# Answer no the expected time is indeed w 10 min j

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Answer: No, the expected time is indeed ¯ W = 10 min.

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J. Virtamo 38.3143 Queueing Theory / Poisson process 15 Explanation for the hitchhiker’s paradox In short, the explanation of the paradox lies therein that the hitchhiker’s probability to arrive during a long interarrival interval is greater than during a short interval . Given the interarrival interval, within that interval the arrival instant of the hitchhiker is uniformly distributed and the expected waiting time is one half of the total duration of the interval. The point is that in the selection by the random instant the long intervals are more frequently represented than the short ones (with a weight proportional to the length of the interval). Consider a long period of time t . The waiting time to the next car arrival W ( τ ) as the function of the arrival instant of the hitchhiker τ is represented by the sawtooth curve in the figure. The mean waiting time is the average value of the curve. odotusaika W í î ì í î ì í î ì í î ì X 1 X 2 X n . . . t W _ ¯ W = 1 t integraldisplay t 0 W ( τ ) 1 t n summationdisplay i =1 1 2 X 2 i (sum of the areas of the triangles; X i is the interarrival time) As t → ∞ the number of the triangles n tends to t/ ¯ X . ¯ W = 1 ¯ X 1 n n summationdisplay i =1 1 2 X 2 i = 1 2 X 2 ¯ X For exponential distribution X 2 = ( ¯ X ) 2 + V[ X ] bracehtipupleft bracehtipdownrightbracehtipdownleft bracehtipupright ( ¯ X ) 2 = 2( ¯ X ) 2 , thus ¯ W = ¯ X
J. Virtamo 38.3143 Queueing Theory / Poisson process 16 Inhomogeneous Poisson process Thus far we have considered a Poisson process with a constant intensity λ . This can be generalized to a so called inhomogeneous Poisson process by letting the intensity to vary in time λ ( t ). (Note. λ ( t ) is a deterministic function of time.) l (t) The probability of an arrival in a short interval of time ( t, t + dt ) is now λ ( t ) dt + o ( dt ). The probability for more than one arrivals is of the order o ( dt ) The expected number of arrivals in the interval ( t, t + dt ) is E[ N ( t, t + dt )] = summationdisplay n =0 n · P { n arrivals in ( t, t + dt ) } = λ ( t ) dt + o ( dt ) Correspondingly, the expected number of arrivals in a finite interval (0 , t ) is E[ N (0 , t )] = E[ integraldisplay t 0 N ( u, u + du )] = integraldisplay t 0 E[ N ( u, u + du )] = integraldisplay t 0 λ ( u ) du (The expectation of a sum is always the sum of the expectations of individual terms, therefore the order of integration and expectation can be interchanged.)

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J. Virtamo 38.3143 Queueing Theory / Poisson process 17 Inhomogeneous Poisson process (continued) In the same way as in the case of an ordinary homogeneous Poisson process, we can derive a differential equation for the generating function G t ( z ) of the counter process N ( t ) (number of arrivals in (0 , t )) of an inhomogeneous Poisson process, d dt G t ( z ) = ( z - 1) λ ( t ) G t ( z ) d dt log G t ( z ) = ( z - 1) λ ( t ) from which we get by integration G t ( z ) = e ( z - 1) integraldisplay t 0 λ ( u ) du Denote the expected number of arrivals in (0 , t ) by a ( t ) a ( t ) = E[ N ( t )] = integraldisplay t 0 λ ( u ) du We see that G t ( z ) is the generating function of a random variable with Poisson distribution.
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