Question

The Riverton Post Office has four stations for service. Customers line up in single file for service on a FIFO basis. The mean arrival rate is 40 per hour, Poisson Distributed, and the mean service time per service is 4 minutes, exponentially distributed.

In evaluating the system's operating characteristic, what decision should be made?

1. Decrease the number of servers
2. Increase the service rate
3. No changes required, system is adequate
4. Increase the number of servers
5. Decrease the arrival rate

A vending machine at City Airport dispenses hot coffee, hot chocolate, or hot tea, in a constant service time of 20 seconds. Customers arrive at the vending machine at a mean rate of 60 per hour (Poisson distributed). Determine the operating characteristics of this system.

How busy are they?

1. 45%
2. 33%
3. 50%
4. 300%

Answer 1

Q1

Arrival rate ()= 40 per hour = 40/60 = 0.666667 per min

Number of server (c)=4

Average service rate ()= 1/4 = 0.25 per min

Calculating service utilization factor for the given Model:

Service utilization factor = Average arrival rate/ (average service rate*number of servers)

Service utilization factor =0.666667 / (4*0.25)

Service utilization factor =0.666667/1

Service utilization factor = 0.666667

Service utilization factor = 66.67%

The system is adequate but poorly utilized.

We will try to reduce the number of servers to 3.

Service utilization factor = 0.666667/(0.25*3)

Service utilization factor =  0.666667/0.75

Service utilization factor = 0.888888

Service utilization factor = 88.88%

If we decrease the number of servers to 3, the Service utilization factor improves to 88.88% with system adequacy. Hence 'Option A - Decrease the number of servers' is the correct answer.

Option B - Increase the service rate - Will further reduce service utilization factor and hence not recommended.

Option C - System is adequate but is poorly utilized. Changes are required to improve utilization

Option D- Increase the number of servers - Will further reduce service utilization factor and hence not recommended

Option E- Decrease arrival rate- Independent variable that cannot be controlled and hence cannot be decreased.

Q2

Arrival rate ()= 60 per hour = 60 /60 = 1 per min

Average service rate ()= 1/20 = 0.05 per sec = 0.05*60 per min = 3 per min

Calculating service utilization factor for the given Model:

Service utilization factor = Average arrival rate/ average service rate

Service utilization factor = 1/3

Service utilization factor = 0.3333

Service utilization factor = 33.33%

Service utilization factor = 33% (Rounding off)

Hence option B is correct.