Dynamic Routing on Arrays References P N K=1 W K . an Easy Calculation Shows That W K = 2 F(k=n) N + O( 1 N 2 ), Where

A measure for asymptotic eeciency of a hypothesis based on the sum of observations. 4] B. Hajek. Hitting-time and occupation-time bounds implied by drift analysis with applications. Dynamic routing on arrays 17 with total expectation less than 0. Hence the tail of the distribution of M s;h is exponentially decreasing. As in Theorem 4.7, we can show that a packet's head generated at relative time bs in the h-th interval is delayed b 1 steps in its interval of origin only if either jM s;h j 1? 0 2 1 or jB s+1;h j + jB s+2;h j + : : :jB s++1;h j 1+ 0 2 1. Again, the probability of either event is e ?((1). The rest of the analysis of the delay distribution can be carried out as before. Similarly, the probability that a given queue contains more than q packet heads is shown to be e ?(q logN) by distinguishing two cases depending on whether the queue has been empty in the last bq logN steps. The analysis can be easily extended to higher dimensions. Details are omitted. 8 Concluding remarks 1. All our results easily generalize to the case when the arrivals obey a Poisson distribution. This is because the generating function of a Poisson distribution obeys the inequality stated in Proposition 3.4 (for a Poisson distribution, it is in fact an equality.) 2. An analogue of Theorem 4.7 is shown in 9] for the ring and the torus when the arrival rate is less than 49% of network capacity. Whether this can be extended to the case when the arrival rate is less than 99% of network capacity remains an intriguing open question. 3. An analogue of Theorem 7.1 holds for the N N torus using a synchronized version of the farthest-rst protocol. We describe this protocol when N is a multiple of b: a packet's head moving in increasing order in i and j can move at location (i; j) at time t only when t is congruent to i+j modulo b. Similar rules apply to packets heading in other directions. When two packets wish to traverse the same edge, the one heading further in that direction has priority. 4. We have shown that the ergodic expected delay for the farthest-rst protocol on an N{ array and on the ring is asymptotically proportional to a negative power of 1 ? (when N goes …