뗚9헂 업뛓싛 업뛓싛쫇컒쏇쎿룶죋뚼뫜쫬쾤뗄쿖쿳ꆣ틲캪죋믲컯믲쫇탅쾢캪쇋뗃떽쒳훖럾컱뇘탫업뛓ꆣ폐튻샠업뛓쫇폐탎뗄ꎬ샽죧퓚쫛욱뒦뗈듽싲욱뗄업뛓ꎬ볓폍햾잰웻뎵뗈듽볓폍뗄업뛓뗈ꎻ뮹폐튻샠업뛓쫇컞탎뗄ꎬ샽죧뗧뮰붻뮻믺뷓떽뗄뗧뮰뫴뷐탅뫅뗄업뛓ꎬ뗈듽볆쯣믺훐탄뒦샭믺뒦샭뗄탅쾢뗄업뛓뗈ꆣ캪쇋탰쫶뗄랽뇣ꎬ업뛓헟컞싛쫇죋ꆢ컯ꆢ믲탅쾢ꎬ틔뫳춳돆캪ꆰ맋뿍ꆱꆣ럾컱헟컞싛쫇죋ꎬ믲쫂컯ꎬ샽죧튻첨뗧ퟓ볆쯣믺튲뿉틔쫇업뛓쾵춳훐뗄럾컱헟ꎬ컒쏇틔뫳춳돆캪ꆰ럾컱풱ꆱꆣ 업뛓쿖쿳쫇컒쏇늻쾣췻돶쿖뗄쿖쿳ꎬ틲캪죋뗄업뛓틢캶ퟅ훁짙쫇샋럑쪱볤ꎻ컯뗄업뛓퓲쮵쏷쇋컯뗄믽톹ꆣ떫쫇업뛓쿖쿳좴컞램췪좫쿻쪧ꎬ헢쫇튻훖쯦벴쿖쿳ꆣ평폚맋뿍떽듯볤룴쪱볤뗄쯦믺탔뫍캪맋뿍럾컱쪱볤뗄쯦믺탔쫇업뛓쿖쿳닺짺뗄풭틲ꆣ죧맻짏쫶뗄솽룶쪱볤쫇만뚨뗄ꎬ컒쏇뻍뿉틔춨맽췗짆낲업살췪좫쿻돽업뛓쿖쿳ꆣ 업뛓싛쫇퇐뺿업뛓쾵춳퓚늻춬뗄쳵볾쿂ꎨퟮ훷튪뗄쫇맋뿍떽듯뗄쯦믺맦싉뫍럾컱쪱볤뗄쯦믺맦싉ꎩ닺짺뗄업뛓쿖쿳뗄쯦믺맦싉탔ꆣ튲뻍쫇튪붨솢랴펳헢훖쯦믺탔뗄쫽톧쒣탍ꆣ퇐뺿뗄ퟮ훕쒿뗄쫇캪쇋퓋폃헢킩맦싉ꎬ뛔쪵볊뗄업뛓쾵춳뗄짨볆폫퓋탐ퟶ돶ퟮ폅뗄뻶닟ꆣ 업뛓싛훐뗄쫽톧쒣탍쫇룹뻝룅싊뫍쯦믺맽돌뗄샭싛붨솢웰살뗄ꎬ컒쏇쿈살쳖싛늴쯉맽돌뫍짺쏰맽돌ꎬ좻뫳,퓙듋믹뒡짏퇐뺿업뛓쾵춳뗄뷡릹벰웤훷튪뗄쫽톧쒣탍ꎬퟮ뫳퇐뺿업뛓쾵춳뗄폅뮯컊쳢ꆣ 늴쯉맽돌뫍짺쏰맽돌 늴쯉맽돌 죧맻폃N(t)뇭쪾퓚[0,t]쪱볤쓚맋뿍떽듯뗄ퟜ쫽ꎬ퓲뛔폚쎿룶룸뚨뗄쪱뿌tꎬN(t)뚼쫇튻룶쯦믺뇤솿ꆣ쯦벴뇤솿ퟥ{N(t)t∈[0,T]}돆ퟷ쫇튻룶쯦믺맽돌ꆣ 죴뛔 <<"<ꎬ폐 tttt12nn+1P(N()=N()=,N()=,",N()= t+i+tititin1n11122nn=P(N()=N()=) (9-1) tn+i1n+ti1nn퓲돆쯦벴맽돌{N(t)t∈[0,T)}캪십뛻뿂럲맽돌ꆣ릫쪽ꎨ9-1ꎩ쯹뇪쪾뗄탔훊돆캪ꆰ컞뫳킧탔ꆱꆣ쯼뗄쪵볊틢틥쫇쮵ꎺ죧맻폃뇭쪾쿖퓚쪱뿌ꎬ뇭쪾캴살쪱ttnn+1뿌ꎬ,,",뇭쪾맽좥뗄튻쾵쇐쪱뿌ꎬ퓲맋뿍떽살뗄맽돌퓚틔잰쯹뒦뗄ힴ첬tttt12n−1n
ꎨ벴맋뿍떽듯쫽ꎩ뛔풤퇔맽돌퓚틔뫳뗄ힴ첬늻웰횱뷓ퟷ폃ꆣ tn죴쯦벴맽돌{N(t)t∈[0,T]}뻍폐ꆰ뛀솢퓶솿탔ꆱꎬ벴뛔죎튻ퟩ<,",(n≥3)ꎬ쯦벴뇤솿N()−N(),N()−N(),",N()−N()쿠뮥뛀ttttttttt12n2132nn−1솢ꎬ쟒뛔죎틢t∈[0,T],폐 k(λt)−λtP(N(t)=k)= k=0,1,2,"" (9-2) ek!웤훐닎쫽λ>0ꎬ퓲돆헢룶맽돌캪늴쯉맽돌ꆣ 뛀솢퓶솿탔쮵쏷퓚뮥늻쿠붻뗄쪱볤쟸볤[,),[,),"",[,)쓚맋뿍떽ttttt1223n−t1nk∞(λt)−λt듯쟩뿶쫇쿠뮥뛀솢뗄ꆣ평폚=1 ∑ek=0k!쯹틔N(t)뗄웚췻횵캪 kk−1∞∞(λt)(λt)−λtλtE(N(t)kλt⋅ ∑∑eek=0!k=1−!i∞(λt)−λtλt= (9-3) ∑ei=0iE(N(t)λ= (9-4) t틲듋ꎬ닎쫽λ뻍쫇떥캻쪱볤볤룴쓚떽듯맋뿍뗄욽뻹쫽ꎬ춬쪱ꎬ컒쏇뮹뿉틔쟳뗃쯦믺뇤솿N(t)뗄랽닮캪 D(N(t)=λt (9-5) 샽 쒳쳬짏컧ꎬ듓10뗣30럖떽11뗣47럖ꎬ쎿룴20쏫훓춳볆튻듎살떽쒳웻뎵햾뗄돋뿍쫽ꎬ릲뗃230룶볇슼쫽뻝ꎬ헻샭뫳뗃떽죧쿂뗄춳볆뷡맻ꎺ 뇭9-1 돋뿍쫽쒿 0 1 2 3 4욵 쫽 100 81 34 9 6 쫔폃튻룶늴쯉맽돌살쏨쫶듋뎵햾돋뿍뗄떽듯맽돌ꎬ늢뻟쳥킴돶쯼뗄룅싊럖늼ꆣ 뷢ꎺ튪킴돶웤룅싊럖늼ꎬ횻탨좷뚨릫쪽ꎨ9-2ꎩ훐뗄닎쫽λ벴뿉ꆣ룹뻝λ뗄틢틥ꎬ횻튪쿈쟳돶쎿20쏫훓뗄욽뻹쫽 1λ=(0×100+1×81+2×34+3×9+4×6)= 230틲듋뿉횪쎿럖훓욽뻹떽듯뗄맋뿍쫽캪
λ=3×=ꎨ죋/럖훓ꎩ 맊쯹쟳뗄돋뿍떽듯맽돌쯹싺ퟣ뗄룅싊럖늼캪 k()−(N(t)=k)= ek!튻냣뗘ꎬ폐죧쿂뷡싛ꎺ 뚨샭1 죴쯦믺맽돌{N(t)t∈[0,T)}싺ퟣ쿂쇐죽룶쳵볾ꎺ (1)뛀솢퓶솿탔ꎺ뛔죎튻ퟩ<<""<(n≥3)ꎬ쯦믺뇤솿ttt12nN()−N(),N()−N(),"",N()−N()쿠뮥뛀솢ꎻ tttttt2132nn−1(2)욽컈탔ꎺ뛔폚[s,s+t]⊂[0,T]ꎬퟜ폐 P[N(s+t)−N(s)=k]=P[N(t)−N(0)=k] ∞웤훐P(N(0)=0)=1,P(N(t)=k)=1ꎻ ∑k=0∞(3)웕뇩탔ꎺ쇮ϕ(t)P(N(t)=k)ꎬ폐 ∑k=2ϕ(t)lim=0 t=0t퓲{N(t)t∈[0,T]}쫇튻룶늴쯉맽돌ꆣ 뛀솢퓶솿탔쮵쏷퓚[0,T]훐뗄죎틢쟸볤[s,s+t]쓚살떽k룶맋뿍헢튻쫂볾폫쟸볤[0,s]쓚살떽맋뿍뗄쟩뿶쿠뮥뛀솢ꎬ벴퓚[0,s]쓚맋뿍살떽뗄쟩뿶쯹ퟷ뗄죎뫎볙뚨쿂ꎬ볆쯣돶살뗄퓚[s,s+t]쓚살떽k룶맋뿍뗄쳵볾룅싊뻹쿠뗈ꆣ춬쪱뿉횪ꎬ뻟폐뛀솢퓶솿탔뗄맽돌뇘좻뻟폐컞뫳킧탔ꆣ 욽컈탔쮵쏷퓚[s,s+t]쓚살떽뗄쫽횵폫쟸볤뎤뛈t폐맘ꎬ뛸폫쪱볤웰뗣s컞맘ꆣ튲뻍쫇쮵ꎬ맽돌뗄춳볆맦싉늻쯦쪱볤뗄췆틆뛸룄뇤ꎬ퓚춬퇹뎤뛈뗄쪱볤볤룴쓚살떽k룶맋뿍뗄룅싊쫇튻룶뎣쫽ꆣ 웕뇩탔뇭쏷ꎬ퓚춬튻쮲볤살떽솽룶믲솽룶틔짏맋뿍쪵볊짏쫇늻뿉쓜뗄ꆣ벴퓚돤럖킡뗄쪱볤볤룴훐ꎬퟮ뛠살떽튻룶맋뿍ꆣ 퓚업뛓싛샯ꎬ뎣냑늴쯉맽돌돆캪늴쯉쇷믲ퟮ볲떥쇷ꎬ닎쫽λ돆캪ퟮ볲떥쇷뗄잿뛈ꆣ쮳뇣쮵튻쿂ꎬ늴쯉맽돌뮹뻟폐뿉횪탔ꆣ벴죧맻{(t)}뫍{(t)}쫇솽룶늴NN12쯉맽돌ꎬ떽듯잿뛈럖뇰캪뫍ꎬ쟒솽룶맽돌쿠뮥뛀솢ꎬ퓲{(t)}+{(t)}죔λλ12N1N2캪튻늴쯉맽돌ꎬ웤떽듯잿뛈캪+ꎬ췆맣듋뷡싛ꎬ퓲룼뛠룶뛀솢뗄늴쯉맽돌뫏λ1λ2늢뫳죔캪튻늴쯉맽돌ꎬ웤떽듯잿뛈캪룷맽돌떽듯잿뛈횮뫍ꆣ 늴쯉맽돌퓚업뛓싛훐웰ퟅ훘튪ퟷ폃ꆣ틲캪늴쯉쇷믲뷼쯆폚늴쯉쇷뗄쪵볊쟩뿶뺭뎣믡폶떽ꎬ늢쟒늴쯉쇷뗄쫽톧뒦샭뫜볲떥ꆣ 뢺횸쫽럖늼뫍낮뛻샊럖늼 죴폃뇭쪾뗚n캻맋뿍쯹탨뗄럾컱쪱볤ꎬ퓲{}쫇튻ퟥ솬탸탔쯦믺뇤솿ꆣ죧τnτn
맻{}훐룷룶쯦믺뇤솿쿠뮥뛀솢ꎬ쟒럾듓쿠춬뗄뢺횸쫽럖늼ꎺ τn−µt1−et≥0P(τ≤t)= (9-6) n0t<0ꎨ웤훐닎쫽µ>0ꎩ웤룅싊쏜뛈몯쫽캪 −µtµ⋅et≥0p(t)= (9-7) 0t<0퓲럾컱쪱볤τ뗄웚췻횵캪ꎺ n+∞+∞−µtE(τ)=tp(t)dt=µtedtn∫∫ −∞0+∞1−µt+∞−µt=−teedt= (9-8) 0∫0µ1퓲폐 µ= (9-9) E(τ)n1폚쫇ꎬ뻍쫇쎿캻맋뿍쯹탨튪뗄욽뻹럾컱쪱볤ꎬ뛸µ뇭쪾떥캻쪱볤쓚쓜놻럾µ컱췪뗄맋뿍욽뻹쫽ꆣ퓚업뛓싛훐춨뎣폃ꆰ욽뻹ꆱ살뇭쪾룅싊싛훐뗄쫽톧웚췻ꆣ 춬쪱뿉쟳뗃τ뗄랽닮캪 n1D(τ)= (9-10) n2µ짨{N(t)}쫇쏨쫶맋뿍떽듯쟩뿶뗄쯦믺맽돌ꎬ틔t뇭쪾뗚n룶맋뿍떽듯뗄쪱n뿌ꎬ퓲 T=t−t캪뗚n캻맋뿍폫쯻뗄잰튻캻맋뿍ꎨ뗚n−1캻맋뿍ꎩ떽듯쪱볤nnn−1뗄쪱볤볤룴ꆣ쿔좻{T}튲쫇튻ퟥ쯦믺뇤솿ꆣ맘폚떽듯볤룴쪱볤{T}폫맋뿍떽듯맽nn돌{N(t)}횮볤뗄맘쾵ꎬ뿉횤뗃죧쿂뗄뷡싛ꎺ 뚨샭2 맋뿍떽듯맽돌{N(t)}쫇튻룶닎쫽캪λ뗄늴쯉맽돌뗄돤럖뇘튪쳵볾캪ꎺ쿠펦뗄맋뿍떽듯볤룴{T}쫇튻ퟥ쿠뮥뛀솢뗄쯦믺뇤솿ꎬ쟒쎿룶쯦믺뇤솿뚼럾듓쿂n쏦뗄뢺횸쫽럖늼ꎺ −µt1−et≥0P(T≤t)= (9-11) n0t<0듋뚨샭쮵쏷ꎬꆰ맋뿍쇷쫇ퟮ볲떥쇷ꆱ폫ꆰ맋뿍떽듯볤룴쿠뮥뛀솢쟒럾듓쿠춬뗄뢺횸쫽럖늼ꆱ쫇뗈볛뗄솽훖쏨쫶랽램ꆣ평폚짏쏦뗄뢺횸쫽럖늼뗄쫽톧웚췻11E(T)=ꎬ쯹틔퓚ퟮ볲떥쇷훐ꎬ맋뿍떽듯쪱볤볤룴뗄욽뻹횵캪ꆣ nλλ
샽 퓚쒳ퟹ듳쟅쟅뿚ꎬ맛달떽26솾떽듯쟅뿚튪맽쟅뗄웻뎵ꎬ웤떽듯쪱뿌볇슼죧쿂ꎨ뾪쪼맛달쪱뿌캪0ꎬ떥캻캪쏫ꎩꎺ 0 15 17 23 24 25 31 39 55 58 62 63 65 68 80 82 85 89 97 99 103111121122 123 133쫔폃튻룶늴쯉맽돌쏨쫶헢룶떽듯맽돌ꎬ늢킴돶뻟쳥뗄룅싊쒣탍ꆣ 1뷢ꎺ틲캪늴쯉쇷훐ꎬ맋뿍떽듯뗄쪱볤볤룴뗄욽뻹횵캪ꎬ쯹틔뿉ퟅ쫖쟳뗃λꎬλ퓙뚨돶룅싊쒣탍ꆣ웻뎵떽듯뗄쪱볤볤룴틀듎캪ꎺ 15 2 6 1 168163412 3 12 2 3 4 8248101110 틲듋ꎬ떽듯볤룴쪱볤뗄욽뻹횵캪 15+2+⋅⋅⋅+10133==ꎨ쏫ꎩ 25251뻍쫇욽뻹쎿룴쏫훓떽듯튻솾웻뎵ꆣ틲캪맋뿍떽듯볤룴욽뻹횵캪ꎬ뛸λλ11뻍쫇늴쯉쇷뗄룅싊쒣탍훐닎쫽ꎬ평=뿉뗃λ==ꎬ벴쎿쏫훓욽λ뻹떽듯뗄웻뎵쫽풼캪ꆣ 폚쫇뿉폃죧쿂뗄늴쯉럖늼살쏨쫶떽듯맽돌{N(t)}ꎺ k()−(N(t)=k)=e k=0,1,2,⋅⋅⋅ k!튲뿉폃죧쿂뗄뢺횸쫽럖늼살쏨쫶웤떽듯볤룴{T}ꎺ n−1−et≥0P(T≤t)= n0t<0맘폚뢺횸쫽럖늼ꎬ탨튪잿뗷튻룶쯼쯹뻟폐뗄ꆰ컞볇틤탔ꆱꆣ쿈뾴튻룶컊쳢ꎬ죧맻맋뿍떽듯볤룴쪱볤럾듓뢺횸쫽럖늼ꎬ욽뻹볤룴쪱볤10쏫ꎬ폖볙짨퓚쒳튻쪱뿌ꎨ죎튻쪱뿌ꎩ살뾼달헢룶떽듯맽돌ꎬ랢쿖ퟮ뫳튻캻맋뿍틑떽듯쇋7쏫ꎬ쓇쎴쿂튻캻맋뿍욽뻹뮹탨뛠뎤쪱볤닅믡떽듯쓘ꎿ믘듰쫇돶죋틢쇏횮췢뗄ꎺ뮹탨튪10쏫ꆣ뾴살틑뺭맽좥쇋뗄7쏫놻틅췼쇋ꎬ맊돆ꆰ컞볇틤탔ꆱꆣ쿂쏦살횤쏷헢튻쳘탔ꎺ 짨떽듯볤룴T럾듓뢺횸쫽럖늼죧릫쪽ꎨ9-11ꎩ쯹쪾ꎬ뛸s캪죎튻쪱뿌ꎬ퓲떽듯볤룴T듳폚뗈폚s뗄룅싊캪ꎺ +∞−λt−λsP(T≥s)=1−P(T≤s)=λedt=e ∫s폖틲캪T≥s폫T≥s+t(t>0)춬쪱랢짺뗄룅싊뗈폚T≤s+t뗄룅싊ꎬ벴 +∞−λt−λ(s+t)P(T≥s,T≥s+t)=P(T≤s+t)=λedt=e ∫s+t떱룸뚨쳵볾T≥s쪱ꎬ쳖싛T≥s+t뗄쳵볾룅싊ꎬ랢쿖
−λ(s+t)P(T≥s,T≥s+t)e−λtP(T≥s+tT≥s)===eP(T≥t) −λsP(T≥s)e평뷡맻횪ꎬ듋쳵볾룅싊폫s컞맘ꆣ헢쮵쏷컞싛퓚늴쯉맽돌뗄쪲쎴쪱뿌ꎬ벴컞싛좡쓄튻쪱뿌캪웰뗣ꎬ뾼달훁쿂튻캻맋뿍떽듯쯹뺭맽뗄쪱볤ꎬ웤룅싊ꎨ벴P(T≥s+tT≥s)ꎩ폫듋쪱뿌횮잰뗄ퟮ뫳튻캻맋뿍쫇쪲쎴쪱뫲떽듯뗄컞맘ꆣ쯼뫍맋뿍쿠볌떽듯뗄볤룴쪱볤뚼럾듓쿠춬닎쫽뗄뢺횸쫽럖늼ꆣ 뮹뿉틔횸돶뗄쫇ꎬ쓜싺ퟣ컞볇틤탔P(T≥s+tT≥s)=P(T≥t)뗄럖늼ꎬ튲횻쓜쫇뢺횸쫽럖늼ꆣ 쿂쏦쳖싛낮뛻샊ꎨErlangꎩ럖늼ꆣ 낮뛻샊럖늼뗄쏜뛈몯쫽쫇 k−1µ⋅(µt)−µtet≥0pt=(k−1)!() (9-12) t<00웤훐닎쫽µ>0ꎬk돆캪뷗쫽(k=0,1,2,⋅⋅⋅)ꆣ 떱k=1쪱ꎬ퓲낮뛻샊럖늼튲뻍쫇뢺횸쫽럖늼ꆣ죴쯦믺뇤솿τ럾듓k뷗낮뛻샊럖늼ꎬ퓲τ뗄웚췻횵뫍랽닮럖뇰캪 kkE(τ)=,D(τ)= (9-13) 2µµ뿉틔횤쏷ꎬ죧맻ξ,ξ,⋅⋅⋅,ξ쫇k룶쿠뮥뛀솢뻟폐쿠춬뗄뢺횸쫽럖늼ꎨ닎쫽캪12kµꎩ뗄쯦믺뇤솿ꎬ퓲쯦믺뇤솿τ=ξ+ξ+⋅⋅⋅+ξ럾듓k뷗낮뛻샊럖늼ꆣ 12k퓚업뛓싛훐ꎬ뎣냑맋뿍떽듯볤룴쪱볤벰뷓쫜럾컱쪱볤폫낮뛻샊럖늼솪쾵웰살ꆣ죧맻맋뿍튪솬탸뷓쫜뒮솪뗄k룶럾컱첨럾컱ꎬ룷럾컱첨뗄럾컱쪱볤쿠뮥뛀솢ꎬ쟒럾듓쿠춬뗄뢺횸쫽럖늼ꎨ닎쫽캪µꎩꎬ쓇쎴맋뿍놻헢k룶럾컱첨럾컱췪ퟜ릲쯹탨뗄쪱볤뻍럾듓낮뛻샊럖늼ꆣ헢샯펦쮵쏷뗄쫇퓚맋뿍뷓쫜솬탸뗄럾컱쪱ꎬ횻폐떱k룶럾컱첨뚼췪돉쇋뛔쒳룶맋뿍뗄럾컱횮뫳ꎬ쿂튻룶맋뿍닅쓜뷸죫헢룶뒮솪럾컱쾵춳ꆣ 짺쏰맽돌 짺쏰맽돌튲쫇튻훖십뛻뿂럲맽돌ꆣ퓚업뛓싛훐뫜뛠업뛓맽돌뫍헢룶맽돌쿠럂ꆣ컒쏇쳘뇰맘탄짺쏰맽돌퓚춳볆욽뫢쪱랴펳돶살뗄컈첬룅싊ꎬ늢횱뷓냑헢훖컈첬룅싊펦폃폚붨솢룷훖업뛓쒣탍ꆣ 1. 짺쏰맽돌뗄뚨틥 튻뛑쾸뻺ꎬ쯦쪱볤췆틆ꎬ폐뗄럖쇑캪솽룶ꎬ폐뗄쯀췶ꎬ뺭맽튻뛎쪱볤횮뫳ꎬ쾸뻺뇤캪뛠짙ꎿ헢훖쾸뻺뗄럖쇑폫쯀췶뗄맽돌뻍쫇뗤탍뗄짺쏰맽돌뗄샽ퟓꆣ짨쎿룶쾸뻺퓚+t쪱볤쓚럖쇑돉솽룶쾸뻺뗄룅싊캪λ+t+0(+t)ꎻ뛸퓚+t쪱볤쓚쯀췶
