On Completion Times of Networks of Sequential and

Concurrent Tasks

Daniel BerleantJianzhong Zhang

Gerald ShebléNSF Workshop on Reliable

Engineering Computing, Sept. 15-17, 2004

Strategy Use intervals to discretize, manipulate,

and generalize probability distributions Others here also use intervals for

this or related purposes Ray Moore, Vladik Kreinovich, Arnold

Neumaier (clouds), Scott Ferson, Fulvio Tonon, Janos Hajagos

On Completion Times of Networks of Sequential and Concurrent Tasks

Time in Sequence and in Parallel

A large set of practical problems involve Subproblems that involve durations Subproblems may be prerequisites of each

other …so one must complete before next begins …total time is the sum of subproblem times

Subproblems might not be prerequisites …so they can be solved concurrently …total time is the maximum of subproblem

times

A Few Example ProblemsTime to…

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)

(D. Berleant and J. Zhang, Bounding the times to failure of

2-component systems, IEEE Trans. Reliability, in press.)

complete an activity in an engineering project= max(times to complete sub-activities)

Uncertainty and Concurrent 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 (non-trivial without independence assumption)

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)

Let’s Focus on Activity Networks

…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)

Activity Network Examples

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

Outline of the Argument

(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(Independent Case)

The joint tableau contains marginals (shaded) & a random set (unshaded)

Two Concurrent Activities(Unknown Dependency)

Sequential Activities

Similar to concurrent tasks, except Use “+” instead of “max”

Extending to Activity Networks I

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