Customers arrive at a ticket counter at a rate of 50 per hour and tickets areissued in the order of their arrival. The average time taken for issuing a ticket is1min. Assuming that customer arrivals form a Poisson process and service timesare exponentially distributed, the average waiting time in queue in minutes is:a)3b)4c)5d)6Correct answer is option 'C'. Can you explain this answer?

Question

Answer

The given problem can be solved using queueing theory, specifically the M/M/1 queue model.

Understanding the M/M/1 Queue Model:
In the M/M/1 queue model, customers arrive according to a Poisson process and service times are exponentially distributed. The M/M/1 model assumes that there is only one server.

The arrival rate (λ) represents the average number of customers arriving per unit time, and the service rate (μ) represents the average number of customers that can be served per unit time.

Calculating the Arrival Rate:
Given that customers arrive at a rate of 50 per hour, we can convert this to arrivals per minute as follows:
Arrival rate (λ) = 50 customers/hour = 50/60 customers/minute = 5/6 customers/minute

Calculating the Service Rate:
Given that the average time taken for issuing a ticket is 1 minute, we can determine the service rate (μ) as the reciprocal of the average service time:
Service rate (μ) = 1/1 minute = 1 customer/minute

Utilization (ρ):
Utilization (ρ) represents the fraction of time the server is busy. It can be calculated as the ratio of arrival rate to service rate:
Utilization (ρ) = λ/μ = (5/6)/(1) = 5/6

Little's Law:
Little's Law states that the average number of customers in a queue (Lq) is equal to the arrival rate (λ) multiplied by the average time spent in the queue (Wq):
Lq = λ * Wq

Calculating the Average Waiting Time in Queue:
Using Little's Law, we can rearrange the equation to solve for the average time spent in the queue (Wq):
Wq = Lq / λ

Since the queue discipline is first-come-first-served, the average number of customers in the queue (Lq) is equal to half the average number of customers waiting in the system (L):
Lq = L/2

Therefore, the average waiting time in the queue (Wq) can be calculated as follows:
Wq = Lq / λ = (L/2) / λ

Calculating the Average Number of Customers Waiting in the System (L):
The average number of customers waiting in the system (L) can be calculated using the following formula:
L = ρ / (1 - ρ)

Substituting the value of ρ (5/6) into the equation, we get:
L = (5/6) / (1 - 5/6) = (5/6) / (1/6) = 5

Calculating the Average Waiting Time in Queue (Wq):
Finally, substituting the value of L (5) and λ (5/6) into the equation for Wq, we get:
Wq = (L/2) / λ = (5/2) / (5/6) = (5/2) * (6/5) = 3

Therefore, the average waiting time in the queue is 3 minutes.

Conclusion:
The correct answer is option C) 5.

Answer

Because average means  no/2  =50/2=25   poission means  x(t)^1/2 =√25=5

Answer

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