Question

Question 3(a):
When customers arrive at Cool's Ice Cream Shop, they take a number and wait to be called to purchase ice cream from one of the counter servers. From experience in past summers, the store's staff knows that customers arrive at a rate of 150 per hour on summer days between 3:00 p.m. and 10:00 p.m., and a server can serve 1 customer in 1 minute on average. Cool's wants to make sure that customers wait no longer than 10 minutes for service. Cool's is contemplating keeping three servers behind the ice cream counter during the peak summer hours.
(i) Will this number be adequate to meet the waiting time policy?
(ii) What will be the probability that 3 to 4 customers in Shop?
(iii) In winter season, arrival rate of customer is reduced to half from 3:00 p.m. and 10:00 p.m. What decision should be taken by the owner according to cost cutting point of view?
Question 3(b):
Analysis of arrivals at a PSO gas station with a single pump (filler) has shown the time between arrivals with a mean of 10 minutes. Service times were observed with a mean time of 6 minutes.
(i) What is the probability that a car will have to wait?
(ii) What is the mean number of customers at the station?
(iii) What is the mean number of customers waiting to be served?
(iv) PSO is willing to install a second pump when convinced that an arrival would expect to wait at least twelve minutes for the gas. By how much the flow of arrivals is increased in order to justify a second booth?

Answer 1

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Explanation:

3a)

Answer

Average arrival rate, λ = 150 per hr.

Average service rate, μ = 1 in 1 minutes = 60 per 60 minutes = 60 per hr.

Number of servers, M = 3

(i)

The required waiting time, W_q = 10 minutes = (1/6) hours

So, the required queue length, L_q = W_q * λ = (1/6)*150 = 25 customers

λ/μ = 150/60 = 2.5

Use the infinite source table for the above value and find the value of L_q for different values of M.

The value of Lq is 3.511 which is quite less compared to the target value of 25 customers. So, the number of servers equal to '3' is an adequate decision.

(ii)

For M=3 and λ/μ=2.5, we have P₀ = 0.045 (check the table of part-i)

P₃ = (λ/μ)³ * P₀ * (1/3!) = (2.5)^3 * 0.045 * (1/6) = 0.117

P₄ = (λ/μ)⁴ * P₀ * (1/3!) * (1/3^(4-3)) = (2.5)^4 * 0.045 * (1/6) * (1/3) = 0.098

So, Prob{having 3 to 4 customers in the system} = P₃ + P₄ = 0.215

(iii)

Average arrival rate, λ = 75 per hr.

λ/μ = 75/60 = 1.25 ~ 1.3

Target L_q is same as before i.e. 25 customers.

The target can be achieved by having 2 servers. So, one server may be laid off to save costs.

3b)

Average arrival rate, λ = 1 per 10 minutes = 6 per 60 minutes = 6 per hr.

Average service rate, μ = 1 in 6 minutes = 10 per 60 minutes = 10 per hr.

(i)

The probability of an arriving car waits = 1 - λ/μ = 1 - 6/10 = 0.4 or 40%

(ii)

The average number of customers in the station, L_s = λ / (μ - λ) = 6 / (10 - 6) = 1.50

(iii)

The average number of customers waiting, L_q = λ² / {μ.(μ - λ)} = 6^2/(10*(10-6)) = 0.9 customrs

(iv)

The required average waiting time (W_q) is 12 minutes or 0.20 hours.

So,

W_q = λ / {μ.(μ - λ)} >= 0.2

or, λ / {10*(10 - λ)} >= 0.2

or, λ = 20 - 2λ

or, λ = 6.667

So, the arrival rate has to be 6.667 per hour (which is 1 customer in 60/6.667 i.e. 9 minutes) or more to justify adding another server.