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Unformatted text preview: Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience LongRun Behavior Examples 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/ 34 Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience LongRun Behavior Examples Definition Transition Diagram Examples of Birth Death Processes Birth and Death Processes The birth and death process is easiest to see with an example. Suppose X ( t ) = the number of people in a system at time t . What is the state space of { X ( t ) , t } ? The state space is the set of nonnegative integers (or some subset thereof) The BD process has several properties when there are n people in the system, new arrivals occur one at a time, the rate of these arrivals is n . when there are n people in the system, people leave one at a time, the rate of these departures is n . the times between arrivals are independent exponentials as are the times between departures. the arrival and service times are independent. The parameters { n , n } and { n , n } are called the birth and death rates, respectively. 2/ 34 Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience LongRun Behavior Examples Definition Transition Diagram Examples of Birth Death Processes Birth and Death Processes The birth and death process is easiest to see with an example. Suppose X ( t ) = the number of people in a system at time t . What is the state space of { X ( t ) , t } ? The state space is the set of nonnegative integers (or some subset thereof) The BD process has several properties when there are n people in the system, new arrivals occur one at a time, the rate of these arrivals is n . when there are n people in the system, people leave one at a time, the rate of these departures is n . the times between arrivals are independent exponentials as are the times between departures. the arrival and service times are independent. The parameters { n , n } and { n , n } are called the birth and death rates, respectively. 2/ 34 Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience LongRun Behavior Examples Definition Transition Diagram Examples of Birth Death Processes Birth and Death Processes The birth and death process is easiest to see with an example. Suppose X ( t ) = the number of people in a system at time t . What is the state space of { X ( t ) , t } ?...
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This note was uploaded on 10/07/2010 for the course ORIE 3510 taught by Professor Resnik during the Spring '09 term at Cornell University (Engineering School).
 Spring '09
 RESNIK

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