RENEWAL-REWARD PROCESSES in Visual Studio .NET

Create Quick Response Code in Visual Studio .NET RENEWAL-REWARD PROCESSES
RENEWAL-REWARD PROCESSES
Denso QR Bar Code Reader In .NET
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET applications.
Also, de ne the random variable TB = the amount of time the process spends in the set B of states during one cycle Note that TB = 0 1 IB (u) du for a continuous-time process {X(t)}; otherwise, TB equals the number of indices 0 k < S1 with X(k) B The following theorem is an immediate consequence of the renewal-reward theorem Theorem 223 For the regenerative process {X(t)} it holds that the long-run fraction of time the process spends in the set B of states is E(TB )/E(C1 ) with probability 1 That is, 1 t t lim
Print Quick Response Code In .NET Framework
Using Barcode maker for VS .NET Control to generate, create Quick Response Code image in .NET framework applications.
t 0 S
Denso QR Bar Code Reader In VS .NET
Using Barcode recognizer for .NET Control to read, scan read, scan image in .NET applications.
IB (u) du =
Print Barcode In .NET Framework
Using Barcode generator for VS .NET Control to generate, create bar code image in Visual Studio .NET applications.
E(TB ) E(C1 )
Bar Code Recognizer In .NET Framework
Using Barcode recognizer for .NET framework Control to read, scan read, scan image in VS .NET applications.
with probability 1
QR Code Drawer In C#
Using Barcode generation for .NET Control to generate, create QR Code JIS X 0510 image in Visual Studio .NET applications.
for a continuous-time process {X(t)} and 1 n n lim
Creating QR-Code In .NET
Using Barcode encoder for ASP.NET Control to generate, create QR Code JIS X 0510 image in ASP.NET applications.
IB (k) =
Drawing Denso QR Bar Code In Visual Basic .NET
Using Barcode creation for VS .NET Control to generate, create QR-Code image in .NET framework applications.
E(TB ) E(C1 )
Create EAN128 In .NET Framework
Using Barcode drawer for .NET framework Control to generate, create EAN 128 image in .NET applications.
with probability 1
Barcode Creator In Visual Studio .NET
Using Barcode printer for VS .NET Control to generate, create bar code image in .NET framework applications.
for a discrete-time process {X(n)} Proof The long-run fraction of time the process {X(t)} spends in the set B of states can be interpreted as a long-run average reward per time unit by assuming that a reward at rate 1 is earned while the process is in the set B and a reward at rate 0 is earned otherwise Then E(reward earned during one cycle) = E(TB ) The desired result next follows by applying the renewal-reward theorem Since E(IB (t)) = P {X(t) B}, we have as consequence of Theorem 223 and the bounded convergence theorem that, for a continuous-time process, 1 t t lim
EAN 13 Drawer In .NET
Using Barcode drawer for .NET Control to generate, create European Article Number 13 image in .NET applications.
t 0 t 0
UPC E Generation In .NET
Using Barcode printer for Visual Studio .NET Control to generate, create GS1 - 12 image in .NET applications.
P {X(u) B} du =
Barcode Drawer In Java
Using Barcode creator for Java Control to generate, create barcode image in Java applications.
E(TB ) E(C1 )
Barcode Drawer In Visual C#
Using Barcode printer for .NET Control to generate, create barcode image in .NET framework applications.
Note that (1/t) P {X(u) B} du can be interpreted as the probability that an outside observer arriving at a randomly chosen point in (0, t) nds the process in the set B In many situations the ratio E(TB )/E(C1 ) could be interpreted both as the longrun fraction of time the process {X(t)} spends in the set B of states and as the probability of nding the process in the set B when the process has reached statistical equilibrium This raises the question whether limt P {X(t) B} always exists This ordinary limit need not always exist A counterexample is provided by
Create ANSI/AIM Code 128 In Visual Studio .NET
Using Barcode drawer for ASP.NET Control to generate, create ANSI/AIM Code 128 image in ASP.NET applications.
RENEWAL-REWARD PROCESSES
Bar Code Encoder In Visual C#.NET
Using Barcode creator for Visual Studio .NET Control to generate, create bar code image in .NET applications.
periodic discrete-time Markov chains; see 3 For completeness we state the following theorem Theorem 224 For the regenerative process {X(t), t T },
Scan Code 3/9 In .NET Framework
Using Barcode recognizer for .NET Control to read, scan read, scan image in Visual Studio .NET applications.
lim P {X(t) B} =
Make Barcode In VB.NET
Using Barcode creation for .NET framework Control to generate, create bar code image in .NET applications.
E(TB ) E(C1 )
Decoding EAN-13 In .NET Framework
Using Barcode scanner for .NET Control to read, scan read, scan image in VS .NET applications.
provided that the probability distribution of the cycle length has a continuous part in the continuous-time case and is aperiodic in the discrete-time case A distribution function is said to have a continuous part if it has a positive density on some interval A discrete distribution {aj , j = 0, 1, } is said to be aperiodic if the greatest common divisor of the indices j 1 for which aj > 0 is equal to 1 The proof of Theorem 224 requires deep mathematics and is beyond the scope of this book The interested reader is referred to Miller (1972) It is remarkable that the proof of Theorem 223 for the time-average limit t limt (1/t) 0 IB (u) du is much simpler than the proof of Theorem 224 for the ordinary limit limt P {X(t) B} This is all the more striking when we take into account that the time-average limit is in general much more useful for practical purposes than the ordinary limit Another advantage of the timeaverage limit is that it is easier to understand than the ordinary limit In interpreting the ordinary limit one should be quite careful The ordinary limit represents the probability that an outside person will nd the process in some state of the set B when inspecting the process at an arbitrary point in time after the process has been in operation for a very long time It is essential for this interpretation that the outside person has no information about the past of the process when inspecting the process How much more concrete is the interpretation of the timeaverage limit as the long-run fraction of time the process will spend in the set B of states! To illustrate Theorem 224, consider again Example 221 In this example we analysed the long-run average behaviour of the regenerative process {X(t)}, where X(t) = 1 if the machine is up at time t and X(t) = 0 otherwise It was shown that the long-run fraction of time the machine is down equals E(D)/[E(U ) + E(D)], where the random variables U and D denote the lengths of an up-period and a down-period This result does not require any assumption about the shapes of the probability distributions of U and D However, some assumption is needed in order to conclude that
Paint GTIN - 128 In Java
Using Barcode printer for Java Control to generate, create EAN / UCC - 14 image in Java applications.
lim P {the system is down at time t} =
E(D) E(U ) + E(D)
(222)
It is suf cient to assume that the distribution function of the length of an up-period has a positive density on some interval We state without proof a central limit theorem for the renewal-reward process