Question

The manager of a grocery store in the retirement community of Kapiri is interested in providing good service to the senior citizens who shop in his store. Presently, the store has a separate check-out counter for senior citizens. On average, 30 senior citizens per hour arrive at the counter according to a Poisson distribution and are served at an average rate of 35 customers per hour with exponential service times. Find the following;

(a) Utilization of the checkout clerk

(b) Number of customers in a system

(c) Number of customers in line

(d) Time spent in the system

(e) Waiting time in line

The manager of Kapiri grocery wants to answer the following question:

(1) What service rate would be required to have customers average only eight minutes in

Answer 1

We have given,

L = Average arrival time = $\lambda$ = 30 per hour

M = Average service rate = $\mu$ = 35 per hour

a.) Utilization of the checkout clerk, $U = \dfrac{L}{M}$ = $\dfrac{30}{35}$ = $85.7$ %

b.) Number of customers in a system,

Firstly we have to find out the Average number of customers or units waiting in line for service

$L_q = \dfrac{\lambda^2}{\mu(\mu-\lambda)}$

$= \dfrac{30\times 30}{35(35-30)}$

$L_q= 5.14$ ,

Now, Average number of customers or units in the system = $L_q + \dfrac{\lambda}{\mu}$

= $5.14+\dfrac{30}{35}$

$= 5.997$

c.) Number of customers in a line $= L_q = 5.14$

d.) Time spent in the system

But before that we have to find out,

The Average time a customer or unit spends waiting in line for service $W_q = \dfrac{L_q}{\lambda}$

Hence, $W_q = \dfrac{5.14}{30} = 0.17$

Now, time spent in the system $W = W_q + \dfrac{1}{\mu}$

= $0.17 + \dfrac{1}{35}$

$= 0.19$

e.) Waiting time in line $= W_q = 0.17$

1.) Service rate would be required to have customers average only eight minutes in = $\dfrac{60}{8} \times 35 = 262.5$