- #1

Mehmood_Yasir

- 68

- 2

## Homework Statement

Pedestrians approach to a signal for road crossing in a Poisson manner with arrival rate ##\lambda## per sec. The first pedestrian arriving the signal pushes the button to start time ##T##, and thus we assume his arrival time is ##t=0##, and he always see ##T## wait time. A GREEN light is flashed after ##T##, and all arrived pedestrians within ##T## must cross. Process repeats. What is the expected wait of the

*LAST*pedestrian crossing the road?

## Homework Equations

##P_k=\frac { {(\lambda T)}^k e^{-\lambda T} } {k!}##

## The Attempt at a Solution

If ##t_l## is the arrival time of last pedestrian, the stay time is T minus ##t_l##. then this stay time is also exponentially distributed with same parameter ##\lambda##. For expectation value, the exponential pdf can be integrated over the interval from 0 to T. I am not sure if this statement is correct that T minus ##t_L## is also exponentially distributed, because ##t_L## is arrival time of last, the next exponential in original may have larger value than ##T##.