The functional equations of undiscounted denumerable state Markov renewal programming

Elke Mann
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Elke Mann: Universität Bonn, Institut für Angewandte Mathematik

A chapter in Semi-Markov Models, 1986, pp 79-96 from Springer

Abstract: Abstract In this paper we want to establish conditions for the existence of a finite solution of the functional equations 1 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeaaaaaaaaa8 % qacaWGNbWaaeWaa8aabaWdbiaadMgaaiaawIcacaGLPaaacqGH9aqp % paWaaCbeaeaapeGaciyBaiaacggacaGG4baal8aabaWdbiaadggacq % GHiiIZcaWGHbWaaeWaa8aabaWdbiaaigdaaiaawIcacaGLPaaaa8aa % beaakmaaxababaGaaGjbV-qacqqHJoWuaSWdaeaapeGaamOAaiabgI % GiolaadMeaa8aabeaakiaaysW7peGaamiCamaabmaapaqaa8qacaWG % PbGaaiilaiaadggacaGGSaGaamOAaaGaayjkaiaawMcaaiaadEgada % qadaWdaeaapeGaamOAaaGaayjkaiaawMcaaiaacYcapaGaaGjbV-qa % caWGPbGaeyicI4SaamysaiaacYcapaGaaGjbVdaa!5E4E! $$ g\left( i \right) = \mathop {\max }\limits_{a \in a\left( 1 \right)} \mathop {\;\Sigma }\limits_{j \in I} \;p\left( {i,a,j} \right)g\left( j \right),\;i \in I,\;$$ 2 % MathType!MTEF!2!1!+- % feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn % hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr % 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 % vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x % fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaabm % aabaGaamyAaaGaayjkaiaawMcaaiabg2da9maaxababaGaciyBaiaa % cggacaGG4baaleaacaWGHbGaeyicI4SaamyqamaaBaaameaacaWGVb % aabeaalmaabmaabaGaamyAaaGaayjkaiaawMcaaaqabaGccaaMe8+a % amWaaeaacaWGYbWaaeWaaeaacaWGPbGaaiilaiaadggaaiaawIcaca % GLPaaacqGHsislcaWG0bWaaeWaaeaacaWGPbGaaiilaiaadggaaiaa % wIcacaGLPaaacaWGNbWaaeWaaeaacaWGPbaacaGLOaGaayzkaaGaey % 4kaSYaaCbeaeaacqqHJoWuaSqaaiaadQgacqGHiiIZcaWGjbaabeaa % kiaadchadaqadaqaaiaadMgacaGGSaGaamyyaiaacYcacaWGQbaaca % GLOaGaayzkaaGaamODamaabmaabaGaamOAaaGaayjkaiaawMcaaaGa % ay5waiaaw2faaiaacYcacaWGPbGaeyicI4Saamysaaaa!6A1D! $$ v\left( i \right) = \mathop {\max }\limits_{a \in {A_o}\left( i \right)} \;\left[ {r\left( {i,a} \right) - t\left( {i,a} \right)g\left( i \right) + \mathop \Sigma \limits_{j \in I} p\left( {i,a,j} \right)v\left( j \right)} \right],i \in I$$ where I is a denumerable set, A(i) is a compact metric space for all i ϵ I, 0

Keywords: Decision Rule; Markov Decision Process; Reward Function; Optimality Equation; Gain Rate (search for similar items in EconPapers)
Date: 1986
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DOI: 10.1007/978-1-4899-0574-1_6

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