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Unformatted text preview: Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Introductory Engineering Stochastic Processes, ORIE 3510 Instructor: Mark E. Lewis, Associate Professor School of Operations Research and Information Engineering Cornell University Disclaimer : Notes are only meant as a lecture supplement not substitute! 1/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs DTMC Example Consider a DTMC with the following onestep transition matrix. P = 1 2 3 4 5 6 7 8 9 1 1 1 2 / 3 1 / 3 1 / 2 1 / 2 1 / 2 1 / 2 0 2 / 3 1 / 3 1 / 2 1 / 2 1 / 4 3 / 4 2/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Transition Diagram 5 2 3 6 7 4 8 9 1 1 1 1 1/2 1/2 2/3 1/3 1/4 3/4 1/3 1/2 1/2 2/3 1/2 1/2 3/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Transition Diagram 5 2 3 6 7 4 8 9 1 1 1 1 1/2 1/2 2/3 1/3 1/2 1/4 3/4 1/3 1/2 1/2 2/3 1/2 3/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Classes and Short Term Calculations Recurrent: { 1 , 2 , 3 } , { 4 , 5 } Transient: { 6 , 7 } , { 8 } , { 9 } Suppose f ( i ) = i for i = 1 , 2 ,..., 9 Hopefully by now we know that we know how to compute E ( f ( X 3 )  X = i ) Compute P 3 by matrix multiplication This yields p (3) ij for all i , j S . Compute E ( f ( X 3 )  X = i ) = j S f ( j ) p (3) ij (this is 1.3891 for i = 7) 4/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Classes and Short Term Calculations Recurrent: { 1 , 2 , 3 } , { 4 , 5 } Transient: { 6 , 7 } , { 8 } , { 9 } Suppose f ( i ) = i for i = 1 , 2 ,..., 9 Hopefully by now we know that we know how to compute E ( f ( X 3 )  X = i ) Compute P 3 by matrix multiplication This yields p (3) ij for all i , j S . Compute E ( f ( X 3 )  X = i ) = j S f ( j ) p (3) ij (this is 1.3891 for i = 7) 4/ 16 Transition Matrix Transition Diagram Classes and Short Term Calculations LongTerm Behavior Longterm Behavior Recurrent States Average Costs Classes and Short Term Calculations Recurrent: { 1 , 2 , 3 } , { 4 , 5 } Transient: { 6 , 7 } , { 8 } , { 9 } Suppose f ( i ) = i for i = 1 , 2 ,..., 9 Hopefully by now we know that we know how to compute E ( f ( X 3 )  X = i ) Compute P 3 by matrix multiplication This yields p (3) ij for all i , j S ....
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