뗄룅싊캪µ+t+0(+t)ꎬ룷룶쾸뻺퓚죎튻쪱볤쓚럖쇑믲쯀췶뚼쫇쿠뮥뛀솢뗄ꆣ죴붫쾸뻺뗄럖쇑믲쯀췶뚼뾴돉쫇쯦믺쫂볾ꎬ쓇쎴퓚+t쪱볤쓚랢짺솽룶쪱볤뗄룅싊22뗈폚쿂쏦죽헟횮튻ꎺ(λ+t+0(+t)),(µ+t+0(+t)),(λ+t+0(λ))⋅(µ+t+0(+t))ꆣ쯹틔퓚+t쪱볤쓚랢짺솽룶믲솽룶틔짏쫂볾뗄룅싊캪0(+t)ꆣ폖짨퓚쪱뿌t폐i룶쾸뻺ꎬ퓲퓚쪱뿌t++t폐i+1룶쾸뻺뗄룅싊캪λ+t+0(+t)ꎬ퓚쪱뿌t++t쓚폐ii−1룶쾸뻺뗄룅싊캪µ+t+0(+t)ꆣ폃ξ(t)뇭쪾헢뛑쾸뻺퓚쪱뿌t뗄룶쫽ꎬ늢뾼i싇퓚[0,T)쪱볤쓚쾸뻺쫽뗄뇤뮯쟩뿶ꆣ틲캪뛔쎿룶만뚨뗄tꎬξ(t)뚼쫇쯦믺뇤솿ꎬ맊{ξ(t)t∈[0,T)}캪튻룶쯦믺맽돌ꎬ폖틲캪쯼뻟폐컞뫳킧탔ꎬ맊튲쫇튻룶십뛻뿂럲맽돌ꎬ냑짏쫶쾸뻺럖쇑뫍쯀췶뗄맽돌돆캪짺쏰맽돌ꆣ 뚨틥ꎺ짨{ξ(t)t∈[0,T)}캪튻룶쯦믺맽돌ꎬ쯦믺뇤솿ξ(t)뗄좡횵벯뫏캪I={0,1,2,⋅⋅⋅,m}ꎨ믲뿉쇐벯I={0,1,2,⋅⋅⋅⋅⋅⋅}ꎩꎬ돆듋벯뫏캪ힴ첬벯ꎬ짨퓚쪱뿌t쪱ξ(t)=jꎬ쓇쎴퓚쪱뿌t++t쪱ꎬξ(t++t)=j+1뗄룅싊캪λ+t+0(+t)ꎬ웤훐jλ>0캪폫t컞맘뗄뎣쫽ꎻ퓚쪱뿌t++t쪱ꎬξ(t++t)캪I훐웤쯼풪쯘뗄룅싊캪j0(+t)ꆣ싺ퟣ짏쫶쯦믺맽돌{ξ(t)t∈[0,T)}돆캪짺쏰맽돌ꆣ 듓짏쏦뗄뚨틥살뾴ꎬ죧맻냑ힴ첬뗄뇤뮯샭뷢캪업뛓쾵춳맋뿍뗄떽듯뫍샫좥ꎬ퓲퓚쪱볤퓶솿+t쓚ꎬ횻믡폐튻룶떽듯ꎬ믲폐튻룶샫좥ꎬ믲볈컞떽듯튲컞샫좥ꎨ듋훖쟩뿶뗄룅싊캪1−λ+t−µ+t−0(+t)ꎩꎬ돽듋틔췢뗄웤쯻떽듯ꆢ샫좥쟩뿶죏캪jj쫇늻뿉쓜랢짺뗄ꆣ늢쟒ꎬ퓚+t쓚떽듯튻룶맋뿍뗄룅싊뫍+t뗄뎤뛌폐맘ꎬ뛸폫웰쪼쪱뿌t컞맘ꎻ떽듯튻룶뗄룅싊뮹폫t쪱뿌뗄맋뿍떽듯쫽j폐맘ꎨ틲λ폫j폐j맘ꎩꎬ뛸뫍t쪱뿌틔잰뗄ힴ첬컞맘ꆣ춬샭ꎬ뿉뷸탐+t쪱볤쓚샫뾪튻룶맋뿍뗄룅싊쟩뿶럖컶ꆣ 2. 짺쏰맽돌뗄컈첬룅싊 뷸튻늽뾼달짺쏰맽돌훐쪱볤T=+∞뗄쟩뿶ꆣ뛔짺쏰맽돌컒쏇룼맘탄뗄쫇벫쿞limP(ξ(t)=j)ꆣ뿉틔횤쏷쿂쏦뗄뚨샭ꆣ T=+∞뚨샭3 쇮 λλ⋅⋅⋅λ01jπ=1,π= (j=1,2,⋅⋅⋅) 0jµµ⋅⋅⋅µ12j늢짨짺쏰맽돌{ξ(t)t≥0}뗄ힴ첬벯캪I={0,1,2,⋅⋅⋅,m}믲I={0,1,2,⋅⋅⋅}ꎬ쓇쎴ꎬ떱쿂쇐쳵볾 ∞∞1 π<+∞,=+∞ ∑∑jλπjj=0j=0싺ퟣ쪱ꎬ뛔폚죎틢헽쫽s뫍죎틢j∈I,i∈Iꎬ뚼폐limP(ξ(t)=j)= t→+∞limP(ξ(s+t)=jξ(s)=i)=P>0ꆣ떱ힴ첬벯캪폐쿞벯I={0,1,2,⋅⋅⋅,m}쪱ꎬ jt→+∞
∞−1 p=(π) 0∑jj=0λj−1 p=πp=p j=1,2,⋅⋅⋅,m jj0j−1µj떱ힴ첬벯캪컞쿞벯I={0,1,⋅⋅⋅}쪱 ∞−1 p=(π) 0∑jj=1λj−1 p=πp=p j=1,2,⋅⋅⋅ jj0j−1µj컒쏇돆P(j=0,1,2,⋅⋅⋅)캪짺쏰맽돌퓚춳볆욽뫢쪱뗄룅싊ꎬ믲돆캪컈첬룅싊ꆣ j뛔폚짺쏰맽돌ꎬ틲캪쪱뿌t쪱ힴ첬캪j뗄룅싊럖늼P(ξ(t)=j) j∈Iꎬ붫쯦쪱볤t뗄뇤뮯뛸뇤뮯ꎬ돆횮캪쮲첬뷢ꆣ쮲첬뷢쫇뫜쓑쟳돶뗄ꎬ벴쪹쟳돶살튲쓑틔샻폃ꆣ틲듋ꎬ퓚쪵볊펦폃훐ꎬ컒쏇룼맘탄뗄쫇컈첬룅싊P,j∈Iꆣ컈첬룅싊뗄몬틥j쫇쮵ꎬ떱퓋탐쪱볤t컞쿞뎤쪱ꎬힴ첬뗄룅싊럖붫늻쯦쪱볤뛸뇤뮯ꎬ듋쪱ힴ첬캪j 뗄룅싊P쫇튻룶뎣쫽ꆣ탨튪횸돶ꎬ쯤좻퓚샭싛짏평뚨샭3횪ꎬ짺쏰맽돌탨뺭컞쿞j뎤뗄쪱볤닅믡뷸죫컈첬ꆣ떫퓚쪵볊펦폃훐ꎬ뛔듳뛠쫽컊쳢살쮵ꎬ쿠펦뗄짺쏰맽돌ퟜ믡뫜뿬듯떽컈첬ꎬ늻탨튪뗈떽t→+∞횮뫳ꆣ 튻냣업뛓쾵춳뷡릹 업뛓쒣탍뷡릹 업뛓쾵춳ꎨ믲돆캪럾컱쾵춳ꎩ뿉폃폚쏨쫶업뛓쟩뿶ꆣ벴맋뿍캪쇋믱뗃쒳훖럾컱뛸떽듯럾컱첨ꎻ죴럾컱풱퓚쎦ꎬ맋뿍늻쓜솢벴믱뗃럾컱뛸폖놻퓊탭업뛓뗈듽ꎬ퓲볓죫뗈듽뛓쇐ꎻ믱뗃럾컱횮뫳퓲솢벴샫뾪쾵춳ꆣ헻룶맽돌죧춼9-1쯹쪾ꆣ 쪵볊짏쎿룶뻟쳥뗄업뛓쾵춳뿉쓜폐쯹늻춬ꎬ떫튻냣뗄업뛓쾵춳뚼뻟폐죽룶튪쯘ꎬ헢뻍쫇쫤죫맽돌ꎻ업뛓맦퓲ꎻ럾컱믺릹ꆣ평헢죽룶튪쯘뇣뿉틔릹퓬돶튻룶튻냣업뛓쾵춳뗄뷡릹쒣탍ꆣ늢평떽듯샫좥1뗈듽쿟듋뗃떽쏨쫶쾵춳퓋탐쟩뿶뗄쫽톧쒣럾컱첨탍ꆣ쿂쏦뛔헢죽룶튪쯘퓙볓틔뻟쳥뗄쳖싛ꆣ 춼9-1 1. 쫤죫맽돌
