A NEW GENERALIZATION OF ERLANG DISTRIBUTION WITH BAYES ESTIMATION

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1 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle NEW GENERLIZTION OF ERLNG DISTRIBUTION WITH BYES ESTIMTION Blal hmad Bhat *Parml Kumar ad Mushtaq hmad Wa 3 Dvso of Socal Scece Faculty of Fsheres Ragl Gaderbal SKUST-Kashmr J&K Departmet of Statstcs Uversty of Jammu Jammu J&K 3 Govermet College for Wome Nawakadal Sragar J&K *uthor for Correspodece BSTRCT The Erlag dstrbuto s the dstrbuto of sum of epoetal varates. The Erlag varate becomes Gamma varate whe ts shape parameter s a teger Evas et al I the lterature varous authors dscussed the propertes ad estmato of Erlog dstrbuto e.g. Harschadra ad Rao 988 Bhattacharyya ad Sgh 994 Wper 998 Ja 00 Nar et al. 003 Sur et al. 009 Damodara et al. 00. I ths paper we propose a ew geeralzato of Erlag dstrbuto the dscuss the Bayesa estmato of Erlag dstrbuto usg dfferet prors. We llustrate the results usg a smulato study as well as by dog real data aalyss. Keywords: Probablty Desty Fucto Bayes Estmator Posteror Dstrbuto Pror Smulato INTRODUCTION The org of queug theory was 909 whe.k. Erlag publshed hs fudametal paper relatg to the study of cogesto telephoe traffc Brockmeyer et al The lterature o the theory of queues ad o the dverse feld of ts applcatos has grow tremedously over the years. The aalyss for such a Erlaga queue s ow folklore the queug lterature. The Erlag dstrbuto s the dstrbuto of sum of epoetal varates. Ths dstrbuto ca be epressed as watg tme ad message legth telephoe traffc. If the durato of dvdual calls are epoetally dstrbuted the the durato of successo of calls s the Erlag dstrbuto. The Erlag varate becomes Gamma varate whe ts shape parameter s a teger Evas et al I the lterature we observe that Harschadra ad Rao 988 dscussed some problems of classcal ferece for the Erlaga queue. Bhattacharyya ad Sgh 994 obtaed Bayes estmator for the Erlaga queue uder two pror destes. Wper 998 studed for Er/M/ ad Er/M/c queues uder Bayesa setup ad estmated equlbrum probabltes of the queue sze ad watg tme dstrbutos usg codtoal Mote-Carlo smulato methods. Ja 00 dscussed the problem of the chage pot for the ter arrval tme dstrbuto the cotet of epoetal famles for the Ek/GIc queug system ad obtaed Bayes estmates of the posteror probabltes ad the postos of chage from the Erlag dstrbuto. Nar et al. 003 studed Erlag dstrbuto as a model for ocea wave perods ad obtaed dfferet characterstcs of ths dstrbuto uder classcal set up. Sur et al. 009 used Erlag dstrbuto to desg a smulator for tme estmato of project maagemet process. Damodara et al. 00 obtaed the epected tme betwee falure measures. Further they showed that the predcted falure tmes are closer to the actual falure tmes. I vew of the lterature avalable o Erlag dstrbuto ad ts applcatos we try to geeralze t from a ew proposed model Blal ad Kha 0 ad dscuss ts Bayes estmato. The probablty desty fucto pdf of Erlag dstrbuto s gve by ep 0. = 0 elsewhere α = 3 β>0 Cetre for Ifo Bo Techology CIBTech 4

2 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle Proposed Method of Geeralzato of Erlag Dstrbuto We ow obta Erlag dstrbuto. by a ew method as gve below: Suppose Fu be ay o-egatve cotuous fucto of u defed the terval 0<u<t ad f α s ay gve postve real umber such that t F u F u 0 du. s fte. The the fucto - F / ; 0 t. F = 0 elsewhere s a probablty desty fucto pdf of X a cotuous radom varable. The rth momet of the dstrbuto about the org s ' EX r r F r F r... I partcular the mea μ ad the varace ' F F ad EX F F F are respectvely gve by - F Takg Fu = ep-uβ - wth u = β ad lettg t. t follows that F e d 0 =..3 Usg.3. we get ep 0 = 0 elsewhere.4 whch s the probablty desty fucto pdf of Erlag dstrbuto wth parameters α ad β. Bayes Estmato of Erlag Dstrbuto Bayesa statstcs s a approach to statstcs whch formally seeks use of pror formato wth the data ad Bayes Theorem provdes the formal bass for makg use of both sources of formato a formal maer. Bayes theorem s stated as Posteror α Lkelhood Pror Cetre for Ifo Bo Techology CIBTech 5.

