Confidence Interval Estimations of the Parameter for One Parameter Exponential Distribution

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1 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Cofdece Iterval Estmatos of the Parameter for Oe Parameter Epoetal Dstrbuto Juthaphor Ssomboothog Abstract The objectve of ths paper was to propose cofdece terval estmato for a oe parameter epoetal dstrbuto Evaluato of the effcecy for ths estmato was proved va theorems ad a smulato study was coducted to compare the coverage probabltes ad epected legths of the three cofdece tervals (, Eact ad Asymptotc cofdece tervals) The results showed that the cofdece terval whch s derved ths paper uses the same formula as for the Eact cofdece terval whch s wdely used Addtoally, the epected legth of the cofdece terval s shorter tha that of the Asymptotc cofdece terval for a small sample sze ad all levels of the parameter ad cofdece coeffcet Furthermore, the three cofdece terval estmatos get systematcally closer to the omal level for all levels of the sample sze ad the parameter I addto, the effceces of the three cofdece terval estmatos seem to be o dfferet for a large sample sze ad all levels of the parameter ad cofdece coeffcet Ide Terms Cofdece terval, estmato, epoetal dstrbuto, coverage probablty, parameter T I INTRODUCTION HE oe parameter epoetal dstrbuto s a cotuous dstrbuto ad s ofte used as a model for durato It s also sutable for the dstrbuto of the tme betwee evets whe the umber of evets ay tme terval s determed usg a Posso process Ths dstrbuto plays a mportat role the formato of models may felds of relablty aalyss research, eg, bologcal scece, evrometal research, dustral ad systems egeerg [], [], [3], [4], [5], [6], [7] I addto, the oe parameter epoetal dstrbuto s also used the theory of watg les or queues whch s appled may stuatos, cludg bakg teller queues, arle check-s ad supermarket checkouts [8], [9], [], [] I addto, Sa ad Dama [] appled the oe parameter epoetal dstrbuto to aalyze queug system wth a epoetal server ad a geeral server uder a cotrolled queue dscple Whether these relablty ad queug aalyss methods ca yeld precse ad accurate results deped o the methods of parameter Mauscrpt receved February 3, 5; revsed July 8, 5 Ths research was facally supported by the Kasetsart Uversty Research ad Developmet Isttute (KURDI) Juthaphor Ssomboothog s wth the Departmet of Statstcs, Faculty of Scece, Kasetsart Uversty, Bagkok 9, Thalad (e-mal: fscjps@kuacth) estmato There are two types of parameter estmato from a probablty dstrbuto, amely pot ad terval estmatos I statstcs, a pot estmato volves the use of observed data from the dstrbuto to calculate a sgle value as the value of parameter ; t s almost certa to be a correct estmato as metoed by Koch ad Lk [3] I ths research, we vestgate the terval estmato of parameter that provdes a rage of values wth a kow probablty of capturg the true parameter The geeral theory of cofdece terval estmato was developed by Neyma [4] who costructed cofdece tervals va the verso of a famly of hypothess tests The wdely used techque of costructg a cofdece terval of the parameter for oe parameter epoetal dstrbuto s based o the pvotal quattes approach whch determes what s kow as a Eact cofdece terval as metoed by Hogg ad Tas [5], ad Casella ad Berger [6] Ths method s vald for ay sample sze as metoed by Geyer [7], Balakrsha et al [8] ad Cho et al [9] Where a large sample sze s appled, a Asymptotc cofdece terval s mostly used to costruct a sequece of the estmator ˆ of wth a desty fucto f ( ; ) that s asymptotcally ormally dstrbuted wth mea ad varace ( ) [4], [], [] I ths study, the cofdece terval estmato s proposed for oe parameter epoetal dstrbuto Ths cofdece terval s derved based o the approach of vertg a test statstc whch has a very strog correspodece betwee hypothess testg ad terval estmato The method s most helpful stuatos where tuto deserts us ad we have o dea as to what would costtute a reasoable set as metoed by Casella ad Berger [6] The effcecy comparsos terms of the coverage probabltes ad the epected legths of the three cofdece tervals are vestgated va the theorem proofs Furthermore, a smulato ca be also performed to carry out effcecy comparsos II MATERIALS AND METHODS A Crteros for the Effcecy Comparso The effcecy comparso crtera amog the three methods of the ( )% cofdece tervals (Eact, Asymptotc ad cofdece tervals) are the coverage probablty ad the epected legth of cofdece terval (Advace ole publcato: 4 November 5)

