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lecturenotes03 - MS&E 223 Simulation Peter J Haas Lecture...

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MS&E 223 Lecture Notes #3 Simulation Generalized Semi-Markov Processes Peter J. Haas Spring Quarter 2009-10 Generalized Semi-Markov Processes (GSMP’s) Ref : Section 1.4 in Shedler or Section 4.1 in Haas 1. Motivation The Markov and semi-Markov models that we have discussed previously do not have sufficient modeling power to capture many of the complex discrete-event stochastic systems that arise in practice. The exponential distributional assumptions of the CTMC model often do not hold; neither does the implicit assumption in the semi-Markov process model that only a single “clock” is running in each state. The GSMP model avoids these restrictive assumptions. See the textbook by Gerald Shedler for a thorough treatment of GSMP’s. Heuristically, a GSMP {X(t): t 0} makes stochastic state transitions when one or more events associated with the occupied state occur. (Associated events = events that can possibly occur in the state = events that are scheduled in the state.) events associated with a state “compete” to trigger the next state transition each event has its own distribution for determining the next state new events can be scheduled at each state transition---a new event with respect to a state transition is an event that is associated with the new state and either (a) is not associated with the old state or (b) is associated with the old state and also triggers the state transition for each new event, a clock is set with a reading that indicates the time until the event is scheduled to occur; when the clock runs down to 0 the event occurs (unless it is cancelled in the interim) clock reading time xx o an old event with respect to a state transition is an event, associated with the old state, that does not trigger the state transition and is associated with the next state; its clock continues to run down a cancelled event with respect to a state transition doesn’t trigger the state transition and is not associated with the next state; its clock reading is discarded clocks can run down at state-dependent speeds 2. GSMP Building Blocks S : a (finite or countably infinite) set of states E = {e 1 , e 2 , . .. , e M }: a finite set of events E(s) : the set of events scheduled to occur in state s S. Of course, E(s) E. We say that event e is active in s if e E(s). Page 1 of 8

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MS&E 223 Lecture Notes #3 Simulation Generalized Semi-Markov Processes Peter J. Haas Spring Quarter 2009-10 p( ; s, E ' s * ) : the probability that the new state is given that the events in E ' s * simultaneously occur in s. If E * = {e*} for some e* E(s), then we simply write p( ; s, e*). ' s r(s, e) : the nonnegative finite speed at which clock for e runs down in state s; typically r(s, e)=1, but can be set to other values in order to model “processor sharing” or “preempt resume” service discipline. (For modeling the latter we allow r(s, e) = 0.) F( ; , , s, E ' s ' e * ) : the distribution function used to set the clock for the new event when the simultaneous occurrence of the events in E ' e * triggers a state transition from s to . ' s
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lecturenotes03 - MS&E 223 Simulation Peter J Haas Lecture...

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