MGO$304$ Homework Assignment $3$

Due Date: $04/09/2024,11$:$59$PM

M$/$M$/1$ Calculations

A grocery has a separate checkout counter for citizens. On average, $40$ citizens per hour arrive at the counter, according to a Poisson process, but $25\%$ of them did not purchase anything and directly walk through the counter without join the service queue. The remaining citizens are served at an average rate of $35$ customers per hour, with exponential service times.

Please find the following operating characteristics $(5$ points in total, $0\mathrm{.}5$ points for each$)$:

a$.$ The arrival rate

$40$ Citizens per hour $=$ amive rate

b$.$ The service rate

$35$ customers per hour $=$ service rate

c$.$ Consumers' inter$-$arrival time

d$.$ Average utilization of the checkout counter.

e$.$ Probability of zero customers in the system.

f$.$ Probability that an arriving customer must wait for the service.

g$.$ Average number of customers in the system.

h$.$ Average number of customers waiting in line.

i$.$ Average time a customer spent in the system.

j$.$ Average waiting time in line.

Bonus Questions $(1$ point, $0\mathrm{.}25$ points for each$)$:

The manager of the grocery store wants answers to the following questions:

k$.$ What service rate would be required so that customers averaged only $8$ minutes in the system?

For the service rate you got from question $k,$ what is the probability of having more than $4$ customers in the system?

$m.$ What service rate would be required to have only a $10$ percent chance of exceeding $4$ customers in the system?