval ras fe aie inde ire Cy equi soalilies ara ished wn sata nba the inl sat bas bee Finely, inn Merk snow ff ler, Chaps 18) che sie Fans tion patie lore are eee since Bo PLP ot) Steet for this the Chopmer-Kolonasory squion, name eat t at = Eres Using 12, Presenting eatin of gener fu ein NEG = Net bade bic 1 aH bel SVT fe Teint tvaeife — B= alta so the pesmi Markov sshict etree ote ar exponcnsal holding tine Fe tie nest uansiion probabil; Py 1. consis List @endioms invalid eal aries ta wanlenteeal¢ a, As bene = (Jeo Eporeromeit 6-8" lity at all wi Ps Yt # ‘wl et w mores Pa the soueiocal peakehility of a tesweit Com j ‘ose, given Ue iene 100 “Th penwrting anise ce lhe hae eastio priis h Wig ave ihe, inom es Uhe eatin EE Posthestiey! = Wa oe tig =~ tinal Iw serene ysl rive a ra Flerval 6 foe, the pokey that i il ocupy a Ua at Une #5 01 — Je ML — roummoxe rearmc Tar ayeeaces 2135 tod the conditional pros his iy that it will alo occupy trunk at time Dees Ve rormsyenng erating futon, with and the ndintors of calls ada Ea, ep VAG DEL aly Clg) g6 + Ae “The generating ft lls ia his intervl 5 his expression ceed to ther lh yumer, sve ls arve inpensently ond eine els erive sith probity ©" Cae the generacing fonction for lls resin in DN NOs se Neen! an expllir—110 gt dy — 1giat +0) ue or brevity (rt # hee bon rite for aR, Tiny the generating Vel fale arving 1 ests DO ate as) eoe EL tata Mb eo + ate = vet apie BO a RO BU RAL ety = aid — et 4 By sna: arzrvent, the genes Gti er rps aon ya was EXE Macinntyt SWE De AHO Nien Fas Dest ete dg) Fo KL gk BOG bat De bad + YMG Baad Fe = Bie = aed 1196 tw wrtr vere tsocaL soeaiaL, verouia 195t 8, Famroren Compara Conation expetatons, Uke ELVE0.NG)] in equation (2), sre aed Jor evsinatign ofthe morents of M(T). They aay be determined fom the transition probability nerstng sw to i agra asa mater any of romvenioes tha! theme ebachs fa et £90 Cast onde (2 FS vo), Shes, om the suri of alatisteal equibium, the call probabilities atthe frst eps fer lent of ils valu, aa already ote his value ray be ak a ra without lea of generality "Thus forthe seve moment is suet detcmne od — EXON = Dips Dirt an PRN) = with write Tae E rately By (12), this ie tne sme ee <1 Ny — tie ep Fale.) Geb ly BL Senet y+ se) on ot = Saisie ee aro) In che gare ay the sm ser creel expat, hati elo) = BENTO 8 is obtain fm Goals ya = Dot DE Pate, dye Mole 35) = Gaol A LIE EED ~ opie tate t aye) + wae) oly + Darla + 0) oli) = HF OTR) + 0) + alta Eee +) 3) "Filly, the thind onder eomration sms out oth, 530) = BINDING + EEF 8 I = 81 Pig) + a) + a Keb) bay be) eet etl Pye by) H+) bax +00) eCoato) slo # et +) Fate oe D9) Ge ob Asillappesr, the areangerent frame in (22), (23 aul (21) eoresponds tw the espatsinn ot uninary moments in tems of cuuncante (ei ints}; (21) coresponds tay — 8+ eB, | ABky = SEE By Sih he Ft cumulane fr the Poison of ea bf 8) en ‘Mowents see stein! for these nests BY integrations noted, equation (2, the fist monn! i fr any dg ne ct ition Shice fete are two waysol orlringHnerecks cheese met i wen = 2 fa {sven 9 eee) ea ( canna ‘The bt step fe by the forma for sovesng she une inion ine cated [aan = Fon Paw ‘The variance o: second ental usenet ie se eeond cumlant fe ischen Ver Cen] = BIGOT) — 8 * 08) =Bfemor—a MS mp ray svtrme meemeaL paver, AcroRER 15 ce there ae. be wit = G wayact ordering # epoch, the third manent may gareny = Sf" af dof deo — =o .e p “a [on Corse 9 Hive dhe st rp nage mesa’ evaluated by use of the Getty af a fam fae = 0b geo) bao =f fam ar faeacr Pe) =a he et tile tee hy scene inversion of uterstion onder, tne ese finaly extn) cae) ond fy = Maen ~ 1 = Bari] — awn) +28 = HWE] ab = ee