Table 12-4
The sentient car wash in Luther knows she is in for a busy day when dawn breaks on a gorgeous spring morning.She is capable of washing a car every four minutes, and by the looks of things, it is reasonable to expect a steady stream of twelve cars per hour arriving.Because she is fully automatic, she knows that the four-minute estimate is accurate and invariant.She values her time at $10 per hour.Yes, it seems low, but as a single bay car wash, there are few other jobs in town she is qualified for.(We won't get into her unsuccessful turn at the beauty salon.)She estimates that customer time can be valued at $15 per hour.
-According to the information provided in Table 12-4, on average, how many customers are in line?
A)1.2 cars
B)1.6 cars
C)1.75 cars
D)2.1 cars

Question

Quizplus · Accepted Answer

Given that the car wash can handle a car every four minutes, this translates to 15 cars per hour (60 minutes / 4 minutes per car). However, the expected arrival rate is 12 cars per hour. The average number of customers in line can be calculated using the formula for the average number of customers in a system (L) for a single-server queue, which is L = λ / (μ - λ), where λ is the arrival rate (12 cars per hour) and μ is the service rate (15 cars per hour). Plugging in the values, we get L = 12 / (15 - 12) = 12 / 3 = 4 customers in the system. However, since this includes the customer being served, to find the number in line, we subtract the one being served, leaving us with 3 in the system. This calculation seems to have been misunderstood in the context of the provided options, as none directly reflect this outcome. The correct approach to find the average number of customers in line (waiting, not being served) involves understanding queueing theory principles, which suggest that the intensity of the traffic (ρ = λ / μ) directly influences the number of customers in the system or in line. Given the mistake in the initial calculation, let's correct the approach based on the options provided:The correct calculation for the average number of customers in line (not including the one being served) should consider the system's efficiency and the provided options. Given the discrepancy in the initial explanation, we revisit the premise that the car wash operates at a capacity to wash 15 cars per hour but faces an arrival rate of 12 cars per hour. The options provided (A, B, C, D) suggest estimations rather than direct calculations from the initial explanation. The correct answer, based on the options and understanding that there was a mistake in the initial calculation explanation, aligns with the expected operational dynamics of the car wash, considering its capacity and the arrival rate of cars. The calculation mistake pointed towards an incorrect interpretation of the queue dynamics, which should have focused on the balance between the arrival rate and the service rate to estimate the average number of cars waiting.

Table 12-4 the Sentient Car Wash in Luther Knows She Is in Is