Question

The probability that k number of vehicles arrive (i.e. cross a predefined line) in time t is given as (λt)^k e^-λt/k! where λ is the average vehicle arrival rate. What is the probability that the time headway is greater than or equal to time t₁?
1. λe^λt₁
2. λe^-t1
3. eλt1
4. e^-λt1

Answer 1

Correct Answer - Option 4 : e^-λt1

Concept:

The Poisson’s model that accounts for nonuniformity of flow which is derived by assuming the random pattern of vehicle arrivals at a specified point is given by

\(P\left( n \right) = \frac{{{{\left( {λ t} \right)}^n} \times {e^{ - λ t}}}}{{n!}}\)..... ( 1 )

where

P(n) = probability of having n vehicles arrive in time t,

n - Number of vehicles in time t

λ = average vehicle flow or arrival rate in vehicles per unit time,

t = time interval

e = base of the natural logarithm (e = 2.718)

Time headway is defined as the time difference between any two successive vehicles when they cross a given point.

Calculation:

Given:

Time Headway is greater than or equal to t₁∴ Number of vehicles arriving is zero in time t₁ ( n = 0)

Equation 1 becomes

\(P\left( 0 \right) = \frac{{{{\left( {λ t} \right)}^0} \times {e^{ - λ t}}}}{{0!}}\)

⇒ P(0) = e^-λt₁