Is waiting time for a cab always memoryless?

Say, a person is a waiting for a cab by a street. We want to model the time she has to wait for the cab to arrive.

There are two cases to consider: (1) she has not called for a cab. (2) she called for a cab

In (1), there is no scheduled arrival of a cab. If a cab happen to go by her, she will take it. We can model this situation using an exponential random variable.

An exponential random variable is continuous random variable with probability density function $f(t) = \lambda e^{-\lambda t}$ . If the average time that the person has to wait is $20$ minutes, we use $\lambda = \frac{1}{20}$ .

By the memoryless property of exponential distribution, we have $P\{X > t+s|X > s\} = P\{X > t\}$ . Giving $s = 40$ and $t = 5$ , we have $P\{X > 45 | X > 40\} = P\{X > 5\}$ . This property might seem surprising. Say, the person begins to wait and determines the probabilities that he needs to wait for at least $5$ minutes, $40$ minutes and $45$ minutes to be $0.78$ , $0.13$ and $0.11$ respectively. Say, he waits and sees no cab for the first $40$ minutes. The probability that he needs to wait for $5$ additional minutes, namely $P\{X > 45 | X > 40\}$ , is same as $P\{X > 5\} = 0.78$ . It might seem unfair that the probability that she sees no cab for $5$ additional minutes is so high given that she already waited for $40$ minutes. One might expect that the cab may come any minute then with a high probability. Can we explain this behavior?

The reason for this memoryless property is that in modeling the waiting time scenario by an exponential distribution, (like in the case of Poisson distribution) we divided time into tiny intervals and assumed that the intervals are independent (or weakly dependent) of each other. Cabs going in all directions, people getting on and off, traffic patterns etc., all cause the probability of a cab arriving in one interval unaffected by whether a cab had arrived or not in an earlier interval. Regardless of the fact that the person waited for $40$ minutes already, the outside traffic system has not changed, and thus we have the memoryless property.

Now, consider case (2) where the person called for a cab. Here, we assume there is a particular cab company that she called the cab from and there is a scheduled time of arrival. Although there is a scheduled time, the arrival time is still not deterministic and hence, can be modeled by a random variable. The question is if the random variable is exponential and in particular, if the waiting time still has the memoryless property? Say, the cab is scheduled to arrive $40$ minutes from when she called. And, she waits for $40$ minutes and the cab hasn’t arrived still. Then, is the probability that the cab arrives in the next minute high, or is the probability, like in case (1), going to remain the same as what it was for the cab to arrive in the first minute when she started waiting? As can be expected, it is the former in this case. We do not have the memoryless property unlike case (1).

Where does the difference between (1) and (2) come from? In modelling case (2), if we divide the time into tiny intervals, we can see that the time intervals are not independent. For example, if the cab from the company arrived in a time interval, the probability that the cab from the company arrives in several following time intervals will diminish. Because of lack of independence between the time intervals, the scenario cannot be modeled as an exponential distribution, and hence, we do not have the memoryless property.