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A part of the ece/cs 541: computer system analysis course at the university of illinois at urbana-champaign. It covers state-based methods for analyzing discrete time markov chains (dtmcs) and continuous time markov chains (ctmcs). Topics include markov processes, state occupancy probability vectors, transition probability matrices, accessibility, flow equations, and event rates.
Typology: Lab Reports
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Module 3, Slide 1
Prof. William H. Sanders and David M. Nicol Department of Electrical and Computer Engineering and
Coordinated Science Laboratory
University of Illinois at Urbana-Champaign
http://www.crhc.uiuc.edu/PERFORM
Module 3, Slide 2
Recall that
availability
quantifies the alternation between proper and improper
service.–
( t
) is 1 if service is proper, 0 otherwise.
( t
)] is the probability that service is proper at time
t.
t ) is the fraction of time the system delivers proper service during [0,
t ].
For many systems, availability is a more “user-oriented” measure thanreliability.
-^
However, it is often more difficult to compute, since it must account for repairand/or replacement.
Module 3, Slide 4
Availability modeling can be done with combinatorial methods, but only withthe independent repair assumption. More accurate modeling with state-basedmethods relaxes the independence assumptions.–
Failures need not be independent. Failure of one component may makeanother component more or less likely to fail.
-^
Repairs need not be independent. Repair and replacement strategies are animportant component that must be modeled in high-availability systems.
-^
High-availability systems may operate in a degraded mode. In a degradedmode, the system may deliver only a fraction of its services, and the repairprocess may start only after the system is sufficiently degraded.
We use random processes to model these systems.
Module 3, Slide 5
Random processes are useful for characterizing the behavior of real systems.A
random process
is a collection of random variables indexed by time.
Example:
( t
) is a random process. Let
(1) be the result of tossing a die. Let
be the result of tossing a die plus
(1), and so on. Notice that time (
One can ask:
( ) [^
]
( )
( )
( ) [^
]^
n
n X E
(^136)
(^136)
Module 3, Slide 7
Recall that for a random variable
, we can use the cumulative distribution
X^
to
describe the random variable.In general, no such simple description exists for a random process.However, a random process can often be described succinctly in various differentways. For example, if
is a random variable representing the roll of a die, and
( t
is the sum after
t rolls, then we can describe
( t
) by
( t
( t
( t
i |
( t
j
i
j
or
( t
1
2
, where the t
’s are independent. i
Module 3, Slide 8
T
If the number of time points defined for a random process, i.e., |
|, is finite or
countable (e.g., integers), then the random process is said to be a
discrete-time
random process
If |
| is uncountable (e.g., real numbers) then the random process is said to be a continuous-time random process
Example: Let
( t
) be the number of fault arrivals in a system up to time
t. Since
t^
is a real number,
( t
) is a continuous-time random process.
Module 3, Slide 10
If the state space
of a random process
is finite or countable
(e.g.,
= {1,2,3,.. .}), then
is said to be a
discrete-state random process
Example: Let
be a random process that represents the number of bad
packets received over a network.
is a discrete-state random process.
If the state space
of a random process
is infinite and uncountable (e.g.,
then
is said to be a
continuous-state random process
Example: Let
be a random process that represents the voltage on a
telephone line.
is a continuous-state random process.
We examine only discrete-state processes in this lecture.
Module 3, Slide 11
Analog signal
A to D converter
Computeravailability
model
round-based
networkprotocolmodel
Time
State Continuous
Discrete
Discrete
Continuous
Module 3, Slide 13
Markov chain
is a Markov process with a discrete state space.
We will always make the assumption that a Markov chain has a state space in{1,2,.. .} and that it is time-homogeneous.A Markov chain is
time-homogeneous
if its future behavior does not depend on
what time it is, only on the current state (i.e., the current value).We make this concrete by looking at a
discrete-time Markov chain
(hereafter
has the following property:
(^
)^
( )
(^
)^
(^
)^
(^
)
(^
)^
( )
(^ ) k ij
O
t
t
i t Xj
k t X P
n O X n t X n t X i t X j k t X P
−
−^
2
1
Module 3, Slide 14
Notice that given
i ,^ j
, and
k
is a number!
can be interpreted as the probability that if
has value
i , then after
k
time-steps,
will have value
j
Frequently, we write
to mean
(^
) k Pij
(^
) k Pij
P^ ij
(^ )
P^ ij
Module 3, Slide 16
Notice that given
i ,^ j
, and
k
is a number!
can be interpreted as the probability that if
has value
i , then after
k
time-steps,
will have value
j
Frequently, we write
to mean
(^
) k Pij
(^
) k Pij
P^ ij
(^ )
P^ ij
Module 3, Slide 17
Let
π
be a row vector. We denote
π
to be the i
i -th element of the vector. If
π
is a
state occupancy probability
vector, then
π
( ki
) is the probability that a DTMC has
value
i (or is in state
i ) at time-step
k
Assume that a DTMC
has a state-space size of
n
, i.e.,
n
}. We say
formally
π
( ki
( k
i
Note that
for all times
k
1
π
i^
k
Module 3, Slide 19
Notice that this resembles vector-matrix multiplication.In fact, if we arrange the matrix
}, that is, if ij
then
p
ij^
, and ij
π
π
, where
π
(0) and
π
(1) are row vectors, and
π
is a
vector-matrix multiplication.The important consequence of this is that we can easily specify a DTMC in terms ofan occupancy probability vector
π
and a transition probability matrix
all for
holds
which,
have We
1
j
n i
ij
i
j^
π
=
π
p^ 1n
p^11 p^ n
p^ nn
Module 3, Slide 20
Given
π
(0) and
, how can we compute
π
( k
We can generalize from earlier that
π ( k
π
( k
Also, we can write
π
( k
π
( k
, and so
π ( k
π ( k
π
( k
2
Similarly,
π
( k
π
( k
, and so
π ( k
π ( k
2
π
( k
3
By repeating this, it should be easy to see that
π ( k
π
k
