# What Is the Expected Wait Time for the Last Pedestrian at a Traffic Light?

• Mehmood_Yasir
In summary, the expected wait of the LAST pedestrian crossing the road is exponentially distributed with the same parameter, and is approximately equal to the time it takes for the button to push and for the first pedestrian to arrive.
Mehmood_Yasir

## 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##.

Mehmood_Yasir said:

## 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##.
You already know the wait time pdf of the kth pedestrian ##f(t|k)## from a previous thread. If you know the probability p(k) that the kth pedestrian is the last pedestrian, then you can get the pdf of the last pedestrian's waiting time.

Mehmood_Yasir said:

## 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?

- - - - -
If this is "merely" for expected values, an easier approach may be to use conditioning (specifically Law of Total Expectation).

e.g. Suppose ##n = 0## arrivals happen in ##(0, t]## -- i.e. no arrivals once the button is pushed. The expected wait time of the last person, i.e. the button pusher, is ## t##.

Now for all other cases, ##n \gt 0##:
you can ignore the button pusher and look at this in terms of 'regular' Poisson mechanics. Conditioning on ##N(t) = n##, all arrivals are uniformly distributed in ##(0, t]##. There are some technical nits on ##n!## orderings but pick one without loss of generality -- because we're actually looking at order statistics here. The easy problem is then to find the CDF of time until first arrival given ##N(t) = n##

You should get (with ##U## referring to the i.i.d. uniform random variables that exist in this conditional world -- there are n of them)

##Pr\{\text{Arrival 1's time} \gt \tau \big \vert N(t) =n \} = 1 - F_{min}U = \big[1 - F_U(u)\big]^n##

(note this was the underpinning of a recent thread: https://www.physicsforums.com/threa...f-ind-r-v-that-follows-distribution-f.946651/ and working through the CDFs for minimum and maximum amongst N i.i.d. random variables -- order statistics-- is an exercise worth doing.)

you can integrate this complementary CDF to get the expected time between Light going off and first 'real' arrival. Thus you have ##E\big[ X_1 \big \vert N(t) =n\big]##, then using a symmetry argument, or the fact that this process is time reversible (why?), you have thus found ##E\big[ X_n \big \vert N(t) =n\big]##.

Putting it all together with Law of Total Expectation (and hopefully note spoiling the result), should give:

##E\big[\text{Last Ped's wait time}\big] = p_\lambda(n=0, t) \Big(t\Big) + \sum_{n=1}^{\infty}p_\lambda(n, t) \Big(E\big[ X_n \big \vert N(t) =n\big]\Big)##

When you're all done, you should see there's some nice structure in the factorials in the denominator that should allow a closed form for the series.

- - - -
What I've said above is all discussed in a lot more detail here:

https://ocw.mit.edu/courses/electri...ring-2011/course-notes/MIT6_262S11_chap02.pdf

which is something I've mentioned to you a while back. I am not supposed to give this whole thing away, though, so I leave the reading and a lot more lifting, to you.

Last edited:
Mehmood_Yasir and tnich

## What are the safety measures for pedestrians at a road crossing?

The safety measures for pedestrians at a road crossing include using designated crosswalks, obeying traffic signals, making eye contact with drivers, and looking both ways before crossing.

## What should a pedestrian do if there is no designated crosswalk?

If there is no designated crosswalk, a pedestrian should find the nearest intersection or a well-lit area to cross the road. They should also make sure to look both ways and wait for a safe gap in traffic before crossing.

## What are some common causes of accidents involving pedestrians at road crossings?

Some common causes of accidents involving pedestrians at road crossings include distracted driving, failure to yield, and poor visibility or lighting. Pedestrians can also contribute to accidents by not following safety measures.

## What should a pedestrian do if they are in a crosswalk and a car is not yielding to them?

If a car is not yielding to a pedestrian in a crosswalk, the pedestrian should stop and make eye contact with the driver to ensure they are seen. If the car still does not yield, the pedestrian should wait for a safe gap in traffic before crossing.

## Are there any specific laws or regulations for pedestrians at road crossings?

Yes, there are laws and regulations for pedestrians at road crossings. These include obeying traffic signals, using designated crosswalks, and yielding to vehicles when crossing outside of a crosswalk. It is important for pedestrians to familiarize themselves with these laws to ensure their safety.

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