Scaling of avalanche queues in directed dissipative sandpiles

We simulate queues of activity in a directed sandpile automaton in 1+1 dimensions by adding grains at the top row with driving rate $0<r\le 1$. The duration of elementary avalanches is exactly described by the distribution $P_1(t)\sim t^{-3\mathrm{/}2}\exp (-1\mathrm{/}{L}_c)$, limited either by the system size or by dissipation at defects $L_c=\min (L,\xi )$. Recognizing the probability $P_1$ as a distribution of service time of jobs arriving at a server with frequency $r$, the model represents a new example of the $<E,1,GI\mathrm{/}\mathrm{\infty }\mathrm{/}1>$ server queue in the queue theory. We study numerically and analytically the tail behavior of the distributions of busy periods and energy dissipated in the queue and the probability of an infinite queue as a function of driving rate.