DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

More documents

Recommendations

Info

4 t Si t ∫ + ( ) i ) t When RA i (t) remains constant S i (t) becomes: RA ( τ dτ = P . (4) i Pi Si ( t) = . (5) RA ( t) Now, consider the example network in Fig. 1. In the network only N 1 has the chance to allocate its resource as it has two residing components. T N1 is invariant to resource allocation and equal to 300 (=100*1+100*2). But, T A1 and T A2 can vary depending on the resource allocation of N 1 . When the resource is allocated equally to the components, both RA A1 (t) and RA A2 (t) are equal to 0.5 initially. As A 1 completes at t=200 (=100*1/0.5), A 2 starts utilizing the resource fully from then, i.e. RA A2 (t)=1 for t≥200. So, A 2 completes 50 tasks at t=200 (=50*2/0.5) and remaining 50 tasks at t=300 (=200+50*2/1). A 3 completes at t=202 (=200+1*2/1) because task inter-arrival time from A 1 is equal to its service time. As A 4 ’s service time is less than task inter-arrival time (=4) for t≤200, A 4 completes 49 tasks at t=200 with one task in queue arriving at t=200. From t=200 task inter-arrival time from A 2 becomes reduced to 2 which is less than A 4 ’s service time. So, tasks become accumulated till t=300 and A 4 completes at t=353 (=200+51*3/1). In this way we trace exact system behavior under three resource allocation strategies as shown in Fig. 2. RA 1 4/5 2/3 1/2 1/3 1/5 1:1 1:2 1:4 0 50 100 150 200 250 w A1 : w A2 1 : 1 1 : 2 1 : 4 T A1 200 300 300 T A2 300 300 250 T A3 202 302 352 T A4 353 303 302.5 T 353 303 352 (a) Completion time 300 Time The network cannot complete at less than t=300 because each of N 1 and N 3 requires 300 CPU time. When the resource is allocated with 1:2 ratio, the completion time T is minimal close to 300. The ratio is proportional to each component’s total required CPU time, i.e., 1:2 ≡ 100*1:100*2. One interesting question is whether the proportional allocation can give the best performance even if the successors have different parameters. i RA 1 1/2 1/3 1/5 0 50 100 150 200 250 300 Time (b) Resource availability of A 1 (c) Resource availability of A 2 Fig. 2. Effects of resource allocation. Depending on the resource allocation of node N 1 , each of components A 1 and A 2 follows different resource availability profile as in (b) and (c). Consequently, the difference results in different completion times as in (a). 4/5 2/3 1:1 1:2 1:4 The answer is yes. If a component A 1 is allocated more resource than the proportional allocation, T A3 is dominated by the maximal of T A1 and A 3 ’s total CPU time. But, the first quantity is less than T N1 and the second quantity is an invariant. So, allocating more resource than the proportional allocation cannot help reducing the completion time of the network. However, if a component is allocated less resource than the proportional allocation, its successor’s task inter-arrival time is stepwise decreasing. As a result, the successor underutilizes resource and can complete later than under the proportional allocation. Therefore, the proportional allocation leads the network to efficiently utilize distributed resources and consequently helps minimizing the completion time of the network, though it is localized independent of the successors’ parameters. B. Optimal Open-loop Policy To generalize the arguments for arbitrary network configurations, we define Load Index LI i which represents component i’s total CPU time required to process its tasks. As a component needs to process its own root tasks as well as incoming tasks from its predecessors, its number of tasks L i is identified as in (6) where i denotes the immediate predecessors of component i. Then, LI i is represented as in (7). ∑ L = rt + L (6) i i i a∈i LI = L P To provide theoretical foundation of optimal resource allocation policy, we convert a network into a network with tasks having infinitesimal processing times. Each root task is divided into r infinitesimal tasks and each P i is replaced with P i /r. Then, the load index of each component is the same as the original network but tasks are infinitesimal. We denote the completion time of the network with infinitesimal tasks as T´. Also, we define a term called task availability as an indicator of relative preference for task arrival patterns. An arrival pattern gives higher task availability than another if cumulative number of arrived tasks is larger or equal over time. A component prefers a task arrival pattern with higher task availability as it can utilize more resource. Consider a network and reconfigure it such that all components have their tasks in their queues at t=0. Each component has maximal task availability in the reconfigured network and the completion time of the reconfigured network forms the lower bound T LB of a network’s completion time T given by: T LB = Max n∈N i i ∑ i∈ K n a LI i (7) . (8) Theorem 1. T´ equals to T LB when each node allocates its resource proportional to its residing components’ load indices as:
