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We need to ensure the clipping region encompasses the old position and the new position of the puck irrespective of the puck s direction, which means we are repainting a bit more than we strictly need to. Note that paint requests may be buffered so that the actual clipping rectangle will be the union of the clipping rectangles requested by each repaint().

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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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

178 179 180

t=a n n b fsg e rst 1 bfsg =bs ; n!

6:68

6:71

.

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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There are various ways in which we could use an image buffer. In this case we have created an image for the background and an image for the puck. On each animation we draw in the background image, then the puck image. To improve performance further, we again call repaint() with a clipping region. It is worth going through the whole PlayingFieldBuffered class there is not much to it, as it only has to add a bit to PlayingField. puckImage and backgroundImage hold the two images:

6:72

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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