Proof and Application of Little’s Formula in Optimizing Bulk-service Multi-server Queue Systems

This paper presents closed-form expressions for the steady-state mean queue length in a complex bulk-service, multi-server queueing system denoted as GI/M a,b /c. In this continuous system, customers arrive according to an arbitrary distribution, and ′′ c ′′ servers operate independently, each serving a group of customers with a size between ′′ a ′′ and ′′ b ′′ . We demonstrate the applicability of Little’s formula to this queueing system, establishing a foundational relationship between the average queue length and the average time customers spend in the system. Our findings provide valuable insights into the system’s performance, and essential for assessing efficiency and optimizing service strategies. Additionally, We formulate an objective function and compute the optimal values of the decision variables (a, b) that maximize the expected profit.