뛔폚쫤죫맽돌ꎬ컒쏇탨튪쇋뷢뗄튻룶랽쏦쫇맋뿍풴뗄쟩뿶ꎨ맋뿍뗄ퟜ쳥뿉쓜쫇폐쿞벯ꎬ튲뿉쓜쫇컞쿞벯ꎩꎬ떫훷튪뗄랽쏦쫇쇋뷢맋뿍떽듯럾컱쾵춳뗄맦싉ꎬ헢훖맦싉훷튪랴펳퓚맋뿍쿠볌떽듯볤룴쪱볤뗄룅싊럖늼짏ꆣ벸훖훷튪쟩뿶죧쿂ꎺ (1)뚨뎤쫤죫ꆣ맋뿍폐맦싉뗄떽듯ꎬ쎿룴쪱볤α떽듯튻캻맋뿍ꆣ샽죧ퟔ뚯짺닺쿟짏뗄ힰ엤볾ꆣ맋뿍쿠볌떽듯뗄볤룴쪱볤{T}뗄럖늼몯쫽쫇ꎺ nt<α0P(T≤t)= nt≥α1(2)ퟮ볲떥쇷ꆣ벴{T}훐룷룶T쿠뮥뛀솢쟒뚼럾듓춬튻뢺횸쫽럖늼ꎺ nn−λt1−et≥0P(T≤t)= n0t<0믲헟쮵ꎬ퓚[0,t)쓚떽듯뗄맋뿍쫽N(t)쿠펦뗄쯦믺맽돌쫇늴쯉맽돌ꎬ벴N(t)뗄룅싊럖늼캪ꎺ k(λt)−λtP(N(t)=k)=e k=0,1,2,⋅⋅⋅ k!(3)k뷗낮뛻샊럖늼ꆣ벴{T}훐룷룶T쿠뮥뛀솢ꎬ쟒뚼뻟폐쿠춬뗄낮뛻샊럖늼nn쏜뛈ꎺ k−1λ(λt)−λtet≥0p(t)= (k−1)!t<00(4)튻냣뛀솢럖늼ꆣ벴훐{T}룷룶T쿠뮥뛀솢ꎬ쟒뚼뻟폐쿠춬뗄룅싊럖늼ꆣ nn1(a)1s(b)1(c)s 춼9-2 2. 럾컱믺릹 맘폚듋튪쯘탨튪쏷좷뗄쫇ꎺ럾컱첨뗄룶쫽ꆢ럾컱첨횮볤뗄뒮늢솪뷡릹ꎨ춼9-2ꎩꆢ럾컱첨캪쎿캻맋뿍럾컱쯹탨뗄쪱볤τ뗄럖늼쟩뿶ꆣ n
쿂쏦뷩짜벸훖뎣볻뗄럾컱쪱볤럖늼ꎺ (1)뚨뎤럖늼 쎿캻맋뿍뗄럾컱쪱볤τ뻹캪뎣쫽β,τ뗄럖늼몯쫽캪 nn0τ<βnP(τ≤t)= n1τ≥βn(2)뢺횸쫽럖늼 {τ}훐룷룶τ쿠뮥뛀솢ꎬ쟒뚼럾듓쿠춬뗄뢺횸쫽럖늼 nn0t<0 P(τ≤t)= n−µt1−t≥0e(3)k뷗낮뛻샊럖늼 {τ}훐룷룶τ쿠뮥뛀솢ꎬ쟒뚼뻟폐쿠춬뗄k뷗낮뛻샊럖nn늼ꎬ웤쏜뛈몯쫽캪 0t<0k−1 p(t)= µ(µt)−µtet≥0(k−1)!(4)튻냣뛀솢럖늼 {τ}훐룷룶τ쿠뮥뛀솢ꎬ쟒뚼뻟폐쿠춬뗄룅싊럖늼ꆣ nn3. 업뛓맦퓲 업뛓맦퓲뿉쏨쫶떽듯뗄맋뿍내헕퓵퇹뗄쮳탲뷓쫜럾컱ꆣ퓚늻춬뗄쪵볊컊쳢훐ꎬ업뛓맦퓲쫇뛠퇹뗄ꆣ튻냣뿉럖캪쯰쪧훆ꆢ뗈듽훆ꆢ믬뫏훆죽샠ꆣ 떱튻캻맋뿍떽듯쪱ꎬ죴쯹폐뗄럾컱첨뻹놻햼폃ꎬ룃맋뿍ퟔ뚯쿻쪧ꎬ뻟폐헢훖쳘뗣뗄업뛓맦퓲돆캪쯰쪧훆ꆣ샽죧ꎬ튻캻맋뿍떽듯쒳튻싃망ꎬ죧맻틑뺭뿍싺ꎬ쯻뻍믡샫뾪헢싃망떽뇰뒦춶쯞ꆣ 맋뿍떽듯쪱ꎬ죴쯹폐뗄럾컱첨뻹놻햼폃ꎬ맋뿍붫업돉뛓컩뗈듽럾컱ꎬ뻟폐헢훖쳘뗣뗄업뛓맦퓲돆캪뗈듽훆ꆣ뷓쫜럾컱뗄듎탲튻냣닉폃쿈떽쿈럾컱맦퓲ꆣ튲뿉틔폐웤쯻맦퓲ꎬ샽죧뫳떽쿈럾컱ꆢ폅쿈좨쿈럾컱ꆢ쯦믺럾컱ꎨ톡좡뗈듽뛓쇐훐죎튻맋뿍뷸탐럾컱ꎩ뗈뗈ꆣ뫳쏦쳖싛뗄업뛓쒣탍뚼닉폃ꆰ쿈떽쿈럾컱ꆱ뗄맦퓲ꆣ 믬뫏훆쫇쯰쪧훆뫍뗈듽훆볦뛸폐횮뗄쟩뿶ꆣ볙뚨럾컱쾵춳뗄죝솿폐쿞ꎬퟮ뛠횻쓜죝쓉k룶맋뿍ꎨ냼삨뗈듽뫍헽퓚뷓쫜럾컱뗄맋뿍ꎩꎬ쓇쎴떱맋뿍떽듯쪱ꎬ랢쿖럾컱쾵춳틑뿍싺ꎬ룃맋뿍붫ퟔ뚯쿻쪧ꎬ럱퓲뻍뷸죫럾컱쾵춳ꎬ헢쫇튻훖쟩뿶ꆣ듋췢ꎬ뮹뿉틔폐맋뿍뗈듽럾컱쪱볤폐쿞뗄쟩뿶ꎬ벴떱뎬맽튻뚨쪱볤쪱ꎬ맋뿍붫ퟔ뚯쿻쪧ꆣ 업뛓쾵춳뗄쫽솿횸뇪 튻룶럾컱쾵춳ꎬ튻랽쏦쫇죧맻럾컱믺릹맽킡뛸늻쓜싺ퟣ맋뿍뗄탨튪ꆣ뻍믡닺짺펵벷쿖쿳늢퓬돉럾컱훊솿뗄쿂붵ꆣ틲듋맋뿍쾣췻믺릹듳킩뫃ꆣ쇭튻랽쏦ꎬ죧맻럾컱믺릹맽듳ꎬ퓲죋솦ꆢ컯솦뗈랽쏦뗄뾪횧튪퓶볓ꎬ늢폐뿉쓜퓬돉풴뗄샋럑ꎬ듓헢랽쏦뗄럖컶살뾴ꎬ짨훃뗄럾컱믺릹맽듳캴뇘쓜쫕떽뫃뗄킧맻ꎬ틲듋쾣췻믺릹킡킩ꆣ퇐뺿업뛓쾵춳뗄쒿뗄뻍쫇튪퓚맋뿍뗄탨튪뫍럾컱믺릹뗄맦쒣횮볤뷸탐좨뫢
뻶닟ꎬ쪹웤듯떽뫏샭뗄욽뫢ꆣ 업뛓싛퇐뺿뗄컊쳢듳쳥럖캪솬솽샠ꆣ뗚튻샠쫇퓚럾컱믺릹짨훃횮잰ꎬ룹뻝맋뿍쫤죫맽돌폫럾컱맽돌뗄튪쟳ꎬ춨맽뛔업뛓쾵춳듓뚨탔떽뚨솿뗄럖컶ꎬ좷뚨쾵춳뗄뷡릹쒣탍ꎬ붨솢맀볆쪵볊쾵춳탔쓜뗄쫽솿횸뇪ꎬ퓚뺭ퟛ뫏럖컶뫳ퟷ돶좨뫢뻶닟ꎬ뚨돶럾컱믺릹ꎨ죧뗧탅뻖ꆢ룛뿚싫춷ꆢ믺뎡엜뗀뗈ꎩ뗄맦쒣ꎻ뗚뛾샠컊쳢쫇뛔틑폐뗄럾컱쾵춳뷸탐뚨탔ꆢ뚨솿벰ퟛ뫏럖컶ꎬ뷸탐ힷퟙ뻶닟ꎬ퇐뺿뛔쾵춳뷸탐ퟮ폅뿘훆뗄닟싔ꎬ쳡룟쾵춳뗄럾컱훊솿ꎬ늢쪹뗃쾵춳퓚듯떽맦뚨훊솿횸뇪뗄잰쳡쿂좡뗃뗄뺭볃킧틦ퟮ듳ꆣ 퇐뺿짏쫶뗄솽샠컊쳢쪱ꎬ뇘탫뛔쾵춳뷸탐뚨솿뗄럖컶ꆣ퓚쾵춳훐쓜릻랴펳쾵춳뷡릹벰탔쓜뗄뿉폃폚뚨솿뗄훷튪쫽솿횸뇪죧쿂ꎺ 뛓뎤ꆪꆪ횸쾵춳훐뗄맋뿍쫽ꎨ냼삨뗈듽럾컱뗄맋뿍뫍헽퓚뷓쫜럾컱뗄맋뿍ꎩꎻ 뗈듽뛓뎤ꆪꆪ쾵춳훐업뛓뗈듽뗄맋뿍쫽ꆣ 뚺쇴쪱볤ꆪꆪ맋뿍퓚쾵춳훐뗄업뛓뗈듽쪱볤폫뷓쫜럾컱쯹폃쪱볤횮뫍ꎻ 뗈듽쪱볤ꆪꆪ맋뿍퓚업뛓뗈쪱쯹폃쪱볤ꆣ헢킩횸뇪뻹캪쯦믺뇤솿ꎬ쟒틀삵폚쪱뿌 믲맋뿍쫽ꆣ죧잰쯹쫶ꎬ컒쏇훷튪늻쫇퇐뺿쮲첬맽돌뗄쫽솿횸뇪ꎬ횻뛔춳볆욽컈ힴ첬쿂뗄컈첬쫽솿횸뇪룐탋좤ꎬ쯹틔퓚뚨솿럖컶업뛓쾵춳뗄탔쓜쪱ꎬ춨뎣쫇횸럖컶쾵춳퓚욽뫢ힴ첬쿂뗄탔쓜ꆣ 캪쇋뚨솿럖컶튻냣뗄쾵춳ꎬ컒쏇폖틽죫쇋쿂쇐뇭쪾업뛓쒣탍뗄럻뫅ꎬ폃쯻쏇붨솢짏쫶뿉쫽솿뮯뗄횸뇪뗄쫽톧쒣탍ꆣ 헢킩럻뫅벰웤뚨틥쫇ꎺ λꆪꆪ떥캻쪱볤쓚욽뻹떽듯뗄맋뿍쫽ꎨ욽뻹떽듯싊ꎩꆣ 1ꆪꆪ욽뻹떽듯볤룴쪱볤ꆣ λµꆪꆪ떥캻쪱볤쓚쫜떽럾컱뗄맋뿍욽뻹쫽ꎨ욽뻹럾컱싊ꎩꆣ 1ꆪꆪ쎿캻맋뿍뗄욽뻹럾컱쪱볤ꆣ µSꆪꆪ럾컱첨룶쫽ꆣ ρꆪꆪ쎿룶럾컱첨뗄럾컱잿뛈ꆣ Pꆪꆪ퓚춳볆욽뫢쪱ꎬ쾵춳훐폐j룶맋뿍뗄룅싊ꆣ jLꆪꆪ뛓뎤뗄웚췻횵ꆣ Lꆪꆪ뗈듽뛓뎤뗄웚췻횵ꆣ qWꆪꆪ뚺쇴쪱볤뗄웚췻횵ꆣ
Wꆪꆪ뗈듽쪱볤뗄웚췻횵ꆣ q늻춬뗄업뛓쾵춳뻟폐늻춬뗄업뛓쒣탍ꎬ틲듋탨튪폃튻킩럻뫅뇭듯늻춬샠탍쒣탍뗄훷튪쳘헷ꆣ헢킩쳘헷뿉쇐캪6쿮ꎺ쫤죫맽돌ꎨ맋뿍떽듯볤룴쪱볤뗄럖늼ꎩꎺ럾컱쪱볤뗄럖늼ꎻ럾컱첨뗄룶쫽ꎨ뛠룶럾컱첨쪱ꎬ볙짨룷룶럾컱첨쫇늢솪뗄ꎬ쎿룶럾컱첨횻뛔떥룶맋뿍뷸탐럾컱ꎩꎻ쾵춳죝솿ꎨ럾컱첨룶쫽볓짏뿉죝쓉뗄뗈듽맋뿍쫽ꎩꎻ맋뿍살풴뗄ퟜ쳥쫽ꎻ업뛓맦퓲ꆣ틲듋ꎬ컒쏇뿉폃럻뫅뇭쪾늻춬뗄업뛓쒣탍ꎬ샽죧ꎬ뿉붫짏쫶6룶쳘헷내쮳탲폃룷ퟔ뗄럻뫅쇐돶ꎬ늢폃ꆰ/ꆱ룴뾪ꎬ벴 쫤죫맽돌/럾컱쪱볤럖늼/럾컱첨룶쫽/쾵춳죝솿/맋뿍풴쫽/업뛓맦퓲ꆣ 평폚놾헂쳖싛뗄컊쳢뚼닉폃쿈떽쿈럾컱맦퓲ꎬ쯹틔퓚쒣탍뗄럻뫅뇭쪾훐늻뇘퓙쇐돶업뛓맦퓲뗄럻뫅ꆣ떱쾵춳죝솿믲맋뿍풴컞쿞쪱ꎬ튲붫쯼쏇듓쒣탍뗄럻뫅훐쪡좥ꆣ폃틔뇭쪾맋뿍떽듯쪱볤볤룴럖늼뫍럾컱럖늼뗄뎣폃럻뫅폐ꎺ Mꆪꆪ쫤죫맽돌캪ퟮ볲떥쇷ꎬ믲럾컱쪱볤캪뢺횸쫽럖늼ꆣ Dꆪꆪ뚨뎤럖늼ꆣ Eꆪꆪk뷗낮뛻샊럖늼ꆣ kGꆪꆪ럾컱쪱볤캪튻냣뛀솢럖늼ꆣ 샽죧M/M/1/1뇭쪾쫤죫맽돌캪ퟮ볲떥쇷ꆢ럾컱쪱볤캪뢺횸쫽럖늼ꆢ떥럾컱첨ꆢ쾵춳죝솿캪1뗄쯰쪧훆업뛓쒣탍ꎻM/G/S/k뇭쪾쫤죫맽돌캪늴쯉쇷ꆢ럾컱쪱볤캪튻냣뛀솢럖늼ꆢs룶럾컱첨ꆢ쾵춳죝솿캪k뗄믬뫏훆업뛓쒣탍ꆣ 뛔폚쯰쪧훆뫍믬뫏훆뗄업뛓쾵춳ꎬ맋뿍퓚떽듯업뛓쾵춳쪱ꎬ랢쿖럾컱첨컞뿕믲헟쾵춳죝솿틑싺ꎬ뻍ퟔ뚯쿻쪧뛸퓙늻뷸죫쾵춳ꆣ틲듋ꎬ떽듯뗄맋뿍늻튻뚨좫뷸죫쾵춳ꆣ캪듋ꎬ틽죫폐킧떽듯싊λꎬ쯼쫇횸떥캻쪱볤쓚욽뻹뷸죫쾵춳뗄맋뿍죋쫽ꎬe뛸떽듯싊λ쫇횸떥캻쪱볤쓚욽뻹살떽쾵춳뗄맋뿍죋쫽(닎뾴춼9-3)ꆣ 랴삡업뛓럾컱ힴ첬1떽듯맋뿍2샫좥맋뿍s 춼 9-3 듋뒦ꎬ뿉틔횸돶ꎬ떱업뛓쒣탍뒦폚욽뫢ힴ첬쪱ꎬ폐쿂쏦뗄샮쳘뛻릫쪽ꎺ L=λW (9-14) qeq
L=λW (9-15) 벸룶업뛓쒣탍 놾뷚붫쳖싛벸룶뻟쳥뗄업뛓쾵춳ꎬ헢킩쾵춳뗄ힴ첬뇤뮯맽돌뚼쫴폚짺쏰맽돌ꆣ놾뷚붫룸돶헢킩쾵춳쯹뛔펦뗄업뛓쒣탍ꆣ헢킩쒣탍뗄쫤죫맽돌뻹캪늴쯉쇷ꆢ럾컱쪱볤캪뢺횸쫽럖늼ꎬ헢킩쫇ퟮ뎣볻뗄업뛓쒣탍ꎬ벴떥뛓ꆢ늢쇐뗄뛠럾컱첨쒣탍뗄뇪ힼ쟩뿶뫍쳘쫢쟩뿶ꎬ샻폃짺쏰맽돌뗄컈첬룅싊ꎬ뿉뗃떽폃틔뇭듯헢킩쒣탍쳘헷뗄쫽솿횸뇪ꆣ M/M/S업뛓쒣탍 놾쒣탍횸뗄쫇헢퇹뗄업뛓쾵춳ꎻ맋뿍떽듯럾듓늴쯉럖늼ꎨ믲떽듯볤룴쪱볤럾듓뢺횸쫽럖늼ꎩꎻ럾컱쪱볤럾듓뢺횸쫽럖늼ꎻ늢쇐뗄럾컱첨뗄룶쫽캪S(S≥1)룶ꎻ쾵춳죝솿뫍맋뿍풴쫽컞쿞ꎻ업뛓맦퓲캪쿈떽쿈럾컱뗄뗈듽훆업뛓쾵춳ꎬ쯹틔M/M/S/∞/∞ꎨ쿈떽쿈럾컱ꎩ쒣탍뗄볲킴ꆣ 떱S=1쪱ꎬ쿠펦뗄M/M/1뿉뾴ퟷM/M/S뗄쳘쫢쟩뿶ꎬ벴M/M/1쫇떥럾컱첨뗄뗈듽훆쒣탍ꎬ죧춼9-2ꎨaꎩ쯹쪾ꆣ떱S>1쪱ꎬ퓲M/M/S캪늢쇐뛠럾컱첨뗄업뛓쒣탍ꎬ죧춼9-2ꎨcꎩ벰춼9-3쯹쪾ꆣ 짏쏦맘폚쾵춳쳘뗣뗄쏨쫶뫜훘튪ꎬ늻뻟놸헢킩쳘뗣뗄쒣탍늻돆캪M/M/S쒣탍ꆣ샽죧떱S룶럾컱첨죔캪늢솪ꎬ떫퓚쎿룶럾컱첨잰뚼탎돉튻룶뗈듽뛓쇐뗄업뛓쾵춳ꎬ죧춼9-4쯹쪾ꎬ쪵볊짏쫇S룶M/M/1업뛓쾵춳ꎬ뛸늻쫇뇪ힼ뗄M/M/S쒣탍ꆣ 쿂쏦쿈쳖싛폫M/M/S쾵춳쿠펦뗄짺쏰맽돌틔벰듋짺쏰맽돌뗄컈첬룅싊ꎬ헢룶룅싊쫇쯹탨쟳돶뗄쫽솿횸뇪횮튻ꆣ 짨떽듯볤룴쪱볤{T}럾듓뗄뢺횸쫽럖늼캪 n0t<0P(T≤t)= n−λt1−t≥0e럾컱쪱볤{τ}럾듓뗄뢺횸쫽럖늼캪 n0t<0P(τ≤t)= n−µt1−t≥0e웤훐닎쫽λ뫍µ죧잰쯹쫶ꎬ뻍쫇욽뻹떽듯싊뫍욽뻹럾컱싊ꆣ
1 2 s춼 폃ξ(t)뇭쪾퓚쪱뿌t쾵춳쓚뻟폐뗄맋뿍쫽ꎬ쯼쫇쯦믺뇤솿ꎬ맊{ξ(t)}쫇튻룶쯦믺맽돌ꎬ웤ힴ첬벯캪I={0,1,2,"}ꆣ헢룶쯦벴맽돌뿉뾴ퟷ쫇튻룶짺쏰맽돌ꎨ쫓맋뿍떽듯캪ꆰ짺ꆱꎬ샫좥캪ꆰ쏰ꆱꎩꆣ틲듋쫤죫맽돌캪λ뗄ퟮ볲떥쇷ꎨ뻟폐뛀솢퓶솿탔뫍욽컈탔ꎩꎬ맊퓚+t쪱볤쓚떽듯튻캻맋뿍뗄룅싊뿉킴돉 λ+t−λ+tP(ξ(t)=1)=e=t(1−λt+0(t)) 1! =λ+t+0(+t) 퓚쫤죫맽돌훐뢺횸쫽럖늼뫍늴쯉럖늼뗄뗈볛탔ꎬ뛔폚뻟폐뢺횸쫽럖늼뗄럾컱쪱볤τ벰쿠펦뗄럾컱맽돌튲뻟폐듋훖탔훊ꆣ틲듋ꎬ퓚+t쪱볤쓚튻룶럾컱첨뛔튻n룶맋뿍뷡쫸럾컱뗄룅싊캪µ+t+0(+t)ꆣ틲캪쾵춳폐S룶럾컱첨ꎬ쯹틔뛔헻룶쾵춳살쮵ꎬ퓚+t쪱볤쓚뷡쫸뛔튻캻맋뿍럾컱뗄룅싊캪Sµ+t+0(+t)ꆣ죧맻퓙ퟷ뷸튻늽뗄럖컶ꎬ뿉췆횪듋짺쏰맽돌폐죧쿂뷡맻ꆣ λ=λj=0,1,2,"j j=1,2,",S jµµ=jj=S+1,S+2,"Sµ컒쏇맘탄뗄쫇듋짺쏰맽돌뗄컈첬룅싊Pꆣ쿖퓚뾼싇뚨샭3뗄쳵볾ꆣ jSjj1≤j≤Sρj!틲듋 π= (9-16) jsSjj>SρS!λ웤훐 ρ= (9-17) Sµ틲듋 ∞Sjssjπρ+ρ∑∑∑j!!j=0j=0j=S+1 ∞s∞111+∑∑∑λπλπλπj=0j=0j=S+1jjjj
s∞1j!!1=+ ∑∑jjSjλρSλρ=0j=+1∞∞1 맊떱ρ<1쪱ꎬπ<+∞,=+∞ ∑∑jλπ=0j=0jj뚨샭3 뗄쳵볾뗃떽싺ퟣꎬ폚쫇떱ρ<1쪱ꎬ평뚨샭3뗄뷡싛뇣뗃떽쇋듋쾵춳뗄컈첬룅싊 ∞S−1jS(Sρ)(Sρ)1−1 P=(π)=(+⋅) (9-18) 0∑∑jj!S!1−ρj=0j=0j(Sρ)1≤j≤SP0j! P=πP= (9-19) jj0SjSρj>SP0S!폐쇋M/M/S쒣탍뗄컈첬룅싊P(j=0,,")헢룶횸뇪ꎬ뇣뿉뗃떽쾵춳퓚욽j뫢ힴ첬쿂뗄뛓뎤웚췻횵Lꎬ뗈듽뛓뎤웚췻횵L틔벰W,W뗈횸뇪뗄볆쯣릫쪽ꆣ qq평L벰L뗄뚨틥ꎬ뿉뗃 q∞ρL=j⋅p=Sρ+⋅p (9-20) ∑js2(1−ρ)j=0∞ρLjp=p (9-21) q∑S+jS2(1−ρ)j=0쇭솽룶쫽솿횸뇪쫇W뫍Wꆣ틲캪업뛓쾵춳뒦폚욽뫢ힴ첬ꎬ평샮쳘뛻릫쪽qꎨꎩꎬꎨꎩ틔벰쒣탍뗄뗈듽훆ꎬ횪떽듯쾵춳뗄맋뿍붫좫늿뷸죫쾵춳ꆣ벴λ=λꎬ쯹틔 eL=λ⋅W=λ⋅We L=λ⋅W=λ⋅Wqeqq11ρ맊 W=L=+p (9-22) s2λµλ(1−ρ)1ρW=L=⋅p (9-23) qqS2λλ(1−ρ)λ쳘뇰뗄ꎬ떱S=1쪱ꎬ뛔폚M/M/1쒣탍뗄횸뇪ꎬ떱ρ=<1쪱ꎬ평릫쪽ꎨ9-8ꎩµ
