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where fy t; : denotes the probability density of Ot with given Y0 y. In order to state the next proposition, let F denote the distribution function of a standard normal variable and introduce the notation m t mb t ma t ; D t 2 D t 2 m t 2 s2 ; D t m t s2 g x gt S t; x ; s D t D t 2 s D t 1 S t;x 2 =2 S t; x F S t; x p e R t; x D t 2p

MIDP 2.0 (JSR 118)

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to 3-PARTITION: let A1 , , Am (triplets) be m disjoint sets such that each Ai , i = 1, , m, contains exactly 3 elements of A, whose sum is B Nodes nx and n0 are allocated to the same processor, which shall be called P0 The remaining nodes n1 , , n3m are allocated in triplets to the processors P1 , , Pm Let ai1 , ai2 , ai3 be the elements of Ai The nodes ni1 , ni2 , ni3 , corresponding to the elements of triplet Ai , are allocated to processor Pi Their entering edges ei1 , ei2 , ei3 , respectively, are scheduled on L0i as early as possible in any order The resulting schedule is illustrated for P0 ,Pi and L0i in Figure 718 What is the length of this schedule The time to execute nx and n0 on P0 is w(nx ) + w(no ) = B + 2 = T On each link L0i the three edges corresponding to the triplet Ai take B time units for their communication (Figure 718) The rst communication starts as early as possible, that is, after 1 time unit when nx nishes After the last communication has nished, only one node remains to be executed on processor Pi ; the other two nodes are already executed during the communication (Figure 718) This is guaranteed by the fact that ai is a positive integer and w(ni ) = 1 for 1 i 3m The execution of this last node takes 1 time unit and, hence, each processor Pi nishes at 1 + B + 1 = T Thus, a feasible schedule S was derived, whose schedule length matches the time bound T ; hence, it is a solution to the constructed C-SCHED instance Conversely, assume that the instance of C-SCHED admits a solution, given by the feasible schedule S with sl(S) T It will now be shown that S is necessarily of the same kind as the schedule constructed previously Nodes nx and n0 must be scheduled on the same processor, say, P0 , since otherwise the communication e0 is remote and takes one time unit As a consequence, the earliest nish time of n0 would be w(nx ) + c(e0 ) + w(n0 ) = B + 3, which is larger than the bound T Since processor P0 is fully occupied with these two nodes until B + 2, all remaining nodes have to be executed on the other processors That means that all corresponding communications are remote.

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where s, D t , ma t , and mb t are de ned by Eqs. (5.2), (5.3), and (5.5), respectively. Proposition 5.4.1. Assume that heavy load condition (5.1) holds for some positive N p constant g. Then the distribution of V0 = N converges weakly as N 4 I. Furthermore, de ning p N h x lim P V0 > x N

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