MS&E 223
Lecture Notes #3
Simulation
Generalized SemiMarkov Processes
Peter J. Haas
Spring Quarter 200910
Generalized SemiMarkov Processes (GSMP’s)
Ref
: Section 1.4 in Shedler or Section 4.1 in Haas
1.
Motivation
The Markov and semiMarkov models that we have discussed previously do not have sufficient modeling
power to capture many of the complex discreteevent 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 semiMarkov 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 transitiona 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 statedependent 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).
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Lecture Notes #3
Simulation
Generalized SemiMarkov Processes
Peter J. Haas
Spring Quarter 200910
•
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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 Spring '09
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 GSMP, Peter J. Haas, Simulation Peter J., Generalized SemiMarkov Processes, Processes Spring Quarter

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