Question

Customers arrive at the first class ticket counter of a theater at the rate of 12 per hour. There is one clerk serving the customers at the rate of 30 per hour. The arrivals are poisson in nature and the service time follows exponential distribution. Find (i) probability that there is no customer at the counter. (ii) probability that there are more than two customers at the counter. (iii) probability that there is no customer waiting to be served. (iv) probability that a customer is being served and nobody is waiting.

Answer 1

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theater at the rate of 12 per hour. There is one clerk serving the customers at the rate of 30 per hour. The arrivals are poisson in nature and the service time follows exponential distribution. Find (i) probability that there is no customer at the counter. (ii) probability that there are more than two customers at the counter.

Answer 2

Answer:

1. (i) probability that there is no customer at the counter = 0.6
2. (ii) probability that there are more than two customers at the counter   0.064
3. (iii) probability that there is no customer waiting to be served = 0.84
4. (iv) probability that a customer is being served and nobody is waiting for = 0.24

Explanation:

Arrival rate λ  =12 per hour

Service rate μ = 30 per hour.

Traffic Intensity P λ/μ= 12 / 30 =  2 / 5

1) λ=12; λ=30; ρ=λ=0.4

=P(0=1−ρ1−0.4=0.6,

So, the probability that there is no customer at the counter is 0.6.

2) p (more than two customers at the counter) = p (three or more customers in the queue)

= $p^{3} =(\frac{2}{3})^{3} = 0.064$

3) p (no customer is waiting to be served) = p ( at most one customer at counter)

= p0 + p1 = 0.6 + 0.6(0.4) = 0.6 + 0.24 = 0.84

4) p (a customer is being served and no body is waiting) = p1 = 0.6* 0.4 = 0.24

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