Markov Jump Introduction
Markov Jumps describe transitions in systems with discrete states and continuous time. More versatile than classic Markov chains, they capture real-world processes like queueing networks and stock prices.
Defining Jump Rates
Each state transition in a Markov jump process has an associated rate, determining how quickly transitions occur. These rates are key to the exponential waiting times between jumps.
Infinitesimal Generator Matrix
The generator matrix defines transition rates for a Markov jump process. Its off-diagonal elements are non-negative, representing the rate from one state to another.
Embedded Markov Chain
By examining the transitions without timing, a Markov jump process reduces to an embedded Markov chain. This chain only considers the sequence of states, not the duration spent in them.
Applications in Queuing Theory
In queuing theory, Markov jump processes model systems like customer service lines, with jumps representing arrivals and departures and rates reflecting service and arrival speeds.
Continuous-Time Birth-Death Processes
A special case of Markov jump processes, birth-death processes, model populations where each state change is either an increment (birth) or a decrement (death) with specific rates.
Solving with Kolmogorov Equations
To analyze Markov jump processes, one uses Kolmogorov's forward and backward equations. They provide probabilities for each state over time, fundamental for understanding the process dynamics.
What does 'Markov Jump' imply?
Discrete states, continuous time transitions
Continuous states, discrete time transitions
Continuous states and time transitions
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