Analyzing congestion at a shrinkwrap machine
Unit loads arrive randomly at a shrinkwrap machine. An average of 15 unit loads arrive per hour at a Poisson rate. The time required for a unit load to be processed through the shrinkwrapping operation is a constant of 2.5 minutes.
The operating characteristics are obtained using Equations 10.172 through 10.175 by letting \lambda=15, \mu=24, \sigma^{2}=0, and \rho=0.625.
L=\rho+\frac{\lambda^{2} \sigma^{2}+\rho^{2}}{2(1-\rho)} (10.172)
L_{q}=\frac{\lambda^{2} \sigma^{2}+\rho^{2}}{2(1-\rho)} (10.173)
W=\frac{1}{\mu}+\frac{\lambda\left(\sigma^{2}+\frac{1}{\mu^{2}}\right)}{2(1-\rho)} (10.174)
W_{q}=\frac{\lambda\left(\sigma^{2}+\frac{1}{\mu^{2}}\right)}{2(1-\rho)} (10.175)
L=0.625+\frac{(15)^{2}(0)+(0.625)^{2}}{2(1-0.625)}=1.1458 unit load
L_{q}=\frac{(15)^{2}(0)+(0.625)^{2}}{2(1-0.625)}=0.5208 unit load
W=\frac{L}{\lambda}=0.07639 hour per unit load
=4.583 minutes per unit load
W_{q}=W-\frac{1}{\mu}=2.083 minutes per unit load
