  This note is concerned with the impact on job latency of splitting a token
bucket into multiple sub-token buckets with equal aggregate parameters and
offered the same job arrival process. The situation commonly arises in
distributed computing environments where job arrivals are rate controlled (each
job needs one token to enter the system), but capacity limitations call for
distributing jobs across multiple compute resources with scalability
considerations preventing the use of a centralized rate control component (each
compute resource is responsible for monitoring and enforcing that the job
stream it receives conforms to a certain traffic envelope). The question we
address is to what extent splitting a token bucket into multiple sub-token
buckets that individually rate control a subset of the original arrival process
affects job latency, when jobs wait for a token whenever the token bucket is
empty upon their arrival. Our contribution is to establish that independent of
the job arrival process and how jobs are distributed across compute resources
(and sub-token buckets), splitting a token bucket always increases the sum of
job latencies in the token buckets, and consequently the average job latency.
