M/M/1 Queue with Loading and Unloading

Consider a single machine and single lot class, but where process time consists of three parts: loading the lot, servicing the lot, and unloading the lot. The dispatch rule is FIFO. We can use the formula given before for the M/G/1 queue to calculate the limiting expected system cycle time, where the process time has three parts,

l E[S₀²] (2 (1-r) )^-1 + m^-1,

Suppose l = 0.05, and the three parts of the process time are distributed exponentially with means 1/m₁ = 1.0, 1/m₂ = 4.0, and 1/m₃ = 2.0. Then

m^-1 = m₁ + m₂+ m₃ + = 7.0.

Now the second moment of the process time,

E[S₀²] = Var[S₀] + E[S₀]² = m₁^-2 + m₂ ^-2 + m₃ ^-2 + m^-2 = 70.0,

since the process time is the sum of three independent exponential random variables. The traffic intensity r = 0.35. Hence the limiting expected system cycle time should be

(0.05 * 70.0)(2*(1-0.35))^-1 + 7.0 » 9.7.