Discrete-time stochastic processes

The conditional probabilities are then pr z z n t0

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Unformatted text preview: event implies that N (t), the number of arrivals by time t, must be at least n; i.e., it implies the event {N (t) ≥ n}. Similarly, {N (t) ≥ n} implies {Sn ≤ t}, yielding the equality in (2.2). This equation is essentially obvious from Figure 2.1, but is one of those peculiar obvious things that is often difficult to see. One should be sure to understand it, since it is fundamental in going back and forth between arrival epochs and counting rv’s. In principle, (2.2) specifies the joint distributions of {Si ; i > 0} and {N (t); t > 0} in terms of each other, and we will see many examples of this in what follows. In summary, then, an arrival process can be specified by the joint distributions of the arrival epochs, the interarrival intervals, or the counting rv’s. In principle, specifying any one of these specifies the others also.1 2.2 Definition and properties of the Poisson process The Poisson process is an example of an arrival process, and the interarrival times provide the most convenient description since the interarrival times are defined to be IID. Processes with IID interarrival times are particularly important and form the topic of Chapter 3. Definition 2.1. A renewal process is an arrival process for which the sequence of interarrival times is a sequence of IID rv’s. Definition 2.2. A Poisson process is a renewal process in which the interarrival intervals have an exponential distribution function; i.e., for some parameter ∏, each Xi has the density fX (x) = ∏ exp(−∏x) for x ≥ 02 . The parameter ∏ is called the rate of the process. We shall see later that for any interval of size t, ∏t is the expected number of arrivals in that interval. Thus ∏ is called the arrival rate of the process. 2.2.1 Memoryless property What makes the Poisson process unique among renewal processes is the memoryless property of the exponential distribution. 1 By definition, a stochastic process is a collection of rv’s, so one might ask whether an arrival process (as a stochastic process) is ‘really’ the arrival epoch process 0 ≤ S1 ≤ S2 ≤ · · · or the interarrival process X1 , X2 , . . . or the counting process {N (t); t ≥ 0}. The arrival time process comes to grips with the actual arrivals, the interarrival process is often the simplest, and the counting process ‘looks’ most like a stochastic process in time since N (t) is a rv for each t ≥ 0. It seems preferable, since the descriptions are so clearly equivalent, to view arrival processes in terms of whichever description is more convenient at the time. 2 With this density, Pr {Xi =0} = 0, so that we can regard Xi as a positive random variable. Since events of probability zero can be ignored, the density ∏ exp(−∏x) for x ≥ 0 and zero for x < 0 is effectively the same as the density ∏ exp(−∏x) for x > 0 and zero for x ≤ 0. 2.2. DEFINITION AND PROPERTIES OF THE POISSON PROCESS 61 Definition 2.3. Memoryless random variables: A non-negative non-deterministic rv X possesses the memoryless property if, for every x ≥ 0 and t ≥ 0, Pr {X > t + x} = Pr {X > x} Pr {X > t} . (2.3) Note that (2.3) is a statement about the complementary distribution function of X . There is no intimation that the event {X > t + x} in the equation has any relation to the events {X > t} or {X > x}. For the case x = t = 0, (2.3) says that Pr {X > 0} = [Pr {X > 0}]2 , so, since X is non-deterministic, Pr {X > 0} = 1. In a similar way, by looking at cases where x = t, it can be seen that Pr {X > t} > 0 for all t ≥ 0. Thus (2.3) can be rewritten as Pr {X > t + x | X > t} = Pr {X > x} . (2.4) If X is interpreted as the waiting time until some given arrival, then (2.4) states that, given that the arrival has not occured by time t, the distribution of the remaining waiting time (given by x on the left side of (2.4)) is the same as the original waiting time distribution (given on the right side of (2.4)), i.e., the remaining waiting time has no memory of previous waiting. Example 2.2.1. If X is the waiting time for a bus to arrive, and X is memoryless, then after you wait 15 minutes, you are no better off than you were originally. On the other hand, if the bus is known to arrive regularly every 16 minutes, then you know that it will arrive within a minute, so X is not memoryless. The opposite situation is also possible. If the bus frequently breaks down, then a 15 minute wait can indicate that the remaining wait is probably very long, so again X is not memoryless. We study these non-memoryless situations when we study renewal processes in the next chapter. For an exponential rv X of rate ∏, Pr {X > x} = e−∏x , so (2.3) is satisfied and X is memoryless. Conversely, it turns out that an arbitrary non-negative non-deterministic rv X is memoryless only if it is exponential. To see this, let h(x) = ln[Pr {X > x}] and observe that since Pr {X > x} is nonincreasing in x, h(x) is also. In addition, (2.3) says that h(t + x) =...
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This note was uploaded on 09/27/2010 for the course EE 229 taught by Professor R.srikant during the Spring '09 term at University of Illinois, Urbana Champaign.

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