For this question, we return to Pat’s, one of the Philadelphia cheesesteak restaurants mentioned in
Problem Set #8 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.1 Suppose, on average, that
51 customers arrive per hour, while the average service time per customer is 48 seconds. Assume
that the single-channel (M/M/1) queueing 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. . .
(i) . . . 24 seconds?
(ii) . . . 42 seconds?
(iii) . . . 1 minute, 15 seconds?
(d) What is the probability that the number of arrivals in a one-minute period is...
(i) . . . no customers?
(ii) . . . one customer?
(iii) . . . two customers?
(iv) . . . more than two customers?
(e) What is the probability that an arriving customer will have to wait for service?
(f) What is the average amount of time, in minutes, that a customer spends waiting in line (i.e.,
before they reach the service window)?
(g) What is the average number of customers waiting in line?
(h) What is the average amount of time, in minutes, that a customer spends in the service system
(i.e., from getting in line to receiving their sandwich)?
(i) What is the average number of customers in the service system?
(j) What is the probability that there are exactly two customers in the service system?