a new (slightly inelegant) stab at eliminating the slow tail problem

script/trapezoid_simulator.rb

@@ -0,0 +1,78 @@
+require 'pp'
+
+def estimate_acceleration_distance(initial_rate, target_rate, acceleration) 
+    (target_rate*target_rate-initial_rate*initial_rate)/(2*acceleration)
+end
+
+def intersection_distance(initial_rate, final_rate, acceleration, distance) 
+    (2*acceleration*distance-initial_rate*initial_rate+final_rate*final_rate)/(4*acceleration)
+end
+
+ACCELERATION_TICKS_PER_SECOND = 20
+
+def trapezoid_params(step_event_count, nominal_rate, rate_delta, entry_factor, exit_factor)
+  initial_rate = (nominal_rate * entry_factor).round
+  final_rate = (nominal_rate * exit_factor).round
+  acceleration_per_minute = rate_delta*ACCELERATION_TICKS_PER_SECOND*60
+
+  accelerate_steps = 
+    estimate_acceleration_distance(initial_rate, nominal_rate, acceleration_per_minute).round;
+  decelerate_steps = 
+    estimate_acceleration_distance(nominal_rate, final_rate, -acceleration_per_minute).round;
+  
+  # Calculate the size of Plateau of Nominal Rate. 
+  plateau_steps = step_event_count-accelerate_steps-decelerate_steps;
+  
+  # Is the Plateau of Nominal Rate smaller than nothing? That means no cruising, and we will
+  # have to use intersection_distance() to calculate when to abort acceleration and start braking 
+  # in order to reach the final_rate exactly at the end of this block.
+  if (plateau_steps < 0)
+    accelerate_steps = 
+      intersection_distance(initial_rate, final_rate, acceleration_per_minute, step_event_count).round
+    plateau_steps = 0;
+  end
+  
+  accelerate_until = accelerate_steps;
+  decelerate_after = accelerate_steps+plateau_steps;
+  {:step_event_count => step_event_count,
+   :initial_rate => initial_rate,
+   :final_rate => final_rate,
+   :nominal_rate => nominal_rate,
+   :rate_delta => rate_delta,
+   :accelerate_until => accelerate_until,
+   :decelerate_after => decelerate_after}
+end
+
+def simulate_trapezoid(params)
+  result = {}
+  rate = params[:initial_rate]
+  step_event = 0.0
+  max_rate = 0
+  while(step_event < params[:step_event_count]) do
+    step_events_in_frame = rate/60.0/ACCELERATION_TICKS_PER_SECOND
+    step_event += step_events_in_frame
+    max_rate = rate if rate > max_rate
+    if (step_event < params[:accelerate_until]) 
+      rate += params[:rate_delta]
+    elsif (step_event > params[:decelerate_after])
+      if rate > params[:final_rate]
+        rate -= params[:rate_delta]
+      else
+        return :underflow_at => step_event, :final_rate => rate, :max_rate => max_rate
+      end
+    end
+#    puts "#{step_event} #{rate}"
+  end 
+  {:final_rate => rate, :max_rate => max_rate}
+end
+
+(10..100).each do |rate|
+  (1..5).each do |steps|
+    params = trapezoid_params(steps*1000, rate*100, 10, 0.1, 0.1)
+    result = simulate_trapezoid(params)
+  #  puts params.inspect
+    line = "#{steps*10} final: #{result[:final_rate]} == #{params[:final_rate]} peak: #{result[:max_rate]} == #{params[:nominal_rate]} d#{params[:nominal_rate]-result[:max_rate]} "
+    line << " (underflow at #{result[:underflow_at]})" if result[:underflow_at]
+    puts line
+  end
+end