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QU 4 Vehicles are passing through a toll gate at the rate of 70 per hour. The average time to pass through the gate is 45 seconds. The arrival rate and service rate follow a Poisson distribution. There is a complaint that the vehicles wait for a long time. The authorities are willing to install one more gate to reduce the average time to pass through the toll gate to 35 seconds if the idle time is less than 9% and the average queue length through the gate is more than 8 vehicles, check whether the installation of the second gate is justified. Soultion: ∆=

Arrival rate =

70 hr

Average time to pass through the gate = 45/sec μ=

3600 80 = 45 hr

= Service time

∆ 70 ρ= = =0.875 μ 80

Waiting no of vehicles in the queue = Lq =

ρ2 0.8752 = =6.125 vehicles 1−ρ 1−0.875

Expected time to pass through the gate = 35 seconds '

μ=

3600 102.85 = 35 hr

ρ '=

∆ 70 = =0.68 μ ' 102.85

Percent of idle time =

1−ρ=32

Average queue length < 8 and the idle time >9 % Therefore, the installation of the new gate is not justified

Explanation: In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time and/or space if these events occur with a known average rate and independently of the time since the last event. The Poisson distribution can also be used for the number of events in other specified intervals such as distance, area or volume. Queuing theory is the mathematical study of waiting lines, or queues. In queuing theory a model is constructed so that queue lengths and waiting times can be predicted. Queuing theory is generally considered a branch of operations research because the results are often used when making business decisions about the resources needed to provide a service. Single queuing nodes are usually described using Kendall's notation in the form A/S/C where A describes the time between arrivals to the queue, S the size of jobs and C the number of servers at the node. Many theorems in queue theory can be proved by reducing queues to mathematical systems known as Markov chains. In this question we are dealing with a single queue for which the arrival rate and service rate both follow a Poisson distribution. It is proposed to add another queue if the wait for the current queue is too much. The installation of the other queue is justified if and only if the two conditions: idle time of the servers is less than 9% and average no of customers in the queue is more than 8 are met. We start solving by initially calculating the traffic intensity -

ρ . This is

calculated by dividing the arrival rate by the service rate. Then we calculate the average number of customers waiting in the queue at 2

any given time using the formula -

ρ 1−ρ . In the question this comes out to

be 6.125 which is approximately equal to 6. Thus, the first condition has failed but we still carry on solving the question. Now, we take the expected time to pass through the toll gate after the installation of the second gate = 35 seconds. We transform this into an appropriate form to compare it to the arrival rate, this comes out 102.85/hour. Now, we calculate the new traffic intensity

ρ'

which comes out to be 0.68.

Now, we find out the percent of idle time which we calculate to be 32%. Thus, this condition is also violated as idle time is greater than 9%. Thus, the installation of the second gate isn’t justified as both the conditions fail.