Formulate and characterize the following queues according to

Formulate and characterize the following queues according to the standard notation. Containers arrive at a small port for loading onto ships at a regular interval, one container every five minutes. There is only a single crane to load the containers and the time needed to load any container is distributed exponentially with mean of 4 minutes. Assume the small port can accommodate up to 10 waiting containers including the one been loaded) at any one time. A small local bank with 10,500 account holders has two service counters in its only banking facility with area for up to 10 customers. Customers arrive at random at a rate of 2 every hour and each is serviced over a period distributed uniformly between 10 an 40 minutes.

Solution

Formulate and characterize the following queues accoding to standard notation.

i) Containers arrive at a small port for loading onto ships at a regular interval one container every 5 minutes.

Let us consider here random variable X is time needed to load any container.

Given that X follows Exponential distribution with mean 4 minutes.

The p.d.f. of Exponential distribution is,

f (x) = * e^-(*x⁾ x 0

Parameter of the Exponential distribution is .

In notation we can write as X ~ Exp()

Here we have given that, mean of the distribution is 4 minutes.

And we know that the mean of the Exponential distribution is ( 1 / ).

Comparing mean with 4.

( 1 / ) =4

= 1/4

Therefore X ~ Exp(1/4)

Also we have given that the capacity of small port is up to 10 containers so n = 10. at any one time.

ii) There is a small local bank which has 10500 account holders and the bank has two service counters that counters area is up to 10 customers and for every hour 2 customers are arrive at random .

Here random variable X is customer is serviced over a period .

Given that X follows Uniform distribution with period between 10 minutes to 40 minutes.

The p.d.f. of Uniform distribution is,

f(x) = 1 / (b - a) a X b

In notation we can write as X ~ U(10,40)

We have given that a = 10 and b = 40.

f(x) = 1 / 30