2 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Let CI L( X), U( X ) be a cofdece terval of a parameter based o the data X havg the omal ( )% level, where L( X ) ad U( X ), respectvely, are the lower ad upper edpots of ths cofdece terval The followg deftos provde the effcecy comparso crteros ths study: Defto The coverage probablty assocated wth a cofdece terval CI L( X), U( X ) for the ukow parameter s measured by P L( X), U( X ) (see [4]) Defto The legth of a cofdece terval, W U( X) L( X ), s smply the dfferece betwee the upper U( X ) ad lower L( X ) edpots of a cofdece terval CI L( X), U( X ) The epected legth of a cofdece terval CI s gve by E (W) [4]) (see [], [3], B Cofdece Iterval Estmatos for Parameter Throughout the followg dscusso, the essetal codtos for ths work are deoted by (A) (A3) as follows: (A) Let X,X,, X be a radom sample of sze from a populato of oe parameter epoetal dstrbuto wth : The probablty parameter where desty fucto of oe parameter epoetal radom varable X s gve by (see [4]) f (; ) e,, otherwse (A) Let ad,,, respectvely, be the th th ad quatles of the ch-square dstrbuto wth degrees of freedom where (A3) Let relato Z be a postve costat whch satsfy the Z Z ad ( ) s the cumulatve dstrbuto fucto of the stadard ormal dstrbuto For, the followg three methods of ( )% cofdece tervals are studed for the effcecy comparsos: ) Eact cofdece terval The cofdece set costructo wth the use of pvotal quattes s called the Eact cofdece terval For, the ( )% Eact cofdece terval for parameter s gve by (see [6]) X X CI, Eact,, where ad hold codto (A),, ) Asymptotc cofdece terval A asymptotc cofdece terval s vald oly for a suffcetly large sample sze For, the ( )% asymptotc cofdece terval for parameter s gve by (see []) X X CI, Asymptotc Z Z Z holds codto (A3) where 3) cofdece terval We propose the cofdece terval whch s derved from usg a verso of a test statstc as show theorem Lemma Let X, X,, X hold codto (A) The acceptace rego for a,, level lkelhood rato test of H : versus H : s gve by * A( ) (,,, ): e k, where * k s a costat chose to satsfy P (X, X,, X ) A( ) Proof Let X, X,, X hold codto (A) Cosder testg H : versus H : wth level (,) where s a fed postve real umber, we have The lkelhood fucto s gve by L( ) f ( ; ) e e Observe that () () (Advace ole publcato: 4 November 5)

3 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Sup L( ) e, (3) sce has the sgle elemet O the other had, oe has Sup L( ) e, (4) X, so that the value mamzes L( ) over sce X s the mamum lkelhood estmator of Combg equatos (3) ad (4), the lkelhood rato test statstc s Sup L( ) e Sup L( ) We do ot reject H : f ad oly f k where k s a geerc costat or * e * k where k k e Thus, the acceptace rego for a,, level lkelhood rato test of H : versus H : s gve by * A( ) (,,, ) : e k, * where k s a costat chose to satsfy P (X, X,, X ) A( ) Lemma Let X be a ch-square radom varable wth degrees of freedom where The probablty desty fucto of X s gve by f() e,, otherwse (5) There ests a momet geeratg fucto of X such that M X (t) for t ( t) Proof Let X be a ch-square radom varable wth degrees of freedom where From (5), we fd the momet geeratg fucto of X as follows: tx t M X (t) E e e e d t e d t e d t e d ( t) ( t) e d ( t) ( t) t ( t) where t or (t) e d t Thus, the momet geeratg fucto of X s gve by M X (t) for t ( t) Lemma 3 Let X,X,, X hold codto (A) The W X s a ch-square radom varable wth degrees of freedom where Proof Let X,X,, X hold codto (A) The the momet geeratg fucto of X s gve by (see [8]) M X (t) t, t where,,, The momet geeratg fucto of W X s (Advace ole publcato: 4 November 5)

