## 2.12.3 Unordered Arrival TimesWe now know that arrivals occur singly in a Poisson process with interarrival times between successive arrivals distributed as negative exponential random variables. But suppose that we turn things around a bit and say that we have waited through the interval [0, t] and that exactly m Poisson arrivals have occurred in [0, t]. And, rather than queue up in some orderly fashion (like first arrivals first in line), all the arrivals are mixed together, such as the students sitting in a classroom or the patients sitting in the waiting room of an outpatient clinic. We wish to derive a property of the Poisson process which in part is responsible for its nickname "most random of random processes." We call the in arrivals the unordered arrivals of a Poisson
process, since they are not ordered in accordance with the
time of their arrival. We wish to determine the probabilistic
behavior of the in unordered arrival times. To do this, we
partition the interval [0, t] into 2m + 1 arbitrary
subintervals a_{i} and b_{i}, as shown in Figure 2.13. Suppose
that we are interested in the event E_{m} that exactly one
arrival occurred in each of the subintervals b_{i} and that
no arrival occurred in any of the subintervals a_{i}. We
wish to calculate the probability of E_{m} occurring, given
that exactly m arrivals occurred in [0, t].(Clearly, the probability of E _{m} is zero for any numbers of arrivals
other than m.) Thus, invoking the definition of conditional
probability, we wantBut E _{m} is the union of 2m + 1 events that, by the Poisson postulates,
are mutually independent. For m of the events, we want the
probability of exactly one arrival in a subinterval b_{k}, and
this is simplyBut suppose that the process had been one in which we took each potential arrival, say a person, and "tossed" him/her "at random" into the interval [0, t], with successive tosses being independent. The probability that any particular arrival "ends up" in interval b _{k} is then simply b_{k}/t, since the arrival time of such a person
is uniformly distributed over [0, t]. The probability that the m arrivals distributed over [0, t] in this way each end
up as the single occupant of one subinterval b_{k} is then simplywhere the factorial term arises from the number of distinct ways in which the arrivals could be situated in the subintervals; that is, there are m! points in the experiment's sample space having the property we desire, and each point has the same probability. Since (2.60) and (2.61) are identical, we thus have derived the following result: the unordered arrival times in a Poisson process
are independently, uniformly distributed over the fixed time interval
of interest. |