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

Abstract

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 = N T 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.