M/M/1 Queues in Series with Scrap

We demonstrate whole-lot scrapping here only, so that we may compare with analytic results for M/M/1 queues in series. Suppose we have three workstations in series, with some scrap occurring after steps one and two. If the interarrival times to the first workstation are exponentially distributed, and all process times are exponentially distributed, then we can analyze the limiting expected time in queue separately for each queue, as before. The only modification is that the arrival rate to each queue changes, as some lots are removed due to scrapping.

Let the arrival rate to workstation one be l₁ = 1.0 lots per hour. If approximately one out of every ten lots is scrapped during processing at workstation one, then the arrival rate to workstation two is l₂ = l₁ * 0.9 = 0.9 lots per hour. If approximately one out of every five lots is scrapped during processing at workstation two, then the arrival rate to workstation three is l₃ = l₂ * 0.8 = 0.72 lots per hour. Suppose that the three processing rates are m₁ = 2.5, m₂ = 2.0, and m₃ = 2.5 lots per hour, and that each workstations consists of a single machine. Then, the total limiting expected system cycle time should be

(m₁ - l₁)^-1 + (m₂ - l₂)^-1 + (m₃ - l₃)^-1 = 0.67 + 0.91 + 0.56 = 2.14 hours