ρ−1뗃p=(1+)=1−ρ맊폐 01−ρp=1−ρ0 (9-24) jp=(1−ρ)⋅ρj=1,2,"j22λρ쟒 L== (9-25) qµ(µ−λ)1−ρλρL== (9-26) µ−λ1−ρλW== (9-27) qµ(µ−λ)µ(1−ρ)11W== (9-28) µ−λµ(1−ρ)샽 쿂쇐쫽뻝쫇떽듯폊뻖뗄맋뿍쫽뫍뛔맋뿍럾컱쪱볤뗄춳볆뷡맻죧뇭2ꆣ틔쎿3럖훓캪튻룶쪱뛎ꎬ춳볆쇋100룶쪱뛎훐맋뿍떽듯뗄쟩뿶틔벰뛔100캻맋뿍럾컱뗄쪱볤ꆣ 뇭2 떽듯쫽 0 1 2 3 4 5 6 쪱뛎쫽 14 27 27 18 9 4 1 럾컱쪱볤ꎨ쏫ꎩ0ꆫ12 12ꆫ2424ꆫ3636ꆫ4848ꆫ6060ꆫ72 72ꆫ84 맋뿍죋쫽 33 22 15 10 6 4 3 96ꆫ108ꆫ120ꆫ150ꆫ180ꆫ럾컱쪱볤ꎨ쏫ꎩ84ꆫ96 108 120 150 180 200 맋뿍죋쫽 2 1 1 1 1 1 짨듋럾컱쾵춳쫇튻룶M/M/1쒣탍업뛓ꎬ쫔쟳맋뿍떽듯폊뻖뫳ꎬ탨튪업뛓뗈듽럾컱뗄룅싊ꆢ늻탨뗈듽뗄룅싊틔벰웤쯼쫽솿횸뇪ꆣ 뷢ꎺ쿈쟳돶쎿쪱뛎쓚떽듯맋뿍뗄욽뻹쫽 0×14+1×27+2×27+3×18+4×9+5×4+6×1= 틲쎿3럖훓캪튻룶쪱뛎ꎬ쯹틔맋뿍욽뻹떽듯싊캪 λ= ꎨ맋뿍/31럖ꎩ퓙볆쯣쎿캻맋뿍쯹탨뗄욽뻹럾컱쪱볤ꎬ뇭룱훐럾컱쪱볤닉폃ퟩ훐횵ꎬ뿉뗃 µ
11=(6×33+18×22+30×15+42×10+54×6+66×4+78×3+90×2+102×1µ100 +114×1+135×1+165×1+190×1) =ꎨ쏫ꎩ ꎨ럖ꎩ 맊듋업뛓쾵춳뗄욽뻹럾컱싊뫍럾컱첨뗄럾컱잿뛈럖뇰캪 λµ=ꎨ맋뿍/럖ꎩ ρ== µ샻폃릫쪽ꎨꎩ~ꎨꎩ뿉뗃쾵춳뒦폚춳볆욽뫢ힴ첬쪱뗄쿂쇐횸뇪ꎺ 쾵춳훐폐쇣캻맋뿍뗄룅싊 P=1−ρ=1−= 0P튲뻍쫇맋뿍떽듯뫳늻탨뗈듽뗄룅싊ꆣ쯹틔맋뿍떽듯뫳탨업뛓뗈듽럾컱뗄룅0싊캪 1−P=P= 0쾵춳훐웚췻뗄맋뿍쫽ꎨ뗈듽뛓뎤뗄웚췻횵ꎩ 22λρL===ꎨ캻맋뿍ꎩ qµ(µ−λ)1−ρ맋뿍퓚쾵춳훐웚췻뚺쇴쪱볤 1W==ꎨ럖ꎩ µ−λ맋뿍퓚쾵춳훐웚췻뗈듽럾컱쪱볤 λW==ꎨ럖ꎩ qµ(µ−λ)샽 짨쒳튽풺뗄췢뿆뮻튩쫒폐2캻떱냠뮤쪿ꎬ뮻튩헟뗄떽듯맽돌뿉죏캪쫇튻룶늴쯉맽돌ꎬ뮻튩쪱볤럾듓뢺횸쫽럖늼ꆣ죧맻욽뻹쎿킡쪱폐20캻늡죋살뮻튩ꎬ쎿캻늡죋뗄뮻튩쪱볤욽뻹캪5럖훓ꆣ쫔쟳쎿튻캻늡죋떽뮻튩욽뻹뮨럑쪱볤ꎬ늢컊쫇럱펦퓶볓튻캻떱냠뮤쪿ꆣ 뷢ꎺ쿖퓚뗄업뛓쾵춳캪M/M/2쒣탍ꎬ탨쟳돶뚺쇴쪱볤뗄웚췻횵Wꆣ평쳢틢뿉횪
2011λ==ꎨ맋뿍/럖ꎩ 뛸=5ꎨ럖훓ꎩ 603µ1맊 µ=ꎨ맋뿍/럖ꎩ 5λλ5뛸 ρ=== S⋅µ2⋅µ6볆쯣W탨쿈쟳돶P=Pꎬ평릫쪽ꎨꎩ횪 S22−1j2(2ρ)(2ρ)1−1P=+⋅ 0∑j!2!1−ρj=01011012−1 =(1++⋅()⋅6)= 6261듺죫릫쪽ꎨꎩ,뿉뗃 2(2ρ)11012 P=P=P=P=()⋅ js202!2611붫P듺죫릫쪽ꎨꎩꎬ뿉뗃 21ρ52 W=+⋅p=5+3×6×× 22µλ(1−ρ)6=ꎨ럖ꎩ 죧맻퓶볓튻캻뮤쪿ꎬ퓲쾵춳캪M/M/3쒣탍ꎬ춬샭뿉쟳뗃 P= P=P==ꎨ럖ꎩ 평뷡맻뾴돶ꎬ퓶볓튻캻뮤쪿뫳ꎬ늢쎻폐쏷쿔쯵뛌튻캻늡죋퓚뮻튩욽뻹뮨럑쪱볤ꎬ쯹틔늻탨튪퓶볓튻캻뮤쪿ꆣ 쿂쏦뛔M/M/S업뛓쒣탍(S≥2)폫S룶M/M/1쒣탍뇈뷏ꆣ 샽 늻럁짨S=2쪱ꎬ뇈뷏M/M/2(λ=5,µ=4)폫솽룶M/M/1쒣탍ꆣ튲뻍쫇튪뇈뷏업튻룶뛓쇐벰업솽룶뛓쇐솽헟럾컱킧맻뗄폅쇓ꆣ 뷢ꎺ업튻룶뛓쇐뗄쟩뿶횸튻룶M/M/2쒣탍ꆣ맘폚듋쒣탍뗄룷쫽솿횸뇪뿉평릫쪽ꎨꎩ~ꎨꎩ쟳뗃ꎬ죧뇭9-3쯹쪾ꆣ죧맻쎿룶럾컱첨잰뚼룷업튻뛓ꎬ쟒1뷸죫뛓쇐뫳뇣볡돖늻뮻ꎬ떽듯뗄맋뿍틔룷뗄룅싊볓죫떽쎿룶뛓훐ꎬ헢뻍쫇업솽2룶뛓뗄쟩뿶ꆣ헢쪱ꎬ풭살뗄M/M/2쾵춳뇤뮯돉쇋솽룶뛀솢뗄M/M/1쾵춳ꎬ늢λ5쟒쎿룶뛓쇐뗄욽뻹떽듯싊λꆢλ캪λ=λ==ꆣ뛔M/M/1쒣탍뗄훷튪횸뇪121222
뿉평릫쪽ꎨꎩ~ꎨꎩ쟳뗃ꎬ죧뇭9-3쯹쪾ꆣ 뇈뷏뇭9-3훐룷횸뇪ꎬ돽L췢ꎬ웤폠뗄횸뇪뻹쿔쪾돶업튻룶뛓ꎨ1ꎩ뇈업솽룶뛓ꎨ2ꎩ킧맻튪뫃ꆣ퓚쯹쇐뗄4룶횸뇪훐ꎬ맋뿍퓚쾵춳훐뗄욽뻹뚺쇴쪱볤Wꎬ쫇뫢솿럾컱훊솿폅쇓뗄튻룶훘튪횸뇪ꆣ퓚솽훖업뛓랽쪽쿂ꎬ업솽룶뛓쪱뗄쎿룶M/M/1뗄W캪뇈업튻룶뛓쪱뗄M/M/2뗄W=튪듳ꆣ 뇭9-3 µ L W 쒣탍 λ LW pp(1)M/M/2 5 4 (2)M/M/1 4 뇈뷏뗄뷡맻쮵쏷ꎬ뺡맜뚼쫇짨훃솽룶럾컱첨ꎬ떫평폚닉폃늻춬뗄업뛓쒣탍ꎬ웤킧맻쫇늻튻퇹뗄ꆣ닉폃벯훐쪹폃뗄랽낸ꎨ듓뛸탎돉뛠럾컱첨업뛓쾵춳ꎩ튪폅폚닉폃럖즢쪹폃뗄랽낸ꎨ냼삨탎쪽짏퓚튻웰ꎬ쪵볊짏쫇럖즢쪹폃뗄랽낸ꎩꎬ퓚뺭볃쫵폯훐돆횮캪ꆰ맦쒣쫕틦ꆱꆣ퓚뾼싇럾컱짨쪩뗄늼뻖폫쪹폃쪱ꎬ탨튪힢틢헢튻틲쯘ꆣ M/M/S/k업뛓쒣탍 M/M/S/k쒣탍캪맋뿍떽듯볤룴쪱볤뫍럾컱쪱볤뻹럾듓뢺횸쫽럖늼ꆢS룶럾컱첨ꆢ쾵춳죝솿캪kꎨ폐쿞룶ꎩ뗄업뛓쾵춳ꆣ떱k=S쪱캪쯰쪧훆업뛓쾵춳ꎻ떱k>S쪱캪믬뫏훆업뛓쾵춳ꆣ죧맻맋뿍떽듯업뛓쾵춳쪱ꎬ쾵춳쓚틑폐k룶맋뿍ꎬ쓇쎴헢캻맋뿍뻍ퟔ뚯쿻쪧ꆣ쿂쏦좷뚨듋쾵춳쫽솿횸뇪뗄맘쾵쪽ꆣ 폃ξ(t)뇭쪾퓚쪱볤t쾵춳쓚뗄맋뿍쫽ꎬ퓲{ξ(t)}쫇튻룶쯦믺맽돌ꎬ웤ힴ첬벯쫇폐쿞벯I={0,1,2,",k}ꆣꎬ뿉횤쏷{ξ(t)}튲쫇튻룶짺쏰맽돌ꎬ늢폐 λ=λj=0,1,2,",k−1j j=1,2,",s jµµ=jj=S+1,s+2,",kSµλ뫍µ뗄틢틥춬잰ꎬ폫M/M/S샠쯆ꎬ평뚨샭ꎨꎩ뇣뿉뗃떽쾵춳뗄컈첬룅싊 S−1ik3i(SP)Sρ−1 p=+) (9-29) 0∑∑i!!i−0i=Sj(SP)j=1,2,",Sp0j p= (9-30) jsjSρj=S+1,S+2,",kp0S!
