At the start of football season, the York student ticket office gets busy the da

Question

At the start of football season, the York student ticket office gets busy the day before the first game. Customers arrive at the rate of four every ten minutes. A ticket seller can service a customer in four minutes. Traditionally, there are two ticket sellers working. The university is considering an automated ticket machine similar to the airlines' e-ticket system. The automated ticket machine can service a customer in 2 minutes. What is the average time in the system for the automated system model?

Explanation / Answer

Ans....
Lambda = average number of customers arriving per unit of time= 4 for 10 minutes= 24 customers arrive in one hour
Mu = average number of customers that can be served per unit of time= 1 customer in 2 minutes= 30 customers in 1 hour
L = lambda / lambda - mu = 24 / 30 - 24= 4
P=Average utilisation=24/30 = 0.8
1. Lq=P*L= 0.8*4 = 3.2. Hence no. of customers in the queue is 3.2
2. Average time waiting in the system = 1/mu -lambda = 1/ 30-24 = 1/6= 0.16
3. Proportion that the system is busy:
W= 1/Mu –Lambda= 1/30-24= 0.16
L=Lambda*W= 24*0.16= 3.84

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