4.4. The m-machine problems with communication delays
4.4.3. The m-machine case
First, the performances of some algorithms are presented without uncertainty.
Specific sensitivity analysis might be performed on these algorithms, but the general results presented afterward are applied to them instead.
4.4.3.1. Some results in a deterministic setting
The main results are given in [CHR 95]; see also [HAN 98, VAR 96]. Outside the two processor case, there are few polynomial problems. Let us consider first the case8 infinite number of machines (an unbounded number of machines is more appropriate).
Without communication, this is the central scheduling problem, often called (somewhat abusively) the PERT problem. Thus, it is sufficient to compute the critical path for finding a minimal makespan schedule. Adding communications changes things. Indeed, with any problem with communication delays, computing the length of a path cannot be done before assignment is made (as this length takes into account the effective communications). The problem is NP-hard for tree precedence graphs and arbitrary communication delays.
Chrétienne [CHR 89] proposes a polynomial algorithm in the case of SCT trees.
The algorithm is adapted from ETF for this particular case. The resulting schedule is linear (two independent tasks are not executed on the same machine), as with algorithms based on clusters. Without SCT hypothesis, list schedules are no longer dominant, due to non-dominance of linear schedules: it might be necessary to delay some task to execute it on the same processor as its successor.
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Many approximation results exist (positive and negative) in the case of m machines and usually in UECT hypothesis; see [CHR 95, HAN 01]. The algorithms are based on ETF list methods or on clusters, such as DSC algorithms of [GER 92]
and the CBoS algorithm from the previous section that can be generalized to m machines. It then admits (see [GUI 97]) an absolute performance bound ω− ω∗ ≤ m−12 . Cluster-based algorithms have the effect of minimizing the number of effective communications, hence they are interesting methods to diminish the uncertainty impact, as shown by the two-machine case.
4.4.3.2. Framework for sensitivity analysis
In this section a sensitivity analysis is presented concerning the communication delays. The precedence graph is arbitrary. For any precedenceTi≺ Tjrepresented by the arcak, an estimated delay˜ckand an interval[αk, βk] are given. By hypothesis, the communication delays (real and estimated) are inside this interval. Denote byα and β then-vectors composed of the lower and upper bounds of these intervals. In the most general case, these quantities might depend on the task assignment, hence the studied schedule, see [SAN 05]. A worst case analysis is made, and sensitivity bounds are computed for given vectorsα and β. The above notations can be used: c = max βk
andc = min αk. The execution durations are fixed, and we use notationp = min pj. Last, by reference to the study of section 4.3.2, a possible definition of the perturbation magnitude is here described asε = c− c.
4.4.3.3. Stability studies
Let us first consider the stability problem, that is the performance loss of some schedule S after some disturbance, see Chapter 1. This loss is measured by the stability ratio (relative stability) st˜c(S) = ωωc˜c(S)(S), and when it does exist by the stability difference or absolute difference (absolute stability) astu˜(S) = ωc(S)− ωc˜(S).
As might be expected, no upper bound ofastc˜(S) independent of the graph size exists, and this remains true in really special cases. [MOU 03] considers for instance the unbounded case and Chrétienne’s algorithm.
THEOREM4.9.– Consider the problemP∞|tree, SCT|Cmax. Leth be the height of the task graph (considering the arcs),l its width, and ε = c− c. Let S be a schedule computed by Chrétienne’s algorithm for delaysc. Then˜
astc˜(S)≤ min(h, l − 1) × ε and this bound is tight.
94 Flexibility and Robustness in Scheduling
From the theorem, there indeed exists an absolute stability bound, but for either height bounded or width bounded graphs. This result is obtained using the hypotheses of unboundedness onm and of linearity of the obtained schedule.
However, it is possible to bound the stability ratio. Let us first note that inside the chosen framework,stc˜(S) = ωωβc˜(S)(S), as the makespan naturally increases with the delays for a fixedS.
Note. Letx and y be two communication vectors.
W (x, y) = 1 +maxp
kyk
minkxk
yk +maxpkyk
THEOREM4.10.– [SAN 05] For any estimated delay vector˜c, st˜c(S)≤ W (˜c, β)
In order to lighten the notations, the fact that the quantitiesW can in fact depend onS is omitted. The expression is much simplified if equal intervals are considered:
[c, c] for all effective communications. The best bound is obtained in the case of equal estimationsc˜k= u≥ c ∀k. In that case,
stu(S)≤ p + c p + u.
4.4.3.4. Sensitivity bounds
Suppose now that an optimal scheduleS is available for the estimated delays ˜c.
Because of the above observations on the stability difference, it seems very difficult to bound the worst case absolute deviation ofS, as was done in section 4.4.2, and bounds on the relative deviation are sought instead. A general bound is proved in [SAN 05].
From it, we can deduce:
THEOREM4.11.– Suppose that∀k, ck ∈ [αk, βk]. If S is optimal for delays α, or if S is optimal for delaysβ, then
W (α, β) = p + maxkβk p + maxkβk· mink αk
βk
is a relative sensitivity bound forS.
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IfS is an optimal schedule for an estimated vector different from α and β, the bound is less interesting (for equal vectorsα and β). As for stability, the expression of the bound is much simpler in the equal interval case:
THEOREM 4.12.– Suppose that∀k, ck ∈ [c, c]. If S is optimal for constant delays u∈ [c, c], then
p + c p + c is a relative sensitivity bound forS.
The bound of theorem4.12 is tight in two particular cases. Let us first consider the problem studied in section 4.4.2; thenp = 1, and in the LCT case (c = 1) any UECT-optimal schedule admits a relative bound of 1+c2 (theorem 4.7). Remember that the CBoS algorithm admits an absolute bound because it is processor-ordered, but theorem4.12 does not assume any hypothesis on S, apart from its optimality in the deterministic case.
Let us now consider the problem for which Chrétienne’s algorithm is optimal, P∞|tree, sct|Cmax. Any vectorc can be chosen for the estimated delays, while the˜ algorithm builds an optimal schedule for these delays as far as the SCT hypothesis is respected:c ≤ p. In particular, theorem 4.12 applies. Furthermore it is easy in that case to show that the bound is tight (for instance, when complete binary trees are considered). This bound is independent of the estimation used to build the schedule.
Let us finally note that this bound, contrary to section 4.3.2, cannot be expressed as a function of a unique parameter describing the size of the disturbance or of the uncertainty; both parametersc and c are necessary. Another difference comes from the nature of the data subject to uncertainty. If it is a processing time as in section 4.3.2, this duration is at least known a posteriori. However, if a communication is not effective in the chosen schedule, the associated real delay will never be known. Hence a posteriori analysis is questionable (and can not help to verify the adequacy of the hypotheses on the delays).