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2 Poisson Processes Math 345 - Stochastic processes - Spring 2020 2 Poisson processes 2.1 Arrival processes Poisson processes are used to model continuous-time arrivals into a system (customers, orders, signals, packets), and can be thought of as continuous-time analogs of the Bernoulli processes. In a Bernoulli process the arrival times are fixed and discrete, while for general arrival processes the arrival times themselves are random variables, and form a random process Definition 2.1. An arrival process is a sequence of increasing rv's 0 < S1 < S2 < : : : , where the inequality Sj < Sj+1 is meant in the sense that the difference Sj+1 − Sj is a positive rv X (i.e. FX (0) = 0). The rv's Sj, j = 1; 2;::: are called arrival epochs, and they give the times of first, second, etc. arrivals. One can use an arrival process to model any random repeating phenomenon, where the epochs Sj will represent the times when the phenomenon occurs. Notice that the process starts at time t = 0, and simultaneous arrivals are ruled out by strict inequalities between the epochs. However, one can assign an additional sequence of random variables to the arrival epochs to track multiple arrivals. For a continuous arrival process the arrival epochs are continuous rv's, and hence the probability of an arrival at any fixed time is zero. To fully describe the arrival process one needs to specify the joint distribution of S1;S2;::: . Instead of defining the arrival process via the arrival epochs, one can also give the interarrival times, just as in the case of the Bernoulli process. For example the first interarrival time is the time until the first arrival occurs, or X1 = S1, the second interarrival time is the time between the first and second arrivals, or X2 = S2 − S1, and in general one has Xj = Sj − Sj−1. Thus we have the following relation between the interarrival times and the arrival epochs: X1 = S1 X = S − S n 2 2 1 X :::::: and Sn = Xj: Xj = Sj − Sj−1 j=1 :::::: In practice, the interarrival times are IID rv's, and it's easier to specify an arrival process via the interarrival times, rather than via the joint distribution of arrival epochs. An alternative way of characterizing an arrival process is through the aggregate number of arrivals, which in the case of continuous arrival processes will be a continuum family of discrete rv's, denoted by fN(t); t > 0g, which track the total number of arrivals up to and including the time t. Thus N(t) = 0 means that no arrival occurred in the interval (0; t]; N(t) = 1 means that one arrival occured in the same interval, etc. We will refer to the family fN(t)g as a counting process. We will take N(0) = 0 with probability 1, since the probability of an arrival occurring at t = 0 is nill. Clearly the counting process fN(t); t > 0g satisfies the property N(τ) ≥ N(t) for τ ≥ t > 0; which is again understood in the sense that the difference is a nonnegative rv. We also have the following equality of events fSn ≤ tg = fN(t) ≥ ng; since the event on the left means that the nth arrival happened at some epoch τ ≤ t, while the event on the right means that at least n arrivals occurred in the interval (0; t] (for a rigorous proof, show that the set on the left is a subsets of the set on the right and vice versa). Thus the joint CDF of the arrival epochs can be found from the joint CDF of the counting process, and vice versa. So an arrival process can be characterized by either arrival epoch, the counting process of aggregate arrivals, or the interarrival times. This then gives the freedom to use whichever description is most convenient for the problem at hand. A particular class of arrival processes are renewal processes. Definition 2.2. An arrival process is called a renewal process, if the interarrival times are positive IID rv's. Poisson processes form a special subclass of renewal processes, which we discuss in detail next. 2.2 Poisson processes Definition 2.3. A Poisson process is a renewal process where the interarrival times are IID exponential rv's, that is, for some λ > 0, called the rate of the process, the interarrival times are represented by IDD rv's fXj; j = 1; 2;::: g with the common density −λt fX (t) = λe ; t ≥ 0: We will shortly see that for an time