Gerard Is Choosing Between Two Methods of Organizing His Bakery

Question

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The Answer is 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:
C_SCounter= C_WCounter +

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: C<sub>SCounter</sub>= C<sub>WCounter</sub> + = + C<sub>SCounter</sub> = + = 1.1429 customers At the cashier step: C<sub>SCashier</sub> = C<sub>W</sub><sub>C</sub><sub>ashier</sub> + = + C<sub>SCashier</sub> = + = 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: T<sub>SCounter</sub> = T<sub>W</sub><sub>C</sub><sub>ounter</sub><sub> </sub><sub>+</sub> = + T<sub>SCounter</sub> = + = 0.1429 hours At the cashier step: T<sub>SCashier</sub> = T<sub>W</sub><sub>C</sub><sub>ashier</sub> + = + T<sub>SCashier</sub> = + = 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. P<sub>0</sub>= P<sub>0</sub> = P<sub>0</sub> = 0.5 The average number of customers in the system for method two is: C<sub>S</sub> = P<sub>0</sub> + C<sub>S</sub> = 0.5 + C<sub>S</sub> = 0.75 customers The average time spent in the bakery for method 2 is: T<sub>S</sub> = P<sub>0</sub> + T<sub>S</sub> = 0.5 + T<sub>S</sub> = 0.0938 hours To summarize: Method 1

=

+

C_SCounter =

+

= 1.1429 customers
At the cashier step:
C_SCashier = C_W_C_ashier +

=

+

C_SCashier =

+

= 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:
T_SCounter = T_W_C_ounter₊

=

+

T_SCounter =

+

= 0.1429 hours
At the cashier step:
T_SCashier = T_W_C_ashier +

=

+

T_SCashier =

+

= 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:
C_S =

P₀ +

C_S =

0.5 +

C_S = 0.75 customers
The average time spent in the bakery for method 2 is:
T_S =

P₀ +

T_S =

0.5 +

T_S = 0.0938 hours
To summarize:

Method 1

Gerard Is Choosing Between Two Methods of Organizing His Bakery

Definitions: