On Completion Times of
Networks of Sequential and
Concurrent Tasks
On Completion Times of
Networks of Sequential and
Concurrent Tasks
Strategy
Daniel Berleant
Jianzhong Zhang
Gerald Sheblé
NSF Workshop on Reliable Engineering
Computing, Sept. 15-17, 2004
…so one must complete before next begins
…total time is the sum of subproblem times
Real-valued times x & y: max(x, y)
Interval-valued times: [max(xl, yl),max(xh,
yh)]
Random times:
Fx(t)Fy(t), if x, y independent
(non-trivial without independence assumption)
Probability box times:
Fx(t)Fy(t), if independent
(D. Berleant and J. Zhang, Bounding the times to failure of
2-component systems, IEEE Trans. Reliability, in press.)
…so they can be solved concurrently
…total time is the maximum of subproblem
times
Uncertainty and Concurrent Times
reach a quorum to begin a meeting
= max(arrival times of necessary participants)
go broke
= max(times to deplete bank accounts)
use up the world’s oil reserves
= max(times to use up oil fields)
failure of a redundant 2-component device
= max(times to failure of the components)
Subproblems might not be prerequisites
Time to…
Subproblems that involve durations
Subproblems may be prerequisites of each
other
Ray Moore, Vladik Kreinovich, Arnold
Neumaier (clouds), Scott Ferson, Fulvio
Tonon, Janos Hajagos
A Few Example Problems
A large set of practical problems involve
Others here also use intervals for this or
related purposes
Time in Sequence
and in Parallel
Use intervals to discretize, manipulate, and
generalize probability distributions
complete an activity in an engineering project
= max(times to complete sub-activities)
Uncertainty and Sequential Times
Real-valued times x & y: x + y
Interval-valued times: [(xl + yl), (xh + yh)]
Can use intervals to discretization & compute:
Random times: sum of R.V.’s – use convolution
(trickier without independence assumption)
Probability box times: sum of uncertain R.V.’s
(trickier without independence assumption)
(non-trivial without independence assumption)
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Let’s Focus on Activity Networks
Activity Network Examples
…a problem in engineering project management
(1) Model time of an activity as a sum of its factors
(2) Time to complete sequential activities is sum
(3) Time to complete concurrent activities is max
Use (1) to help compute (2)
Use (2) & (3) to help compute activity networks
(such a network is a DAG)
Outline of the Argument
Previous Work
A large literature on activity networks exists
Careers have been based on the topic
Many aspects have been examined in detail
Correlations have been addressed
Lack of knowledge about dependency has been
less examined
Problems involving severe uncertainty present new
challenges & opportunities
The work closest to the present paper models
uncertainty with band copulas
Two Concurrent Activities
(Independent Case)
(1) Time to complete concurrent activities is max
(2) Compute activity time as the sum of its factors
(3) Time to complete sequential activities is sum
(2) and (3) are computed the same way
Use (1) & (3) to help compute activity networks
(such a network is a DAG)
Two Concurrent Activities
(Unknown Dependency)
The joint tableau contains marginals (shaded) & a random set (unshaded)
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Extending to Activity Networks I
Sequential Activities
Similar to concurrent tasks, except
Use “+” instead of “max”
Three sequential activities:
Add two, add result and third activity
Problem: partial result is not a CDF
It is a “probability box”
Solution:
convert a p-box to set of pairs (p, i)
This set can be a tableau marginal
Extending to Activity Networks II
Three concurrent activities:
Compute two, then compute result and
third
(Same strategy as sum of 3)
Extending to Activity Networks III
Propagate sums & maxes throughout the DAG
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