λ웤훐ρ=ꎬ뛔폚죝솿폐쿞뗄쒣탍늻뇘폐ρ<1뗄쿞훆ꎬ틲캪늻뿉쓜돶쿖컞Sµ쿞뎤뗄업뛓쟩뿶ꎬL뫍L뿉평웤몬틥횱뷓볆쯣ꎬ폐 qkL=p (9-31) ∑jj=0k−s L=jp (9-32) q∑S+jj=0맋뿍떽듯뛸쓜뷸죫쾵춳뗄룅싊캪1−Pꎬ맊쾵춳뗄폐킧떽듯싊캪 k λ=λ(1−P) (9-33) ek퓙룹뻝샮쳘뛻릫쪽뇣뿉쟳뗃W뫍Wꆣ q쳘뇰뗄ꎬ떱S=1쪱ꎬ뿉뗃M/M/1/k쒣탍뗄쫽솿횸뇪죧쿂ꎺ 1−ρρ≠1k+11−ρP= (9-34) 01ρ=1k+1jρ(1−ρ)ρ≠1k+11−ρ P= (9-35) j1ρ=1k+1kk+1ρ(1−(k+1)⋅ρ)+ρρ≠1k+1(1−ρ)(1−ρ)L= (9-36) kρ=122k+1ρ(k+ρ)ρ≠1−k+11−ρ1−ρL= (9-37) qk(k−1)ρ=12(k+1)샽 쒳뗧뮰컊톶뒦짨놸폐6늿뗧뮰ꎬ욽뻹쎿럖훓폐4듎컊쳢톶뗧뮰ꎨ냼삨뷓춨뗄뫍캴뷓춨뗄ꎩꎬ컊톶떽듯럾듓늴쯉럖늼ꎬ쎿듎춨뮰뗄욽뻹쪱볤캪럖훓ꎬ춨뮰쪱볤럾듓뢺횸쫽럖늼ꆣ쫔컊듲떽컊톶뒦뗄뗧뮰쓜뷓춨뗄룅싊캪뛠짙ꎿ 뷢ꎺ평쳢틢횪ꎬ헢쫇튻룶M/M/6/6업뛓쾵춳뗄쒣탍ꆣ떱k=S=6쪱ꎬ헢쫇튻룶쯰쪧훆쾵춳ꆣ늢쟒욽뻹떽듯싊λ=4ꎬ틲욽뻹럾컱쪱볤
1= 쯹틔욽뻹럾컱싊µ=2 µλ41맊럾컱잿뛈ρ=== Sµ6×23붫ρ벰S듺죫릫쪽ꎨꎩ뫍ꎨꎩ뗃 −111i6(6×)(6×)533P+=0∑i!i6!i=0 16(6×)j(Sρ)3P=P=P=P=⋅P=!6!틲캪떱k=S=6쪱ꎬ6룶뗧뮰뻹놻폃ꎬ틲듋ꎬ탂떽듯뗄컊톶뗧뮰늻쓜놻뷓춨ꎬ쯹틔P=P뻍쫇탂살떽뗄뗧뮰캴쓜놻뷓춨뗄룅싊ꆣ뛔폚럾컱쾵춳살쮵ꎬ폐돆ퟷk6쯰쪧룅싊ꎬ쯼쫇쯰쪧훆쾵춳뗄훘튪횸뇪ꆣ틲듋ꎬ컊톶뗧뮰쓜놻뷓춨뗄룅싊캪 1−P= 6뛔폚쯰쪧훆쾵춳평릫쪽ꎨꎩ쮳뇣뮹뿉틔쟳돶쾵춳뗄폐킧떽듯싊캪 λ=λ(1−P)=4×= e6샽 쒳볓폍햾짏폐튻룶볓폍뇃ꎬ뛸쟒뎡뗘뫜킡ꎬퟮ뛠횻쓜춣럅5솾웻뎵ꎨ냼삨헽퓚볓폍뗄튻솾ꎩꆣ짨웻뎵떽듯볤룴뫍볓폍쪱볤뚼럾듓뢺횸쫽럖늼ꎬ욽뻹쎿2럖훓떽듯튻솾웻뎵ꎬ쎿솾웻뎵볓폍욽뻹탨튪2럖훓ꆣ쫔컊ꎺ살볓폍뗄웻뎵떽듯볓폍햾쪱쓜솢벴볓폍뗄룅싊캪뛠짙ꎿ볓폍햾뎡뗘폐뿕뗄룅싊캪뛠짙ꎿ쪵볊볓폍뗄웻뎵퓚볓폍햾욽뻹뚺쇴뗄쪱볤쫇뛠짙ꎿ 뷢ꎺ듋캪튻룶M/M/1/5업뛓쒣탍ꆣ 11평쳢틢횪k=5,S=1,=2(럖)ꎬ=2ꎨ럖ꎩ λµ11λ퓲λ=ꎨ솾/럖ꎩꎬµ=ꎨ솾/럖ꎩꎬρ==1 22µ벴욽뻹럾컱잿뛈캪1ꆣ 평릫쪽ꎨꎩ뫍ꎨꎩ훐 ρ=1뗄쟩뿶 뿉횪ꎬ쾵춳뗄컈첬룅싊캪
1111P==ꎬ쇭췢P=P=P=P=P== 012345k+16k+16웤훐P캪쾵춳훐컞맋뿍뗄룅싊ꎬP캪쾵춳훐컞뿕캻뗄룅싊ꆣ 05뛔폚믬뫏훆쾵춳(k>S)ꎬ평릫쪽ꎨꎩꎬ틲캪폐킧떽듯싊캪 115λ=λ(1−P)=λ(1−P)=×(1−)=ꎨ솾/럖훓ꎩ ek52612퓙폃샮쳘뛻릫쪽벰릫쪽ꎨꎩ훐ρ=1뗄쟩뿶ꎬ뇣뿉쟳뗃쪵볊볓폍뗄웻뎵퓚볓폍햾욽뻹뚺쇴뗄쪱볤 kL2W===6ꎨ럖훓ꎩ λλee5쯹틔뿉솢벴볓폍뗄룅싊캪P=ꎬ랢쿖뎡뗘폐뿕뗄룅싊캪1−P=ꎬ욽뻹056뚺쇴쪱볤캪6럖훓ꎬ M/M/S/m/m업뛓쒣탍 잰쏦쳖싛뗄업뛓쒣탍훐ꎬ맋뿍풴뚼쫇컞쿞뗄ꆣ쿖퓚튪쳖싛뗄쫇튻훖룼쳘쫢뗄쟩뿶ꎬ벴업뛓쾵춳뻟폐S룶럾컱첨ꎬ떫맋뿍풴캪폐쿞쫽m,웤쯻맦뚨폫M/M/S/k쒣탍뗄맦뚨춬퇹ꎬ헢샯k=mꆣ헢훖쾵춳훐죧맻틑폐쇋m룶맋뿍 ꎬ뻍늻믡퓙폐탂뗄맋뿍떽듯ꎬ돽럇쾵춳훐뗄맋뿍뗃떽럾컱뗄뫳폖략믘맋뿍풴ꎬ쾵춳닅뿉쓜폐맋뿍볌탸떽살ꆣ웤쒣탍죧춼9-5쯹쪾ꆣ 1맋뿍s럾컱쾵춳략믘맋뿍 춼9-5 폐쿞맋뿍풴뗄럾컱쾵춳뗄뗤탍샽ퟓ쫇믺웷뾴맜컊쳢ꎨ벴믺웷틲맊헏춣믺듽탞컊쳢ꎩꆣ쿖퓚짨폐S룶릤죋릲춬뾴맜m첨믺웷ꎨm≥S,m첨믺웷뻍쫇맋뿍풴쫽ꎩꆣ쎿떱믺웷랢짺맊헏쪱ꎬ뻍솢벴폐튻캻릤죋뢺퓰탞샭ꎬ쪹웤믖뢴짺닺ꆣ쯹틔ꎬ돶맊헏뗄믺웷뻍쫇튪쟳믱뗃럾컱뗄맋뿍ꎬ벴맋뿍떽듯뻍쫇믺웷돶쇋맊헏ꎻ릤죋뻍쫇럾컱첨ꆣ떱S룶릤죋럖뇰퓚탞샭S첨돶맊헏뗄믺웷쪱볤ꎬ탂랢짺맊헏뗄믺웷뻍횻쓜
뗈듽릤죋탞샭ꆣ헢샯볙뚨ꎺ ꎨ1ꎩ쎿첨믺웷솬탸헽뎣퓋뗄쪱볤뚼럾듓닎쫽캪λ뗄뢺횸쫽럖늼ꎬ쎿첨믺1웷욽뻹솬탸퓋쪱볤캪,λ뻍쫇튻첨믺웷퓚떥캻퓋쪱볤쓚랢짺맊헏뗄욽뻹듎λ쫽ꆣ ꎨ2ꎩ쎿첨믺웷뗄탞뢴쪱볤뚼럾듓닎쫽캪µ뗄뢺횸쫽럖늼ꎬ릤죋탞샭믺웷뗄1욽뻹쪱볤캪ꆣ µꎨ3ꎩ쎿첨믺웷퓚죎뫎쪱뛎쓚솬탸퓋뗄쪱볤폫릤죋탞뢴쪱볤뇋듋뛀솢ꆣ 죴틔ξ(t)뇭쪾퓚쪱볤t늻퓚헽뎣퓋뗄믺웷쫽ꎬ쯼쫇튻룶쯦믺뇤솿ꎬ퓲{ξ(t)}캪튻쯦믺맽돌ꎬ쯼뗄ힴ첬벯 I={0,1,",m}ꆣ뿉틔횤쏷{ξ(t)}쫇튻룶짺쏰맽돌ꎬ 쟒폐ꎺ λ=(m−j)λj=0,1,2,",m−1j j=1,2,",S jµµ=jj=S+1,S+2,",mSµ폚쫇룹뻝뚨샭3뇣뿉틔뗃떽쾵춳뗄컈첬룅싊ꎺ −1jjSmmλj!λp=+ (9-38) 0∑∑j−SjµS!Sµj=0j=S+1jmλ⋅pj=1,2,",S0jµ p= (9-39) jjmj!λj=S+1,S+2,",m⋅p0j−SjS!Sµmm−S퓙룹뻝L=P뫍L=jP뗃떽L벰L뗄릫쪽ꆣ탨힢틢뗄쫇ꎬ쿖퓚맋∑∑jqS+jqj=0j=0뿍욽뻹떽듯싊늻쫇λ뛸쫇λ=(m−L)ꆣ펦폃샮쳘뛻릫쪽뮹뿉떼돶쟳W벰W뗄볆eq쯣릫쪽ꆣ 쳘뇰뗄ꎬ떱S=1쪱ꎬ폐쿂쫶떥럾컱첨쾵춳뗄횸뇪릫쪽캪ꎺ −1jmm!λp= (9-40) 0∑(m−i)!µi=0jm!λp=⋅p (9-41) j0(m−j)!µ
