M/M/1 Queues in Series

Consider a sequence of three workstations, with each lot having to visit the workstations exactly once, in sequence. If the interarrival time to the first machine is exponentially distributed, and process times at all machines are exponentially distributed, then according to Section III.4 of Asmussen (1987), the limiting input process to each queue is Poisson (interarrival times are exponentially distributed), and we may calculate limiting expected time in queue separately for each queue. Suppose the arrival rate to the system is l = 1.0 wafers per hour, and the service rates at the three workstations are m₁ = 2.5, m₂= 2.0, and m₃ = 2.5 wafers per hour. For simplicity, suppose each workstation has exactly one machine. Thus, the total limiting expected time to pass through the three workstations should be

(m₁ - l)^-1 + (m₂ - l)^-1 + (m₃ - l)^-1 = 0.67 +1.0 + 0.67 = 2.33 hours