Oil Change Center Queue 

Cars arrive at an oil change center following a Poisson process at the rate of four per hour. There is a single available mechanic, and the time taken to perform an oil change is exponentially distributed, with an average of 12 minutes.

In[1]:=

X \[ScriptCapitalP] = QueueingProcess[4, 60/12];

Steady-state distribution for the system.

In[2]:=

X \[ScriptCapitalD] = StationaryDistribution[\[ScriptCapitalP]];

Distribution for the number of cars in the system.

In[3]:=

X DiscretePlot[PDF[\[ScriptCapitalD], k], {k, 1, 20}, ExtentSize -> 0.5, PlotRange -> All]

Out[3]=

Probability that there are more than three cars at the oil change center.

In[4]:=

X NProbability[x > 3, x \[Distributed] \[ScriptCapitalD]]

Out[4]=

Mean and variance for the number of cars in the system.

In[5]:=

X {Mean[\[ScriptCapitalD]], Variance[\[ScriptCapitalD]]}

Out[5]=