Question

A supermarket has three cashiers. The supermarket manager implements a policy that if there are 1 to 3 customers in the supermarket, then only one cashier is on duty. If there are 4 to 6 customers in the supermarket then two cashiers will be on duty. The three cashiers will be on duty if there are more than 6 customers in the supermarket. The arrival of customers to the supermarket follows the Poisson process with a rate of 10 people / hour. The service time of each cashier is exponentially distributed with an average of 12 minutes per customer. This problem can be modeled as a birth process and a death process. a. Draw a transition diagram of this problem that is supplemented by birth rates and mortality rates for each state. b. Determine the long-term opportunity there are n customers in the system (Pn) c. Determine the expected number of unemployed cashiers.

Answer 1

When there are 1 to 3 customers in the supermarket, then only one cashier is on duty. Also If there are 4 to 6 customers in the supermarket then two cashiers will be on duty. The three cashiers will be on duty only if there are more than 6 customers in the supermarket.

The poisson distribution is used for the arrival of customers and exponential process for the service time provided to the customers. The expected number of unemployed cashiers can be still caluclated by designing a model which will contain both the use of poisson's as well as the exponential distribution.The answer for the expected number which is also known as mean will be 8.