Question

2) (18 pts. In the central railway station 15 computerized reservation counters are available. A customer can book his/her ticket for any train on any day in any one of these computerized reservation counters. The average time spent per customer by each clerk is 5 minutes. Average arrivals per hour during three types of activity periods have been calculated and customers have been surveyed to determine how long they are willing to wait during each type of period. Type of period Arrivals / hour | Customer's acceptable average waiting time Peak 110 15 minutes Normal 10 minutes Low 5 minutes Making suitable assumptions on this queueing process, determine how many counters should be kept open during each type of period. 60 a) List your assumptions. b) Determine how many counters should be kept open during each type of period (peak, normal, low).

Answer 1

ANSWER:

The following assumptions can be made for doing calculations in this queuing model.

1. The averages arrival rate being λ. The arrival of customers follows the Poisson probability law.
2. The average service rate is μ. The service time has an exponential distribution.
3. A customer can book his/her ticket on any train in any day on any of these counters i.e. there are as many queues as the service stations, the queues as the service stations, the queues are selected at random by the arrivals and there is no line switching. Hence the system can be thought of and analyzed as the one composed of X different single server queuing systems.

Thus, the arrival rate

λ = 110/X during peak period
λ = 60/X during the normal period
λ = 30/X during the low period

Service rate for all periods = μ = 60/5 = 12 per hour.

1. Customers can wait for a period of 15 minutes at peak hours.i.e. 0.25 hours.

average waiting time =>    

W. b, %3D μ(μ- λ)

W_q = 0.25 , λ = 110/X and μ=12.

putting these value we will find the value of X which will be 12.22.

so 13 counters must be kept open to ensure that the average waiting time does not exceed 15 minutes.

2.

Customers can wait for a period of 10 minutes at peak hours.i.e. 0.167 hours.

average waiting time =>    

W. b, %3D μ(μ- λ)

W_q = 0.167 , λ = 60/X and μ=12.

putting these value we will find the value of X which will be 7.5.

so 8 counters must be kept open to ensure that the average waiting time does not exceed 10 minutes.

3.

Customers can wait for a period of 5  minutes at peak hours.i.e. 0.083 hours.

average waiting time =>    

W. b, %3D μ(μ- λ)

W_q = 0.083 , λ = 30/X and μ=12.

putting these value we will find the value of X which will be 5.

so 5 counters must be kept open to ensure that the average waiting time does not exceed 5 minutes.

So the no of Counter for (13,8,5 ) for peak, Normal, Low Period.