5 LI i w i( t ) = wi = ω n( i ) for all t ≥ 0 , (9) LI ∑ p∈K n( i ) p T LB s ωn + ωn = Max n∈N ω n s ∑ i∈ K n LI i (12) where n(i) denotes a node in which component i resides. Theorem 2. T s´ equals to T s LB under proportional allocation. Proof. RA i (t) is more than or equal to assigned weight proportion as: w i ( t ) RA ( t ) ≥ for t ≥ 0 . (10) ω i n( i ) Proof. RA i (t) becomes: RA i( t ) ≥ ω w ( t ) n( i ) i + ω s n( i ) for t ≥ 0 . (13) Suppose a component i receives its tasks at a constant interval of T LB /L i . Then, under proportional allocation, S i (t) is less than or equal to T LB /L i over time as shown in (11). P i = ∫ wi = ω n( i ) LB T ⇒ L i t+ Si( t ) t S ( t ) = i RA ( τ )dτ ≥ ≥ S ( t ) i i LI ∑ i LI p∈K n( i ) p ∫ for t ≥ 0 t+ Si( t t ) LI Si( t ) ≥ T w i( t ) dτ ω n( i ) i LB S ( t ) i (11) So, any component can complete by T LB and generate tasks at a constant interval of T LB /L i from t=T LB /L i (first task generation time) under proportional allocation when it receives tasks at a constant interval of T LB /L i from t=0 (first task arrival time). As tasks are infinitesimal and root tasks increase task availability, each component can receive infinitesimal tasks at a constant interval in 0≤t≤T LB or more preferably, and complete at less than or equal to T LB . So, the network completes at T LB under proportional allocation. From Theorem 1 we can conjecture that a network can achieve a performance close to T LB under proportional allocation in the limit of large number of tasks. We propose the proportional allocation as an optimal resource allocation policy. Though the proportional allocation is localized, the network can maximize the utilization of distributed resources and achieve desirable performance. Coordinated resource allocation throughout the network emerges as a result of using the load index as global information. If nodes do not follow the proportional allocation policy, some components can receive their tasks less preferably resulting in underutilization and consequently increased completion time as have shown in the previous subsection. Another important property of the proportional allocation policy is that it is itself adaptive. Suppose there are some stressors sharing resources with the components. We denote ω s n as the amount of shared resource by a stressor in node n. Then, the lower bound performance T LB s under stress is given by (12). We denote the completion time under stress as T s´. Then, (11) results in (14) under proportional allocation. LB Ts L i ≥ S ( t ) for t ≥ 0 (14) i Therefore, the network completes at T LB s under proportional allocation. Theorem 2 depicts that the proportional allocation policy is optimal independent of the stress environments. Though we do not consider them explicitly, the policy gives lower bound performance adaptively. This characteristic is especially important when the system is vulnerable to unpredictable stress environments. Modern networked systems can be easily exposed to various adverse events such as accidental failures and malicious attacks, and the space of stress environment is high-dimensional and also evolving [26]-[28]. C. Adequacy criterion The arguments we have made hold in the limit of large number of tasks. As the term “large” is obscure we need to give it a concrete definition. We define it with an adequacy criterion, by which one can evaluate if the desirable properties of the proportional allocation hold for a given network. For this purpose we characterize upper bound performance of a network under proportional allocation. Theorem 3. Under proportional allocation a network’s upper bound T UB of completion time T is given by: T UB = T LB + Max Max e∈E j∈Se ∑ i∈j [ P ∑ LI i p∈K n( i) p / LI i ] , (15) where E denotes a set of components which have no successor and S e a set of task paths to component e. A task path to component e is a set of components in a path from a component with no predecessor to component e and does not include component e. Proof. From (11) we can induce the lowest upper bound S i UB of S i (t) as:
Page 1 and 2:
Ultra*Log PSU/IAI Final Report for
Page 3 and 4:
Contents Contents .................
Page 5:
Executive Summary Ultra*Log is a De
Page 8 and 9:
2.3 Gnanasambandam, N., Lee, S., Ku
Page 10 and 11:
6 Characterization and analysis of
Page 12 and 13:
timizing simultaneously the link de
Page 14 and 15:
where ∆ 1 (j) ≥ 0 and ∆ 2 (i,
Page 16 and 17:
Table 2. GA Results Agent N A D max
Page 18 and 19:
connected component, in which a pat
Page 20 and 21:
Random Small-world Scale-free with
Page 22 and 23:
Growth mechanisms Start with a smal
Page 24 and 25:
Table 2. The proposed network’s c
Page 26 and 27:
¡ ¡ ¢ £ ¤ ¥ ¦ £ § ¨ © ¥
Page 28 and 29:
© © ¤ ¢ ¨ ¤ £ ¦ ¨ © §
Page 30 and 31:
DMAS Controller. The functional uni
Page 32 and 33:
1000 500 0 -500 -1000 0.2 0.4 0.6 0
Page 34 and 35:
tems. Proceedings of the Second Joi
Page 36 and 37:
Proceedings of the 1st Open Cougaar
Page 38 and 39:
Page 40 and 41:
Page 42 and 43:
Page 44 and 45:
1 SITUATION IDENTIFICATION USING DY
Page 46 and 47:
3 3.2 Behavior In SSC society an ag
Page 48 and 49:
5 All the behavior parameters may n
Page 50 and 51:
Estimating Global Stress Environmen
Page 52 and 53:
chaotic deterministic time series.
Page 54 and 55:
100% 1 95% 2 14 15 64% TAO 4 64% 62
Page 56 and 57:
interactions as a dynamical system
Page 58 and 59: ehavior states under varying system
Page 60 and 61: Extensive testing and validation of
Page 62 and 63: 5 Conclusions and Future Research T
Page 64 and 65: 2 to function critically even under
Page 66 and 67: 4 points into a corresponding inner
Page 68 and 69: 6 stresses well. The Warehouse 1 ag
Page 70 and 71: © £ ¨ ¥ ¤ § ¢ ¥ ©
Page 72 and 73: Table 1: Notation Symbol Descriptio
Page 74 and 75: 4 Conclusions and Future Work 4.1 C
Page 76 and 77: O, U Stresses Physical/Infrastructu
Page 78 and 79: Manuscript for IEEE Transactions on
Page 106 and 107: 2 discuss problem domain and in Sec
Page 110 and 111: 6 S UB i ∑ = P LI / LI . (16) i p
Page 112 and 113: 8 policies while underutilizing in
Page 114 and 115: architecture based on both the Grid
Page 116 and 117: to machines (network topology) and
Page 118 and 119: component is a member of one of the
Page 120 and 121: denotes the immediate predecessors
Page 122 and 123: limit of large number of tasks. If
Page 124 and 125: Min-min heuristic algorithm Step 1:
Page 126 and 127: We set up eight different experimen
Page 128 and 129: 0. 4 8057 8237 0.5 6439 6635 0.6 53
Page 130 and 131: pp. 191-200. [3] I. Foster, C. Kess
Page 132 and 133: Technology, Cambridge, MA, 1995. [2
Page 134 and 135: Manuscript for IEEE TRANSACTIONS ON
Page 156 and 157: Coordinating Control Decisions of S
Page 158 and 159:
directly. It has coarser scale. It
Page 160 and 161:
Understanding Agent Societies Using
Page 162 and 163:
• For tasks, the UID of the direc
Page 164 and 165:
for tracking the dependencies betwe
Page 166 and 167:
within the monitored enclave. With
Page 168 and 169:
Figure 1. Agent Hierarchy in CPE So
Page 170 and 171:
Figure 3. The World Model CPY Agent
Page 172 and 173:
Table 2. TechSpecs: Infrastructure
Page 174 and 175:
terms of the average waiting times
Page 176 and 177:
Acknowledgements The work described
Page 178 and 179:
key realization to tackle this prob
Page 180 and 181:
are or ignored perturbations. The e
Page 182 and 183:
sustained oscillations. If the inst
Page 184 and 185:
solved by biological systems for li
Page 186 and 187:
non-linear and non-stationarity. Ne
Page 188 and 189:
D k : Outflow from high priority qu
Page 190 and 191:
the system starts to perform self-s
Page 192 and 193:
sustained in a supply chain. Resour
Page 194 and 195:
which are necessary to make good mo
Page 196 and 197:
focusing on the computational compl
Page 198 and 199:
line of research that deals with ne
Page 200 and 201:
ealizable. But the inherent complex
Page 202 and 203:
Min, H. and Zhou, G., 2002, Supply
Page 204 and 205:
In the rest of this paper we report
Page 206:
shows relatively irregular behavior
show all

DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

Create successful ePaper yourself

Delete template?

Save as template?