µL=m−(1−p) (9-42) 0λ L=L−(1−p) (9-43) q0샽 4쏻릤죋뾴맜10첨믺웷ꎬ쎿첨믺웷욽뻹쎿맽냫킡쪱뻍튪탞샭튻듎ꎬ쎿듎탞샭탨튪10럖훓ꆣ짨믺웷솬탸퓋쪱볤뫍탞샭쪱볤뻹럾듓뢺횸쫽럖늼ꆣ쫔쟳믺웷랢짺맊헏늻쓜십짏뗃떽탞샭뗄룅싊ꎬ틔벰L,L,W,W뗈쫽솿횸뇪ꆣ qq뷢ꎺ내쳢틢횪ꎬ헢쫇튻룶M/M/S/m/m쾵춳ꆣ 111S=4,m=10, λ=,µ=,ρ=ꆣ 30103듺죫릫쪽ꎨꎩꎬꎨꎩ뿉뗃쾵춳뗄컈첬룅싊ꎺ 012341010101010λλλλλ P=+++++ 00µ1µ2µ3µ4µ5671010105!λ6!λ7!λ+++ 234!4µ4!4µ4!4µ8910−1010108!λ9!λ10!λ ++ 4564!4µ4!4µ4!4µ=(1++5++++++-1 ++) -1=() =10λ뛸P=⋅P== 1001µ210λ춬샭P=⋅P=5⋅P= 2002µ P= P= 34 P= P= 56P= P= 78P= P= 90P= P= 100퓙볆쯣L뫍L q
10 L=jP= ∑jj=010−4 L=j⋅P=P+2P+3P+4P+5P+6P q∑4+j5678910j=0 = 뛸맋뿍떽듯싊캪 1 λ=λ(m−L)=(10−)= e30평샮쳘뛻릫쪽쟳뗃W벰W뗃ꎺ ===ꎨ럖ꎩ λ===ꎨ럖ꎩ qλ랢짺맊헏뗄믺웷늻쓜십짏믱뗃탞샭뗄룅싊캪 1−(P+P+P+P)= 0123샽 쒳웳튵폐쒳훖탍뫅믺웷죴룉첨ꎬ쯼쏇뗄솬탸릤ퟷ쪱볤럾듓춬튻닎쫽λλ뗄뢺횸쫽럖늼ꆣ릤죋탞샭쪱볤럾듓춬튻닎쫽µ뗄뢺횸쫽럖늼ꆣ짨=ꆣ쿖폐µ솽룶랽낸ꎺ랽낸ꋱ캪3쏻릤죋룷ퟔ뛀솢뾴맜믺웷ꎬ쎿죋뾴맜6첨믺웷ꆣ랽낸ꋲ캪3쏻릤죋릲춬뾴맜20첨믺웷ꎬ떱믺웷탨튪캬탞쪱ꎬ횻튪3쏻릤죋훐폐죋뿕쿐ꎬ뻍뿉틔뗃떽탞샭ꆣ쫔뇈뷏솽룶랽낸뗄폅쇓ꆣ 뷢ꎺ랽낸ꋱ쿔좻쫇탎돉쇋3룶M/M/1/6/6업뛓쒣탍ꆣ뾼달튻쿂듋쒣탍뗄폐맘쫽솿횸뇪ꆣ λ틑횪=ꎬ평릫쪽ꎨꎩ벰ꎨꎩ뿉쟳뗃쾵춳뗄컈첬룅싊죧쿂ꎺ µP=, p=, P= 012 P=, P=, P= 345 P=. 6평릫쪽ꎨꎩ벰릫쪽ꎨꎩ뗃 L= L= q뛸폐킧떽듯싊캪
λ=λm−L=λ6−=λ ()()e평샮쳘뛻릫쪽뗃랽낸ꋱ뗄탨탞믺웷욽뻹탞샭쪱볤캪 == ()qλλeλ퓙살뾼달랽낸ꋲꎬ헢쫇튻룶M/M/3/20/20업뛓쒣탍ꆣ=ꎬ뿉쟳뗃폐µ맘쫽솿횸뇪죧쿂ꎺ 쾵춳뗄컈첬룅싊쫇 P=, P=, P=, P= 0123 P=, P=, P=, P= 4567 P=, P=, P=, P= 891011p=, 뛸p,p,...,p뚼벸뫵뗈폚쇣ꆣ듓뛸뿉뗃 1213142020L=p= ∑jj=020−3Ljp= q∑3+jj=0퓚M/M/3/20/20훐뗃폐킧떽듯싊캪 λ=λ20−=λ ()e쯹틔랽낸ꋲ뗄믺웷뗈듽탞샭쪱볤 ꋲ= ()λ 좡솽룶랽낸뗄욽뻹뗈듽탞샭쪱볤횮뇈캪 Wꋱ()=÷= Wꋲλλ()q평뇈횵뿉뗃랽낸ꋲ뗄믺웷뗈듽탞샭쪱볤뇈랽낸ꋱ뗄튪킡ꎬ쯹틔랽낸ꋲ폅폚랽낸ꋱꆣ헢뿉뾴ퟷ쫇잰쏦쯹쫶맘폚벯훐쪹폃뗄랽낸평폚럖즢쪹폃뗄랽낸헢튻뷡싛폖튻샽횤ꆣ 튻냣럾컱쪱볤뗄M/G/1업뛓쒣탍 잰쏦뗄죽룶업뛓쒣탍뻹쫇늴쯉쫤죫맽돌ꎬ럾컱쪱볤럾듓뢺횸쫽럖늼뗄업쇐쾵춳ꆣ쿖퓚붫럾컱쪱볤럾듓뗄럖늼탎쪽폨틔삩듳ꎬ늻ퟷ죎뫎쿞훆ꎬ벴럾듓죎뫎튻룶
럖늼뻹뿉ꎬ컞싛헢훖럖늼쫇럱쓜킴돶럖늼몯쫽뗄맘쾵쪽ꆣ M/G/1쒣탍쫇떥럾컱첨뗄뗈듽업뛓쾵춳ꎬ쫤죫맽돌캪늴쯉맽돌ꎬ뛸룷맋뿍뗄럾컱쪱볤쫇쿠뮥뛀솢쟒뻟폐쿠춬럖늼뗄쯦믺뇤솿ꆣ 볙뚨쫤죫맽돌쫇틔λ캪닎쫽뗄늴쯉맽돌ꎬ럾컱쪱볤τ웚췻횵뫍랽닮럖뇰n1λEτ=뫍 Dτꎬ럾컱잿뛈캪ρ=ꆣ떱ρ<1쪱ꎬ뿉뗃컈첬쿂뗄뷡싛ꎺ ()()nnµµP=1−ρ (9-44) 022ρ+λDτ()nL=ρ+ (9-45) 21−ρ()22ρ+λDτ()nL= (9-46) q21−ρ()펦폃샮쳘뛻릫쪽ꎨꎩ벰ꎨꎩ뮹뿉뗃컈첬쿂뗄 LLq W= W= qλλ샽 짨ퟶ쎿쳗컷럾탨튪틀듎뺭맽쯄뗀릤탲ꆣ쒳럾ힰ뗪횻폐튻캻ퟶ컷럾뗄럾ힰ쪦뢵ꆣ짨맋뿍잰살뚨훆컷럾뗄맽돌캪늴쯉맽돌ꎬ욽뻹쎿훜살떽7죋ꎨ쎿죋뚨훆튻쳗컷럾ꎬ쟒짨쎿훜릤ퟷ6쳬ꎬ쎿쳬릤ퟷ10킡쪱ꎩꆣ쎿뗀릤탲쯹탨쪱볤럾듓춬닎쫽뗄뢺횸쫽럖늼ꎬ욽뻹탨튪2킡쪱ꆣ쫔컊튻캻맋뿍듓뚩믵떽ퟶ뫃튻쳗컷럾욽뻹탨튪뛠짙쪱볤ꎿ 1234뷢ꎺ짨 X,X,X,X 럖뇰쫇닃러캪뗚n캻맋뿍러훆컷럾쪱퓚쯄뗀릤탲짏nnnn쯹뮨뗄쪱볤ꆣ쯼쏇쫇쿠뮥뛀솢쟒럾듓쿠춬닎쫽뗄뢺횸쫽럖늼뗄쯦믺뇤솿ꆣ맊캪뗚1234n캻맋뿍러훆튻쳗컷럾쯹탨뗄ퟜ쪱볤 τ=X+X+X+X쫇럾듓4뷗낮뛻샊nnnnn럖늼뗄쯦믺뇤솿ꆣ i평쳢틢횪ꎬ쯦벴뇤솿X(i=1,2,3,4)뗄럖늼몯쫽캪 n−1/2t1−et≥0iP(X≤t)= n0t<0퓙평k뷗낮뛻샊럖늼훐쯦믺뇤솿뗄웚췻횵벰랽닮릫쪽ꎨꎩ뗃 44E(τ)==8 D(τ)==16 nn211221틲듋ꎬ쎿캻맋뿍뗄욽뻹럾컱쪱볤캪=Eτ=8ꆣ쇭췢ꎬ평쳢틢뮹횪 ()nµ
7λ== 60λ7퓲 ρ==×8= µ60평릫쪽ꎨꎩ뿉뗃욽뻹뛓뎤캪 +×16()60 L=+= 2×1−()ퟮ뫳평샮쳘뛻릫쪽 뿉횪쎿캻맋뿍캪뗃떽튻쳗컷럾욽뻹쯹탨쪱볤캪 W== (킡λ쪱) 벴듳풼캪훜ꆣ
