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At a self service store, cashier can serve 10 customers in 5 minutes as per exponential distribution. On an
average 15 customers arrive every 10 minutes as per Poission distribution. Probability that the cashier would be idle is_____.
(Important - Enter only the numerical value in the answer)
Correct answer is '0.25'. Can you explain this answer?
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Solution:

Given, cashier can serve 10 customers in 5 minutes as per exponential distribution.

On average, 15 customers arrive every 10 minutes as per Poisson distribution.

The probability that the cashier would be idle can be calculated as follows:

Step 1: Find the arrival rate (λ) using the Poisson distribution.

λ = 15 customers/10 minutes

λ = 1.5 customers/minute

Step 2: Find the service rate (μ) using the exponential distribution.

μ = 10 customers/5 minutes

μ = 2 customers/minute

Step 3: Find the utilization factor (ρ) using Little's law.

ρ = λ/μ

ρ = 1.5/2

ρ = 0.75

Step 4: Find the probability that the cashier would be idle using the formula:

P(0 customers in the system) = (1-ρ)

P(0 customers in the system) = (1-0.75)

P(0 customers in the system) = 0.25

Therefore, the probability that the cashier would be idle is 0.25.
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At a self service store, cashier canserve 10 customers in 5 minutes asper exponential distribution. On anaverage 15 customers arrive every10 minutes as per Poissiondistribution. Probability that thecashier would be idle is_____.(Important - Enter only the numerical value in the answer)Correct answer is '0.25'. Can you explain this answer?
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