3. At a border inspection station, vehicles arrive at the rate of 10 per hour in
ID: 389765 • Letter: 3
Question
3. At a border inspection station, vehicles arrive at the rate of 10 per hour in a Poisson distribution. For simplicity in this problem, assume that there is only one lane and one inspector, who can inspect vehicles at the rate of 12 per hour in an exponentially distributed fashion.(USE MODEL 1)
a. What is the average length of the waiting line?
b. What is the average time that a vehicle must wait to get through the system?
c. What is the utilization of the inspector?
d. What is the probability that when you arrive there will be three or more vehicles ahead of you?
Explanation / Answer
Following details are provided :
Arrival rate of vehicles ( @ 10 per hour ) = a = 10 per hour
Service rate ( i.e. inspection rate ) of vehicles = s = 12 per hour
= Average length of line of vehicles waiting for getting inspected + average length of the line of vehicles getting inspected
= a^2/ S x ( s -a ) + a/s
= ( a^2 + as – a^2) / s ( s – a )
= as/s x ( s -a )
= a/ ( s -a )
= 10/2
= 5
Average length of the waiting line = 5 vehicles
= Average time vehicle takes to wait in the queue + Average time the vehicle takes to go through inspection
= a/S x ( S – a ) + 1/S
= ( a + s – a ) / s x ( s – a )
= 1/ ( s – a )
= 1/( 12 – 10 )hour
= ½ hour
= 30 minutes
Average time a vehicle must wait to go through the system = 30 minutes
Probability that there will be 1 vehicle waiting = ( a/s) x Po = ( 10/12) x Po = 0.8333x 0.1666 = 0.1388
Probability that there will be 2 vehicles waiting = ( a/s ) ^2 x Po = 0.8333x 0.8333 x 0.1666 = 0.1157 ( rounded to 2 decimal places )
Therefore, probability that there will be less than 3 vehicles waiting
= Po + P1 + P2
= 0.1666 + 0.1388 + 0.1157
= 0.4211
Hence, probability there will be three or more vehicles ahead of you
= 1 – probability that there will be less than 3 vehicles ahead of you
= 1 – 0.4211
= 0.5789
Probability that there will be three or more vehicles ahead of you = 0.5789
Average length of the waiting line = 5 vehicles
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