M/M/1 Queues in Series with Rework

We test whole-lot rework here only, so that we may compare with analytic results for M/M/1 queues in series. Suppose we have two workstations in series, with some possibility of rework at step one. By rework, we mean that there is a certain probability that for every lot leaving step one, it may have to visit a third machine for processing, then pass through step one again. Suppose all three workstations have processing rate μ = 2.0 lots per hour. Let the arrival rate into the system be λ = 1.0 lot per hour. However, that is not the effective arrival rate that workstation one sees, due to the possibility of rework. Also suppose that each workstation contains a single machine.

Let the probability of rework be p = 0.20, that is, approximately one out of every five lots will be reworked. Lots that are reworked visit a rework workstation, then return to workstation one. Lots that are not reworked visit workstation three, then exit the system. To calculate the analytic limiting expected system cycle time, we need first to calculate the effective arrival rate to all three workstations, given that rework can occur. First, the arrival rate to workstation one is incoming lots at a rate of λ, plus reworked lots at a rate of (pλ), plus those lots that are reworked twice at a rate of (p-2λ), etc. Thus,

  • λ2 = Ε pn = (1 - p)-1 = 1.25 lots per hour.
  • The arrival rate to the rework workstation is the probability of rework times the effective arrival rate into workstation one,
  • λr = pλ1 = 0.25 lots per hour.
  • The arrival rate to workstation two is the exit rate of workstation one (that must be its arrival rate λ1) minus the rate of lots being reworked (λr),
  • λ2 = λ1 - λr = 1.0 lot per hour.
  • Finally, the expected system cycle time is the expected number of visits to each workstation times the expected time at each workstation, added together,
  • λ1 (λ (μ - λ1))-1 + λr (λ (μ - λr))-1 + λ2 (λ (μ - λ2))-1 = 2.81 hours.