Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 3 / 27K / a / Solution

Let regeneration cycles have lengths $C_i$, contain $A_i$ arrivals, and accrue queue-length rewardFubini's geometric identity writes this area as the sum, over customers in the cycle, of their time in the system, up to boundary terms whose contribution vanishes over many cycles. The renewal-reward theorem therefore givesHence Little law is