4 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 tw M (t) E e W t X E e t X E e t X E e t M X ( t) ( t) for t (6) ( t) We refer from Lemma, equato (6) s a momet geeratg fucto of a ch-square radom varable wth degrees of freedom where Thus, W X a ch-square radom varable wth degrees of freedom where Theorem Let X,X,, X hold codto (A) For, the ( )% cofdece terval for parameter s gve by X X CI,,, where ad,, hold codto (A) Proof Let X,X,, X hold codto (A) From Lemma, we kow that the acceptace rego gves the ( )% cofdece set * C(,,, ) : e k ( 8 ) The epresso of C(,,, ) equato (8) depeds o,,, oly throug, thus the cofdece terval ca be epressed the form of equato (9) C : L U (7) (9) where L ad U are fuctos whch satsfy the costrats equatos () ad () as follows: P (X, X,, X ) A( ) () ad L U e e () L U a We set L where costats a b ad U Substtutg equato () equato (), results a b a b e e From equatos (9) ad (), the ( )% b () (3) cofdece terval for parameter s gve by : a b where costats a b satsfy the followg (4) X X P P a b a b X X P b a X P b W a where W (5) X From Lemma 3, we kow that W s a chsquare radom varable wth degrees of freedom where For the equal-tals probablty, the costats a ad b are equal to ad,,, respectvely, whch hold codto (A) Therefore, the ( )% cofdece terval for parameter s gve by X X CI,,, (Advace ole publcato: 4 November 5)

5 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Remark The ( )% cofdece terval for parameter uses the same formula as the ( )% Eact cofdece terval III EVALUATION OF EFFICIENCY FOR THE PROPOSED CONFIDENCE INTERVAL I ths secto, we vestgate the performaces of the three cofdece tervals (Eact, Asymptotc ad cofdece tervals) whch are gve the theorems 3 33 As a result of Remark, the followg theorems are proved merely for the Asymptotc ad cofdece tervals: Theorem 3 Let X,X,, X hold codto (A) The CI ad CI deote the Asymptotc Asymptotc ad cofdece terval, respectvely The coverage probabltes of CI Asymptotc ad CI satsfy equato (6) for all levels of the parameter ad the cofdece coeffcet Asymptotc P CI P CI (6) Proof Let X, X,, X hold codto (A) The, the mea ad varace of X are gve by E(X ) ad V(X ), respectvely Frst, we cosder the coverage probablty assocated wth a cofdece terval CI whch s deoted by X X P CI P,,, X X P,, X P,, X From Lemma 3, we kow that dstrbuted wth degrees of freedom The, s a ch-square P CI (7) Lkewse, the coverage probablty assocated wth a cofdece terval CI Asymptotc s gve by X X P CI Asymptotc P, Z Z X X P Z Z Z Z P X X P Z Z Usg the cetral lmt theorem, we kow that appromately stadard ormally dstrbuted The, Asymptotc X P CI (8) From equatos (7) ad (8), we obta the followg result: Asymptotc P CI P CI for all levels of the parameter ad the cofdece coeffcet Proposto 3 Let X, X,, X hold codto (A) The epected legth of a Asymptotc cofdece terval s gve by E W C where Asymptotc C Z Z ad Z for all levels of the parameter ad the cofdece coeffcet Proof Let X X CI, Asymptotc Z Z s be the Asymptotc cofdece terval Usg the Defto, the legth of CI Asymptotc s deoted by W Asymptotc X X Z Z X Z Z (Advace ole publcato: 4 November 5)

6 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 X Z Z The epected legth of a Asymptotc cofdece terval s gve by E W E X Asymptotc Z Z E X Z Z C E X where C Z Z ad Sce E X where, the Z E W C Asymptotc for all levels of the parameter ad the cofdece coeffcet Proposto 3 Let X,X,, X hold codto (A) The epected legth of a cofdece terval s gve by E W D where D,, for all levels of the parameter ad the cofdece coeffcet X X Proof Let CI, be the,, cofdece terval Usg the Defto, the legth of CI s deoted by X X W,, X,, X,, The epected legth of a cofdece terval s gve by E W E X,, E X DE X,, where D,, Sce E X where, the E W D for all levels of the parameter ad the cofdece coeffcet Theorem 3 Let ad hold codto (A), ad Z,, holds codto (A3) If there Z Z Z est a sample sze such that ad,, the parameter ad the cofdece coeffcet, the E W E W Asymptotc Proof Let ad,, levels of the cofdece coeffcet Supposg there est a sample sze such that Z ad,, Z for all levels of Z for all Z From Propostos 3 ad 3, the dfferece betwee the epected of W ad that of W Asymptotc ca be wrtte the form of equato (9) E W E W Asymptotc D C D C (9) The term D C equato (9) becomes D C Z Z,, Z Z,, (Advace ole publcato: 4 November 5)

