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H t; r : D r ht=D1 r :
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6:65
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for n nb < na or Ca 0; for n na < nb or Cb 0; for n na nb ; Ca > 0; Cb > 0; 6:66 6:67 E e rH t =a etr=a E e rN t =a :
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Proof. See Cohen [21]. To give the reader insight into the approach, we sketch the proof for the M =G=1 case with heavy-tailed service time distribution as given in
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6.6 HEAVY TRAFFIC LIMIT THEOREM FOR THE WORKLOAD PROCESS
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Eq. (6.49) (see also Cohen [20]). This is a case with n nb < na . In this M=G=1 case, E e skt so E e snt erst 1 1 bfsg =bs : Hence, putting s rD r ; t t=D1 r ; 6:70 6:69
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t=a n n b fsg e rst 1 bfsg =bs ; n!
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it follows from Eqs. (6.65) and (6.69) that E e
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For the term in square brackets, we have the representation (6.49). It can be veri ed that gb arD r = 1 r 3 0 for r 4 1. The fact that D r is the unique zero of the contraction equation (6.55) with the property that D r 5 0 for r 4 1 nally implies that lim E e rN t;r =a er t=a :
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The convergence in distribution of N t; r follows after application of the convergence theorem for Laplace Stieltjes transforms [20]. The statements concerning H t; r follow immediately from those for N t; r . j REMARK 6.6.2. The distribution of the stochastic variable N t ; t ! 0, and similarly of H t ; t ! 0, is a n-stable distribution (cf. Samorodnitsky and Taqqu [33, p. 5]). The process fN t ; t ! 0g is a process with stationary independent increments. As is evident from its LST representation, it is self-similar with index 1=n, that is, N bt and b1=n N t ; b > 0, have the same distribution for every t > 0. Note that fH t ; t ! 0g is not self-similar. In Samorodnitsky and Taqqu [33] this process fN t ; t ! 0g with 1 < n < 2 is called a n-stable Levy motion with independent self-similar increments; for n 2, it is the Brownian motion. In the M =G=1 case, it is veri ed in Cohen [20] (using the fact that the process fht ; t ! 0g has stationary independent increments) that the nite-dimensional distributions of the fN t; r ; t ! 0g process converge to those of the fN t ; t ! 0g process.
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The proof of Theorem 6.6.1 for the GI =G=1 case proceeds in principle along the same lines as sketched above for M =G=1. However, the proof of the weak convergence of N t; r to N t is more complicated. Let us now turn to the workload processes fvt ; t ! 0g and the contracted and scaled fw t; r ; t ! 0g. We again discuss the M =G=1 case. It follows from formula (4.99) of Cohen [15, p. 262] that, for Re s ! 0, Re y > 0, I
e yt E e svt jv0 0 dt
1 y s 1 r 1 bfsg =bs   I yt 1 s e P vt 0jv0 0 dt
6:73
with (cf. Formula (4.94) of Cohen [15, p. 262] and Eq. (6.29)) I
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1 r 1 1 r 1 : y 1 r rE e yP y E e yP V 6:74
Having established this relation with P V , one can now use Theorem 6.3.7 that speci es the heavy traf c behavior of DP r P V ; note that DP r 1 r D r D1 r . Again apply the transformation (6.70), with Re r ! 0 and 0 < 1 r ( 1, and also y oD1 r ; Hence, I
Re o ! 0: