Before revision:  An earlier bound for a single service tier is extended to the case of multiple service tiers  .
After revision:  An earlier bound for a single service tier is extended to the case of multiple service tiers . For the context of scheduling microservices in a private setting, the framework could be used to determine the required VM resources for a token-bucket constrained workload stream. Finally, we note that the looser Markov inequality may be useful in settings where the job service times are dependent .