7 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Z Z,, I equato (), we kow that ad () Z, Z,, the D C () Sce ad substtutg equato () equato (9), the we obta E W E W Asymptotc That s E W E W Asymptotc Lemma 3 Let W be a ch-square radom varable wth degrees of freedom where The values of, ad whch hold codto (A) They ca be, wrtte the forms of () Z, whe Z () () Z, whe Z (3) whe Z holds codto (A3) as a sample sze creases for all levels of the cofdece coeffcet Proof Let W be a ch-square radom varable wth degrees of freedom for, the mea ad varace of W are gve by E(W) ad V(W) 4, respectvely The codto (A) holds The, the proof of () s as follows: P W (4), From equato (4), the ch-square dstrbuto (after stadardzato) teds to the stadard ormal dstrbuto as creases []; that s, W E(W) lm P V(W) E(W), V(W) E(W), lm (5) V(W) By substtutg E(W) ad V(W) 4 equato (5), the we obta,, lm P Z (6) Also, by usg codto (A3), we have P Z Z Z (7) ad Z As creases, equato (6) s appromately equal to equato (7), we obta equato (8), Z (8) After rearragg equato (8), we coclude that Z, As a result of, we also obta, Z That s, the sample sze must be greater tha Z Lkewse, the proof of () s as follows: P W (9), From equato (9), the ch-square dstrbuto (after stadardzato) teds to the stadard ormal dstrbuto as creases []; that s, W E(W) lm P V(W) E(W), V(W) E(W), lm (3) V(W) By substtutg E(W) ad V(W) 4 equato (3), the we obta,, lm P Z (3) Also, by usg codto (A3), we have P Z Z Z (3) As creases, equato (3) s appromately equal to, equato (3), we obta Z (33) After rearragg equato (33), we coclude that Z, as creases for all levels of As a result of, we also obta, Z That s, the sample sze must be greater tha Z (Advace ole publcato: 4 November 5)

8 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Theorem 33 If codtos (A) (A3) hold ad there ests a sample sze such that, the () Z E W E W Asymptotc, () E W teds to decrease ad () E W teds to decrease Asymptotc whe sample sze creases for all levels of the parameter ad the cofdece coeffcet Proof Assume codtos (A) - (A3) hold Whe a sample sze such that creases for all levels of the Z cofdece coeffcet, the proofs of () () are as follows: () To prove Proposto 3 that E W E W, we use Asymptotc E W D (34) where D (35),, Substtutg equatos () ad (3) from Lemma 3 equato (35), we obta that D Z Z Z Z (36) Substtutg equatos (36) equato (34), we obta E W whch s Z Z the appromate formula as for Proposto 3 E W Asymptotc Thus, E W E W as creases Asymptotc for all levels of ad () We kow that from codto,, (A) The, there est a costat D whch s show Proposto 3 such that D for all levels of ad,, Cosder equatos () ad (3) from Lemma 3 such that Z Z, (37) Z Z, (38) where Z s a postve costat whch depeds o ad Z for codto (A3) holds From equatos (37) ad (38), the values of, ad, ted to crease whe creases Therefore, the value of D teds to decrease as creases,, That s, E W D teds to decrease as creases for all levels of, ad () There ests a costat C whch s show Proposto 3 such that C Z Z Z Z where Z s a postve costat whch depeds o for codto (A3) holds Therefore, the value of C teds to Z Z decrease as creases That s, E W C Asymptotc teds to decrease as creases for all levels of, ad IV SIMULATION RESULTS Ths secto provdes a smulato study for the coverage probabltes ad epected legths of the three cofdece tervals (Eact, Asymptotc ad cofdece tervals) Ne populatos were each geerated of sze N =, the form of a oe parameter epoetal dstrbuto wth = 5,,, 5, 7,, 3, 5 ad For each populato, sample szes of =,, 3, 4, 5, 6, 7, 8, 9,, 5 ad, were radomly geerated 5, tmes From each set of samples, we the used the three methods to costruct the 95% cofdece terval for the parameter I ths secto, the case of sample sze Z where Z 3846 s coducted to guaratee that the results of the smulato study coform to the results theorems 3 33 whch are metoed secto III The results from the smulato are preseted Fgs ad Fg shows that the Eact, Asymptotc ad cofdece tervals acheve coverage closest to the omal level (95) o average for all levels of ad sample sze Ths smulato result coforms to the results theorem 3 I addto, Fg shows that the Asymptotc cofdece terval has the wdest epected legth of cofdece terval for a small sample sze Ths cocluso coforms to the results theorem 3 Furthermore, the epected legths of the three cofdece tervals do ot seem to be dfferet for a large sample sze ad all levels of I addto, they ted to decrease whe the sample sze creases for all levels of Ths smulato result coforms to the results theorem 33 (Advace ole publcato: 4 November 5)

