Question

Consider a queueing system with one server and infinite capacity. Suppose the arrival rate at the queue is 36 customers per hour, there are 9 customers on average in the system at any given time, and a server can serve 40 customers per hour. a) What is the average waiting time before the customer begins service (in minutes)? b) On average, how many customers are waiting to be served?

Answer 1

a) The average waiting time before the customer begins service is 13.5 minutes.

b) Number of customers are waiting to be served is 8.1.

Explanation:

Given : Consider a queuing system with one server and infinite capacity. Suppose the arrival rate at the queue is 36 customers per hour, there are 9 customers on average in the system at any given time, and a server can serve 40 customers per hour.

To find :

a) What is the average waiting time before the customer begins service (in minutes)?

b) On average, how many customers are waiting to be served?

Solution :

A queueing system with one server and infinite capacity.

Let
`\lambda` be the arrival rate of customers i.e.
`\lambda=36/hr`

L be the average number of customers in the system i.e. L=9

`\mu` be the number of customers a server can serve i.e.
`\mu=40/hr`

Average utilization of system is given by,

`P=(\lambda)/(\mu)`

`P=(36)/(40)`

`P=(9)/(10)`

`P=0.9`

Average time spent waiting in the system is given by,

`W=(1)/(\mu-\lambda)`

`W=(1)/(40-36)`

`W=(1)/(4)`

`W=0.25`

a) The average waiting time before the customer begins service is given by,

`A_w=P* W`

`A_w=0.9* 0.25`

`A_w=0.225\ hr`

Converting into minutes,

1 hour = 60 minutes

0.225 hour =
`0.225* 60` minute

0.225 hour = 13.5 minute

The average waiting time before the customer begins service is 13.5 minutes.

b) Number of customers are waiting to be served is given by,

`n=P* L`

`n=0.9* 9`

`n=8.1`

Number of customers are waiting to be served is 8.1.