Analyzing congestion at an in-process inspection station

Parts are supplied by a conveyor to an inspection station from a numerically controlled milling machine. Because of the insignificant variation in machining time and conveyor speed, parts arrive at the inspection station in a deterministic fashion at a rate of six per minute.

Question

Analyzing congestion at an in-process inspection station

Parts are supplied by a conveyor to an inspection station from a numerically controlled milling machine. Because of the insignificant variation in machining time and conveyor speed, parts arrive at the inspection station in a deterministic fashion at a rate of six per minute.

Accepted Answer

Inspection is performed manually, and the inspection times are exponentially distributed with a mean of [latex]7.5[/latex] seconds. In this situation, [latex]\lambda=6, \mu=8, c=1[/latex], and [latex] ho=0.75[/latex]. From Figure [latex]10.67, heta=0.55 .[/latex] Therefore, if an estimate for [latex] heta[/latex] is desired such that [latex]\left| heta_{k}- heta_{k-1} ight|<[/latex] [latex]0.001[/latex], then, for the example with [latex] ho=0.75[/latex], the following iterative process can be used:

[latex] heta_{k}=e^{-\left(1- heta_{k-1} ight) / 0.75}[/latex]

Calculations show that if [latex] heta_{0}=0.55[/latex], then [latex] heta_{1}=0.5488116, heta_{2}=0.5479427[/latex], and [latex] heta_{3}=0.5473083[/latex]. Therefore, substituting [latex]0.5473083[/latex] for [latex] heta[/latex] in Equations [latex]10.176[/latex] through [latex]10.179[/latex] yields

[latex]\begin{aligned} L =\frac{ heta}{1- heta}\end{aligned}[/latex] (10.176)

[latex]\begin{aligned} L_{q} =\frac{ heta^{2}}{1- heta}\end{aligned}[/latex] (10.177)

[latex]\begin{aligned} W =\frac{1}{\mu(1- heta)}\end{aligned}[/latex] (10.178)

[latex]\begin{aligned} W_{q} =\frac{ heta}{\mu(1- heta)} \end{aligned}[/latex] (10.179)

[latex]\begin{gathered}L=\frac{0.5473083}{0.4526917}=1.209 ext { parts } \end{gathered}[/latex]

[latex]\begin{gathered}L_{q}=\frac{0.5473083}{0.4526917}=0.662 ext { part }\end{gathered}[/latex]

[latex]\begin{aligned} W =\frac{1}{8(0.4526917)}=0.276 ext { minute per part } \end{aligned}[/latex]

[latex]\begin{aligned} W_{q} &=\frac{0.5473083}{8(0.4526917)}=0.151 ext { minutes per part } \end{aligned}[/latex]

Question 10.70: Analyzing congestion at an in-process inspection station Par...

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