THE SEMI-MARKOV DECISION MODEL in .NET

Maker QR-Code in .NET THE SEMI-MARKOV DECISION MODEL
THE SEMI-MARKOV DECISION MODEL
Denso QR Bar Code Scanner In .NET
Using Barcode Control SDK for VS .NET Control to generate, create, read, scan barcode image in .NET framework applications.
pij (a) =
QR Code ISO/IEC18004 Generation In .NET Framework
Using Barcode generation for .NET Control to generate, create QR Code ISO/IEC18004 image in Visual Studio .NET applications.
j = i, a A(i) and i I , ( / i (a))pij (a), ( / i (a))pij (a) + [1 ( / i (a))], j = i, a A(i) and i I
Recognize QR Code JIS X 0510 In VS .NET
Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET framework applications.
This discrete-time Markov decision model has the same class of stationary policies as the original semi-Markov decision model For each stationary policy R, let g i (R) denote the long-run average cost per time unit in the discrete-time model when policy R is used and the initial state is i Then it holds for each stationary policy R that i I (713) gi (R) = g i (R), This result does not require any assumption about the chain structure of the Markov chains associated with the stationary policies However, we prove the result (713) only for the unichain case Fix a stationary policy R and assume that the embedded Markov chain {Xn } in the semi-Markov model has no two disjoint closed sets Denote by Xn the state at the nth decision epoch in the transformed discretetime model It is directly seen that the Markov chain {Xn } is also unichain under policy R The equilibrium probabilities j (R) of the Markov chain {X n } satisfy the equilibrium equations j (R) =
Create Barcode In .NET
Using Barcode generator for .NET framework Control to generate, create bar code image in VS .NET applications.
i (R)p ij (Ri ) i (R)
Bar Code Scanner In .NET Framework
Using Barcode scanner for .NET framework Control to read, scan read, scan image in .NET applications.
pij (Ri ) + 1 j (R), i (Ri ) j (Rj )
Print QR Code 2d Barcode In Visual C#
Using Barcode encoder for .NET Control to generate, create Quick Response Code image in VS .NET applications.
j I
Making QR In VS .NET
Using Barcode maker for ASP.NET Control to generate, create QR image in ASP.NET applications.
Hence, letting uj = j (R)/ j (Rj ) and dividing by , we nd that uj =
QR Code Printer In VB.NET
Using Barcode drawer for Visual Studio .NET Control to generate, create Denso QR Bar Code image in Visual Studio .NET applications.
ui pij (Ri ),
Print Bar Code In VS .NET
Using Barcode generation for VS .NET Control to generate, create barcode image in .NET applications.
j I
UPC-A Supplement 2 Creation In VS .NET
Using Barcode encoder for Visual Studio .NET Control to generate, create UPC-A Supplement 2 image in .NET applications.
These equations are precisely the equilibrium equations for the equilibrium probabilities j (R) of the embedded Markov chain {Xn } in the semi-Markov model The equations determine the j (R) uniquely up to a multiplicative constant Thus, for some constant > 0, j (R) = j (R) , j (Rj ) j I
European Article Number 13 Generation In VS .NET
Using Barcode generator for .NET framework Control to generate, create EAN13 image in VS .NET applications.
j I j (Rj ) j (R)
EAN-8 Maker In Visual Studio .NET
Using Barcode generation for Visual Studio .NET Control to generate, create EAN 8 image in Visual Studio .NET applications.
Since j I j (R) = 1, it follows that = (713) now follows easily We have g(R) =
Generate EAN128 In Visual Studio .NET
Using Barcode generator for ASP.NET Control to generate, create EAN 128 image in ASP.NET applications.
j I
Encoding Bar Code In .NET
Using Barcode printer for ASP.NET Control to generate, create barcode image in ASP.NET applications.
The desired result
Draw Bar Code In Java
Using Barcode generation for Java Control to generate, create barcode image in Java applications.
cj (Rj ) j (R) = cj (Rj ) j (R)/
Create Code39 In Visual C#.NET
Using Barcode creator for .NET framework Control to generate, create Code39 image in .NET applications.
j I
Scan Bar Code In VS .NET
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications.
j I
Encode Data Matrix 2d Barcode In .NET Framework
Using Barcode creator for ASP.NET Control to generate, create DataMatrix image in ASP.NET applications.
cj (Rj ) j (R) j (Rj ) j (Rj )
Generating Barcode In Visual C#
Using Barcode generator for VS .NET Control to generate, create barcode image in .NET applications.
j (Rj ) j (R)
Code 128 Code Set A Encoder In Visual C#
Using Barcode generation for .NET Control to generate, create Code 128B image in VS .NET applications.
j I
SEMI-MARKOV DECISION PROCESSES
and so, by Theorem 711, g(R) = g(R) Thus we can conclude that an average cost optimal policy in the semi-Markov model can be obtained by solving an appropriate discrete-time Markov decision model This conclusion is particularly useful with respect to the value-iteration algorithm In applying value iteration to the transformed model, it is no restriction to assume that for each stationary policy the associated Markov chain {X n } is aperiodic By choosing the constant strictly less than mini,a i (a), we always have pii (a) > 0 for all i, a and thus the required aperiodicity
ALGORITHMS FOR AN OPTIMAL POLICY
In this section we outline how the algorithms for the discrete-time Markov decision model can be extended to the semi-Markov decision model Policy-iteration algorithm The policy-iteration algorithm will be described under the unichain assumption This assumption requires that for each stationary policy the embedded Markov chain {Xn } has no two disjoint closed sets By data transformation, it is directly veri ed that the value-determination equations (632) for a given stationary policy R remain valid provided that we replace g by g i (Ri ) The policy-improvement procedure from Theorem 621 also remains valid when we replace g by g i (R i ) Suppose that g(R) and i (R), i I , are the average cost and the relative values of a stationary policy R If a stationary policy R is constructed such that, for each state i I , ci (R i ) g(R) i (R i ) +
j I
pij (R i ) j (R) i (R),
(721)
then g(R) g(R) Moreover, g(R) < g(R) if the strict inequality sign holds in (721) for some state i which is recurrent under R These statements can be veri ed by the same arguments as used in the second part of the proof of Theorem 621 Under the unichain assumption, we can now formulate the following policyiteration algorithm: Step 0 (initialization) Choose a stationary policy R Step 1 (value-determination step) For the current rule R, compute the average cost g(R) and the relative values i (R), i I , as the unique solution to the linear equations i = ci (Ri ) g i (Ri ) +