3 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle The pror s the probablty of the parameter ad represets what was thought before seeg the data. The lkelhood s the probablty of the data gve the parameter ad represets the data ow avalable. The posteror represets what s thought gve both pror formato ad the data just see. I may practcal stuatos the formato about the shape ad scale parameters of the samplg dstrbuto s avalable a depedet maer. Therefore here t s assumed that the parameters α ad β are depedet a pror ad a pror dstrbutos chose ths paper s Trucated Posso dstrbuto as a pror for shape parameter ad Iverted Gamma dstrbuto as a pror for scale parameter. The loss fucto cosdered ths paper s squared error loss fucto. The squared error loss fucto for the shape parameter c ad the scale parameter b are defed as L 3. L 3. whch s symmetrc ad ad represet the true ad estmated values of the parameters. Posteror Dstrbutos uder Dfferet Iformatve Prors The posteror dstrbutos usg dfferet formatve prors for ukow parameters α shape ad β scale are derved the followg subsequet subsectos. The probablty desty fucto pdf of Erlag dstrbuto s gve by ep = 0 elsewhere α = 3 β>0 Let X X... X be a radom sample from the Erlag dstrbuto the lkelhood fucto of the sample observatos: :... s defed as L ; - ep α = 3 β>0 3.4 Whe Shape Parameter α s Ukow ad Scale Parameter β s Kow We assume pror for shape parameter α the trucated Posso dstrbuto gve by ep- g c; 3.5 ep α = 3 β>0 By combg the lkelhood fucto ad the pror desty the posteror dstrbuto of α gve data s l g ; 3.6 ep l ep α = 3 The Bayes estmator uder squared error loss fucto wth the pror g ; s gve by Cetre for Ifo Bo Techology CIBTech 6

4 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle Cetre for Ifo Bo Techology CIBTech 7 l ep l ep 3.7 The posteror varace of Bayes estmator s gve by l ep l ep C Var. 3.8 Whe Scale Parameter β s Ukow ad Shape Parameter α s Kow We choose Iverted Gamma dstrbuto the pror for scale parameter β as gve by ep- ; - - g 3.9 β>0 α β >0. The posteror desty after combg lkelhood ad pror s gve by 0 ep g 3.0 The Bayes estmator uder squared loss fucto s gve by 3. The posteror varace of Bayes estmator s gve by. Var 3. Smulato Study: I the smulato study we have chose =50 for several values of parameters. The smulato program was wrtte S-Plus/R software. The results obtaed y usg smulato study s preseted below:

5 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle Table : Bayes Estmators ad ther Varaces uder Trucated Posso Dstrbuto for =50 α = α =3 α =6 α =9 V V V Cetre for Ifo Bo Techology CIBTech 8 V Table : Bayes Estmators ad ther Varaces uder Iverted Gamma Dstrbuto for =50 β= β =3 β =6 β =9 V V V V Read Data alyss To check the valdty of the model we cosder the survval tme weeks for 0 male rats Lawless 003 that were eposed to a hgh level of radato. The data s ad 69. The three goodess of ft test reveal that Kolmogrov-Smrov test: Test statstc: wth p-value derso-darlg test: Test statstc: Ch-square test: Test statstc: wth p-value From above tests t s evdet that the Erlag dstrbuto wth parameters α =0 ad β =.9 fts the data set well. It s observed that results from ths data aalyss echo the same patter as foud the smulato study. The results obtaed are agreemet wth the earler studes. REFERENCES Bhat B ad Kha B 0. Geeralzato of Gamma Dstrbuto. dvaces ppled Research Bhattacharyya SK ad Sgh NK 994. Bayesa estmato of the traffc testy M/Ek/ queue. Far East Joural of Mathematcal Sceces Brockmeyer E Halstorm HL ad Jeso 948. The Lfe ad Works of. K. Erlag. Trasactos. of the Dash cademy of Techcal Sceces 77.

6 Iteratoal Joural of Iovatve Research ad Revew ISSN: Ole Ole Iteratoal Joural valable at 06 Vol. 4 prl-jue pp.4-9/bhat et al. Research rtcle Damodara D Gopal G ad Kapur PK 00. Bayesa Erlag software relablty model. Commucato Depedablty ad Qualty Maagemet Evas M Hastgs N ad Peacock B 000. Statstcal Dstrbutos thrd edto US New York Joh Wley ad Sos Ic. Harschadra K ad Rao SS 988. ote o statstcal ferece about the traffc testy parameter M/Ek/ queue. Sakhya B Ja S 00. Estmatg the Chage Pot of Erlag Iterarrval Tme Dstrbuto INFOR- OTTW Techology Publcatos Uversty of Toroto Press Caada US. Lawless JF 003. Statstcal Models ad Methods for Lfe Tme Datasecod edto Wley New York US. Haq ad Dey S 0. Bayesa estmato of Erlag dstrbuto uder dfferet pror dstrbutos. Joural of Relablty ad Statstcal Studes 4. Nar UN Muraleedhara G ad Kurup PG 003. Erlag dstrbuto model for ocea wave perods Joural of Ida Geophyscal Uo Sur PK Bhusha B ad Jolly 009. Tme estmato for project maagemet lfe cycles: smulato approach. Iteratoal Joural of Computer Scece ad Network Securty Wper MP 998. Bayesa aalyss of Er/M/ ad Er/M/C queues. Joural of Statstcal Plag ad Iferece Cetre for Ifo Bo Techology CIBTech 9