Twelve customers arrive at the lunch counter at George's Happy Hog per hour and select the hickory smoked meat du jour along with a side and soda for $5. The server doubles as the cashier and can take and fill an order in three minutes on average. What is the lowest number n such that the likelihood of n or more customers in line and receiving service is 0? Assume that the arrival and service times can be assumed as exponentially distributed.
A) one
B) two
C) three
D) four

Question

Quizplus · Accepted Answer

The system can be modeled as an M/M/1 queue, where the arrival rate λ = 12 customers/hour and the service rate μ = 20 customers/hour (since 60 minutes / 3 minutes per customer = 20). The number of customers in the system (n) includes those being served. For an M/M/1 queue, the probability that there are n customers in the system is (1 - ρ)ρ^n, where ρ = λ/μ. Here, ρ = 12/20 = 0.6. The likelihood of n or more customers in line and receiving service approaches 0 as n approaches infinity. However, in practical terms, for the question asking for the lowest number n such that the likelihood is 0, it misunderstands how probabilities in such queues work, as the probability never truly hits 0 for any finite n. The question seems to aim at understanding queue dynamics rather than pinpointing a specific n where probability is exactly 0, which is not achievable. However, interpreting the intent as finding a point where the system can no longer serve customers faster than they arrive, we look for stability in the system, which is already given by the rates. The question's phrasing might be misleading, but given the options, the closest interpretation would be to consider when the system is effectively unable to handle the queue, which doesn't exactly apply here as described.

Twelve Customers Arrive at the Lunch Counter at George's Happy