A certain small grocery store has a single checkout stand with a full-time cashier. Customers arrive at the stand “randomly”

(i.e., a Poisson input process) at a mean rate of 30 per hour. When there is only one customer at the stand, she is processed by the cashier alone, with an expected service time of 1.5 minutes. However, the stock boy has been given standard instructions that whenever there is more than one customer at the stand, he is to help the cashier by bagging the groceries. This help reduces the expected time required to process a customer to 1 minute. In both cases, the service-time distribution is exponential.

(a) Construct the rate diagram for this queueing system.

(b) What is the steady-state probability distribution of the number of customers at the checkout stand?

(c) Derive L for this system. (Hint: Refer to the derivation of L for the M/M/1 model at the beginning of Sec. 17.6.) Use this information to determine Lq, W, and Wq.