Scaling of avalanche queues in directed dissipative sandpiles

Using numerical simulations and analytical methods we study a two-dimensional directed sandpile automaton with nonconservative random defects (concentration c) and varying driving rate I. The automaton is driven only at the top row and driving rate is measured by the number of added particles per time step of avalanche evolution. The probability distribution of duration of elementary avalanches at zero driving rate is exactly given by P-1 (t,c)=t(-3/2) exp[t ln(1-c)]. For driving rates in the interval 0<r less than or equal to 1 the avalanches are queuing one after another, increasing the periods of noninterrupted activity of the automaton. Recognizing the probability P-1 as a distribution of service time of jobs arriving at a server with frequency r, the model represents an example of the class [E,1,GI/infinity,1] server queue in the queue theory. We study scaling properties of the busy period and dissipated energy of sequences of noninterrupted activity. In the limit c-->0 and varying linear system size L much less than 1/c we find that at driving rates r less than or equal to L-1/2 the distributions of duration and energy of the avalanche queues are characterized by a multifractal scaling and we determine the corresponding spectral functions. For L much greater than 1/c increasing the driving rate somewhat compensates for the energy losses at defects above the line r similar to root c. The scaling exponents of the distributions in this region of phase diagram vary approximately linearly with the driving rate. Using properties of recurrent states and the probability theory we determine analytically the exact upper bound of the probability distribution of busy periods. In the case of conservative dynamics c=0 the probability of a continuous flow increases as F(infinity)similar to r(2) for small driving rates.