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only computation levels are affected here, but this is a general problem that can occur for most level based priority metrics, especially in regular task graphs or for integer task graph weights Scheduling algorithms therefore use additional criteria to decide the node order For example, in a bottom level based order, such a tie breaker can be the top level of the nodes But often ties are broken randomly Example Priority Schemes Given the enormous number of proposed scheduling algorithms, the following list of priority schemes represents only a small, though important part, of the existing ones In the HLF (highest level rst) (Hu [93]) and HLFET (highest level rst with estimated times) (Adem et al [2]) algorithms, the nodes are ordered according to the computation bottom level blw The CP/MISF (critical path/most immediate successors rst) algorithm by Kasahara and Nartia [102] is also blw based but breaks the ties by giving precedence to the node with the higher number of successors Wu and Gajski [207] propose the MCP (modi ed critical path) heuristic, where the nodes are ordered by their bl and for nodes with equal bl, the bl values of the successors are considered and so on A critical path based order is proposed by Ahmad and Kwok [4, 114], where non-CP nodes are scheduled according to their bl and ties are broken with tl Sinnen and Sousa analyze [177] various priority schemes of two different types: (1) node orders according to the (computation) bottom level augmented with metrics based on the entering communication of a node; and (2) critical path based orders as proposed by Ahmad and Kwok [4, 115] Dynamic Level or critical path based priority schemes turn into dynamic priorities when they are recalculated in each scheduling step based on the current partial schedule Section 44 explained how the allocated path length is calculated given the (partial) processor allocation of the nodes For instance, the allocated critical path might be recalculated for each partial schedule It should be noted that the allocated bottom level bl(n, Scur ) of a node n remains unaltered in list scheduling until n is scheduled; that is, bl(n, Scur ) = bl(n) the bottom level of n depends only on its descendants (see Eq (443)) In general, recalculating the levels or the CP in each step of scheduling multiplies the costs for the level or CP determination by a factor of |V| However, an ef cient implementation might only update the levels for those nodes that are affected by the scheduling of a node Start Times Apart from node orders based on task graph characteristics, priority schemes often use the potential start time of free nodes as a preference criterion The node priority thereby becomes a function of the processor Choosing the node processor pair as discussed in Section 512 is such a priority scheme Another technique selects the free node in each step that has the earliest DRT, assuming all communications are remote, sometimes called ready node (Yang and Gerasoulis [210]).



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The max-min fair allocation is unique and such that no ow rate l, say, can be increased without having to decrease that of another ow whose allocation is already less than or equal to l Max-min fairness can be achieved exactly by centralized or distributed algorithms, which calculate the explicit rate of each ow However, most practical algorithms sacri ce the ideal objective in favor of simplicity of implementation [1] The simplest rate sharing algorithms are based on individual ows reacting to binary congestion signals Fair sharing of a single link can be achieved by allowing rates to increase linearly in the absence of congestion and decrease exponentially as soon as congestion occurs [6].





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It has recently been pointed out that max-min fairness is not necessarily a desirable rate sharing objective and that one should rather aim to maximize overall utility, where the utility of each ow is a certain nondecreasing function of its allocated rate [15, 18] General bandwidth sharing objectives and algorithms are further discussed in Massoulie and Roberts [21] Distributed bandwidth sharing algorithms and associated mechanisms need to be robust to noncooperative user behavior A particularly promising solution is to perform bandwidth sharing by implementing per ow, fair queueing The feasibility of this approach is discussed by Suter et al [29], where it is demonstrated that an appropriate choice of packets to be rejected in case of congestion (namely, packets at the front of the longest queues) considerably improves both fairness and ef ciency 1642 Randomly Varying Traf c.

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