Question

The arrival rate of customer at a banking counter follows a ppoisson distribution with a mean of 30 per hours.The service rate of the counter clerk also follows poisons distribution with mean of 45 per hour.1)What is the probability of having zero customer in the system.2)What is the probability of having 8 customer in the system.3)What is the probability of having 12 customers in the system

Answer 1

Given :-

Arrival rate $(\lambda) = 30$ per hours.

Service rate $(\mu) = 45$ per hours.

$$\begin{array}{l} \rho = \frac {\lambda}{\mu} \\ = \frac {3 0}{4 5} = 0. 6 6 6 6 \\ \end{array}$$

Hence,

Probability of having "n" customers is

$$\begin{array}{l} P = \rho^{n} (1 - \rho) \\ = (0. 6 6 6) ^ {n} (1 - 0. 6 6 6) \\ = (0. 6 6 6) ^ {n} (0. 3 3 3 3) \\ \end{array}$$

1) What is the probability of having "zero" customer in the system

Hence, "0" customer have

$$\begin{array}{l} P = (0. 6 6 6) ^ {0} (0. 3 3 3 3) \\ P = 0. 3 3 3 \\ \end{array}$$

2) What is the probability of having "8" customer in the system

Hence, 8 customer have

$$\begin{array}{l} P = (0. 6 6 6) ^ {8} (0. 3 3 3 3) \\ P = 0. 0 1 2 8 8 9 \\ \end{array}$$

3) What is the probability of having "12" customer in the system

Hence, 12 customer have

$$\begin{array}{l} P = (0. 6 6 6) ^ {1 2} (0. 3 3 3 3) \\ P = 0. 0 0 0 2 5 3 8 2 0 1 3 \\ \end{array}$$

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