If service time is exponentially distributed with a mean of 15 per minute, determine - 1. Probability that the booking clerk waits for the customer. - 2. Probability that there are at least 3 customers in the queue. - 3. Average number of customers in system - 4. average time spent in the queue. - 5. Probability that the customer is served within four minutes. -

Mumbai University > Mechanical Engineering > Sem 7 > Operations Research

Marks: 10 Marks

Year: Dec 2015

μ=15 per minute,λ=10 per minute.

1. Probability that the booking clerk waits for the customer: $p_0=1-\dfrac{λ}{μ}=1-\dfrac{10}{15}=0.3333$

2. Probability that there are at least 3 customers in the queue: $p(≥n)=\bigg(\dfrac{λ}{μ}\bigg)^n=\bigg(\dfrac{10}{15}\bigg)^3=0.2963$

3. Average number of customers in system : $L_s=\dfrac{λ}{μ-λ}=\dfrac{10}{15-10}=2$

4. average time spent in the queue: $W_s=\dfrac{λ}{μ} \dfrac{1}{μ-λ}=\dfrac{10}{15} \dfrac{1}{15-10}=0.1333 mi \\ ns=8 \sec$

5. Probability that the customer is served within four minutes: $=(μ-λ) e^{-(μ-λ)^t}=(15-10) e^{-(15-10)^4} \\ =0$