Auto repair shop with 1 mechanic only, one phase repair, any car can come in. Ar
ID: 419735 • Letter: A
Question
Auto repair shop with 1 mechanic only, one phase repair, any car can come in. Arrival rate of cars: Poisson distribution, mean = 3 and STD = 1 Service rate: Negative exponential distribution, mean = 3.2 and STD= 0.6 First come first serve rule Find the following: a- Average number of cars in the system (waiting and being served). b- Average time a car spends in the system (waiting time plus service time). c- Average number of the cars waiting in the queue. d- Average time a car spends waiting in the queue. e- Utilization factor of the mechanic. f- Probability of 0 cars in the whole place (shop and parking lot). g- Probability of more than 9 cars in the whole place.
Explanation / Answer
The given case is a G/G/1 queuing system. Below is the given information -
Arrival Rate ? = 3
STD ?a = 1 => Ca2 = ?a2/(1/?)2 = 1/(1/3)2 = 9
Service Rate µ = 3.2
STD ?s = 0.6 => Cs2 = ?s2/(1/µ)2 = 0.62/(1/3.2)2 ?= 3.69
e) Utilization ? = ?/µ = (3/3.2) = 0.9375
c) Average number of cars in the queue Lq = ?2*(1 + Cs2)*(Ca2+ ?2*Cs2)/2*(1 ? ?)*(1 + ?2Cs2)
Lq = (0.9375)2*(1+3.69)(9+(0.9375)2*3.69)/2*(1-0.9375)*(1+(0.9375)2*3.69) = 95
Lq = 95
d) Average time a car spends waiting in the queue Wq = Lq/? = 95/3 = 31.67
b) Average time a car spends in the system = W = Wq + 1/µ = 31.67+(1/3.2) = 31.98
a) Average number of cars in the system = L = ?W = 3*31.98 = 95.95
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