9 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 Coverage probablty ϴ = 5 Eact Asymptotc Coverage probablty ϴ = Sample Sze () Coverage probablty ϴ = Coverage probablty Sample Sze () ϴ = 5 Coverage probablty Sample Sze () ϴ = Sample Sze () Sample Sze () 5 Coverage probablty ϴ = 5 Coverage probablty ϴ = 5 Coverage probablty Sample Sze () 4 ϴ = Sample Sze () Coverage probablty Sample Sze () ϴ = Sample Sze () 5 5 Fg Coverage probabltes of the three cofdece tervals for = 5,,, 5, 7,, 3, 5 ad (Advace ole publcato: 4 November 5)

10 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 5 ϴ = 5 Epected legth 5 Eact Asymptotc Epected legth ϴ = Sample Sze () 5 Epected legth ϴ = Epected legth Sample Sze () ϴ = 5 Epected legth Sample Sze () ϴ = Sample Sze () Sample Sze () 5 5 ϴ = 5 5 ϴ = 5 Epected legth 5 Epected legth 5 Epected legth Sample Sze () ϴ = 7 Epected legth Sample Sze () ϴ = Sample Sze () Sample Sze () 5 Fg Epected legths of the three cofdece tervals for = 5,,, 5, 7,, 3, 5 ad V DISCUSSION The cofdece terval whch s derved ths paper uses the same formula as for the Eact cofdece terval whch s wdely used Ths smulato study ad the proof theorem 3 foud that the coverage probabltes of the ad Eact cofdece tervals were close to the omal level for all levels of sample sze as metoed by Geyer [7], Balakrsha et al [8], Cho et al [9] ad Jag ad Wog [5] I addto, the study usg the smulato dataset ad the proof theorem 33 foud that the Asymptotc cofdece terval was very effcet (short epected legth of cofdece terval) for a large sample sze, as metoed by Mukhopadhyay [4], Cho et al [9], Mood et al [] ad Shawesh [] (Advace ole publcato: 4 November 5)

11 IAENG Iteratoal Joural of Appled Mathematcs, 45:4, IJAM_45_4_3 VI CONCLUSION The cofdece terval s derved based o vertg a test statstc approach whch s most helpful stuatos where we have o good dea about a parameter After the proof, we foud that the cofdece terval gave the same formula as for the Eact cofdece terval whch s most commoly used A smulato study was performed to guaratee the theoretcal results 3 33 that are preseted ths artcle ad to compare the effceces of the three methods Eact, Asymptotc ad cofdece tervals terms of the coverage probabltes ad epected legths of cofdece terval The comprehesve comparso results showed that for all levels of ad, the epected legth of cofdece terval s shorter tha that of the Asymptotc cofdece terval for a small sample sze whch satsfy the codtos that Z ad Z For a large,, Z sample sze such that Z, there seemed to be o dfferece the effcecy of the three methods for all levels of ad [4] J Neyma, Outle of a theory of statstcal estmato based o the classcal theory of probablty, Phlosophcal Trasactos of the Royal Socety Seres A, vol 36, pp , 937 [5] RV Hogg, ad EA Tas, Probablty ad Statstcal Iferece, 6th ed, New Jersey: Pretce Hall, [6] G Casella, ad RL Berger, Statstcal Iferece, d ed, Pacfc Grove: CA Dubury, [7] CJ Geyer, Stat 5 otes: more o cofdece tervals, [4//5]; avalable from: wwwstatumedu/geyer/old / 5/ 3otes/cpdf, 3 [8] N Balakrsha, E Cramer ad G Ilopoulos, O the method of pvotg the CDF for eact cofdece tervals wth llustrato for epoetal mea uder lfe-test wth tme costrats, Statstcs ad Probablty Letters, vol 89, pp 4 3, 4 [9] Y Cho, H Su ad K Lee, Eact lkelhood ferece for a epoetal parameter uder geeralzed progressve hybrd cesorg scheme, Statstcal Methodology, vol 3, pp 8 34, 5 [] AM Mood, FA Graybll, ad DC Boes, Itroducto to the Theory of Statstcs, 3rd ed, Sgapore: McGraw Hll, 974 [] MOAA Shawesh, Adjusted cofdece terval for the populato meda of the epoetal dstrbuto, Joural of Moder Appled Statstcal Methods, vol 9, o, pp , [] L Barker, A comparso of e cofdece tervals for a Posso parameter whe the epected umber of evets s 5, The Amerca Statstca, vol 56, o, pp 85 89, [3] M B Swft, Comparso of cofdece tervals for a Posso mea Further cosderatos, Commucatos Statstcs Theory ad Methods, vol 38, pp , 9 [4] V V Patl, ad HV Kulkar, Comparso of cofdece tervals for the Posso mea, REVSTAT Statstcal Joural, vol, o, pp 7, [5] L Jag ad ACM Wog, Iterval estmatos of the twoparameter epoetal dstrbuto, Joural of Probablty ad Statstcs, vol, do: 55//734575, ACKNOWLEDGEMENTS Ths research was facally supported by the Kasetsart Uversty Research ad Developmet Isttute (KURDI) The author would lke to thak the head of the Departmet of Statstcs, Faculty of Scece, Kasetsart Uversty for supplyg the facltes ecessary for the research REFERENCES [] J Baks, Prcples of Qualty Cotrol, New York: Joh Wley, 989 [] NL Johso, S Kotz, ad N Balakrsha, Cotuous Uvarate Dstrbutos, d ed, New Jersey: Joh Wley, 994 [3] N Balakrsha, ad AP Basu, The Epoetal Dstrbuto: Theory Methods ad Applcatos, Amsterdam: Gordo ad Breach Publshers, 995 [4] N Mukhopadhyay, Probablty ad Statstcal Iferece, New York: Marcel Dekker, Ic, [5] DH Besterfeld, Qualty Improvemet, 9th ed, New Jersey: Pretce Hall, [6] DC Motgomery, Itroducto to Statstcal Qualty Cotrol, 7th ed, New York: Joh Wley, [7] W Suryakat, Y Areepog, S Sukparugsee ad G Mttelu, Aalytcal method of average ru legth for tred epoetal AR() processes EWMA procedure, IAENG Iteratoal Joural of Appled Mathematcs, vol 4, o 4, pp 5 53, [8] C Forbes, M Evas, N Hastgs, ad B Peacock, Statstcal Dstrbutos, 4th ed, New Jersey: Joh Wley, [9] M Havv, Queues: A Course Queueg Theory, New York: Sprger, 3 [] UN Bhat, A Itroducto to Queueg Theory, New York: Brkhäuser Bosto, 8 [] OC Ibe, Markov Processes for Stochastc Modelg, USA: Academc Press, 9 [] S Sa ad OA Dama, The M/G/ queue wth heterogeeous servers uder a cotrolled servce dscple: statoary performace aalyss, IAENG Iteratoal Joural of Appled Mathematcs, vol 45, o, pp 3 4, 5 [3] GS JR Koch, ad RF Lk, Statstcal Aalyss of Geologcal Data, New York: Dover Publcato, Ic, (Advace ole publcato: 4 November 5)