Technical Note—Maximum and Minimum Service Times for Batches of Jobs with an Unknown Number of Servers

We consider the maximum and minimum times to service all jobs in a batch of random size at a facility having an unknown number of servers, where all servers have the same service-time distribution, and the batch size N and the service times {T_i, i = 1, 2, …, N} are mutually independent random variables. For arbitrary service-time distribution and arbitrary batch-size probability mass function, the batch service time is bounded above by ∑_i=1^i=NT_i and below by max{T_i, i = 1, 2, …, N}. We obtain the expected minimum batch service time for the special cases of exponential and uniform service-time distributions, and Poisson, binomial, and negative-binomal probability mass functions for batch size. Separate results are stated for the batch size restricted to be positive.