interval of size t the expected number of arrivals will be λt, justifying the use of the term rate for λ. We first discuss an importatnt property of exponential rv's which will be crucial for under- standing the significance of Poisson processes in modeling various processes (such as in queuing theory). Definition 2.4. A rv X has the memoryless property, if X is a positive rv for which Pr(fX > t + xg) = Pr(fX > xg)Pr(fX > tg); for all x; t ≥ 0: (1) The memoryless property can be expressed via the use of the complimentary CDF of the rv, and will take the form c c c FX (t + x) = FX (x)FX (t): We observe that if (1) holds, then Pr(fX > t + X and X > tg) Pr(fX > t + Xg) Pr(fX > t + x j X > tg) = = = Pr(fX > xg): Pr(fX > tg) Pr(fX > tg) This can be interpreted as follows: If X measures the waiting time (say in units of minutes) until an arrival (say, of a bus), then the knowledge that an arrival hasn't happened in the first t minutes doesn't influence the probability of waiting an additional x units of time, i.e., the probability that one has to wait at least x more minutes is the same, whether one has already waited t minutes, or no. Hence the name of the property - memoryless. Notice also that the use of t + x on the left of (1) has nothing to do with x or t, it's simply the numeric sum of the two. In other words, for a memoryless rv one for example will have Pr(fX > 7g) = Pr(fX > 4g)Pr(fX > 3g) = Pr(fX > 5g)Pr(fX > 2g) = Pr(fX > 6g)Pr(fX > 1g) = ::: 2 We next observe that an exponential rv is memoryless, as the complimentary CDF of an expo- nential rv is 1 c −λt −λx Pr(fX > xg) = FX (x) = λe dt = e ; ˆx which will obviously satisfy the memoryless property. Conversely, if X is a memoryless continuous rv, then it has to have exponential PDF. Indeed, if we denote h(x) = ln(Pr(fX > xg), then the memoryless property implies that h(x + t) = h(x) + h(t): It's then easy to see that h(x) has to be a linear function over positive integers, since h(k) = h(1) + ··· + h(1) = kh(1) for any k = 1; 2;:::: | {z } k times This can be trivially extended to first fractional numbers 1=n and then to all rationals m=n. To extend this property to irrationals as well one can use the density of rationals on the real line and the additional property of h(x) of being nonincreasing, since Pr(fX > xg) is, that is, h(y) ≤ h(x) for y > x. The monotonicity then implies that nothing crazy can happen over the irrationals, which are limit points for the set of rationals. Thus, h must be linear with a zero vertical intercept, since h(0) = 0. Then we must have h(x) = −λx for some λ > 0, as h must be zero or negative, since it's a logarithm of numbers less than or equal to 1. But then this show that if X is memoryless, then there is a λ > 0, such that c −λx FX (x) = Pr(fX > xg) = e > 0; for all t ≥ 0: −λx But then the CDF of X is FX (x) = 1 − e , and differentiating the last expression one can obtain the exponential PDF. We also observe, that if a memoryless process is discrete (think of restricting arrival times to k−1 integer values, for example), then it has to have the geometric PMF, that is PX (k) = p(1 − p) . One can see that a geometric rv is memoryless by computing its complimentary CDF, 1 X p(1 − p)m F c (m) = Pr(fX > mg) = p(1 − p)j−1 = = (1 − p)m; X 1 − (1 − p) j=m+1 c where we used the formula for the sum of a geometric progression. But then it's clear that FX (m) = eln(1−p)·m, where ln(1 − p) acts like the −λ for the exponential rv. So following the same ideas as in the previous proof, one can show that a memoryless discrete rv must be geometric. We will shortly see that the Bernoulli process, which has IID geometrically distributed inter- arrival times, can be thought of as a discrete-time analog of the Poisson process, which has IID exponential rv's as the interarrival times. Hence it's not surprising that in both cases the interarrival times are memoryless. The memoryless property can be generalized to the entire Poisson process, and we explore this by looking at interarrival times starting from some cutoff time t. Theorem 2.5. For a Poisson process of rate λ, and any given time t > 0, the length of the interval from t until the first arrival after t is a positive rv Z with CDF 1 − e−λz for z ≥ 0.
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