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23.07.2021 •
Mathematics
Cars arrive at an automatic car wash system every 10 minutes on average. The cars inter-arrival times are exponentially distributed. Washing time for each is 6 minutes per car and is purely deterministic (i.e., the waiting line system is M/D/c). Assuming that the car wash has a single bay to serve the cars, what is the average number of cars waiting in line (L.)?
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Ответ:
the average number of cars waiting in line L
is 0.45
Step-by-step explanation:
Given the data in the question;
Cars arrive at an automatic car wash system every 10 minutes on average.
Car arrival rate λ = 1 per 10 min = [ 1/10 × 60 ]per hrs = 6 cars per hour
Washing time for each is 6 minutes per car
Car service rate μ = 6min per car = [ 1/6 × 60 ] per hrs = 10 cars per hour
so
P = λ/μ = 6 / 10 = 0.6
Using the length of queue in M/D/1 system since there is only one service bay;
L
=
[ P² / ( 1 - P ) ]
so we substitute
L
=
[ (0.6)² / ( 1 - 0.6 ) ]
L
=
[ 0.36 / 0.4 ]
L
=
[ 0.9 ]
L
= 0.45
Therefore, the average number of cars waiting in line L
is 0.45
Ответ: