Question

In a grocery store, there is one cashier counter. Customers arrive at the cashier counter according to a Poisson process. The arrival rate is 30 customers per hour. The service time is exponentially distributed. The mean service time is 1 minute 30 seconds.

1. What is the expected number of customers waiting in the system.

2. What is the expected waiting time.( unit in minutes)

3. What is the utilization rate (unit in %) of the cashier?

Answer 1

Arrival rate() = 30 per hour

It take 1 minutes 30 seconds or 90 seconds to server one customer. So in one hour or 3600 seconds number of customers served = 3600/90 = 40

So service rate () = 40 per hour

1) Expected number of customers waiting in the system =  /(-) = 30/40-30 = 30/10 = 3

2) Expected waiting time = [/(-)] / ]

= [30/(40-30)]/30

= (30/10)/30

= 3/30

= 0.1 hours

= 6 minutes

3) Average utilization = / = 30/40 = 0.75= 75%