For this question, we return to Pat’s, one of the Philadelphia cheesesteak restaurants mentioned in Problem Set 17 about market share analysis. Cheesesteak service at Pat’s is limited to a single walk-up window: as a customer, you get into a single-file line and, upon reaching the service window, you order your sandwich, pay for it, receive it, and leave. Suppose, on average, that 48 customers arrive per hour, while the average service time per customer is 40 seconds. Assume that the single-channel (M/M/1) queuing model is appropriate for this scenario. (Include at least three decimal places in your answers.)
(a) Indicate the average arrival rate λ in customers per minute.
(b) Indicate the average service rate μ in customers per minute.
(c) What is the probability that the service window completes a customer’s order within
18 seconds?
45 seconds?
1 minute, 12 seconds?
(d) Determine the probability that the number of arrivals in a one-minute period is
no customers?
one customer?
two customers?
more than two customers?
(e) What is the probability that an arriving customer will have to wait for service?
(f) What is the average number of customers waiting in line?
(g) What is the average amount of time, in minutes, that a customer spends waiting in line (i.e., before they reach the service window)?
(h) What is the average number of customers in the service system?
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