Vehicles arrive at a toll plaza at the rate of 120 per hour. There are three boo
ID: 2084639 • Letter: V
Question
Vehicles arrive at a toll plaza at the rate of 120 per hour. There are three booths, each capable of servicing 60 cars per hour. Assuming a Poisson distribution for arrivals and an exponential distribution for the service time, determine the average time spent by a car in the plaza. Choose the closest answer to your calculated answer. Be sure to convert your answer to the given units. 40 seconds 52.3 seconds 1 minute 1 minute 26.7 seconds 2 minutes 2 minutes 33.3 seconds 3 minutes 3 minutes 46.7 seconds 4 minutesExplanation / Answer
The incoming vehicle rate (u) is 120 V/hour or 2 V/minute. Hence the probability of arriving no vehicle in 1 min P(0) can be explained as;
P(x) = (u^x * exp^(-u))/fact(x) as it follows the piossion distribution.
Then P(0) = (2^0)*(exp^(-2))/fact(0) = 0.135.
Similarly the probability of arriving 1 vehicles in one min can be expreesed as;
P(1) = (2^1)*(exp^(-2))/fact(1) = 0.271
Therefore the probability of vehicles arriving less than or equal to zero is given by;
P(x<=0) = P(0) = 0.135
Similarly the probability of vehicles arriving less than or equal to one is given by;
P(x<=1) = P(0)+P(1) = 0.135+0.271 = 0.406.
Therefore the no of intervals in an hour when no vehicle will be come at the plaza will be
F(x) = P(0)*60 = 0.135*60 = 8.12.
Now a toll plaza zone can be classified as queueing and merging area. All vehicles are queued to pay the toll. Each one is waiting for there turn to pay the toll at the booth. More no of toll plaza can actually reduce the waiting time of each vehicle. Queueing theory can be applied to find out the waiting time of each vehicle in the toll plaza. Then the time waiting at tollplaza (W) can be represented as;
W = 1/(ua - theta/T) sec/V
where ua is the service rate at each tollplaza (V/hr), theta is the arriving rate of vehicles (V/hr) and T is the no of tollPlaza.
Putting the above values in the above equation, we get
W = 1/(60 - (120/3)) = 1/56 * 3600 = 1 min 4.28 sec.
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