Analysis of asynchronous binary arbitration on digital-transmission-line buses

The widely used asynchronous binary arbitration scheme on the digital-transmission-line model for buses is analyzed. A common misconception is that in this model, binary arbitration settles in at most four units of bus-propagation delay, regardless of the number of arbitration buses. This conjecture is disproved by showing that the settling time of binary arbitration depends on the arrangement of modules on the bus lines. An arrangement of modules on m buses for which binary arbitration settles only after m/2 units of bus-propagation delay is presented. It is also proved that for any arrangement of modules on m buses, binary arbitration settles in at most m/2+2 units of bus-propagation delay. It is then shown that for linear arrangements of modules in increasing order of priorities and equal spacings between modules, three units of bus-propagation delay are necessary and sufficient for binary arbitration to settle