Gerard is choosing between two methods of organizing his bakery

Method one will require a counter person to wait on each customer individually before they proceed to the cashier to pay for their purchases. In this system, customers will arrive at the rate of 8 per hour and can be served by the counter person at the rate of 15 per hour. The cashier will be capable of serving customers at the rate of 20 per hour. For simplification purposes, assume that the arrival rate at the cashier phase is identical to the arrival rate at the first phase and that these waiting lines can be analyzed independently and combined.
Method two will have the two employees wait on customers as they selected their pastries and also collect their money once their selections have been made. Customers still arrive at the rate of 8 per hour but now the total service time (pastry selection and payment collection) takes 5 minutes per customer. Calculate the average number of customers in the bakery and the average time spent in the bakery for both systems and make your recommendations to Gerard.
What will be an ideal response?

Answer: Method one is a two-phase, single server system with an arrival rate into both phases of 8 customers per hour and a service rate at the first phase of 15 customers per hour and the second phase of 20 customers per hour.
The average number of customers in the bakery will be the sum of the customers at the counter and at the cashier.
At the counter step:
CSCounter = CWCounter + = = +
CSCounter = + = 1.1429 customers

At the cashier step:
CSCashier = CWCashier + = = +
CSCashier = + = 0. customers

So the total number in the bakery is 1.1429 + 0.666 = 1.81 customers

The average time spent in the bakery will be the sum of the times spent waiting and being served in both phases.

At the counter step:
TSCounter = TWCounter + = = +
TSCounter = + = 0.1429 hours

At the cashier step:
TSCashier = TWCashier + = = +
TSCashier = + = 0.0833 hours

So the total time in the bakery is 0.1429 + 0.0833 = 0.2262 hours

Method two is a multiple server system with an arrival rate of 8 customers per hour and a service rate of 12 customers per hour. This analysis requires calculating the probability of zero customers in the system.

P0 =

P0 =

P0 = 0.5

The average number of customers in the system for method two is:

CS = P0 +

CS = 0.5 +

CS = 0.75 customers

The average time spent in the bakery for method 2 is:

TS = P0 +

TS = 0.5 +

TS = 0.0938 hours

To summarize:
Method 1 Method 2
Number of Customers 1.81 customers 0.75 customers
Time in System 0.2262 hours 0.0938 hours

Method 1