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CTMCIII-beamer

# CTMCIII-beamer - Birth and Death Processes Communication...

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Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience Long-Run 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

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Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience Long-Run 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 0 } ? The state space is the set of non-negative integers (or some subset thereof) The B-D 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 λ n . when there are n people in the system, people leave one at a time, the rate of these departures is μ n . the times between arrivals are independent exponentials as are the times between departures. the arrival and service times are independent. The parameters { λ n , n 0 } and { μ n , n 0 } are called the birth and death rates, respectively. 2/ 34
Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience Long-Run 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 0 } ? The state space is the set of non-negative integers (or some subset thereof) The B-D 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 λ n . when there are n people in the system, people leave one at a time, the rate of these departures is μ n . the times between arrivals are independent exponentials as are the times between departures. the arrival and service times are independent. The parameters { λ n , n 0 } and { μ n , n 0 } are called the birth and death rates, respectively. 2/ 34

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Birth and Death Processes Communication and Irreducibility for CTMCs Recurrence and Transience Long-Run 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 0 } ? The state space is the set of non-negative integers (or some subset thereof) The B-D 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 λ n .
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